Contrôle optimal de l'attitude d'un lanceur

Université Pierre et Marie Curie Laboratoire Jacques-Louis LionsParis 6 UMR 7598

Thèse de Doctorat de

l'Université Pierre et Marie Curie

Présentée et soutenue publiquement le 1er juillet 2016

pour l'obtention du grade de

Docteur de l'Université Pierre et Marie Curie

Spécialité : Mathématiques Appliquées

par

Jiamin ZHU

Contrôle optimal de l'attitude d'un lanceur

après avis des rapporteurs

M. Jean-Baptiste CaillauMme. Monique Chyba

devant le jury composé de

M. Bernard Bonnard ExaminateurM. Jean-Baptiste Caillau RapporteurM. Max Cerf ExaminateurM. Jean-Michel Coron ExaminateurM. Joseph Gergaud ExaminateurM. Thomas Haberkorn Examinateur

M. Emmanuel Trélat Directeur de Thèse

École Doctorale de Sciences Mathématiques Faculté de Mathématiquesde Paris Centre ED 386 UFR 929

Jiamin ZHU :

Sorbonne Universités, UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-LouisLions, F-75005, Paris, France.Adresse électronique: [email protected], [email protected]

[email protected]

[email protected]

Remerciements

Soyons reconnaissants aux personnes qui nous donnent du bonheur;elles sont les charmants jardiniers par qui nos âmes sont euries.

Marcel Proust

Il me sera très dicile de remercier tout le monde car c'est grâce à l'aide de nom-breuses personnes que j'ai pu mener cette thèse à son terme.

Je tiens tout d'abord à exprimer mes plus sincères remerciements à Emmanuel Trélatqui a été pour moi un directeur de thèse attentif et disponible malgré ses nombreusescharges. Ses encouragements, sa patience et sa conance en moi m'ont permis d'arriverau bout de cette thèse. Sa compétence, sa rigueur scientique et sa clairvoyance m'ontbeaucoup appris. Ils ont été et resteront des moteurs pour mon travail de chercheuse.

J'adresse également mes plus vifs remerciements à Max Cerf, avec qui j'ai beaucoupdiscuté sur l'aspect numérique de ce travail. Son point de vue d'ingénieur m'a souventdirigé vers d'autres méthodes qui m'ont permis de mieux aborder les dicultés que j'aieues dans la résolution numérique. En travaillant avec lui, j'ai pu énormément améliorermes compétences en programmation et aussi naliser un logiciel utile dans la pratique.

Je suis très reconnaissante envers mes deux rapporteurs les professeurs Jean-BaptisteCaillau et Monique Chyba d'avoir accepté de relire ma thèse. Ils ont fait des remarquestrès pertinentes qui m'ont permis d'améliorer mon manuscrit. Je remercie en particulierJean-Baptiste pour les discussions qu'on a eues et pour m'avoir conseillé d'assister à denombreuses conférences pendant ces dernières années.

Je tiens à remercier chaleureusement les professeurs Jean-Michel Coron, JosephGergaud, et Thomas Haberkorn d'avoir accepté de participer à mon jury de thèse etd'y avoir consacré du temps.

Je souhaite remercier toute l'équipe administrative et informatique du laboratoireJL Lions et de l'école doctorale (ED386) pour leur aide et leur patience. Je remercieparticulièrement le professeur Yvon Maday, qui a été le directeur du laboratoire en2012, de m'avoir aidé à faire connaissance avec mon directeur de thèse.

Je remercie tous les thésards du bureau 302 15-25 pour la bonne ambiance de travailet d'entraide. Ce bureau "international" a été très animé avec de nombreuses discus-sions en français entre Abdellah, Simona Oana, Carlo, Philippe, Pierre-Henri, Pierre etOlivier, celles en chinois entre Chaoyu, Long, Shuyang et moi, et celles en anglais entrenous tous.

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Je remercie également les thésards de ma "famille" scientique, surtout Maxime,Jérémy, Michaël et Zheng, pour les discussions, les conférences, les workshops, les mini-cours, les repas, et les soirées qu'on a eus ensemble.

Je remercie aussi tous les thésards et les autres membres du laboratoire JL Lions,notamment ceux avec qui j'ai eu l'occasion de travailler et les autres simplement pourles bons moments partagés à la cantine universitaire les midis ou au bar ou sur les quaisde la Seine les vendredis soir.

Je remercie autant tous mes amis qui m'ont permis d'oublier quelques instants letravail: les Grillon pour m'avoir hébergée pendant trois ans et avoir joué le rôle dema famille française; Anne-Laure pour les "Pizza night" tous les samedis soir et pourm'avoir "forcée" à parler français et m'avoir convertie aux cartes Magic; les membres deJKA Paris pour les nombreux entraînements, stages, et soirées qu'on a eus ensemble;les bons ami(e)s chinois(es) pour les repas chinois et pour les bons moments passésensemble où j'ai pu parler ma langue maternelle; Erica pour m'avoir écoutée, pouravoir partagé des histoires de vie depuis l'an dernier et de m'avoir aidée quand j'étaisen train de chercher mon postdoc; enn, Etienne pour sa présence spéciale à la n dema thèse et qui m'a permis de ne pas écrire trop de fautes dans ces remerciements !

Enn, j'adresse toute mon aection à mes parents qui m'ont soutenue pendantces années. Etant lle unique, je sais que mon absence a été longue et qu'ils se sontbeaucoup inquiétés pour moi à cause des huit mille kilomètres de distance. J'aimeraissimplement leur dire : je vous aime, et ne vous inquiétez pas puisque je sais ce que jeveux faire dans la vie et qu'une personne munie d'un rêve est plus forte.

Publications of Jiamin Zhu

• J. Zhu, E. Trélat, M. Cerf, Minimum time control of the rocket attitude reori-entation associated with orbit dynamics, SIAM J. Cont. Optim., 2016, 54(1):391-422.

• J. Zhu, E. Trélat, M. Cerf, Planar tilting maneuver of a spacecraft: singular arcsin the minimum time problem and chattering, Discrete Cont. Dynam. Syst. Ser.B, 2016, 21(4): 1347-1388.

• J. Zhu, E. Trélat, M. Cerf, Minimum time-energy pull-up maneuvers for airbornelaunch vehicles, Preprint Hal, 2016, 24 pages.

• P. Guzmàn, J. Zhu, Exact boundary controllability of a microbeam model, Journalof Mathematical Analysis and Applications, 2015, 425(2): 655-665.

• M. Xu, J. Zhu, T. Tan, S. Xu, Equilibrium congurations of the tethered three-bodyformation system and their nonlinear dynamics, Acta Mechanica Sinica, 2012,28(6): 1668-1677.

• M. Xu, J. Zhu, T. Tan, S. Xu, Application of Hamiltonian structure-preservingcontrol to formation ying on a J2-perturbed mean circular orbit, Celestial Me-chanics and Dynamical Astronomy, 2012, 113(4): 403-433.

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Contents

1 Preliminaries 171.1 Geometric Optimal Control Theory . . . . . . . . . . . . . . . . . . . . 171.2 Numerical Methods in Optimal Control . . . . . . . . . . . . . . . . . . 241.3 Numerical Continuation . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2 Problem Formulation and Chattering 322.1 Physical Problem and Model . . . . . . . . . . . . . . . . . . . . . . . . 332.2 Optimal Control Problems . . . . . . . . . . . . . . . . . . . . . . . . . 402.3 Chattering phenomenon . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3 Minimum Time Tilting Problem (MTTP) 453.1 Geometric Analysis of Chattering . . . . . . . . . . . . . . . . . . . . . 483.2 Application to (MTTP) . . . . . . . . . . . . . . . . . . . . . . . . . . 593.3 Chattering Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4 Minimum Time Control Problem (MTCP) 884.1 Some General Results for Bi-Input Control-Ane Systems . . . . . . . 894.2 Geometric Analysis of the Extremals of (MTCP) . . . . . . . . . . . . 974.3 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . . . . . . 1054.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

5 Application to the Airborne Launchers 1215.1 Pull-Up Maneuver Problem . . . . . . . . . . . . . . . . . . . . . . . . 1235.2 Application of the PMP . . . . . . . . . . . . . . . . . . . . . . . . . . 1245.3 Resolution algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1275.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133

Bibliography 156

Introduction Générale

Problème. Au cours des dernières décennies, les problèmes de contrôle optimald'attitude (voir, par exemple, [7, 41, 85, 95, 97]) et de transfert orbital (voir, par exem-ple, [18, 16, 23, 74, 60, 103, 111]) d'un vaisseau spatial (lanceur, satellite, etc.) ont étéabondamment étudiés. Le problème de contrôle d'attitude consiste souvent à réorien-ter le vaisseau spatial en temps minimal, c'est-à-dire de chercher une loi de commande(direction et module du couple au cours du temps) minimisant la durée nécessaire àl'accomplissement de la man÷uvre. Le problème de transfert orbital consiste plutôt àdéplacer le vaisseau spatial d'une orbite initiale jusqu'à une orbite nale en minimisantla consommation d'énergie ou de carburant. Dans la littérature, ces deux problèmessont toujours considérés indépendamment l'un de l'autre. En eet, pour un satellite, onconsidère les commandes d'attitude et d'orbite séparément, puisque le couplage entreattitude et trajectoire est en général négligeable. Néanmoins, pour un lanceur (LV),la trajectoire1 est aussi contrôlée par l'attitude et il y a donc un fort couplage avecl'attitude: pour modier la trajectoire ou suivre une trajectoire nominale, il faut desforces extérieures permettant de réaliser la bonne accélération, et donc nous avons be-soin d'agir sur la direction de l'axe longitudinal du LV (voir Fig. 2.2 (a)), c'est-à-dire,nous avons besoin d'agir sur l'attitude du LV, ce qui consiste à contrôler l'angle entrela direction de la propulsion et la direction de l'axe longitudinal du LV (nous appelonscet angle la dérivation propulseur). L'état global du système est alors constitué nonseulement des traditionnelles variables permettant de décrire sa trajectoire au coursdu temps (en tant que point matériel), mais aussi des variables décrivant l'attitude del'engin (variables d'Euler, ou quaternions par exemple). L'objectif est alors de déter-miner une loi de commande de la dérivation propulseur permettant de conduire le LVd'un état initial donné à une conguration nale souhaitée. Ces préoccupations nousamènent au problème couplé: le problème de contrôle optimal associé au système mod-élisé par cette dynamique couplée. Bien que le problème couplé (voir, par exemple,[51, 62, 109]) ait été étudié dans de nombreux travaux précédents (voir, par exemple,[66, 52, 62]), il ne semble pas que ce problème ait été étudié dans le cadre général de la

1 Le terme orbite est utilisée pour décrire le déploiement d'un satellite, tandis que pour un LV,on utilise plutôt le terme trajectoire. La trajectoire décrit la position et la vitesse d'un LV.

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théorie du contrôle optimal jusqu'à présent.Dans le présent manuscrit, nous étudions le problème du temps minimal du contrôle

et du guidage2 d'un LV. Plus précisément, notre travail s'oriente pour la majeure partievers la recherche d'une loi de commande de la dérivation propulseur minimisant le tempsde man÷uvre, soumis au système couplé, lequel contient la dynamique et la cinématiquede l'attitude, ainsi que la dynamique de trajectoire, et les conditions terminales surl'attitude (vitesse angulaire et angles d'Euler) et sur la direction de vitesse. Ce problèmeest noté (MTCP) dans le manuscrit. Dans (MTCP), la position du LV est supposéeconstante puisque le changement de position n'a qu'une inuence mineure sur la vitesseet sur l'attitude (ceci est vérié quand on étudie le problème d'un lanceur aéroportédans le dernier chapitre, où la position est également considérée dans le système).

Dicultés et analyse théorique du problème. Le problème est dicile à résoudreà cause du couplage de l'attitude et de la trajectoire. Ce système vit en dimension 8 etson intégration numérique est délicate car le système contient à la fois une dynamiquelente (trajectoire) et une dynamique rapide (attitude). Il est particulièrement impor-tant de bien comprendre ce couplage en vue de construire une méthode de résolutionnumérique ecace. L'idée est de dénir un problème de départ simple puis d'utiliserune méthode de continuation.

Mais en fait, la diculté essentielle dans ce problème est l'occurrence du phénomènede chattering. Chattering se rapporte à la commande qui commute une innité de foissur un intervalle de temps compact. Un tel phénomène apparaît en général lorsqu'onconnecte des arcs bangs avec un arc singulier d'ordre supérieur (voir, par exemple,[43, 73, 112, 113]).

Nos premières investigations numériques pour résoudre (MTCP) nous ont conduità considérer un problème plus simple, qui décrit le mouvement d'un LV dans un plan.Dans ce problème simplié (noté (MTTP)), nous cherchons à minimiser le tempsde man÷uvre pour un système ane mono-entrée. Remarquons que (MTTP) estintéressant non seulement par rapport au phénomène de chattering, mais aussi parceque le modèle du problème correspond bien à la pratique (les man÷uvres pendant laphase de l'ascension atmosphérique sont généralement planaires). Nous avons utilisé etgénéralisé des résultats de M.I. Zelikin et V.F. Borisov [112, 113] pour appréhender lephénomène de chattering et démontrer l'optimalité locale des extrémales de chattering.

En appliquant le Principe du Maximum de Pontryagin (PMP, voir [84]) et la théoriedu contrôle optimal géométrique (voir [2, 93, 104]), nous avons établi un résultat d'ex-istence du phénomène de chattering pour une classe de systèmes anes bi-entrée, etappliqué ensuite à (MTCP). Plus précisément, en utilisant les conditions nécessairesd'optimalité de Goh et de Legendre-Clebsch généralisée, nous avons démontré qu'ilexiste des arcs de chattering localement optimaux lorsqu'on connecte des arcs réguliersà un arc singulier d'ordre deux.

2 En aéronautique, le système de contrôle fournit la commande d'attitude et le système de guidageproduit la commande de la trajectoire.

Introduction Générale 9

Méthodes numériques. Notre objectif a été de concevoir un algorithme (une méth-ode de résolution) pour résoudre automatiquement notre problème, i.e., avec lequel lasolution peut être obtenue en donnant simplement au programme les paramètres duLV et les conditions terminales souhaitées. De plus, la solution numérique doit êtreobtenue rapidement et avec grande précision.

Rappelons qu'il existe en contrôle optimal deux types principaux de méthodesnumériques: les méthodes indirectes et les méthodes directes3 (voir, par exemple,[6, 99, 104]). Les méthodes directes consistent à discrétiser l'état et le contrôle, etréduire ainsi le problème de contrôle optimal à un problème d'optimisation non linéaire(programmation non linéaire) en dimension nie sous contraintes. Les méthodes in-directes consistent à résoudre numériquement le problème aux valeurs limites obtenuen appliquant le PMP, au moyen d'une méthode de tir (tir simple et tir multiple).L'avantage des méthodes indirectes est leur rapidité et leur précision.

La diculté essentielle des méthodes de tir (qui reposent sur une résolution de typeNewton) est de savoir les initialiser de façon à assurer leur convergence. Il est bien connuqu'il est en général nécessaire de les combiner avec d'autres approches théoriques ounumériques (voir [104]), par exemple, la méthode de continuation, qui est un outil puis-sant en combinaison avec le PMP. L'idée des méthodes de continuation, ou d'homotopie,est de déformer le problème en un problème qui soit plus simple à résoudre, pour en-suite revenir petit à petit, en suivant un chemin, vers le problème initial. Elle estutilisée dans [29, 49, 77] pour résoudre des problèmes de transfert orbital diciles.Dans [20, 27, 47, 71], plusieurs continuations sont utilisées pour introduire l'eet atmo-sphérique et les termes concernant les contraintes sur l'état, an de résoudre le problèmede l'ascension atmosphérique d'un LV à partir d'une solution quasi analytique.

Dans ce manuscrit, nous employons aussi les méthodes de continuation, qui s'avèrentfort ecaces pour notre problème. Grâce à cette approche, l'algorithme que nous avonsdéveloppé peut résoudre le problème sans estimation initiale des inconnues fournies parl'utilisateur.

Lors de la conception de la méthode de résolution pour (MTCP), étant donné que ladynamique de la trajectoire est beaucoup plus lente que celle de l'attitude (première dif-culté), nous partons d'une solution explicite d'un problème simplié, appelé problèmed'ordre zéro (noté par (POZ)), qui ne considère que la trajectoire et prend l'attitudecomme contrôle. Ensuite, nous utilisons une série de continuations pour revenir à lavraie dynamique avec les conditions terminales de (MTCP).

En considérant le phénomène de chattering (seconde diculté) qui apparaît dansle système complet, nous introduisons dans la chaîne de continuations un problèmerégularisé (noté (RMTCP)K): d'abord avec deux paramètres de continuation, nouspassons de la solution de (POZ) à la solution de (RMTCP)K . Ensuite la solution de(MTCP) est approchée par la déformation d'un troisième paramètre de continuation.

Notons que le chattering pose problème pour la convergence des méthodes. Dans

3 Il existe également des méthodes mixtes qui discrétisent les conditions nécessaires obtenues duPMP et utilisent un solveur d'optimisation à grande échelle (voir, par exemple, [5]).

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[11], il est démontré que l'occurrence de chattering est la cause du caractère mal-posédes méthodes de tir (matrice jacobienne non inversible) pour des problèmes mono-entrée. Selon [113], la diculté est notamment liée à l'intégration numérique du sys-tème hamiltonien qui est discontinu, parce que les solutions de chattering aggravent leserreurs d'estimation et d'approximation au cours de l'intégration numérique. À causedu chattering, la méthode de continuation combinée avec la méthode de tir échouepour (MTCP) pour certaines conditions terminales, pour lesquelles la trajectoire op-timale contient un arc singulier d'ordre supérieur. Dans ce cas-là, nous proposons desstratégies sous-optimales en utilisant des méthodes directes, où le contrôle de chatter-ing est approché par un contrôle constant par morceaux comportant un certain nombreN d'arcs bangs. Nous proposons également une stratégie sous-optimale pour stopperla procédure de continuation avant son échec causé par le chattering. Cette dernièreméthode s'est révélée plus rapide que la précédente, et apparaît comme une alternativeintéressante dans la pratique.

Finalement, on applique notre méthode de résolution au contrôle optimal d'un LVaéroporté. Le but de ce problème est de réaliser une man÷uvre de pull-up en temps-énergie minimal (notons ce problème (PUMP)). Dans cette application, nous rajou-tons plusieurs paramètres de continuation pour introduire la position, les forces aéro-dynamiques (surtout la portance et la traînée), et les contraintes sur l'état. Il est bienconnu que les contraintes sur l'état sont diciles à traiter en appliquant le principe dumaximum, puisque le vecteur adjoint devient une mesure de Radon. Nous avons plutôtutilisé une méthode de pénalisation (voir, par exemple, [105, Chapter 7],[36, 63, 76])pour traiter ces contraintes sur l'état.

Structure du manuscrit. Ce manuscrit est structuré en cinq chapitres. Le pre-mier présente quelques aspects du contôle optimal géométrique (conditions nécessairesd'optimalité et notion de crochet de Lie) et les méthodes numériques (méthodes indi-rectes et directes, méthodes d'homotopie) que nous utilisons.

Le second chapitre introduit les problèmes de man÷uvre qu'on se propose de ré-soudre ainsi que la modélisation physique choisie. Nous formulons deux problèmes decontrôle optimal (MTTP) et (MTCP). (MTCP) est un problème de temps minimalde guidage et de contrôle d'un LV, dans lequel le mouvement du LV est décrit par lacinématique et la dynamique d'attitude, mais aussi par sa dynamique de trajectoire.Il s'agit d'un système ane bi-entrée, le contrôle prenant des valeurs dans le disqueeuclidien unité. (MTTP) est la version planaire de (MTCP), c'est-à-dire, les anglesde lacet et de rotation sont constants dans (MTTP). C'est un système ane mono-entrée. Nous rappelons également le problème de Fuller, exemple typique bien connudans lequel la solution présente le phénomène de chattering.

Le troisième chapitre est consacré à l'étude de (MTTP), du point de vue à la foisthéorique et numérique, avec un accent particulier sur le phénomène de chattering.Nous commençons à démontrer qu'il existe des arcs de chattering optimaux quand une

Introduction Générale 11

jonction singulière se produit. Ces études sont basées sur le PMP et sur les résultatsde M.I. Zelikin et V.F. Borisov. Nous donnons ensuite des conditions susantes surles valeurs terminales pour lesquelles les solutions optimales ne contiennent pas d'arcsingulier, et sont bang-bang avec un nombre ni de commutations. De plus, nous met-tons en ÷uvre des stratégies sous-optimales en remplaçant le contrôle de chattering parun contrôle constant par morceaux avec un nombre xé d'arcs bangs. Des simulationsnumériques illustrent nos résultats.

L'avant-dernier chapitre résout (MTCP), par une étude géométrique des extrémalesobtenues par le PMP. Nous commençons par montrer un résultat général pour les sys-tèmes anes bi-entrée, donnant des conditions susantes sous lesquelles le phénomènede chattering se produit. Nous montrons comment ce résultat s'applique à (MTCP).Notre analyse révèle l'existence d'arcs singuliers d'ordre supérieur dans la synthèse op-timale, qui provoque l'occurrence du phénomène de chattering, c'est-à-dire, d'un nom-bre inni de commutations lorsqu'on connecte des arcs réguliers avec un arc singulier.Nous classions ensuite les extrémales régulières de (MTCP) selon leur ordre de con-tact avec la surface de commutation. Grâce à cette analyse théorique préliminaire, nousconcevons ensuite et mettons en ÷uvre des méthodes de résolution directes et indirectesecaces, combinées à des continuations numériques.

Le dernier chapitre présente une application de la méthode de résolution développéedans le chapitre précédent. Nous formulons le problème de contrôle optimal (PUMP),dans lequel nous cherchons à réaliser une man÷uvre de pull-up en temps-énergie min-imal pour un LV aéroporté. Nous appliquons la méthode de tir multiple, la méthodede continuation Prédicteur-Correcteur, des changements de coordonnées, et des lissagesdes champs de vecteurs, an d'améliorer la stabilité numérique et la robustesse de laméthode. Ce chapitre peut être vu comme une validation de l'ecacité et de la ro-bustesse de notre méthode de résolution.

General Introduction

Problem. The optimal control of attitude reorientation (see, e.g., [7, 41, 85, 95, 97])and orbit transfer (see, e.g., [18, 16, 23, 74, 60, 103, 111]) for spacecraft (launch vehicle,satellite, etc.) have been extensively studied in the past few decades. The optimalcontrol problem of attitude reorientation is mainly devoted to determine how to changethe pointing direction of the spacecraft in minimal time, while the optimal controlproblem of orbit transfer focuses mostly on how to move the spacecraft from one orbitor point to another orbit or point by minimizing energy or fuel consumption. In theliterature, these two optimal control problems are always considered separately. Fromthe engineering point of view, for a satellite, it is appropriate to design separately thecontrol laws for the attitude movement and for the orbit movement, since the couplingof attitude and orbit is negligible. However, for a launch vehicle (LV), the trajectory 4 iscontrolled by its attitude and thus there is a strong coupling with the attitude: to modifythe trajectory or to follow a nominal trajectory, a proper acceleration is generated byexterior forces, and so one needs to act on the direction of the longitudinal axis of theLV, i.e., to act on the attitude, which consists of controlling the angle between thethrust direction and the direction of the longitudinal axis of the LV (we call this anglethe propulsive angle). Thus, the state of the system consists not only of the traditionalvariables that describe the trajectory (considering the LV as a mass point), but also ofthe variables describing the attitude (e.g. Euler angles, or quaternions). The objectiveis then to determine a control law of the propulsive angle that drives the LV from agiven initial state to a desired nal conguration. These considerations lead us to acoupled problem: the optimal control problem associated to a system modeled by thecoupled dynamics. Though the control of the coupled problem (see, e.g., [51, 62, 109])was also studied in many previous works (see, e.g., [66, 52, 62]), it does not seem thatthe problem has been investigated in the general optimal control framework so far.

In this work, we study the minimum time control problem for the control andguidance5 of a LV. More precisely, a major part of our work is to nd a control law

4 The term orbit" is used for satellites to describe their deployment, while for a LV, the termtrajectory" is used instead.

5 In aeronautics, the guidance of a LV is the command of trajectory motion, and the control of aLV is the command of attitude motion.

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General Introduction 13

of the propulsive angle minimizing the maneuver time, subject to the coupled system,which consists of the dynamics and the kinematic of attitude, and also of the dynamicsof trajectory, and the terminal conditions on the attitude (angular velocity and Eulerangles) and on the direction of the velocity. This problem is denoted by (MTCP) inthis manuscript. In (MTCP), the position of the LV is supposed constant due to itsminor inuence on the velocity and the attitude (this is veried when we study theproblem for airborne launchers in the last chapter, where the position is also taken intoconsideration in the system).

Diculties and theoretical analysis of the problem. The problem is dicultto solve due to the coupling of the attitude with the trajectory. The system is ofdimension 8 and its numerical integration is delicate since the system contains bothslow (trajectory) and fast (attitude) dynamics. This observation will be particularlyimportant in order to design appropriate numerical approaches. The idea is to dene astarting problem and then to use some appropriate numerical continuation.

However, the essential diculty of this problem is the chattering phenomenon. Chat-tering means that the control switches an innite number of times over a compact timeinterval. Such a phenomenon typically occurs when trying to connect bang arcs with ahigher-order singular arc (see, e.g., [43, 73, 112, 113]).

Our rst numerical investigations for solving (MTCP) lead us to the study of asimpler problem, which describes the movement of the LV in a plane. In this simplerproblem (denoted by (MTTP)), the objective is to minimize the maneuver time for asingle-input control ane system. Note that (MTTP) is of interest not only becauseof the chattering phenomenon, but also because the model is close to real life (themaneuvers during the launcher ascent phase are in general planar). We have usedthe results of M.I. Zelikin and V.F. Borisov [112, 113] to understand the chatteringphenomenon and to prove the local optimality of the chattering extremals.

By using the Pontryagin Maximum Principle (in short, the PMP) and the geometricoptimal control theory (see [2, 93, 104]), we have established an existence result of thechattering phenomenon for a class of bi-input control ane systems and we have appliedthe result to (MTCP). More precisely, based on Goh and generalized Legendre-Clebschconditions, we have proved that there exist optimal chattering arcs when connectingthe regular arcs with a singular arc of order two.

Numerical Approaches Our objective has been to design an algorithm (a resolu-tion method) to solve automatically our problem, i.e., with which the solution can beobtained by simply providing to the program the LV parameters and the desired termi-nal conditions. Moreover, the numerical solution should be obtained rapidly and witha great precision.

Recall that there are two main types of numerical methods for solving optimalcontrol problems: indirect methods and direct methods6 (see, e.g., [6, 99, 104]). The

6 There exist also mixed methods that discretize the PMP necessary conditions and use then a

14

direct methods consist of discretizing the state and the control and thus of reducing theproblem to a nonlinear optimization problem (nonlinear programming) with constraints.The indirect methods consist of numerically solving a boundary value problem obtainedby applying the PMP, by means of a shooting method (single and multiple shooting).The advantages of the indirect methods are their rapidity and their numerical accuracy.

The essential diculty of shooting methods (which rely on a Newton type resolu-tion) is to initialize them successfully. It is well known that in general they have to becombined with other theoretical or numerical approaches (see [104]), such as continua-tion method, which is a very powerful tool to be combined with the PMP. The idea ofnumerical continuation, or homotopy method, is to solve a dicult problem step by stepstarting from a simpler problem by parameter deformation. In [29, 49, 77], the continu-ation method is used to solve dicult orbit transfer problems. In [20, 27, 47, 71], severalcontinuations are used to introduce the atmospheric eects and the path constraint-related terms for solving the endo-atmospheric LV ascent problem starting from a nearlyanalytic solution.

In this manuscript, we also use a continuation method, which proves to be veryecient for our problem. Thanks to this method, the algorithm we developed can solvethe problem without any user-supplied initial guesses to the unknowns.

When designing the resolution approach for (MTCP), based on the fact that tra-jectory movements are much slower than attitude movements (rst diculty), we startfrom the explicit solution of a simplied problem, called problem of order zero (denotedby (POZ)), which considers only the trajectory while taking the attitude as controlinput. Then, we use successive continuations to retrieve the true dynamics and terminalconditions of (MTCP).

Considering the chattering (second diculty) that occurs in the complete system,we introduce the regularized problem (denoted by (RMTCP)K) into the continuationprocedure: with two continuation parameters, we passed from the solution of (POZ)to the solution of (RMTCP)K ; the solution for (MTCP) is approached with a thirdcontinuation parameter.

Note that chattering raises serious issues for the convergence of numerical meth-ods. In [11], it is shown that the presence of chattering arcs may imply ill-posedness(non-invertible Jacobian) of shooting methods for single-input problems. According to[113], the diculty is in particular due to the numerical integration of the discontinuousHamiltonian system because the chattering solutions worsen the approximation and er-ror estimates for standard numerical integration methods. Due to chattering, numericalcontinuation combined with shooting fails for (MTCP) for certain terminal conditionsfor which the optimal trajectory contains a singular arc of higher-order. In that case, wepropose sub-optimal strategies by using direct methods, where the chattering controlis approximated by a piecewise constant control with a xed number N of bang arcs.We also design a sub-optimal strategy by stopping the continuation procedure beforeits failure due to chattering. The latter strategy happens to be faster than the former,

large-scale optimization solver (see, e.g., [5]).

General Introduction 15

and turns out to be an interesting alternative for real-life implementation.

Finally, we apply our resolution method to the optimal control of airborne launchers.The aim of this problem is to realize a minimum time-energy pull-up maneuver (denotethis problem by (PUMP)). In this application, we add several additional continuationparameters to introduce the position, the aerodynamic forces (lift and drag), and somestate constraints. It is well known that state inequality constraints are dicult to treatusing the maximum principle, because the adjoint vector becomes a Radon measure.We rather use a penalty method (see e.g. [105, Chapter 7], [36, 63, 76]) to treat thestate constraints.

Structure of the manuscript. This manuscript is structured into ve chapters.The rst chapter presents some aspects of geometric optimal control theory (necessaryoptimality conditions and concept of Lie bracket) and numerical methods (indirect anddirect methods, homotopy methods) that we use in this work.

The second chapter introduces the maneuver problems that we want to solve and thephysical model. We formulate the optimal control problems (MTTP) and (MTCP).(MTCP) is a time minimum problem for the guidance and control of a LV, whosemotion is described by its attitude kinematics and dynamics but also by its trajectorydynamics. The dynamical system of the problem can be written as a bi-input controlane system with the control taking values in the Euclidean unit disk. (MTTP) is theplanar version of (MTCP), in which the yaw and rotation angles are kept constant.The dynamical system of this problem is a single-input control ane system. We recallalso the Fuller's problem, which is a very well known example in which the optimalsolution is chattering.

The third chapter is devoted to the study of (MTTP), from the theoretical aswell as from the numerical point of view, with a particular focus on the chatteringphenomenon. We begin by proving that there exist optimal chattering arcs when asingular junction occurs. This is based on the PMP and on results by M.I. Zelikin andV.F. Borisov. We then give sucient conditions on the terminal values under whichthe optimal solutions do not contain any singular arc, and are bang-bang with a nitenumber of switchings. Moreover, we implement sub-optimal strategies by replacing thechattering control with a piecewise constant control having a prescribed number N ofbang arcs. Numerical simulations illustrate our results.

The fourth chapter solves (MTCP), based on a geometric study of the extremalscoming from the application of the PMP. We start by establishing a general resultfor the bi-input control ane system, providing sucient conditions under which thechattering phenomenon occurs. We show how this result can be applied to (MTCP).Our analysis reveals the existence of higher-order singular arcs in the optimal synthe-sis, causing the occurrence of chattering, i.e., of an innite number of switchings whenconnecting bang arcs with a singular arc. We classify then the regular extremals of(MTCP) according to their order of contact with the switching surface. Thanks to

16

this preliminary theoretical analysis, we design and implement ecient direct and in-direct numerical methods, combined with numerical continuation, in order to computenumerically the optimal solutions.

The last chapter presents an application of our resolution method developed in theprevious chapter. We formulate the optimal control problem (PUMP), in which weconsider a a minimum time-energy pull-up maneuver for airborne LVs. We implementmultiple shooting, PC continuation, changes of reference frame, and smoothing of vectorelds to improve numerical stability and robustness of our method. This chapter canbe seen as a validation of the eciency and the robustness of our resolution method ona real-life problem.

Chapter 1

Preliminaries

Contents

1.1 Geometric Optimal Control Theory . . . . . . . . . . . . . . 17

1.1.1 Optimal Control Problem . . . . . . . . . . . . . . . . . . . . 18

1.1.2 First and Second Order Optimality Conditions . . . . . . . . 19

1.1.3 Lie Derivative and the Lie Bracket . . . . . . . . . . . . . . . 22

1.2 Numerical Methods in Optimal Control . . . . . . . . . . . . 24

1.2.1 Indirect methods . . . . . . . . . . . . . . . . . . . . . . . . . 24

1.2.2 Direct methods . . . . . . . . . . . . . . . . . . . . . . . . . . 25

1.3 Numerical Continuation . . . . . . . . . . . . . . . . . . . . . 26

1.3.1 Discrete Continuation . . . . . . . . . . . . . . . . . . . . . . 29

1.3.2 Predictor-Corrector continuation . . . . . . . . . . . . . . . . 29

1.1 Geometric Optimal Control Theory

Geometric optimal control (see e.g. [2, 93, 104]) is the combination of classical optimalcontrol with geometric methods in system theory, with the ultimate objective of derivingoptimal synthesis results. More precisely, by combining the knowledge inferred fromthe Pontryagin Maximum Principle (PMP) with geometric considerations (such as theuse of Lie brackets, of subanalytic sets, of dierential geometry on manifolds, and ofsymplectic geometry and Hamiltonian systems), one aims to describe in a precise waythe structure of optimal trajectories. The foundations of geometric control can be datedback to the Chow's theorem and to [22], where Brunovsky found that it was possibleto derive regular synthesis results by using geometric considerations for a large classof control systems. Typically, apart from the ultimate goal of providing a complete

17

18 1.1. Geometric Optimal Control Theory

optimal synthesis, one of the objectives of geometric control is to derive higher-orderoptimality conditions in order to restrict more the set of candidate optimal trajectories.

1.1.1 Optimal Control Problem

Let M (resp. N) be a smooth manifold of dimension n (resp. m), and let M0 and M1

be two subsets of M , and let U be a subset of N . In this chapter, we consider a generalnonlinear optimal control problem P0, that is to minimize the cost functional

C(tf , u) =

∫ tf

0

f 0(x(t), u(t))dt+ g(tf , x(tf )),

such thatx(t) = f(x(t), u(t)), (1.1.1)

andx(0) ∈M0, x(tf ) ∈M1, (1.1.2)

where f : M ×N 7→ TM , f 0 : M ×N 7→ R, and g : R×M 7→ R are smooth, and wherethe controls are bounded and measurable functions dened on [0, tf (u)] of R+, takingvalues in U . The nal time tf can be xed or not. Denote by U the set of admissiblecontrols such that the corresponding trajectories steer the system from an initial pointof M0 to a nal point in M1.

Denition 1.1.1. The end-point mapping E : M×R×U of the system is dened byE(x0, tf , u) = x(x0, tf , u), where t 7→ x(x0, t, u) is the trajectory solution of the controlsystem (1.1.1) associated to u such that x(x0, 0, u) = x0

Assume moreover that U is endowed with the L∞ topology, then the end-pointmapping is smooth, and in terms of the end-point mapping, the optimal control problemunder consideration can be written as the innite-dimensional minimization problem

minC(tf , u) |x0 ∈M0, E(x0, tf , u) ∈M1, u ∈ L∞([0, tf ];U).

Denition 1.1.2. Assume that M0 = x0. A control u dened on [0, tf ] is saidto be singular i the dierential ∂E

∂u(x0, tf , u) is not of full rank. The trajectory x(·)

associated with a singular control u is called singular trajectory.

We recall as well the important concept of local optimality.

Denition 1.1.3. (Local optimality)

• If the nal time tf is xed, then x(·) is said to be locally optimal in L∞ topology(resp. in C0 topology), if it is optimal in a neighborhood of u in L∞ topology(resp. in a neighborhood of x(·) C0 topology)• If the nal time tf is not xed, then

Chapter 1: Preliminaries 19

a trajectory x(·) is said to be locally optimal in L∞ topology if, for everyneighborhood V of u in L∞([0, tf + ε], U), for every real number η so that|η| ≤ ε, for every control v ∈ V satisfying E(x0, tf +η, v) = E(x0, tf , u) thereholds C(tf + η, v) ≥ C(tf , u).

moreover, a trajectory x(·) is said to be locally optimal in C0 topology if,for every neighborhood W of x(·) in M , for every real number η so that|η| ≤ ε, for every trajectory x(·), associated to a control v ∈ V on [0, tf + η],contained in W , and satisfying x(0) = x(0) = x0, x(tf + η) = x(tf ), thereholds C(tf + η, v) ≥ C(tf , u).

The C0 local optimality (resp. L∞ topology) is called strong local optimality (resp.weak local optimality).

1.1.2 First and Second Order Optimality Conditions

Lagrangian multipliers rule We consider the simplied problem P0 with M = Rn,M0 = x0, M1 = x1, and U = Rm. According to the well-known Lagrangianmultipliers rule (and using the C1 regularity of the data), if x ∈ M is optimal thenthere exists a nontrivial couple (ψ, ψ0) ∈ Rn × R such that

ψ.dEx0,tf (u) + ψ0dCtf (u) = 0, (1.1.3)

where dE(·) and dC(·) denote the Fréchet derivative of E(·) and C(·), respectively. De-ne the Lagrangian by Ltf := ψEx0,tf (u) +ψ0dCtf (u). This rst order necessary condi-

tion can be written in the form∂Ltf∂u

(u, ψ, ψ0) = 0. Note that the control constraints canbe taken into account directly in the Lagrangian with additional Lagrangian multiplier[23]. However, this method will lead to weaker results than the PMP.

Dening as usual the intrinsic second-order derivative Qtf of the Lagrangian as

the Hessian∂2Ltf∂2u

(u, ψ, ψ0) restricted to the subspace ker∂Ltf∂u

. It is well known thata second-order necessary condition for optimality is the non positivity of Qtf (withψ0 ≤ 0), and a second-order sucient condition for local optimality is the negativedeniteness of Qtf .

Pontryagin Maximum Principle The Pontryagin Maximum Principle (in short,PMP, see [84]), for the problem P0, is recalled in the following statement.

Theorem 1.1.1. If the trajectory x(·), associated to the optimal control u on [0, tf ],is optimal, then it is the projection of an extremal (x(·), p(·), p0, u(·)) where p0 ≤ 0, andp(·) : [0, tf ] 7→ T ∗x(t)M is an absolutely continuous mapping (called adjoint vector) with

(p(·), p0) 6= 0, such that

x(t) =∂H

∂p(x(t), p(t), p0, u(t)), p(t) = −∂H

∂x(x(t), p(t), p0, u(t)), (1.1.4)


almost everywhere on [0, tf ], where

H(x, p, p0, u) := 〈p, f(x, u)〉+ p0f 0(x, u),

is the Hamiltonian, and there holds

H(x(t), p(t), p0, u(t)) = maxv∈U

H(x(t), p(t), p0, v), (1.1.5)

almost everywhere on [0, tf ]. If moreover, the nal time tf to reach the target M1 is notxed, then one has

maxv∈U

H(t, x(t), p(t), p0, v) = −p0∂g

∂t(tf , x(tf )). (1.1.6)

In addition, if M0 and M1 (or just one of them) are submanifolds of M locally aroundx(0) ∈ M0 and x(tf ) ∈ M1, then the adjoint vector can be built in order to satisfy thetransversality conditions at both extremities (or just one of them)

p(0) ⊥ Tx(0)M0, p(tf )− p0 ∂g

∂x(tf , x(tf )) ⊥ Tx(tf )M1, (1.1.7)

where TxM0 (resp., TxM1) denote the tangent space to M0 (resp., M1) at the point x.

Denition 1.1.4. The quadruple (x(·), p(·), p0, u(·)) is called the extremal lift ofx(·). An extremal is said to be normal (resp., abnormal) if p0 < 0 (resp., p0 = 0).

Remark 1.1.1. The proof of the PMP is generally based on the needle-like variantsand with a conic implicit function theorem (see e.g. [2, 54]).

Second Order Optimal Conditions

(1) Legendre type conditions. The Legendre type conditions are second-orderoptimality conditions. Consider the optimal control problem P0, we have the standardstatement as follows (see e.g. [2, 10, 13]).

Lemma 1.1.1. (Legendre condition)

• If a trajectory x(·), associated to a control u, is optimal on [0, tf ] in L∞ topology,

then the Legendre condition holds along every extremal lift (x(·), p(·), p0, u(·)) ofx(·), that is

∂2H

∂u2(x(·), p(·), p0, u(·)).(v, v) ≤ 0, ∀v ∈ Rm.

• If the strong Legendre condition holds along the extremal (x(·), p(·), p0, u(·)), thatis, there exists ε0 > 0 such that

∂2H

∂u2(x(·), p(·), p0, u(·)).(v, v) ≤ −ε0‖v‖2, ∀v ∈ Rm,


then there exists ε1 > 0 such that x(·) is locally optimal in L∞ topology on [0, ε1].If the extremal is moreover normal, i.e., p0 6= 0, then x(·) is locally optimal in C0

topology on [0, ε1].

We say that we are in the regular case whenever the strong Legendre condition holdsalong the extremal. An extremal (x(·), p(·), p0, u(·)) is said totally singular whenever∂2H∂u2 (x(·), p(·), p0, u(·)) = 0. Then, we have the following generalized condiiton [2, 50].

Lemma 1.1.2. (Goh and Generalized Legendre condition)

• If a trajectory x(·), associated to a piecewise smooth control u, and having a totallysingular extremal lift (x(·), p(·), p0, u(·)), is optimal on [0, tf ] in L

∞ topology, thenthe Goh condition holds along the extremal, that is

∂H∂ui

,∂H

∂uj = 0,

where ·, · denotes the Poisson bracket on T ∗M . Moreover, the generalized Leg-endre condition holds along every extremal lift (x(·), p(·), p0, u(·)) of x(·), that is

H, ∂H∂u

.v, ∂H∂u

.v+ ∂2H

∂u2.(u, v),

∂H

∂u.v ≤ 0, ∀v ∈ Rm.

• If the Goh condition holds along the extremal lift (x(·), p(·), p0, u(·)), if the strongLegendre condition holds along the extremal (x(·), p(·), p0, u(·)), that is, there existsε0 > 0 such that

H, ∂H∂u

.v, ∂H∂u

.v+ ∂2H

∂u2.(u, v),

∂H

∂u.v ≤ −ε0‖v‖2, ∀v ∈ Rm,

and if moreover the mapping ∂f∂u

(x0, u(0)) : Rm 7→ Tx0M is one-to-one, then thereexists ε1 > 0 such that x(·) is locally optimal in L∞ topology on [0, ε1].

Here we have recalled the Legendre type condition. However, in our problem, thedynamic systems are control ane systems, and these conditions are trivially satisedalong the singular extremal with equality. Thus, we need more general optimalityconditions, see chapter 3 and chapter 4 for details.

We recall as well the notion of the conjugate points, since the method developedby Zelikin and Borisov (see chapter 3 for details) is actually essentially related to theconjugate points.

(2) Conjugate point. Consider here the simplied problem P0 with M = Rn, M0 =x0, M1 = x1, and U = Rm. Under strict Legendre assumption, i.e., to assumethat the Hessian ∂2H

∂u2 (x, p, p0, u) is negative denite, the quadratic form Qtf is negativedenite whenever tf > 0 is small enough.


Denition 1.1.5. The conjugate time is dened as the inmum of times t > 0 suchthat Qt has a nontrivial kernel. Denote the rst conjugate time along x(·) by tc.

The extremals are locally optimal (in L∞ topology) as long as we do not encounterany conjugate point. Dene the exponential mapping expx0(t, p0) := x(t, x0, p0), wherethe solution of (1.1.4) starting from (x0, p0) at t = 0 is denoted as (x(t, x0, p0), p(t, x0, p0)).Then, the time tc is a conjugate time along x(·) i the mapping expx0(tc, ·) is not animmersion at p0, i.e., the dierential of the mapping expx0(tc, ·) is not injective (see e.g.[2, 15] for better results).

1.1.3 Lie Derivative and the Lie Bracket

Let Ω be an open and connected subset inM , and denote the space of all innitely oftencontinuously dierentiable functions on Ω by C∞(Ω). Let X ∈ C∞(Ω) be a vector eld.X can be seen as dening a rst-order dierential operator from the space C∞(Ω)into C∞(Ω) by taking at every point q ∈ Ω the directional derivative of a functionϕ ∈ C∞(Ω) in the direction of the vector eld X(q), i.e.,

X : C∞(Ω)→ C∞(Ω), ϕ 7→ Xϕ,

dened by(X.ϕ)(q) = ∇ϕ(q) ·X(q).

We call (X.ϕ)(q) the Lie derivative of the function ϕ along the vector eld X, andgenerally one denote the operator by LX , i.e., LX(ϕ)(q) := (X.ϕ)(q).

Denition 1.1.6. (Lie bracket) The Lie bracket of two vector elds X and Y denedon a domain Ω is the operator dened by the commutator

[X, Y ] = X Y − Y X = XY − Y X.

The Lie bracket actually denes a rst-order dierential operator. Given a functionϕ, then we have

[X, Y ](ϕ) = X(Y.ϕ)− Y (X.ϕ)

= X(∇ϕY )− Y (∇ϕX)

= ∇(∇ϕY )X −∇(∇ϕX)Y

= ∇(∇ϕ)((Y,X)− (X, Y )) +∇ϕ(DY ·X −DX · Y )

= ∇ϕ(DY ·X −DX · Y ).

Therefore, if X : Ω → M, z 7→ X(z), and X : Ω → M, z 7→ Y (z), are coordinatesfor these vector elds, then

[X, Y ](z) = DY (z) ·X(z)−DX(z) · Y (z).


Lemma 1.1.3. Let X, Y , and Z be three C∞ vector elds dened on Ω, and let α,β be smooth functions on Ω. The Lie bracket has the following properties

• [·, ·] is bilinear operator• [X, Y ] = −[Y,X]• [X + Y, Z] = [X,Z] + [Y, Z]• (Jacobi identity) [X, [Y, Z]] + [Y, [Z,X]] + [Z, [X, Y ]] = 0• [αX, βY ] = αβ[X, Y ] + α(LXβ)Y − β(LY α)X

Proof 1.1.1. The rst three properties can be easily done by using the denitionof the Lie bracket, while the last one follows from the product rule of the vector eld(X(α(Y )) = (∇Xα)(Y ) + α(∇XY )), i.e.,

[αX, βY ] = (αX(βY ))− (βY (αX))

= α(X.β)Y + β(XY ) − β(X.α)Y + α(Y X)= αβ[X, Y ] + α(LXβ)Y − β(LY α)X.

These properties show that the vector elds (as dierential operators) form a Liealgebra. A Lie algebra over R is a real vector space G together with a bilinear operator[·, ·] : G×G → G such that for allX, Y, Z ∈ G we have [X, Y ] = −[Y,X] and [X+Y, Z] =[X,Z] + [Y, Z].

Now returning to the problem P0. Assume that f(x(t), u(t)) = f0(x(t))+u(t)f1(x(t)),f 0(x(t), u(t)) = 1, and g(tf , x(tf )) = 0, and dene a C1 function by

h(t) = 〈p, Z(x)〉,

then, the dierential of this function can be written by

h(t) = 〈p, Z(x)〉+ 〈p,DZ(x)x〉= −〈p(Df0(x) + uDf1(x)), Z(x)〉+ 〈p,DZ(x)(f0(x) + uf1(x))〉= 〈p,DZ(x)f0(x)−Df0(x)Z(x)〉+ u〈p,DZ(x)f1(x)−Df1(x)Z(x)〉= 〈p, [f0, Z](x)〉+ u〈p, [f1, Z](x)〉.

(1.1.8)

Let us recall as well the concept of the Poisson bracket. The Poisson bracket isvery related to the Hamiltonians. In the canonical coordinates z = (x, p), given two C1

functions α1(x, p) and α2(x, p), the Poisson bracket takes the form

α1, α2(x, p) =∂α2

∂x

∂α1

∂p− ∂α1

∂x

∂α2

∂p.

According to (1.1.8), taking

α1(x(t), p(t)) = H(x(t), p(t)), α2(x(t), p(t)) = h(x(t), p(t)),

24 1.2. Numerical Methods in Optimal Control

we have

h(t) = H, h(x(t), p(t)) = h0, h(x(t), p(t)) + uh1, h(x(t), p(t)), (1.1.9)

where h0(x(t), p(t)) := 〈p(t), f0(x(t))〉 and h1(x(t), p(t)) := 〈p(t), f1(x(t))〉.Throughout this thesis, we adopt the usual notations

adf0.f1 = [f0, f1], resp. adh0.h1 = h0, h1,

andadif0.f1 = [f0, adi−1f0.f1], resp. adih0.h1 = h0, adi−1h0.h1.

We will see in the following chapters that equations (1.1.8)-(1.1.9) help us to formulatethe derivative of the switching function in a more general and important format.

1.2 Numerical Methods in Optimal Control

As we have mentioned, the numerical approaches in optimal control are usually dis-tinguished between two kinds: direct and indirect methods. Indirect methods consistof solving numerically the boundary value problem derived from the application of thePMP. Direct methods consist of solving a nonlinear optimization problem with con-straints yield from discretizing the state and the control.

1.2.1 Indirect methods

The indirect methods consist of numerically solving a boundary value problem obtainedby applying the PMP, by means of a shooting method. Here we recall two shootingmethods, the simple shooting method and the multiple shooting method. (See e.g.[105]). Consider in this section the problem P0.

Simple shooting method. According to (1.1.5), we may be able to write the optimalcontrol as a function of the state and the adjoint variable (x(t), p(t)). Let z(t) =(x(t), p(t)), the extremal system (1.1.4) can be written with the form z(t) = F (z(t)).The initial and nal conditions (1.1.2), the transversality conditions (1.1.7), and thetransversality condition of the Hamiltonian (1.1.6) can be written with the form ofR(z(0), z(tf ), tf ) = 0. Therefore, we get a two boundary value problem

z(t) = F (t, z(t)), R(z(0), z(tf ), tf ) = 0.

Let z(t, z0) be the solution of the Cauchy problem z(t) = F (t, z(t)), z(0) = z0. Thenthis two boundary value problem is equivalent to determine the zero of the equationR(z0, z(tf , z0), tf ) = 0. This problem can be solved by iterative methods, for example,a Newton type method.


Multiple shooting method. Compared with a single shooting method, the multipleshooting has a better numerical stability. It consists in dividing the interval [0, tf ] intoN subintervals [ti, ti+1] and in considering as unknowns the values of zi := (x(ti), p(ti))at the begining of each subinterval. The application of the PMP to the optimalcontrol problem yields a multi-point boundary value problem, consisting of ndingZ := (p(0), tf , zi), i = 1, · · · , N − 1 such that the dierential equation

zi(t) = F (t, z(t)) =

F0(t, z(t)), t0 ≤ t ≤ t1,

F1(t, z(t)), t1 ≤ t ≤ t2,

· · · ,FN−1(t, z(t)), tN−1 ≤ t ≤ tN ,

and the constraints

x(0) ∈M0, x(tf ) ∈M1, p(0) ⊥ Tx(0)M0,

p(tf )− p0 ∂g

∂x(tf , x(tf )) ⊥ Tx(tf )M1, H(tf ) = 0, z(t−i ) = z(t+i ), i = 1, · · · , N − 1,

are satised. This problem can also be solved by iterative methods. The nodes ofthe multiple shooting method may involve the switching times (at which the switchingfunction changes its sign), and the junction times (entry, contact, or exit times) with aboundary arcs. In this case it is required to have an a priori knowledge of the structureof the optimal solution.

In chapter 4, we implement our numerical resolution method with the single shoot-ing method, while in chapter 4, we use the multiple shooting method to improve thenumerical stability of our method. In the absense of any information on the optimalsolution, we implement the multiple shooting method with a regular subdivision, i.e.,ti = i

tfN

for i = 1, · · · , N − 1. However, we observe also that the node points shouldnot be too much, since the more node points are, the smaller the convergence domainof the shooting method is (due to the higher dimension of the shooting function).

1.2.2 Direct methods

There exist many direct methods, and they are the most evident methods when oneaddresses practically an optimal control problem. By discretizing both the state andthe control, the problem reduces to a nonlinear optimization problem in nite dimen-sion, or nonlinear programming problem. There exist many ways to carry out suchdiscretizations. The choice of the discretization method is depending on the problemunder consideration.

Consider a very simple way of such a discretization. Consider a subdivision 0 =t0 < t1 < · < tN = tf of the interval [0, tf ]. We discretize the controls such thatthey are piecewise constant on this subdivision with values in U . Meanwhile, choose a

26 1.3. Numerical Continuation

discretization of the dierential equations, for example an explicit Euler method. Thenby setting hi = ti+1 − ti, we get xi+1 = xi + hif(ti, xi, ui). Moreover, we discretize thecost by choosing a quadrature procedure. Then, this discretization reduces the optimalcontrol problem P0 to a nonlinear programming problem of the form

minC(x0, · · · , xN , u0, · · · , uN)|xi+1 = xi + hif(ti, xi, ui),

ui ∈ U, i = 1, · · · , N − 1, x0 ∈M0, xN ∈M1.

From a more general point of view, a nite dimensional representations of the controland of the state has to be chosen such that the dierential equation, the cost, and allconstraints under consideration can be expressed in a discrete way.

The numerical resolution of a nonlinear programming problem is standard. Forexample, one can use gradient methods, penalization, quasi-Newton, dual methods,etc. (see e.g. [9, 48, 61, 99]). There exist also many ecient optimization routines suchas IPOPT [108] and MUSCOD-II [37].

Note that there are many possible variants of direct methods such as the colloca-tion methods, the spectral or pseudo-spectral methods, the probabilistic approaches,etc. Another approach to optimal control problems that can be considered as a directmethod, consists of solving the Hamilton-Jacobi equation satised by the value func-tion which is of the form ∂S

∂t+ Hr(x,

∂S∂x

) = 0. The value function is the optimal costfor the optimal control problem of reaching a given point. See [94] for some numericalmethods.

In chapter 3, we propose a sub-optimal solution for the problem (MTTP) by using adirect method. The direct method is implemented with the modeling language AMPL(an automatic dierentiation software, see [42]) and an expert optimization routineIPOPT. With such tools, it is very simple to implement successfully with only few linesof code dicult (nonacademic) optimal control problems within a reasonable time ofcomputation. In chapter 4, for the problem (MTCP), we implement a direct methodthrough the powerful software BOCOP (see [12]), which provides an even easier way ofsolving optimal control problems with Ipopt solver and sparse exact derivatives com-puted by ADOL-C. Note that with BOCOP, one can also choose the time discretizationmethod such as Euler (explicit & implicit), Midpoint, Gauss II, Lobatto III C, etc.

1.3 Numerical Continuation

The basic idea of numerical continuation is to solve a dicult problem step by stepfrom a simpler problem by parameter deformation. The theory of the continuationmethod and many algorithms are well-developed (see e.g. [1, 87]). Here we introducebriey two continuation algorithms. The rst one is the discrete continuation which isthe simplest one that reveals the basic idea of the continuation. The second one is thePredictor-Corrector continuation (in short, PC continuation).


When combining with the shooting problem derived from the PMP, the methodconsists of deforming the problem into a simpler one (that we can easily solve) andthen of solving a series of shooting problems step by step to come back to the originalproblem. According to the research paper [104], the choice of the continuation param-eter may require considerable physical insight into the problem, since it is generallydone according to an intuition or the physical meaning of the system parameters. Thecontinuation parameter can be both a physical parameter or an articial one.

Let nh = dim(Z). Dene a continuation (a deformation)

G : Rnh × [0, 1] 7→ Rnh ,

such thatG(Z, 0) = G0(Z), G(Z, 1) = G1(Z),

where G1 : Rnh 7→ Rnh is the smooth map that one wants to know its zero points, andG0 : Rnh 7→ Rnh is a smooth map having known zero points.

One expects to follow a zero path starting from a point Z0 such that G(Z0, 0) = 0.A zero path is a curve c(s) ∈ G−1(0) where s represents the arc length.

Existence results We present briey some results of the existence of the zero paths.Local existence of the zero paths is answered by the implicit function theorem. Toensure the existence of the zero paths, some regularity assumptions are needed, as inthe following statement.

Theorem 1.3.1. (Existence of the zero paths) Let Ω be an open bounded subset ofRnh and let the mapping G : Ω× [0, 1] 7→ Rnh be continuously dierentiable such that:

• ∀(Z, λ) ∈ (Z, λ) ∈ Ω× [0, 1]|G(Z, λ) = 0, the Jacobian matrix

G′ = [∂G

∂Z1

, · · · , ∂G∂Znh

,∂G

∂λ],

is of maximum rank nh;• ∀Z ∈ Z ∈ Ω|G(Z, 0) = 0 ∪ Z ∈ Ω|G(Z, 1) = 0, the Jacobian matrix G′ =

[ ∂G∂Z1

, · · · , ∂G∂Znh

] is of maximum rank nh;

Then (Z, λ) ∈ Ω × [0, 1]|G(Z, λ) = 0 consists of the paths that is either a loop inΩ × [0, 1] or starts from a boundary point of Ω × [0, 1] and ends at another boundarypoint of Ω× [0, 1].

The proof is in [46] (Theorem 2.1 in the reference). The possible paths and impos-sible paths are shown in the Figure 1.1 (see [46, 49]). The zero path is dieomorphicto a circle or the real line.

Now let us explain the regularity assumptions from the optimal control point ofview. Consider the simplied problem P0 by assuming that M = Rn, M0 = x0,


0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 10

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Figure 1.1: Possible zero paths (left) and impossible zero paths (right).

M1 = x1 and U = Rm. Assume that the real parameter λ ∈ [0, 1] is increasedmonotonically from 0 to 1. Then, we are to solve a family of optimal control problemsparameterized by λ, i.e.,

minEx0,tf ,λ

(uλ)=x1

Ctf ,λ(u),

where E : Rn × R× L∞(0,+∞;R) is the end-point mapping (see Denition 1.1.1).According to the Lagrange multipliers rule, especially the rst order condition

(1.1.3), if uλ is optimal, then there exists (ψλ, ψ0λ) ∈ Rn × R\(0, 0) such that

ψλdEx0,tf ,λ(uλ) + ψ0λdCtf ,λ(u) = 0.

Assume that there are no minimizing abnormal extremals in the problem, then wecan assume ψ0

λ = −1 since (ψλ, ψ0λ) is dened up to a multiplicative scalar. Dene the

Lagrangian byLtf ,λ(u, ψ) := ψλEx0,tf ,λ(u)− Ctf ,λ(u),

then we seek (uλ, ψλ) such that

G(λ, u, ψ) :=

(ψλdEx0,tf ,λ(u)− dCtf ,λ(u),

Ex0,tf ,λ(u)− x1

)=

(∂Ltf ,λ

∂u(u, ψ),

Ex0,tf ,λ(u)− x1

)= 0.

Let (λ, uλ, ψλ) be a zero of G and assume that G is of class C1. Then accordingto the Theorem 1.3.1, we require the Jacobian of G with respect to (u, ψ) at point(λ, uλ, ψλ) to be invertible. More precisely, we have the Jacobian of G as(

Qtf ,λ dEx0,tf ,λ(u)∗

dEx0,tf ,λ(u) 0

), (1.3.1)

where Qtf ,λ is the Hessian∂2Ltf ,λ

∂2u(u, ψ, ψ0) restricted to ker

∂Ltf ,λ

∂u, and dEx0,tf ,λ(u)∗ is

the transpose of dEx0,tf ,λ(u).


It is easy to see that the matrix (1.3.1) is invertible i the linear mapping dEx0,tf ,λ(u)is surjective and the quadrative form Qtf ,λ is nondegenerate. By denition, the mappingdEx0,tf ,λ(u) is surjective means that the control u is not singular (see Denition 1.1.2).The non-degeneracy of Qtf ,λ is related to the concept of conjugate point (see Denition1.1.5).

In fact, Theorem 1.3.1 ensures local feasibility of the continuation method, but thepath may not be globally dened for ∀λ ∈ [0, 1]: it could reach some singularity orwander o to innity before reaching λ = 1.

The rst possibility can be eliminated by assuming the absence of minimizing sin-gular trajectory and of conjugate point over all the domain under consideration andfor every λ ∈ [0, 1]. Recall that the zero path can be parameterized by the arc legth s.Let c(s) := (Z(s), λ(s)) be the zero path such that G(c(s)) = 0. [26] indicates that ifλ = λ(s) is a turning point of order one (λ′(s) = 0, λ′′(s) 6= 0), then the corresponding -nal time tf is a conjugate time, and that the corresponding point Ex0,tf ,λ(u(x0, p0, tf , λ))is the conrresponding conjugate point. There the end-point mapping has been imple-mented with the exponential mapping Ex0,tf ,λ(u) := expx0,λ(tf , p0) with initial condition(x(0), p(0)) = (x0, p0).

Then for eliminating the second possibility, one needs to provide sucient conditionsensuring that the tracked paths remain bounded. This means that we have to ensurethat the initial adjoint vectors pλ(0) that are computed along the continuation procedureremain bounded, uniformly with respect to the parameter λ. Further, this means thatwe have to ensure the properness of the exponential mapping. According to [17, 106],if the exponential mapping is not proper, then there exists an abnormal minimizer. Bycontraposition, if one assumes the absence of minimizing abnormal extremals, then therequired boundedness follows.

1.3.1 Discrete Continuation

Denote the continuation parameter by λ, then by discretizing λ by 0 = λ0 < λ1 < · · · <λnl = 1 and solving a sequence of problems G(Z, λi) = 0, i = 1, · · · , nl, one may be ableto nd the zero points of G1(Z). The reason is that if the increment 4λ = λi+1 − λi issmall enough, then the solution Zi associated to λi such that G(Zi, λi) = 0 is generallyclose to the solution of G(Z, λi+1) = 0. The discrete continuation method is explainedin the Algorithm 1.

1.3.2 Predictor-Corrector continuation

Note that the parameter λ can be ill suited as a parametrization for the zero curve(Z, λ), and the arclength s which is a natural parameter for the curve can be a betterchoice of parametrization.

Let us parameterize the zero curve by arc length s and denote the zero curve bych(s) := (Z(s), λ(s)). Denote ∂G(Z(s),λ(s))

∂(Z,λ)and dch(s)

dsby JG and t(JG), respectively.


Result: The solution of the discrete continuationinitialization Z = Z0, λ0 = 0, 4λ ∈ (4λmin,4λmax);while λ ≤ 1 and 4λmin ≤ 4λ ≤ 4λmax do4λ = min(4λ, 1− λ);λ = λ+4λ;Find the solution Z such that G(Z, λ) = 0;if successful then

Z = Z;λ = λ;

else4λ = 4λ/2;

end

endif successful then

The discrete continuation is successful;else

The discrete continuation is failed;end

Algorithm 1: Discrete continuation

Dierentiating G(Z(s), λ(s)) = 0 with respect to s, we have

JG t(JG) = 0, ‖t(JG)‖ = 1, ch(Z(0), 0) = (Z(0), 0).

Assume that ch(s) is not critical and assume that we know a point of this curve(Z(si), λ(si)). Then we can predict a zero point (Z(si+1), λ(si+1)) by

(Z(si+1), λ(si+1)) = (Z(si), λ(si)) + hs t(JG), (1.3.2)

where hs is a given step size of s. When the step size hs is suciently small, thepoint (Z(si+1), λ(si+1)) may be very close to the solution point (Z(si+1), λ(si+1)) =G−1(ch(si+1)) = 0, and thus makes the Newton type iterative method (serves as acorrector) easier to converge. The basic idea of the PC methode is given in the followingAlgorithm 2.

In the chapter 4, the present PC continuation is used for solving the (MTCP). Inchapter 5, dierent from the dierential continuation proposed in [25] which consistsof carefully integrating the zero curve, we implement a Predictor-Corrector method(combined with the multiple shooting method) which is computational less heavy for ourproblem. As mentioned, the optimal solutions in our problem can contain a singular arcand thus the chattering arcs, thus the assumption of non minimizing singular trajectoryis not satised. To overcome this problem, we introduce the regularized problem, inwhich there is no minimizing singular trajectories. Then let the continuation procedure


Result: The solution of the PC continuationinitialization Z = Z0, hs > 0, λ0 = 0, 4λ ∈ (4λmin,4λmax);while λ ≤ 1 and 4λmin ≤ 4λ ≤ 4λmax do

(Predictor) Predict a point (Z, λ) according to (1.3.2);(Corrector) Find the solution (Z, λ) to G(Z, λ) = 0;if successful then

(Z, λ) = (Z, λ);else

Choose a new steplegth hs;end


The continuation is successful;else

The continuation is failed;end

Algorithm 2: Prediction-Corrector continuation

passing from the regularized problem, we are able to obtain either an optimal solutionof the (MTCP) when the singular arc is absent, or a sub-optimal solution when asingular arc appears.

Chapter 2

Problem Formulation and Chattering

Contents

2.1 Physical Problem and Model . . . . . . . . . . . . . . . . . . 33

2.1.1 Coordinate Systems . . . . . . . . . . . . . . . . . . . . . . . 33

2.1.2 Attitude Movement . . . . . . . . . . . . . . . . . . . . . . . . 35

2.1.3 Trajectory Movement . . . . . . . . . . . . . . . . . . . . . . 37

2.1.4 Equations in the Orbit Frame . . . . . . . . . . . . . . . . . . 39

2.2 Optimal Control Problems . . . . . . . . . . . . . . . . . . . . 40

2.2.1 Minimum Time Tilting Problem (MTTP) . . . . . . . . . . 40

2.2.2 Minimum Time Control Problem (MTCP) . . . . . . . . . . 41

2.3 Chattering phenomenon . . . . . . . . . . . . . . . . . . . . . 42

The full physical model of our problem is a six-degree-of freedom (6DOF) dynamicalsystem. However, it is too complicated to analyse. Therefore, we simplify the fullphysical model by adopting some reasonable assumptions. With the simplied model,we formulate the minimum time control problem (MTCP). (MTCP) has a controlane system of dimension eight with a control of dimension two taking values in theunit Euclidean disk. In order to have a better understanding of the optimal solutionsof (MTCP), we simplied again the problem to the minimum time tilting problem(MTTP), which is a planar version of (MTCP).

In this chapter, we rst establish the physical model of our problem, then we for-mulate the optimal control problems (MTTP) and (MTCP). The analysis of thesetwo problems are given in the next two chapters. We recall also the Fuller's problem,which is a very well-known example in which the optimal solution is chattering.

32

Chapter 2: Problem Formulation and Chattering 33

2.1 Physical Problem and Model

In this section, we rst dene the coordinates systems, and then we establish the equa-tions of the attitude movement and of the trajectory movement. The model consists ofeleven ordinary dierential equations: three for the components of the position vector,three for the components of the velocity vector, three for the Euler angles and two forthe components of the angular velocity vector.

2.1.1 Coordinate Systems

Throughout the manuscript, we make the following assumptions:

• The Earth is a sphere and is xed in the inertial space, i.e., the angular velocity~ωei of the Earth is zero.• The LV engine cannot be shut o during the ight and the module of the thrustforce is constant, taking its maximum value, i.e., T = Tmax.• The mass m of the LV remains constant during the maneuver.

The unit single-axis rotation maps Ri(σ): R → R3×3, for σ ∈ R, i = x, y, z aredened by

Rx(σ) =

1 0 00 cosσ sinσ0 − sinσ cosσ

, Ry(σ) =

cosσ 0 − sinσ0 1 0

sinσ 0 cosσ

,

Rz(σ) =

cosσ sinσ 0− sinσ cosσ 0

0 0 1

.

For a given vector x ∈ R3, taking Ri(σ)x means to rotate the vector x with respectto the axis i by an angle of σ. With this denition, we next introduce the coordinateframes that will be used throughout the manuscript.

The Earth frame SE = (xE, yE, zE) is xed around the center of the Earth O. Theaxis zE points to the North pole, and the axis xE is in the equatorial plan of the Earthpointing to the equinox.

The launch frame (reference frame) SR = (xR, yR, zR) is xed around the launchpoint OR (where the maneuver starts). The axis xR is normal to the local tangentplane, pointing to the launch direction, and the axis zR points to the North. As shownin Figure 2.1 (a), the launch frame is derived from the Earth frame by two ordered unitsingle-axis rotations Rz(`r1) and Ry(−`r2),

SERz(`r1)−−−−→ Ry(−`r2)−−−−−→ SR

where `r1 and `r2 are the longitude and latitude of the launch point, respectively.

34 2.1. Physical Problem and Model

Figure 2.1: Coordinate systems and relations.

The orbit frame So = (xo, yo, zo) is xed around the mass center of LV. The axisxo is normal to the local tangent plane and the axis zo is pointing to the direction ofthe velocity of the LV. As shown in Figure 2.1 (b), the orbit frame can be derived fromthe launch frame SR through rotating around the yR axis by angles òi, i = 1, 2, 3, i.e.,

SRRx(−ò1)−−−−−→ · Ry(−ò2)−−−−−→ So.

The transformation matrix from SR to So is LoR = Ry(−ò2)Rx(−ò1). Let ~r be thevector from the origin point OR of the frame SR towards the center point Ob of the LV,and denote its projection in SR by (rx, ry, rz)

>. Then, according to Figure 2.1, we have

tan ò1 =ryrx, tan ò2 =

√r2y + r2

z

rx +RE

,

where RE is the radius of the Earth. The advantage of using this frame is that thegravity acceleration vector ~g points always to −xo direction. Actually, the vector ~gdepends on the position of the LV. More precisely, in the So frame, we have (~g)o =

(−g, 0, 0)> where g = ‖~g‖. Setting ~RE = (RE, 0, 0)>, we dene ~rd(t) := ~r(t) + ~RE, andthus (~g)R is given by

(~g(~r))R = LRo(~g0‖~rd(0)‖‖~rd(t)‖

)o, (2.1.1)

where ~g0 is the gravity acceleration at the starting point of the maneuver (origin pointof the frame SR).

The body frame Sb = (xb, yb, zb) is dened as follows. The origin of the frame Ob

is xed around the mass center of the LV, the axis zb is along the longitudinal axis ofthe LV, and the axis xb is in the cross-section (see Figure 2.2 (a)). The body frame canbe derived by three ordered unit single-axis rotations from the launch frame, as shownin Figure 2.2 (b),

SRRy(θ)−−−→ Rx(ψ)−−−→ Rz(φ)−−−→ Sb


where θ is the pitch angle, ψ is the yaw angle and φ is the roll angle. Therefore, thetransformation matrix from SR to Sb is

LbR =Rz(φ)Rx(ψ)Ry(θ)

=

cos θ cosφ+ sin θ sinψ sinφ cosψ sinφ − sin θ cosφ+ cos θ sinψ sinφ− cos θ sinφ+ sin θ sinψ cosφ cosψ cosφ sin θ sinφ+ cos θ sinψ cosφ

sin θ cosψ − sinψ cos θ cosψ

,

(2.1.2)

and the transformation matrix from Sb to SR is LRb = L−1bR = L>bR.

The velocity frame Sv is xed around the mass center of LV. The axis xv is parallelto the velocity vector, and the axis zv is normal to the velocity, pointing to the directionof the lift force ~L. This frame can be derived by two unit single-axis rotations from thelaunch frame (see Figure 2.1 (c)),

SRRx(κ)−−−→ Ry(ξ)−−−→ Sv.

We consider the projection of ~v in the SR frame, (~v) = (vx, vy, vz)>. Since vx =

v cos ξ, vy = v sin ξ sinκ, vz = −v sin ξ cosκ, we have cos ξ = vx/v, and tanκ = −vy/vzif sin ξ 6= 0. For convenience of setting initial values in the numerical simulations, wedene θv and ψv as the pitch" and yaw" angles of the velocity vector, and thus wehave vx = v sin θv cosψv, vy = −v sinψv, and vz = v cos θv cosψv. Then, we easily getthat tan θv = 1/ tan ξ cosκ and sinψv = − sin ξ sinκ. Moreover, the transfer matrixfrom Sv to SR is LRv = L>Rv = Rx(κ)>Ry(ξ)

>.

Throughout the manuscript, for any vector x ∈ R3, (x)w is the projection of thisvector in the frame Sw ∈ SE, SR, SoSb, Sv. For example, the index (·)b means thatthe vector is projected into the body frame Sb.

2.1.2 Attitude Movement

Attitude dynamic equations The attitude dynamics is written in vectorial formin the body frame Sb as

d

dt(I~ω)b = −(~ω)b ∧ (I~ω)b + ( ~M)b, (2.1.3)

where I is the inertia matrix, ~ω is the absolute angular velocity vector, i.e., the angularvelocity with respect to the inertial space, and ~M is the control torque introduced bythe LV thrust.

Setting (I)b = diag(Ix, Iy, Iz), (~ω)b = (ωx, ωy, ωz)>, and ( ~M)b = (Mx,My,Mz)

>,


Figure 2.2: Thrust in the body Frame

equation (2.1.3) gives Ixωx = (Iy − Iz)ωyωz +Mx,

Iyωy = (Iz − Ix)ωxωz +My,

Izωz = (Ix − Iy)ωxωy +Mz.

(2.1.4)

The control torque ~M is the cross product of the thrust vector ~T and its momentarm ~L. The moment arm is the vector from the center of mass Ob to the force actingpoint OF , given here by (~L)b = (0, 0,−l)>, where l is the distance between the pointsOb and OF . Moreover, as shown in Figure 2.2 (a), the thrust force vector is

(~T )b = (−T sinµ cos ζ,−T sinµ sin ζ, T cosµ)>, (2.1.5)

where T = Tmax, µ ∈ [0, µmax], and ζ ∈ [−π, π]. The control torque is then

( ~M)b = (~L)b ∧ (~T )b = (−T l sinµ sin ζ, T l sinµ cos ζ, 0)>.

Assume that Ix = Iy, ωz(0) = 0, and let b = Tmaxl/Ix. Then (2.1.4) gives

ωx = −b sinµ sin ζ, ωy = b sinµ cos ζ,

with ωz(t) ≡ 0.Let µmax denote the maximum allowed value of the angle µ. According to the

parameters of the LV engine, µmax is less than 10 and thus the error between sinµ andµ is less than 0.5%. Therefore, in the model we make the approximation sinµ ' µ andwe dene u1 = µ cos ζ, and u2 = µ sin ζ and µ = µ/µmax with b = bµmax. Hence

ωx = −bu2, ωy = bu1. (2.1.6)

Note that since µ ∈ [0, µmax], we have u21 + u2

2 ≤ µ2 ∈ [0, 1].


Attitude kinematics equations Since ~ω is the angular velocity vector of the LVwith respect to the inertial space, it is equal to the sum of the angular velocity ~ωbRof the LV with respect to the launch frame, of the angular velocity ~ωRE of the launchframe with respect to the Earth frame, of the angular velocity ~ωEe of the Earth framewith respect to the Earth, and of the angular velocity ~ωei of the Earth with respect tothe inertial space. According to the rst assumption and denitions of the frames, it iseasy to see that the last three terms are zero, and thus ~ω = ~ωbR. Therefore, based onthe denition of the body frame, the relationship between angular velocity and Eulerangles (with respect to the launch frame SR) areωxωy

ωz

= LbR

0

θ0

+

ψ cosφ

−ψ sinφ0

+

00

φ

,

where LbR is given by (2.1.2). Then the equations of the attitude kinematics are

θ = (ωx sinφ+ ωy cosφ)/ cosψ,

ψ = ωx cosφ− ωy sinφ,

φ = (ωx sinφ+ ωy cosφ) tanψ.

(2.1.7)

Therefore, the two equations of (2.1.6) and the three equations of (2.1.7) describe theattitude movement.

Remark 2.1.1. If ψ = π/2+kπ, k ∈ N, then the Euler angles dened above are notwell dened (usual singularities of the Euler angles). We assume in this manuscript thatthe maneuvers are small enough, so that these singularities will not be encountered.1

2.1.3 Trajectory Movement

To describe the trajectory movement, we need the trajectory kinematics equation, whichcan be written in vectorial form by

(~r)R = (~v)R,

where ~v is the velocity vector with respect to the launch frame SR. This equation canbe rewritten as

rx = vx, ry = vy, rz = vz. (2.1.8)

Note that ~v is a relative velocity, the equation of the trajectory dynamics in vectorial

1In fact, for a LV, the attitude maneuvers are generally planar, i.e., ψ(0) ≈ 0, ψ(tf ) ≈ 0, and nosingularities occur. Moreover, even though a singularity would occur, we could handle the situation bychoosing a new launch frame S′

R in which the new yaw angle is close to zero, i.e., ψ′(0) ≈ 0, ψ′(tf ) ≈ 0.


form is written by

d(m~v)

dt=m~g + ~T + ~TA − 2m~ωR ∧ ~v −m~ωR ∧ (~ωR ∧ ~r), (2.1.9)

where ~ωR is the angular velocity of the launch frame SR with respect to the inertialspace. The fourth term of the right-hand side of (2.1.9) is due to the rotation of SR,the fth term is the Coriolis force, and the last term is the centrifugal force. Thesethree terms are equal to zero because the launch frame SR is xed in the inertial frame.According to the assumptions and the denition of frame SR, these three terms equalto zero because the launch frame SR does not rotate with respect to the inertial space.Then, in the launch frame SR, equation (2.1.9) takes the form

md(~v)Rdt

= m(~g)R + LRb(~T )b + LRv(~TA)v, (2.1.10)

where (~T )b is given by (2.1.5) and (~g)R is calculated by (2.1.1). Note that, in general,the aerodynamical force ~TA, which consists of the lift force and the drag force, is rstprojected into the velocity frame Sv and then into frame SR with LRb. We considerexponential atmospheric density model. Then, the drag and lift forces are expressed inthe velocity frame Sv by ( ~D)v = (−1

2ρv2SCx, 0, 0)>, (~L)v = (0, 0, 1

2ρv2SCz)

>, where Sis the reference surface, Cx and Cz are aerodynamical coecients depending mostly onthe angle of attack α and the relative velocity of the LV with respect to the wind. Theangle of attack α, when the air wind is set to zero, is dened as the angle between thevelocity and the LV's longitudinal axis. The aerodynamic coecients are approximatedby Cx = Cx0 + Cxαα

2, Cz = Cz0 + Czαα, where Cx0, Cxα, Cz0 and Czα are constantcoecients. Then, by using the transfer matrix LRv, we get

( ~D)R = (Dx, Dy, Dz)> =

1

2ρSCxv

2(− cos ξ,− sin ξ sinκ, sin ξ cosκ)>,

(~L)R = (Lx, Ly, Lz)> =

1

2ρSCzv

2(sin ξ,− cos ξ sinκ, cos ξ cosκ)>.(2.1.11)

Finally, the equation of the trajectory dynamics (2.1.10) becomes

vx = a sin θ cosψ + gx + (Dx + Lx)/m,

vy = −a sinψ + gy + (Dy + Ly)/m,

vz = a cos θ cosψ + gz + (Dz + Lz)/m,

(2.1.12)

where a = Tmax/m is constant.


2.1.4 Equations in the Orbit Frame

Attitude kinematics. Recall that the Euler angles in equation (2.1.7) are the Eulerangles dened between Sb and SR. We can also dene by θr, ψr, φr the Euler angles byrotating the orbit frame So to the body frame Sb, i.e.,

SoRy(θr)−−−−→ Rx(ψr)−−−−→ Rz(φr)−−−−→ Sb.

Then, similarly we can derive the attitude kinematics

θr = (ωx sinφ+ ωy cosφ)/ cosψ + ωoRx,

ψr = ωx cosφ− ωy sinφ+ ωoRy,

φr = (ωx sinφ+ ωy cosφ) tanψ + ωoRz,

(2.1.13)

where (~ωoR)b = (ωoRx, ωoRy, ωoRz)> is the angular velocity of the orbit frame So with

respect to the launch frame SR, expressed in the body frame Sb.

Trajectory dynamics. We can also express the trajectory dynamics equation (2.1.9)in the orbit frame So by

d(m~v)odt

=(m~g)o + Lob(~T )b + Lov(~TA)v +m(~ωo ∧ ~v)o, (2.1.14)

where ~ωo is the angular velocity of the orbit frame with respect to the inertial space,L>ob = Lbo = Rz(φr)Rx(ψr)Ry(θr), and Lov = LobLbRLRv.

Planar tilting model. Assume that the aerodynamical forces are equal to zero,the gravity acceleration vector is constant, and ψr ≡ 0, φr ≡ 0. Dene (~v)o =(vrx, vry, vrz)

>, then we can get from (2.1.13) and (2.1.14) that

vrx = a sin θr + cv2rz − g0,

vrz = a cos θr − cvrxvrz,θr = ωy − cvrz,

where c = 1/‖~rd(0)‖ is assumed to be a positive constant. Therefore, together with theattitude dynamics, we have the model for the planar tilting problem

vrz = a cos θr − cvrxvrz,vrx = a sin θr + cv2

rz − g0,

θr = ωy − cvrx,ωy = bu1.

(2.1.15)

40 2.2. Optimal Control Problems

In the launch frame SR we have a slightly simpler model (with c = 0) due to the factthat the directional change of the gravity acceleration vector is ignored in that case.Hence, the model (2.1.15) is indeed more precise.

2.2 Optimal Control Problems

2.2.1 Minimum Time Tilting Problem (MTTP)

The minimum time planar tilting problem of a spacecraft consists of controlling thespacecraft, with certain prescribed terminal conditions on the attitude angles, the ac-celerations, and the velocity direction, while minimizing the maneuver time and keepingconstant the yaw and rotation angles.

Let us formulate the minimum time planar tilting maneuver problem (pitching move-ment of the LV). In this problem, we restrict our study to the planar case, in the sensethat the LV movement remains in a plane.

For convenience, we set x1 = vrz, x2 = vrx, x3 = θr, x4 = ωy and u = u1. Denotingby x = (x1, x2, x3, x4) the state variable, the system (2.1.15) can be written as thesingle-input control-ane system

x = f0(x) + uf1(x), (2.2.1)

where f0 and f1 are the smooth vector elds on R4 dened by

f0 = (a cosx3 − cx1x2)∂

∂x1

+ (a sinx3 + cx21 − g0)

∂

∂x2

+ (x4 − cx1)∂

∂x3

,

f1 = b∂

∂x4

.

(2.2.2)

The terminal conditions are as follows: (i) all initial variables are xed; (ii) the nalvalues of the variables x3 and x4 are prescribed, and we require that, at the nal timetf (which is let free), the velocity vector ~v(tf ) be parallel to the longitudinal axis zb(tf )of the LV.

Dene the ight path angle γ := arctan(x2/x1), and since v =√x2

1 + x22, we have

γ = (a sin(x3 − γ)− g0 cos γ)/v + cv cos γ, v = a cos(x3 − γ)− g0 sin γ. (2.2.3)

The nal condition (~v(tf ))R ∧ (zb(tf ))R = ~0 is then written as γ(tf ) = x3(tf ). In termof v and γ, the velocity components x1 and x2 are x1 = v cos γ and x2 = v sin γ. Weset v(0) = v0 and γ(0) = γ0.

Minimum time planar tilting problem. Let x0 ∈ R4, and let v0, γ0, x30, x40 andx3f be real numbers. In terms of the variables x = (x1, x2, x3, x4), the initial point is


a b c vm (m/s) ωmax (rad/s) θmax(rad)Value 12 0.02 1× 10−6 5000 0.3 π

Table 2.1: System parameters.

dened by x0 = (x10, x20, x30, x40), with x10 = v0 cos γ0 and x20 = v0 sin γ0, and the naltarget is the submanifold of R4 dened by

M1 = (x1, x2, x3, x4) ∈ R4 | x2 cosx3f − x1 sinx3f = 0, x3 = x3f , x4 = 0. (2.2.4)

We consider the optimal control problem, denoted in short (MTTP), of steering thecontrol system (2.2.1) from x(0) = x0 to the nal target M1 in minimal time tf , underthe control constraint u(t) ∈ [−1, 1].

Note that we study this problem with the parameters of Ariane 5 launchers (seeTable 2.1). The modulus of the velocity v takes values in [0, vm], and for the pitchangle and the angular velocity we have the estimate |x3| ≤ θmax and |x4| ≤ ωmax.

2.2.2 Minimum Time Control Problem (MTCP)

In this problem, we assume that the gravity acceleration ~g is constant, i.e., ~g = ~g0, and~L = ~D = ~0. Moreover, since we do not have nal conditions on the position vector ~r,we can consider only the equations of the attitude movement (2.1.6)-(2.1.7) and of thetrajectory dynamics (2.1.12), i.e.,

vx = a sin θ cosψ + gx, vy = −a sinψ + gy, vz = a cos θ cosψ + gz,

θ = (ωx sinφ+ ωy cosφ)/ cosψ, ψ = ωx cosφ− ωy sinφ,

φ = (ωx sinφ+ ωy cosφ) tanψ, ωx = −bu2, ωy = bu1.

(2.2.5)

Dening the state variable x = (vx, vy, vz, θ, ψ, φ, ωx, ωy), we write the system (??) asthe bi-input control-ane system

x = f(x) + u1g1(x) + u2g2(x), (2.2.6)

where the controls u1 and u2 satisfy the constraint u21 + u2

2 ≤ 1, and the vector elds f ,g1 and g2 are dened by

f =(a sin θ cosψ + gx)∂

∂vx+ (−a sinψ + gy)

∂

∂vy+ (a cos θ cosψ + gz)

∂

∂vz

+ (ωx sinφ+ ωy cosφ)/ cosψ∂

∂θ+ (ωx cosφ− ωy sinφ)

∂

∂ψ

+ tanψ(ωx sinφ+ ωy cosφ)∂

∂φ, g1 = b

∂

∂ωy, g2 = −b ∂

∂ωx.

(2.2.7)

42 2.3. Chattering phenomenon

Let vx0 , vy0 , vz0 , θ0, ψ0, φ0, ωx0 , ωy0 , θf , ψf , φf , ωxf and ωyf be real numbers. Theinitial conditions are xed to

vx(0) = vx0 , vy(0) = vy0 , vz(0) = vz0 ,

θ(0) = θ0, ψ(0) = ψ0, φ(0) = φ0, ωx(0) = ωx0 , ωy(0) = ωy0 .(2.2.8)

The desired nal velocity is also required to be parallel to the axis zb, according to(~v(tf ))R ∧ (zb(tf ))R = ~0. Therefore, the nal conditions are

vzf sinψf + vyf cos θf cosψf = 0, vzf sin θf − vxf cos θf = 0,

θ(tf ) = θf , ψ(tf ) = ψf , φ(tf ) = φf , ωx(tf ) = ωxf , ωy(tf ) = ωyf .(2.2.9)

The parallel condition (~v(tf ))R∧(zb(tf ))R = ~0 refers to the rst two equalities of (2.2.9).

Remark 2.2.1. It is indeed natural to consider ~v(tf )//zb(tf ) as a terminal condi-tion, because the LVs are usually planned to maintain a small angle of attack along theight. The zero angle of attack condition ensures that the aerolift is null in order toavoid excessive loading of the structure (see e.g. [8]).

Minimum time control problem (MTCP). We set

x0 = (vx0 , vy0 , vz0 , θ0, ψ0, φ0, ωx0 , ωy0) ∈ R8,

and we dene the target set (submanifold of R8)

M1 = (vx, vy, vz, θ, ψ, φ, ωx, ωy) ∈ R8 | vz sinψf + vy cos θf cosψf = 0,

vz sinψf + vy cos θf cosψf = 0, θ = θf , ψ = ψf , φ = φf ,

ωx = ωxf , ωy = ωyf.(2.2.10)

The minimum time control problem (MTCP) consists of steering the bi-input control-ane system (2.2.6) from x(0) = x0 to the nal target M1 in minimum time tf , withcontrols satisfying the constraint u2

1 + u22 ≤ 1.

2.3 Chattering phenomenon

Let us recall that we speak of a chattering phenomenon (sometimes also called a Fuller'sphenomenon), when the optimal control switches an innite number of times over acompact time interval. It is well known that, if the optimal trajectory of a givenoptimal control problem involves a singular arc of higher order, then no connectionwith a bang arc is possible and the bang arcs asymptotically joining the singular arcmust chatter. On Figure 2.3(b), the control is singular over (t1, t2), and the control u(t)with t ∈ (t1−ε1, t1)∪(t2, t2+ε2), ε1 > 0, ε2 > 0 is chattering. The corresponding optimaltrajectory is called a chattering trajectory. On Figure 2.3(a), the chattering trajectory


oscillates around the singular part and nally gets o" the singular trajectory withan innite number of switchings.

(a) Chattering trajectory

singular part

chattering parts

(b) Chattering control

u

tt1 t2

x(t1)x(t2)

Figure 2.3: An illustration of chattering phenomenon.

To better explain the chattering phenomenon, we recall the well-known Fuller'sproblem (see [43, 73]), which is the optimal control problem

min

∫ tf

0

x1(t)2 dt,

x1(t) = x2(t), x2(t) = u(t), |u(t)| ≤ 1,

x1(0) = x10, x2(0) = x20, x1(tf ) = 0, x2(tf ) = 0, tf free.

We dene ξ =(√

33−124

)1/2

as the unique positive root of the equation ξ4+ξ2/12−1/18 =

0, and we dene the sets

Γ+ = (x1, x2) ∈ R2 |x1 = ξx22, x2 < 0,

R+ = (x1, x2) ∈ R2 | x1 < −sign(x2)ξx22,

Γ− = (x1, x2) ∈ R2 |x1 = −ξx22, x2 > 0,

R− = (x1, x2) ∈ R2 | x1 > −sign(x2)ξx22.

Then the optimal synthesis of the Fuller's problem is the following (see [43, 93, 110]).The optimal control is given in feedback form by

u∗ =

1 if x ∈ R+

⋃Γ+,

−1 if x ∈ R−⋃

Γ−.

The control switches from u = 1 to u = −1 at points on Γ− and from u = −1 tou = 1 at points on Γ+. The corresponding trajectories crossing the switching curves Γ±transversally are chattering arcs with an innite number of switchings that accumulatewith a geometric progression at the nal time tf > 0.

The optimal synthesis for the Fuller's problem is drawn on Figure 2.4. The solutions

44 2.3. Chattering phenomenon

of the Fuller's problem are chattering solutions since they switch transversally on theswitching curves Γ± until nally reaching the target point on the singular surface denedby the union of all singular solutions. In fact, the optimal control of the Fuller's problem,

−5 0 5−5

0

5

x1

x2

u=1

Γ−

u = −1

Γ+

An optimal trajectory

Figure 2.4: Optimal synthesis for the Fuller's problem.

denoted as u∗, contains a countable set of switchings of the form

u∗(t) =

1 if t ∈ [t2k, t2k+1),

−1 if t ∈ [t2k+1, t2k+2],

where tkk∈N is a set of switching times that satises (ti+2 − ti+1) < (ti+1 − ti), i ∈ Nand converges to tf < +∞. This means that the chattering arcs contain an innitenumber of switchings within a nite time interval tf > 0.

In (MTTP), the chattering phenomenon is dened the same as in the Fuller'sproblem. However in (MTCP) where the control input is of dimension two, we denethe chattering in a slightly dierent way, see details in the fourth chapter.

Chapter 3

Minimum Time Tilting Problem

(MTTP)

Contents

3.1 Geometric Analysis of Chattering . . . . . . . . . . . . . . . . 48

3.1.1 Application of the Pontryagin maximum principle . . . . . . . 48

3.1.2 Computation of Singular arcs . . . . . . . . . . . . . . . . . . 49

3.1.3 Geometric Analysis of the Chattering Phenomenon . . . . . . 53

3.2 Application to (MTTP) . . . . . . . . . . . . . . . . . . . . . . 59

3.2.1 Extremal Equations . . . . . . . . . . . . . . . . . . . . . . . 59

3.2.2 Lie Bracket Conguration of the System . . . . . . . . . . . . 60

3.2.3 Singular Extremals . . . . . . . . . . . . . . . . . . . . . . . . 60

3.2.4 Optimality Status of Chattering Extremals . . . . . . . . . . 64

3.3 Chattering Prediction . . . . . . . . . . . . . . . . . . . . . . . 69

3.3.1 Flat-Earth Case c = 0 . . . . . . . . . . . . . . . . . . . . . . 69

3.3.2 Non-Flat Case c > 0 . . . . . . . . . . . . . . . . . . . . . . . 75

3.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . 77

3.4.1 Chattering Prediction . . . . . . . . . . . . . . . . . . . . . . 78

3.4.2 Sub-Optimal Strategies . . . . . . . . . . . . . . . . . . . . . 83

Introduction

State of the art (MTTP) under consideration in this chapter is however more re-lated to the well-knownMarkov-Dubins problem (in short, MD problem) and to variants

45

46

of it. Indeed, if the system were to be directly controlled by the variable x3, then, bytaking the target manifold to be a single point (x(tf ) = xf ) and letting a = b = 1,c = 0, g0 = 0, the system (2.2.1) would be written as

x1 = cosx3, x2 = sinx3, x3 = u,

and therefore, (MTTP) coincides with the MD problem, which was rst settled in [75]and was analyzed in detail by Dubins and many others (see, e.g., [38, 86, 100]). It hasbeen shown that the optimal strategy for the MD problem consists in rst reachingthe singular arc with a single bang arc, then, in following this singular arc until one issuciently close to the nal target, and nally, in leaving the singular arc in order toreach the target with a single bang arc. Recall that the cost for the MD problem is theminimum time.

If we assume that g0 6= 0, i.e., if we have the system

x1 = cosx3, x2 = sinx3 − g0, x3 = u,

then (MTTP) coincides with the Zermelo-Markov-Dubins problem (in short, ZMDproblem) with constant wind eld (wx, wy) = (0,−g0) (see, e.g., [4, 79, 83, 102]). Theoptimal strategy of this problem consists of a nite number of bang and singular arcs.Both the MD and the ZMD problems may involve a singular arc because the singularcontrols of these problems are of intrinsic order one (see further in the present chapterfor this notion).

However, this is not the case for (MTTP) for which the singular control is ofintrinsic order two. In this sense, a problem closer to (MTTP) (with a = b = 1,c = 0, g0 = 0) is the Markov-Dubins problem with angular acceleration control (inshort, MDPAAC) (see [67, 101]). In that problem, the model is a dynamic extensionof the MD system, given by

x1 = cosx3, x2 = sinx3, x3 = x4, x4 = u,

The existence of a chattering phenomenon for MDPAAC was rst put in evidence in[101]. Although the optimality status of these chattering arcs remains unclear, thediscussion of the chattering phenomenon brings interesting issues for the analysis of thepresent (MTTP).

The system we consider here can also be seen as a variation of the MD system, withnonconstant wind and controlled by the inertial control. Thus, we expect the solutionof (MTTP) to share properties similar to MDPAAC (in particular, chattering), MDand ZMD (in view of the global behavior of the solution).

In fact, using [112], we will be able to prove the existence and the optimality of thechattering phenomenon in (MTTP). The chattering phenomenon (also occurring inMDPAAC) is caused by singular controls of intrinsic order two. It makes the optimalsynthesis for (MTTP) essentially dierent from that of the MD or ZMD problem.

Chapter 3: Minimum Time Tilting Problem (MTTP) 47

However, in some sense the optimal solution of (MTTP) consists as well of threepieces: the rst piece consists of bang arcs to reach the singular arc, the second piece isa singular arc, and the third piece consists of a succession of bang arcs nally reachingthe target submanifold.

Since the chattering phenomenon causes diculties in practical use, we will alsoprovide sucient conditions on the terminal conditions, under which the chatteringarcs do not appear in the optimal solution. This prediction result will be useful inorder to decide which numerical method (either direct, or indirect, or sub-optimal) isthe most appropriate.

The analysis of chattering arcs is challenging. Based on a careful analysis of theFuller problem, M.I. Zelikin and V.F. Borisov obtained a geometric portrait of solutionsin the vicinity of the second order singular solutions (see [112, 113]). These solutionsare called chattering solutions. Using their results, we will be able to prove rigorouslythe existence and optimality of chattering solutions in our (MTTP).

The basic idea of their approach to provide sucient conditions for optimality isbased on the following well-known sucient optimality condition (see e.g. [2]):

Let M be a smooth manifold of dimension n, and let T ∗M be its cotangentbundle, endowed with its canonical symplectic structure. If a submanifold Lof T ∗M generated by a given Hamiltonian system on T ∗M is Lagrangian,then a nice regular projection of trajectories of L onto M can also beseen, by canonical injection, as a Lagrangian submanifold of T ∗M , and thetrajectories are locally optimal in C0 topology.

Recall that a submanifold L of a smooth manifold M is said to be Lagrangianif∮γp dx = 0 for every piecewise smooth closed contour γ on the manifold. Hence,

the manifold consisting of the solutions of a Hamiltonian system with transversalitycondition (p dx = 0 on the target manifold) is Lagrangian.

Hence, the problem of proving the local optimality of a solution comes down toconstructing a Lagrangian submanifold. The usual way to construct a Lagrangiansubmanifold is to integrate backward in time the Hamiltonian system from the targetpoint. However, this is not applicable for the chattering arcs because the control is notanymore piecewise constant and the length of switching intervals goes to zero at thesingular junction.

In order to overcome this aw of the usual approach, M.I. Zelikin and V.F. Borisovproposed an explicit procedure to construct Lagrangian submanifolds lled by chatter-ing trajectories. The main diculty of this construction procedure is to analyze theregularity of the projections of the extremal lifts to the state space.

Structure of the chapter The chapter is structured as follows.In Sections 3.1.1 and 3.1.2, the PMP is applied and an usual way to compute

singular controls are recalled. Section 3.1.3 is devoted to recall some results of [112, 113],

48 3.1. Geometric Analysis of Chattering

explaining geometric features of the chattering phenomenon, based on a semi-canonicalform of the Hamiltonian system (see Section 3.1.3) along singular extremals of order two,with the objective of showing how these theoretical results can be applied in practice.

The non-singular (bang-bang) extremals of (MTTP) are analyzed in Section 3.2.1,and the Lie bracket conguration is given in 3.2.2. We prove in Section 3.2.3 that thesingular controls for (MTTP) are of intrinsic order two, which implies the existenceof chattering arcs. Based on the results of M.I. Zelikin and V.F. Borisov, we prove inSection 3.2.4 that the optimal chattering arcs of (MTTP) are locally optimal in C0

topology.In Section 3.3, we provide, for the cases with c = 0 and c > 0 respectively, sucient

conditions on the terminal values under which the optimal solutions do not containany singular arc, and do not chatter. Numerical simulations, in Section 3.4.1, illustratethese conditions.

Since chattering is not desirable in view of practical issues, we propose some sub-optimal strategies in Section 3.4.2, by approximating the chattering control with piece-wise constant controls. Our numerical results provide evidence of the convergence ofsub-optimal solutions to optimal solutions (but this convergence is not analyzed fromthe theoretical point of view in the present chapter).

3.1 Geometric Analysis of Chattering

Let M be a smooth manifold of dimension n, and let M1 be a submanifold of M . Weconsider on M the minimal time control problem

min

∫ tf

0

1dt,

x(t) = f0(x(t)) + u(t)f1(x(t)), |u(t)| ≤ 1,

x(0) = x0, x(tf ) ∈M1, tf ≥ 0 free,

(3.1.1)

where f0 and f1 are two smooth vector elds on M . Since the system and the in-stantaneous cost are control-ane, and the control constraint is compact and convex,according to classical results (see, e.g., [28, 105]), there exists at least one optimalsolution (x(·), u(·)), dened on [0, tf ].

3.1.1 Application of the Pontryagin maximum principle

According to the PMP (see Theorem 1.1.1), there must exist an absolutely continuousmapping p(·) dened on [0, tf ] (called adjoint vector), such that p(t) ∈ T ∗x(t)M for everyt ∈ [0, tf ], and a real number p0 ≤ 0, with (p(·), p0) 6= 0, such that

x(t) =∂H

∂p(x(t), p(t), p0, u(t)), p(t) = −∂H

∂x(x(t), p(t), p0, u(t)), (3.1.2)


almost everywhere on [0, tf ], where

H(x, p, p0, u) = 〈p, f0(x)〉+ u〈p, f1(x)〉+ p0 (3.1.3)

is the Hamiltonian of the optimal control problem (3.1.1), and (the nal time tf beingfree)

H(x(t), p(t), p0, u(t)) = max−1≤v(t)≤1

H(x(t), p(t), p0, v(t)), (3.1.4)

almost everywhere on [0, tf ]. In addition, since tf is not xed, we have

H(x(t), p(t), p0, u(t)) = 0,

almost everywhere on [0, tf ]. Moreover, we have the transversality condition

p(tf ) ⊥ Tx(tf )M1, (3.1.5)

where Tx(tf )M1 denotes the tangent space to M1 at the point x(tf ).

We dene the functions

h0(x, p) = 〈p, f0(x)〉, h1(x, p) =∂H

∂u(x, p, p0, u) = 〈p, f1(x)〉. (3.1.6)

It follows from (3.1.4) that u(t) = sign(ϕ(t)), whenever ϕ(t) = h1(x(t), p(t)) 6= 0. Forthis reason, the function ϕ is also called the switching function.

Bang arcs. We say that the trajectory x(·) restricted to a sub-interval I of [0, tf ] isa bang arc if u(t) is constant along I, equal either to +1 or to −1. We say that thetrajectory is bang-bang on [0, tf ] if it is the concatenation of bang arcs.

Singular arcs. If ϕ(t) = h1(x(t), p(t)) = 0 along a sub-interval I of [0, tf ], then therelation (3.1.4) does not allow to directly infer the control, and in that case we speakof a singular arc, or of a singular extremal.

This denition of the singular control is indeed equivalent to Denition 1.1.2. It iswell known that a trajectory x(·) is singular on [0, tf ] if and only if it has an extremallift (x(·), p(·), p0, u(·)), satisfying (3.1.2) and h1(x(t), p(t)) = 0 on [0, tf ] (see [13, 105]).

3.1.2 Computation of Singular arcs

In order to compute a singular control, the usual method consists of dierentiatingrepeatedly the relation

ϕ(t) = h1(x(t), p(t)) = 0 (3.1.7)

with respect to time, until the control appears in a nontrivial way. Using the Hamil-tonian system (3.1.2), such derivations are done thanks to Poisson brackets and Lie


brackets. By dierentiating (3.1.7) a rst time (along the interval I), we obtain

0 = ϕ(t) = h0, h1(x(t), p(t)) = 〈p(t), [f0, f1](x(t))〉, (3.1.8)

which is a new constraint. Dierentiating a second time, we obtain

0 = ϕ(t) = h0, h0, h1(x(t), p(t)) + u(t)h1, h0, h1(x(t), p(t))

= 〈p(t), [f0, [f0, f1](x(t))〉+ u(t)〈p(t), [f1, [f0, f1](x(t))〉,

in which the control now appears in a nontrivial way provided that

h1, h0, h1(x(t), p(t)) > 0.

The latter condition is known as strengthened Legendre-Clebsch condition (in short,SGLCC). Under this condition, we can indeed compute the singular control as

u(t) = −h0, h0, h1(x(t), p(t))

h1, h0, h1(x(t), p(t)).

It can be noted that the rst derivative of ϕ(·) does not make appear the control.Hence, two derivations in time are at least necessary in order to make appear the controlin a nontrivial way. Such controls are also said to be of minimal order, and actuallythis property is generic (see [19, 30]). Hereafter, due to the fact that optimal singulararcs have to appear with an even number of derivations, we also say that such singulararcs are of intrinsic order one.

If h1, h0, h1(x(t), p(t)) = 0 identically on I, then the above computation doesnot suce and we need to dierentiate more. In that case, we see that we have twoadditional constraints:

h0, h0, h1(x(t), p(t)) = 〈p(t), [f0, [f0, f1]](x(t))〉 = 0, (3.1.9)

andh1, h0, h1(x(t), p(t)) = 〈p(t), [f1, [f0, f1]](x(t)) = 0, (3.1.10)

for every t ∈ I.Let us recall the concept of the order of a singular control. Roughly speaking, it is

the rst integer m such that the control u appears in a nontrivial way in the (2m)th-derivative of the switching function ϕ(·) (see [93, 113]).

Denition 3.1.1. The singular control u (along the sub-interval I) is said to be oflocal order k if the conditions

• the rst (2k− 1)-th time derivatives of the switching function ϕ(t) do not dependon the control u and

ϕ(i)(x(t), p(t)) = 0, i = 0, 1, · · · , 2k − 1,


• the 2k-th time derivative of the switching function ϕ(t) depends on the control ulinearly and

∂

∂uϕ(2k)(x(t), p(t)) 6= 0

hold along the sub-interval I. If moreover the Lie brackets [f1, [adif0.f1]], i = 0, · · · , 2k−2, are identically equal to zero (over the whole space), then the singular control u is saidto be of intrinsic order k.

Remark 3.1.1. If a singular control u is of local order two, then the conditions(along I)

∂

∂uϕ(2)(t) = 〈p(t), [f1, adf0.f1](x(t))〉 = 0,

and

∂

∂uϕ(3)(t) = 2〈p(t), [f1, ad2f0.f1](x(t))〉+ u(t)〈p(t), [f1, [f1, adf0.f1]](x(t))〉 = 0,

are additional constraints that must be satised along the singular arc. In contrast, if uis of intrinsic order two, then these conditions are trivially satised since [f1, adf0.f1] ≡0 and [f1, ad2f0.f1] ≡ 0. In the present chapter, we are in the situation of singular arcsof intrinsic order two, and we will then focus on that case.

Actually, we did not consider, in the above denition, the case where the rst nonzeroderivative is of odd order. Indeed, it is known that the order must be even (hence thedenition obtained by halving it). This fact is due to the following well-known result,usually referred to as Kelley's condition for singular extremals of local order k (see[58, 64]):

If a trajectory x(·), associated with a singular control u(·), is locally time-optimal on [0, tf ] in L

∞ topology, then the generalized Legendre-Clebsch con-dition (GLCC)

(−1)k∂

∂u

d2kh1

dt2k≤ 0,

is satised along the extremal.

Therefore, the GLC for a singular control of local order 2 is

〈p(t), [f1, ad3f0.f1](x(t)) + [f0, [f0, [f1, [f0, f1]]]](x(t)) + [f0, [f1, ad2f0.f1]](x(t))〉 ≤ 0,

and if the singular control of intrinsic order 2, then this condition takes the simplerform

〈p(t), [f1, ad3f0.f1](x(t))〉 ≤ 0.

Turning back to the previous computation, if the singular control is of intrinsic order


two, then by dierentiating ϕ(t) = h0, h0, h1(x(t), p(t)), we get

0 = ϕ(3)(t) = h0, ad2h0.h1(x(t), p(t)) + u(t)h1, ad2h0.h1(x(t), p(t))

= 〈p(t), [f0, ad2f0.f1](x(t))〉+ u(t)〈p(t), [f1, ad2f0.f1](x(t))〉,

which, using the fact that [f1, ad2f0.f1] ≡ 0, leads to the additional constraint

h0, ad2h0.h1(x(t), p(t)) = 〈p(t), [f0, ad2f0.f1](x(t))〉 = 0. (3.1.11)

Dierentiating again, we get

0 = ϕ(4)(t) = h0, ad3h0.h1(x(t), p(t)) + u(t)h1, ad3h0.h1(x(t), p(t))

= 〈p(t), [f0, ad3f0.f1](x(t))〉+ u(t)〈p(t), [f1, ad3f0.f1](x(t))〉.

By denition, we have 〈p(t), [f1, adf 30 .f1](x(t))〉 6= 0, and thus the singular control is

u(t) = − ad4h0.h1(x(t), p(t))

h1, ad3h0.h1(x(t), p(t)), (3.1.12)

which is smooth.

Remark 3.1.2. Along such a singular arc of intrinsic order two, the singular controlis given by (3.1.12) and the constraints (3.1.7), (3.1.8), (3.1.9), (3.1.11) must be satisedalong the arc. Recall that the constraint (3.1.10) is trivially satised since [f1, [f0, f1]] ≡0 according to Denition 3.1.1.

In this chapter, we are actually concerned with optimal singular trajectories ofintrinsic order two, which cause the occurrence of a chattering phenomenon in ourproblem. Let us recall the following result (see [58, 78, 112]).

Lemma 3.1.1. We assume that the optimal solution x(·) of the optimal controlproblem (3.1.1) involves a singular arc (on a sub-interval I) of intrinsic order two, forwhich the strengthened generalized Legendre-Clebsch condition (SGLCC)

∂

∂u

d4h1(t)

dt4= h1, ad3h0.h1(x(t), p(t)) < 0

holds true along an extremal lift. If we have |u(t)| < 1 along the singular arc, then thesingular arc cannot be matched directly with any bang arc. In particular, if I is a propersubset of [0, tf ], then the optimal solution chatters, in the sense that there is an innitenumber of bang arcs accumulating at the junction with the singular arc.

Although this result is known, we will provide a short proof of it when analyzingour spacecraft problem in Section 3.2.3.


Remark 3.1.3. Note that the Fuller problem (see section 2.3) can be adapted to t inthe framework above, although this is not a minimum time problem. Actually, it sucesto add the objective as a third state variable x3, evolving according to x3 = x2

1/2, andthen the Fuller problem can be interpreted, by uniqueness of the solution, as a minimumtime problem with the vector elds f0(x) = (x2, 0, x

21/2)> and f1(x) = (0, 1, 0)>. The

corresponding singular extremal is therefore given by u = 0, x1 = x2 = p1 = p2 = p0 = 0and p3 < 0 being constant. The solutions of the Fuller problem are optimal abnormalextremals for this three-dimensional problem. Moreover, it is easy to see that u = 0 isa singular control of intrinsic order two, along which the SGLCC is satised (p3 < 0).Then Lemma 3.1.1 can be applied.

3.1.3 Geometric Analysis of the Chattering Phenomenon

In this section, we recall some results on chattering solutions established in [112, 113].Since these references are not always easy to read, our objective is also to provide a morepedagogical exposition of these results and to show how they can be used in practice.

Recall that a chattering solution is the optimal solution corresponding to the chat-tering control which switches an innite number of times over a compact time interval.

3.1.3.1 Semi-Canonical Form

The semi-canonical form (see [65, 112]) is a way of writing the Hamiltonian system(3.1.2) in a neighborhood of its singular arcs, which will be used later to analyze thesolutions near (in C0 topology) singular arcs of intrinsic or local order two. The mainidea is to design a variable change that leads to a form involving the switching functionand its derivatives directly as variables. This makes the analysis of the extremals nearthe singular arcs more convenient.

Let x(·) be an optimal trajectory of (3.1.1) on [0, tf ], and let (x(·), p(·), p0, u(·)) bean extremal lift (coming from the PMP). We assume that x(·) involves a singular arcof intrinsic second order two, along the sub-interval I, satisfying the SGLCC.

The Hamiltonian (3.1.3) can be rewritten as H = h0 + uh1 + p0, with h0 and h1

dened by (3.1.6). We dene the new coordinates

z1 = h1, z2 = h(1)1 = h0, h1, z3 = h

(2)1 = ad2h0.h1, z4 = h

(3)1 = ad3h0.h1,

(3.1.13)and using that [f1, [f0, f1]] ≡ 0 and that h1, ad3h0.h1 < 0 along I, we have

z1 = z2, z2 = z3, z3 = z4, z4 = α(x, p) + uβ(x, p),

where α = ad4h0.h1 and β = h1, ad3h0.h1 < 0.Note that z1 is chosen as the switching function ϕ(t) = h1(x(t), p(t)) and zi is chosen

as the (i−1)-th derivative of the switching function. In fact, using that [f1, [f0, f1]] ≡ 0


and using Jacobi's identity, we have

h1, h0, h0, h1 = −h0, h0, h1, h1 − h0, h1, h1, h0= h0, h1, h0, h1 ≡ 0.

This, together with β < 0, indicates that the singular control considered here is ofintrinsic order two and satises the GLCC. By denition, we have zi = 0, i = 1, 2, 3, 4,along such a singular arc.

Assume that ∂(z1,z2,z3,z4)∂(x,p)

is of full rank 4, thus we infer that z1, z2, z3, z4 are func-tionally independent in the neighborhood of the extremal lift (x(·), p(·)), along [0, tf ].We complement z = (z1, z2, z3, z4) with w = (w1, · · · , w2n−4) ∈ R2n−4 such that theJacobian matrix of the mapping (x, p) 7→ (z, w) is nondegenerate, i.e.,

det

(D(z, w)

D(x, p)

)6= 0,

along the extremal. Since our point of view is local, we assume that (x, p) and (z, w) livein R2n. The Hamiltonian system (3.1.2) can be rewritten, locally along the extremal,as

z1 = z2, z2 = z3, z3 = z4, z4 = α(z, w) + uβ(z, w), w = F (z, w, u), (3.1.14)

and the extremal control is given by

u(t) =

1 if z1(t) > 0,

−α/β if z1(t) = 0,

−1 if z1(t) < 0.

Accordingly, we dene the singular surface (smooth manifold consisting of singularextremals of second order) as

S = (z, w) | (z1, z2, z3, z4) = (0, 0, 0, 0),

and the switching surface as

Γ = (z, w) | z1 = 0.

If a trajectory z(·) is a solution of (3.1.14), then a straightforward calculation yieldsthat zλ = Gλ(z(t/λ)) is also a solution of (3.1.14), for any number λ > 0, where

Gλ(z(t

λ)) =

(λ4z1

(t

λ

), λ3z2

(t

λ

), λ2z3

(t

λ

), λz4

(t

λ

)). (3.1.15)

This is an important property for the Fuller problem (self-similar solutions).


The system (3.1.14) is useful in order to analyze the qualitative behavior of solutionsnear the singular surface consisting of singular extremals of intrinsic order two. Toinclude some Hamiltonian systems having singular arcs of local order two, we consider asmall perturbation of the system (3.1.14) in the neighborhood of a given point (0, w0) ∈S, given by

z1 = z2 + F1(z, w, u),z2 = z3 + F2(z, w, u),z3 = z4 + F3(z, w, u),z4 = α(w) + uβ(w) + F4(z, w, u),w = F (z, w, u),

(3.1.16)

with Fi(z, w, u) = o(zi+1), i.e.,

limλ→0+

λ−(5−i)|Fi(Gλ(z(t/λ)), w, u)| < +∞, i = 1, 2, 3, 4. (3.1.17)

The system (3.1.16)-(3.1.17) is called a semi-canonical form.

Remark 3.1.4. The variables (z, w) can be chosen dierently from (3.1.13) in orderto get a simpler local system (3.1.16). This is why this form is called semi-canonical,and not canonical. Moreover, this change of variable is not unique.

3.1.3.2 Geometry of Chattering Extremals

The rst result concerns the existence of chattering solutions. In contrast to Lemma3.1.1, this result can also be applied to the case of singular arcs of local order two,and it describes the phase portrait of optimal extremals in the vicinity of a manifold ofsingular arcs of order two.

Recall that the singular surface S for the system (3.1.16) is of codimension 4. Thesurface S satises four constraints z1 = 0, z2 = 0, z3 = 0, z4 = 0 corresponding re-spectively to null derivatives of the switching functions ϕ(i), i = 0, 1, 2, 3. Consideringa point (0, w0) ∈ S, if β(w0) < 0 and |α(w0)| < −β(w0), there exists a neighborhoodof this point in which the singular extremals passing through it satisfy the GLCC andthe singular control |u| = | − α(w)/β(w)| < 1 is admissible. The following propo-sition indicates that, for any point in such a neighborhood, there exists a family ofchattering extremals coming into this point, and there is another family of chatteringextremals emanating from this point. Note that a family of chattering extremals is aone-parameter family, with the parameter λ dened in (3.1.15).

Proposition 3.1.1. [Bundles with chattering bers] Consider the system (3.1.16),in an open neighborhood of the point (0, w0). If β(w0) < 0 and |α(w0)| < −β(w0), thenthere exists an open neighborhood O of w0 in R2n−4 such that, for any w ∈ O, there aretwo one-parameter families of chattering extremals intersecting only at the point (0, w).

The extremals of the families ll two manifolds N+w and N−w , each of them being

of dimension 2 and homeomorphic to R2, coming respectively into and out of the point(0, w). The switching points of N±w ll two piecewise-smooth curves Γ±w.


The union ∪w∈ON±w of all those submanifolds is endowed with the bundle structurewith base O and two-dimensional piecewise smooth bers lled by chattering extremals.

Figure 3.1: Phase portrait of optimal extremals near the singular surface.

Figure 3.1 illustrates Proposition 3.1.1. The extremals living in the submanifoldsN+w and N−w are chattering. More precisely, the extremals in N+

w reach (0, w) (innite time) with innitely many switchings, and the extremals in N−w leave (0, w) withinnitely many switchings. The submanifoldsN±w can be seen as two-dimensional bers.

Proof 3.1.1. The complete proof of Proposition 3.1.1 is done in [112]. Let ushowever sketch the main steps. Assume that z2 > 0.

1. Prove that there exist self-similar solutions (i.e., the one-parameter family of chat-tering solutions) for the unperturbed system (3.1.14) using the Poincaré mappingΦ of the switching surface to itself.

2. Prove that the points on S are the stable points of Φ Φ, by calculating the eigen-values of d(Φ Φ)(0, w0). Applying the invariant manifold theorem, there exists aone-dimensional ΦΦ-invariant submanifold transversal to S and passing throughthe point (0, w0). The restriction of ΦΦ to this submanifold is a contracting map-ping. It follows the existence of a two-dimensional manifold N+

w0in the (z, w)-

space, lled by chattering extremals entering into (0, w0). Moreover, the smoothdependence theorem leads to the bundle structure of ∪w0N+

w0.

3. Prove that for the small perturbation system (3.1.16), the Poincaré mapping Φ iswell dened and smooth at the points in the neighborhood of N+

w0. Using similar

techniques as in the rst and second steps, prove that the solutions of the perturbedsystem have the same structure than that of the unperturbed system.

When z2 < 0, another two-dimensional manifold N−w0in (z, w)-space lled by chattering

extremals that coming out of the point (0, w0) can be found and ∪w0N−w0is also endowed

with a bundle structure.

The subbundles described in Proposition 3.1.1 are given by

Σ± = ∪w∈ON±w ,


where the subbundle Σ+ (resp., Σ−) is lled by chattering arcs that come into (resp.,come out of) the singular surface. Moreover, we denote the switching surfaces asΓ± = ∪w∈OΓ±w .

Note that it suces to consider only the subbundle Σ+, since the properties of Σ−

can be obtained similarly. We consider the canonical projection π : Σ+ 7→ O from thesubbundle to the base.

3.1.3.3 Optimality Status

We now raise the question of knowing whether these chattering extremals are optimal ornot. Let us consider again the Fuller problem to give an intuitive idea. Using (3.1.13),we choose the new variables z = (p2,−p1,−2x1,−2x2) and then clearly the singularsurface coincides with the origin. According to Proposition 3.1.1, there are two integralsubmanifolds of dimension 2 that are lled by chattering extremals coming into andout of the origin within nite time, with innitely many switchings.

We consider the canonical projection π∗ : (z, w) → x from the (z, w)-space to thex-space (state space). It is known that the extremals ll a Lagrangian submanifold inthe (z, w)-space. Their projection on the state space are the trajectories, of which wewould like to ensure their local optimality status. According to the conjugate pointtheory (see [2, 15]), it suces to ensure that the projection π∗ be regular along theLagrangian manifold (in other words, we require that its dierential be surjective alongthat manifold). Note that we can consider as well the projection from the (x, p)-spaceto the x-space, instead of π∗, because the coordinate change (x, p) 7→ (z, w) is bijectivein the neighborhood of a point (x, p) ∈ S. Indeed, this coordinate only needs to beregular for providing the regularity of projection from (x, p)-space to x-space.

As illustrated on Figure 3.2(a), the above regularity condition ensures that thetrajectories in the x-space do not intersect each other before reaching the target pointor submanifold, and thus ensures to avoid the loss of local optimality of the trajectoriesat the intersection point (i.e., the conjugate point). Figures 3.2(b) and 3.2(c) showthe optimal synthesis of the chattering trajectories π∗(N+

w ) and π∗(N−w ) for the Fullerproblem, respectively. These chattering solutions do not intersect and they are locallyoptimal.

Let M1 be a target submanifold contained in the projection of the singular sur-face π∗S. For any point x ∈ M1, we dene its lift (x, p(x)) satisfying (x, p(x)) ∈ S,H(x, p(x)) = 0 and p(x) dx = 0 (transversality condition). The union N of all suchpoints (x, p(x)) must be transversal to the ow of the singular extremals in S. Thus,the singular extremals reaching the submanifold N ll a submanifold N∗. In short, thesubmanifold N is a lift of the target M1 that intersects with the singular extremals.

It is easy to see that the submanifold N∗ is Lagrangian. Hence the subbundleπ−1(N∗) is Lagrangian as well. Therefore, according to the theory on Lagrangianmanifolds and sucient optimality conditions, it suces to check the regularity of theprojection π∗ restricted to π−1(N∗).


Figure 3.2: (a) Illustration of sucient optimality condition; (b)-(c) Optimal synthesisof the Fuller problem.

The following proposition provides sucient optimality conditions (see [113]) whenthe submanifolds N and N∗ are of dimension n− 3 and n− 2 respectively.

Proposition 3.1.2. Consider the subbundle π−1(N∗) of the bundle Σ+. Assumethat the restriction of the projection π∗ on any smooth part of the bundle π−1(N∗) isregular and can be regularly extended to boundary points of the smooth part. Assumethat the target manifold M1 is connected. Then the projection of the solutions of thesystem (3.1.16) lling π−1(N∗) are locally optimal in C0 topology.

The target submanifold has to be chosen adequately and must be of order n − 3in order to use this proposition. This condition on the dimension is used to take intoaccount the two-dimensional bers mentioned in Proposition 3.1.1.

Figure 3.3: Illustration of Proposition 3.1.2.

As shown in Figure 3.3, due to the endowed bundle structure, for every given initialpoint (z0, w0) in the neighborhood of the singular surface S in (z, w)-space, there is aneighborhood V of the point (z0, w0) such that all extremals starting from the pointsinside V reach a point on N∗ in nite time with innitely many switchings. Then,these extremals reach the target manifold N along the singular extremals in N∗. Ifthe projection π∗ is regular, then the projected trajectories in the x-space are locallyoptimal in C0 topology.


The condition of being a regular projection is the most dicult one to check. Weset

Σ∗ = π−1(N∗), Γ∗ = Σ∗ ∩ Γ+, S0 = S ∩ H = 0.

In [113], the authors provide the following sucient condition for having a regularprojection of Σ∗ into the x-space.

Lemma 3.1.2. Let L be the vector space spanned by the vector ∂/∂z3 and by thevectors of the tangent plane to the switching surface Γ∗. Assume that the restriction ofdπ∗ to L is surjective. Then, the restriction of π∗ to Σ∗ is regular as well.

Remark 3.1.5. Lemma 3.1.2 indicates that dπ. ∂∂z3

should be transversal to the tan-gent plane to the switching surface of the chattering family generated by the submanifoldN . Note that, at the points of the curve N , the tangent plane of the switching surfaceΓ∗ consists of three types of vectors: the nonsingular velocity vector (the time derivativeof the state x asscociated to a nonsingular control), the singular velocity vector (thetime derivative of the state x asscociated to the singular control) and the tangent vectorto the curve N .

3.2 Application to (MTTP)

In this section, we analyze the bang-bang, singular and chattering extremals of (MTTP).We will see that, when the strategy involves a singular arc, then this singular arc isof intrinsic order two, and according to the previous section, this causes a chatter-ing phenomenon. We will prove that chattering extremals are locally optimal in C0

topology.

3.2.1 Extremal Equations

The Hamiltonian of (MTTP) is of the form H = h0 + uh1 + p0, where h0 = 〈p, f0(x)〉and h1 = 〈p, f1(x)〉 = bp4, and the adjoint vector p = (p1, p2, p3, p4) satises the adjointequations

p1 = c(p1x2 − 2p2x1 + p3),

p2 = cp1x1,

p3 = a(p1 sinx3 − p2 cosx3),

p4 = −p3.

(3.2.1)

Since b > 0, we infer from the maximization condition of the PMP that u(t) =sign(p4(t)), provided that ϕ(t) = bp4(t) 6= 0 (bang arcs). The nal condition x(tf ) ∈M1

yields the transversality condition

p1(tf ) cos(γf ) + p2(tf ) sin(γf ) = 0. (3.2.2)

60 3.2. Application to (MTTP)

3.2.2 Lie Bracket Conguration of the System

Before proceeding with the analysis of singular extremals, it is very useful to computethe Lie brackets of the vector elds f0 and f1 dened by (2.2.2). This is what we callthe Lie bracket conguration of the control system (2.2.1).

Lemma 3.2.1. We have

f0 = (a cosx3 − cx1x2)∂

∂x1

+ (a sinx3 + cx21 − g0)

∂

∂x2

+ (x4 − cx1)∂

∂x3

,

f1 = b∂

∂x4

, [f0, f1] = −b ∂

∂x3

,

[f0, [f0, f1]] = −ab sinx3∂

∂x1

+ ab cosx3∂

∂x2

, [f1, [f0, f1]] ≡ 0,

ad3f0.f1 = − ab((x4 − 2cx1) cosx3 + cx2 sinx3)∂

∂x1

− ab sinx3(x4 − 3cx1)∂

∂x2

− abc sinx3∂

∂x3

,

[f1, [f0, [f0, f1]] = [f0, [f1, [f0, f1]] = [f1, [f1, [f0, f1]] = 0,

ad4f0.f1 = ab((−4cx1x4 + cg0 + x24 + 4c2x2

1 − c2x22) sinx3 − 2ac+ 4ac cos2 x3

+ (cx1 − 2x4)cx2 cosx3)∂

∂x1

+ ab(−c2x1x2 sinx3 + 4ac sinx3 cosx3

+ (−x24 + 6cx1x4 − 7c2x2

1) cosx3)∂

∂x2

+ abc(3cx1 cosx3 − 2x4 cosx3

− cx2 sinx3)∂

∂x3

,

[f1, ad3f0.f1] = −ab2 cosx3∂

∂x1

− ab2 sinx3∂

∂x2

,

anddim Span(f1, [f0, f1], [f0, [f0, f1]]) = 3.

It follows from this lemma that the Poisson brackets

h1, h0, h1, h1, h0, h0, h1

are identically equal to 0. This is the main reason why we will have singular extremalsof higher order, as shown in the next section.

3.2.3 Singular Extremals

In this section, we compute all possible optimal singular extremals arcs. Later on, weare going to provide sucient conditions on the initial conditions, under which the


optimal strategy of (MTTP) does not involve (optimal) singular arcs. Before that, letus rst assume that singular arcs do exist, and let us establish some necessary conditionsalong them.

Lemma 3.2.2. Let x(·) be a singular arc, dened on the sub-interval (t1, t2), andlet (x(·), p(·), p0, u(·)) be an extremal lift. Then:

• along that singular extremal, we must have (omitting t for readability)

p1

(a− cx1x2 cosx3 − (g0 − cx2

1) sinx3

)+ p0 cosx3 = 0,

p2


1) sinx3

)+ p0 sinx3 = 0,

p3 = p4 = 0,

(3.2.3)

and

us =c

2b

((−cx2

2 + 2x1x4 − 3cx21 + g0) sin 2x3

+ 2cx1x2 cos 2x3 + 4a cosx3 − 4x2x4 cos2 x3

);

(3.2.4)

• p0 6= 0 (in other words, there is no abnormal singular extremal), and then we setp0 = −1;• the four constraints (3.2.3) are functionally independent;• one has |u(t)| < 1, for almost every t ∈ (t1, t2) (in other words, any singular arcis admissible);• u is of intrinsic order two;• the SGLCC along the singular extremal reads

a− cx1x2 cosx3 − (g0 − cx21) sinx3 > 0. (3.2.5)

In particular, the last item of the lemma states that optimal singular arcs, if theyexist, must live in the region of the state space R4 dened by (3.2.5). The third item ofthe lemma implies that the singular extremals of the problem are in a submanifold ofcodimension 4, i.e., the singular surface of (MTTP) is of codimension 4. Recall thatthe functions yi(x, p) = 0, i = 1, · · · , k dened on a submanifold D ⊂ R2n are saidfunctionally independent if the Jacobian matrix ∂(y1,··· ,yk)

∂(x,p)(dimension k × 2n) is of full

rank k.

Proof 3.2.1. Along the interval I = (t1, t2) on which the singular arc is dened,the switching function ϕ(t) = h1(x(t), p(t)) = bp4(t) must be identically equal to zero.Dierentiating with respect to time, we get that h0, h1 = −bp3 = 0 along I.

Dierentiating again, we get ϕ(t) = h0, h0, h1 + uh1, h0, h1 = 0, and sincethe Poisson bracket h1, h0, h1 is identically equal to 0 (see Lemma 3.2.1), we haveh0, h0, h1 = ad2h0.h1 = −ab(p1 sinx3 − p2 cosx3) = 0 along I (and the equationh1, h0, h1 = 0 does not bring any further information).


Dierentiating again, we get ϕ(t) = h0, h0, h0, h1+uh1, h0, h0, h1 = 0,and there, again from Lemma 3.2.1, the Poisson bracket h1, h0, h0, h1 is identi-cally equal to 0 (and thus brings no additional information). Hence

ad3h0.h1 =− ab(x4(p1 cosx3 + p2 sinx3)

+ cp1x2 sinx3 − 3cp2x1 sinx3 − 2cp1x1 cosx3

)= 0,

(3.2.6)

which gives a new constraint. Finally, a last derivation yields

ad4h0.h1 + uh1, ad3h0.h1 = 0,

and since h1, ad3h0.h1 6= 0, we infer that

u = − ad4h0.h1

h1, ad3h0.h1:= us,

along I. Here, we have ad4h0.h1 = 〈p, ad4f0.f1(x)〉, and

h1, ad3h0.h1 = 〈p, [f1, ad3f0.f1](x)〉 = −ab2(p1 cosx3 + p2 sinx3).

Then, the singular control in feedback form (3.2.4) follows by considering the constraintsϕ(t) = ϕ(t) = ϕ(t) = 0, i.e.,

p3 = p4 = 0, p1 sinx3 − p2 cosx3 = 0.

They are functionally independent because dim Span(f1, [f0, f1], [f0, [f0, f1]]) = 3 (seeLemma 3.2.1). Moreover, using the fact that H ≡ 0 along an extremal, we infer therelations (3.2.3). Setting

y1 = p1


1) sinx3

)+ p0 cosx3,

y2 = p2


1) sinx3

)+ p0 sinx3,

we have rank∂(y1,y2,p3,p4)∂(x,p)

= 4, provided that p0 6= 0 and p1 6= 0, p2 6= 0. This impliesthat these four functions are functionally independent. If p1 = 0 or p2 = 0, then itis easy to see that p1 = p2 = p0 = 0, which violates the PMP. Hence p1 6= 0 andp2 6= 0. If p0 were to be zero, then it would follow from p1 6= 0 and p2 6= 0 thaty = a − cx1x2 cosx3 − (g0 − cx2

1) sinx3 ≡ 0 along I. Dierentiating, we get y ≡ 0 andy = αc(x) +ucβc(x) ≡ 0. Here, αc(·) and βc(·) are two functions of x and uc representsthe control u that appears in the expression of y. By substituting p1 sinx3 = p2 cosx3

into (3.2.6), we get

y3 = −x4 + cx1(2 + sin x23)− cx2 sinx3 cosx3 = 0.

Then, setting y4 = uc − us = −αc(x)/βc(x)− us(x), we check that y = 0, y = 0, y3 = 0


and y4 = 0 are four functionally independent constraints on the x-space. Hence, thisconstraint of codimension 4 implies that the trajectory must be reduced to a singleton,and moreover, there must hold u = us(x) = 0. However, us(x) = 0 is another inde-pendent state constraint, and so the number of constraints along the abnormal extremalhas exceeded the dimension of the extremal (x, p)-space. Therefore p0 6= 0.

Using the numerical values of Table 2.1, we have

|u| ≤ c

2b

(4a+ 6vmωmax + cv2

m

)≤ 0.3, (3.2.7)

and thus |u| < 1. Hence, for (MTTP), we have, along any singular extremal arc,

∂

∂u

dk

dtkh1 = 0, k = 0, 1, 2, 3,

∂

∂u

d4

dt4h1 = β(x, p) = −ab2(p1 cosx3 + p2 sinx3) 6= 0,

and then, according to Denition 3.1.1, the singular solutions (which are admissiblefrom (3.2.7)) are of intrinsic order two. The SGLCC for (MTTP) is written here asβ(x, p) < 0, and hence, using (3.2.3) and taking p0 = −1, we obtain (3.2.5).

Corollary 3.2.1. For (MTTP), any optimal singular arc cannot be connected witha nontrivial bang arc. We must then have chattering, in the following sense. Let u be anoptimal control, solution of (MTTP), and assume that u is singular on the sub-interval(t1, t2) ⊂ [0, tf ] and is non-singular elsewhere. If t1 > 0 (resp., if t2 < tf) then, forevery ε > 0, the control u switches an innite number of times over the time interval[t1 − ε, t1] (resp., on [t2, t2 + ε]).

Proof 3.2.2. This result follows from Lemma 3.1.1 and Lemma 3.2.2. However,the proof is simple and we provide hereafter the argument.

It suces to prove that the existence of an extremal consisting of the concatenation ofa singular arc of higher order with a non-singular arc violates the PMP. The reasoninggoes by contradiction. Assume that t1 > 0 and that there exists ε > 0 such that u(t) = 1over (t1 − ε, t1). By continuity along the singular arc, we have ϕ(t1) = ϕ(1)(t1) =ϕ(2)(t1) = ϕ(3)(t1) = 0, and it follows from the SGLCC β(x, p) < 0 that

0 = ϕ(4)(t+1 ) = ad4h0.h1(t1) + h1, ad3h0.h1(t1)u(t+1 )

> ad4h0.h1(t1) + h1, ad3h0.h1(t1)u(t−1 ) = ϕ(4)(t−1 ),

and hence the switching function t 7→ ϕ(t) = h1(x(t), p(t)) has a local maximum att = t1 and thus is nonnegative over (t1 − ε, t1), provided that ε > 0 is small enough. Itfollows from the maximization condition of the PMP that u(t1) = −1 over (t1 − ε, t1).This contradicts the assumption.


3.2.4 Optimality Status of Chattering Extremals

In this section, we analyze the optimality status of chattering extremals in (MTTP).

Lemma 3.2.3. Assume that x3 6= π/2 + kπ, k ∈ Z. The Hamiltonian system,consisting of (2.1.15) and (3.2.1), can be written as a small perturbation system, in theform (3.1.14), as

z1 = z2,

z2 = z3,

z3 = z4 + F3(z, w, u),

z4 = α0(w) + uβ0(w) + F4(z, w, u),

w = F (z, w, u),

(3.2.8)

where u ∈ [−1, 1] and

limλ→0+

F3(Gλ(z), w, u)

λ(5−3)= 0, lim

λ→0+

F4(Gλ(z), w, u)

λ(5−4)<∞, (3.2.9)

by choosing new variable (z, w) as

z1 = p4, z2 = p(1)4 , z3 = p

(2)4 , z4 = p

(3)4 + ac sinx3p3,

w1 = a(p1 sinx3 + p2 cosx3),

w2 = a(x4 cosx3 + cx2 sinx3 − 2cx1 cosx3)p1 + a sinx3(−x4 + 3cx1)p2,

w3 = p2/p1,

w4 = x1

(3.2.10)

in the neighborhood of the singular surface dened by z = 0 in the (z, w)-space. Inaddition, the SGLCC for system (3.2.8) yields

w1w3 > 0. (3.2.11)

Proof 3.2.3. We have proved that p1 6= 0 and p2 6= 0 along the singular arc. Then,from x3 6= π/2 + kπ, k ∈ Z we can prove that the Jacobi matrix of this variable change

is of full rank by direct calculations, i.e., rank(D(z,w)D(x,p)

)= 8. After some manipulations,

we can express (x, p) by the new variables (z, w) chosen in (3.2.10), as


x1 =w4, x4 = 3cw4 −z4 + w2

w3(w1 − z3), p3 = −z2, p4 = z1,

x2 =w1w2 + w1z4 + w2z3 + z3z4 − cw3(w4w

21 − w4z

23)

cw23(w1 − z3)2

+w1w2 − w1z4 − w2z3 + z3z4

c(w1 − z3)2,

x3 =− 2arctan

(w1 + z3 ± (w2

1w23 + w2

1 − 2w1w23z3 + 2w1z3 + w2

3z23 + z2

3)

(w3w1 − w3z3)

),

p1 =∓√w2

1w23 + w2

1 − 2w1w23z3 + 2w1z3 + w2

3z23 + z2

3/(2aw3),

p2 =∓√w2

1w23 + w2

1 − 2w1w23z3 + 2w1z3 + w2

3z23 + z2

3/(2a).

(3.2.12)

Although this variable transformation is not one to one in the whole (x, p)-space, itdoes not matter, because the semi-canonical system we use is a local system and so wejust need to consider separately the domain x3 ∈ D1 = (−π/2, π/2) and x3 ∈ D2 =(−π,−π/2) ∪ (π/2, π).

The manifold of singular trajectories specied by z = 0 can be written as

S = (x, p)|p3 = 0, p4 = 0, p2 = p1 tanx3, x4 = cx1(2 + sin2 x3)− cx2 sinx3 cosx3.

Dierentiating (z, w) dened in (3.2.10) with respect to time with the help of (2.1.15)and (3.2.1), we get the system in form,

z1 = z2,

z2 = z3,

z3 = z4 + F3(x, p),

z4 = A(x, p) +B(x, p)u,

w = F (x, p, u),

(3.2.13)

with

u =

1 if z1 > 0,

−1 if z1 < 0,

−A(x, p)/B(x, p) if z1 = 0,

whereA(x, p) = h0(x, p), z4, B(x, p) = h1(x, p), z4 = β(x, p)/b.

Note that here we have u = −A/B = us, where us is given in (3.2.4). Hence we infer|u| < 1. By substituting (3.2.12) into (3.2.13), we can obtain f3(z, w), A(z, w), B(z, w)


and F (z, w, u). Then we expand A(z, w) and B(z, w) in the vicinity of S by

A(z, w) = A(0, w) +∞∑k=1

∂kA

∂zk(0, w)

zk

k!, B(z, w) = B(0, w) +

∞∑k=1

∂kB

∂zk(0, w)

zk

k!.

By taking α0(w) = A(0, w), β0(w) = B(0, w), system (3.2.8) is derived. We can seethat system (3.2.8) is a small perturbation of system (3.1.14) since condition (3.1.17)holds, i.e., condition (3.2.9) holds. Moreover, the SGLCC (3.2.11) is derived from

β0(w) = −bw1(1 + w23)

2w3

< 0,

and (3.2.13) is transformed into a small perturbation system of form (3.2.8).

The functions F3(z, w), α0(w) and F (z, w, u) are dierent for x3 ∈ D1 and x3 ∈D2. However, we will see next that this dierence does not have any inuence in thedemonstration of the optimality result of chattering extremals.

Corollary 3.2.2. For (MTTP), there exist two subbundles Σ+ and Σ− having thesingular surface S as a base, and two bers N+ and N− of dimension two lled bychattering solutions.

Proof 3.2.4. It suces to apply Lemma 3.2.3 and Proposition 3.1.1.

We dene S0 = S ∩H ≡ 0. Let us consider an optimal solution x(·) of (MTTP),and let us assume that x(·) contains a singular arc dened on (t1, t2). Let

M∗1 = x2 = Ψ1(x1) ∩ π∗(S0),

be the submanifold where the extremals come into and out of the image of the singularsurface π∗(S0), as shown in Figure 3.4. Here Ψ(·) is a function of x1.

Figure 3.4: Illustration of M∗1 .

In the sequel, we want to analyze the optimality status of the chattering solutionswith the target submanifold M∗

1 . The optimality status of the chattering solutionsstarting from the submanifold M∗

1 can be analyzed similarly by considering the sub-bundle Σ−.


We denote by N1 the lift ofM∗1 in (x, p)-space by associating x ∈M∗

1 with the point(x, p(x)) that belongs to S0 and satises the transversality condition p1 = −p2Ψ′1(x1)(following from (3.1.5)).

Lemma 3.2.4. The submanifold N1 is a Lagrangian submanifold of R8 and is ofcodimension 7. Moreover, the function Ψ1(·) can be chosen such that the submanifoldN1 is transversal to the velocity vector of the singular extremals in S.

Proof 3.2.5. From the denition of N1 and Lemma 3.2.2, we infer that (x, p(x))satises

p1 =cosx3

a− cx1x2 cosx3 − (g0 + cx21) sinx3

,

p2 =sinx3

a− cx1x2 cosx3 − (g0 + cx21) sinx3

,

p3 = 0, p4 = 0,

x2 −Ψ1(x1) = 0,

− x4 + cx1(2 + sin2 x3)− cx2 sinx3 cosx3 = 0,

Ψ′1(x1) tanx3 + 1 = 0.

Then, the x-component of the tangent vector to N1 can be written as

v1 =

(1,∂x2

∂x1

,∂x3

∂x1

,∂x4

∂x1

)>,

where

∂x2

∂x1

= Ψ′1(x1),∂x3

∂x1

= −Ψ′′1(x1)

Ψ′1(x1)sinx3 cosx3,

∂x4

∂x1

= c(2 + sin2 x3) +c

4

Ψ′′1(x1)

Ψ′1(x1)

(2x1(2 + sin 2x3) sin 2x3 − x2 sin 4x3

).

Therefore, the 1-form ω = pdx = p1dx1+p2dx2+p3dx3+p4dx4 vanishes on every tangentvector to the submanifold N1. Thus N1 is of codimension 7 and it is Lagrangian.

Moreover, the x-component of the velocity on the singular trajectories is

v2 = (x1, x2, x3, x4) = (a cosx3 − cx1x2, a sinx3 + cx21 − g0, x4 − cx1, bus)

>,

with us = −α0(w)/β0(w). Hence, to provide transversality, it suces to choose the

function Ψ1 such that v1 and v2 are not proportional, e.g., Ψ′1 6=a sinx3+cx2

1−g0

a cosx3−cx1x2.

It follows from this lemma that the submanifold N∗ lled by singular extremalscoming into N1 is Lagrangian. According to Proposition 3.1.2, it suces to prove theregularity of the projection π∗ on π−1(N∗) using Lemma 3.1.2.


We denote by v3 the nonsingular velocity vector and by vk the derivative of theprojection π∗ of ∂/∂z3. We set V = (v1, v2, v3, vk).

Theorem 3.2.1. If the function Ψ1(·) is chosen such that

detV 6= 0, (3.2.14)

then the chattering solutions of (MTTP) are locally optimal in C0 topology.

Proof 3.2.6. On S0 we have

dπ∗(∂

∂z3

)=∂x1

∂z3

∂

∂x1

+∂x2

∂z3

∂

∂x2

+∂x3

∂z3

∂

∂x3

+∂x4

∂z3

∂

∂x4

,

where

∂x1

∂z3

= 0,∂x2

∂z3

=w2w

23 − 2cw1w4w3 + 3w2

cw21w

23

,

∂x3

∂z3

= − 2w3

w1(1 + w23),

∂x4

∂z3

= − w2

w21w3

,

and hence it follows that

dπ∗(∂

∂z3

)=

(0,w2w

23 − 2cw1w4w3 + 3w2

cw21w

23

,− 2w3

w1(1 + w23),− w2

w21w3

)>.

Denote dπ∗(

∂∂z3

)as vk. Using (3.2.3) and (3.2.10) we can get vk(w) as a vector

depending on state variable x, i.e. vk(x).

According to Lemma 3.1.2 and Remark 3.1.5, if v1, v2, v3 and dπ∗(

∂∂z3

)are linearly

independent, then the projection is regular. In our problem, we have that the nonsingularvelocity vector associated with u = 1 is

v3 = (x1, x2, x3, x4) = (a cosx3 − cx1x2, a sinx3 + cx21 − g0, x4 − cx1, b)

>.

Therefore, if the condition (3.2.14) is satised on the points of N1, then, the curveN1 has been chosen such that the conditions of Proposition 3.1.2 are fullled and so itgenerates the eld of locally optimal chattering solutions in C0 topology.

Remark 3.2.1. The condition (3.2.14) in Theorem 3.2.1 is always satised if onechooses an appropriate function Ψ1(·). Indeed, we have

detV =4∑i=1

vi,1Di,1,


where Di,1 is the (i, 1) minor. Some calculations show that D4,1 = 0, and hence (3.2.14)becomes detV = D1,1 − Ψ′1D2,1 − Ψ′′1/Ψ

′1D3,1 6= 0. It suces to ensure that Di,1,

i = 1, 2, 3, do not vanish simultaneously. We prove this fact by contradiction: otherwise,it is easy to check that they yield three independent constraints in the x-space, andmoreover, they are independent of the constraints y3 = 0 and x2 = Ψ1(x1) for thesingular surface S. In this case, the number of constraints is larger than the dimensionof the x-space. Therefore, Di,1, i = 1, 2, 3 do not vanish simultaneously.

3.3 Chattering Prediction

Since the chattering phenomenon causes deep diculties for practical implementation,due to the fact that an innite number of control switchings within nite time cannot berealized in real-life control strategies, in this section our objective is to provide preciseconditions under which we can predict that an optimal singular arc does not appear,and thus there is no chattering arcs.

A maneuver with γf ≥ γ0 (resp., γf < γ0) is said to be a anticlockwise maneuver(resp., a clockwise maneuver). In practice, the values of x30 and x3f are chosen in(0, π/2), and the values of γ0, x40 are chosen such that |γ0− x30| ≤ 0.1 and |x40| ≤ 0.1.

We distinguish between those two maneuvers because of the gravity force imposed tothe spacecraft. We will see further, in the numerical results, that the clockwise maneuveris easier to perform than the anticlockwise maneuver, in the sense that the clockwisemaneuver is shorter (in time), and there is less possibility of encountering a singular arc.This fact is due to the nonlinear eects caused by the gravity force. Indeed, intuitively,the gravity force tends to reduce the value of x2, and hence the value of tan γ = x2/x1

tends to get smaller. Then the spacecraft velocity turns naturally to the ground (pitchdown) under the eect of the gravity. This tendency helps the clockwise maneuver tobe easier.

Although we have set c = 10−6 (see Table 2.1), we have c ≤ 10−6 in real-life, sincec = 1/r where r is a distance not less than the radius of the Earth. The case c = 0corresponds to a at-Earth case, and the case c ∈ (0, 10−6] will be referred to as thenon-at case.

3.3.1 Flat-Earth Case c = 0

We can see from (2.2.3) that γ is much smaller than x3 and x4 given by (2.2.1). There-fore, the main factor that aects the total maneuver time is the time to change γ fromγ(0) = γ0 to γ(tf ) = γf = x3f . In order to shorten the maneuver time, it is required tokeep γ as large as possible.

To this aim, we consider the time minimum control problem in which x3 is seen as

70 3.3. Chattering Prediction

a control. We call this problem the problem of order zero, i.e.,

min tf s.t.

x1 = a cosx3, x2 = a sinx3 − g0,

x1(0) = v0 cos γ0, x2(0) = v0 sin γ0,

x2(tf )− x1(tf ) tan γf = 0. (3.3.1)

The optimal solution of this problem is easy to compute (explicitly) with the PMP. Theoptimal control on [0, tf ] is given by

x3(t) = x∗3 =

γf + π/2, if x20 cos γf ≤ x10 sin γf ,

γf − π/2, if x20 cos γf > x10 sin γf ,

and the maneuver time is

tf =(x10 tan γf − x20) cos γfa sin(x∗3 − γf )− g0 cos γf

.

Moreover, the adjoint vector is given by

(p1, p2) = (sin γf ,− cos γf )p0

a sin(x∗3 − γf )− g0 cos γf.

Remark 3.3.1. From this expression, we see that g0 makes the anticlockwise ma-

neuver slower, i.e., tf ≥ (x10 tan γf−x20) cos γfa sin(x∗3−γf )

, and the clockwise maneuver faster, i.e.,

tf ≤ (x10 tan γf−x20) cos γfa sin(x∗3−γf )

. Therefore, the clockwise maneuver is easier to perform than

the anticlockwise maneuver thanks to the gravity, which corresponds to intuition, assaid at the beginning of this section.

Turning back to (MTTP) in the at-Earth case, and considering the nal condition(3.3.1) instead of (2.2.4), then from Lemma 3.2.2, the singular surface is given by

S = (x, p) |x3 = x∗3, x4 = p3 = p4 = 0,

p1 = −p0 cosx∗3/(a− g0 sinx∗3), p2 = tanx∗3p1.

It is interesting to see that, the solution of the problem of order zero coincides with thesingular solution of problem (MTTP) in the at-Earth case. We have the followingresults.

Lemma 3.3.1. Let x(·) be an optimal solution of (MTTP) in the at-Earth case,associated with the control u. If x(·) contains at most one point of S2 = (x, p)|x3 = x∗3,then the control u is bang-bang and switches at most two times.

Proof 3.3.1. If u(·) is singular, then (x(·), p(·)) is contained in S ⊂ S2. From


the denition of S, it is easy to prove that x(t1) 6= x(t2) for any t1 6= t2 in [0, tf ],which means that (x(t1), p(t1)) and (x(t2), p(t2)) are two dierent points of S2. Thiscontradicts the condition that x(·) contains at most one point of S2. Therefore, u(·) isbang-bang.

It suces to prove that if x(·) contains at most one point of S2, then ϕ(t), t ∈ [0, tf ],remains of constant sign and has at most one zero. Indeed, if this is true, then ϕ(t) =−p3(t) is monotone, and it follows that the rst derivative of the switching function ϕ(t)has at most two zeros, which means that the control u has at most two switchings. Let usprove this fact by contradiction. If there exists t1 ∈ [0, tf ] such that (x(t1), p(t1)) ∈ S2,using transversality condition (3.2.2), i.e;, p1 + tan γfp2 = 0, and p1 6= 0, p2 6= 0, wehave

ϕ(t1) = −p3(t1) = −a(p1 sinx3 − p2 cosx3) = ap2 cos(x3 − γf )/ cos γf = 0.

From the continuity of ϕ(·), we get that there exist two times τi < t1 and τj > t1such that ϕ(τi)ϕ(τj) < 0. It follows that x3(t1) and x3(t2) are on dierent sides ofx∗3 = γf ± π/2, i.e., (x3(t1) − x∗3)(x3(t2) − x∗3) < 0. However, we know that x30 andx3f = γf are on the same side of x∗3, i.e., (x30 − x∗3)(x3f − x∗3) < 0, and hence theremust exist another time t2 at which ϕ(t2) = 0 ((x(t2), p(t2)) ∈ S2) in order to allow thetrajectory to reach the terminal submanifold. This is a contradiction.

We denote a bang arc with u = 1 (resp. u = −1) as A+ (resp. A−), and we denotea chattering arc and a singular arc by Ac and As, respectively. Let Fx3 be the unionof all trajectories x(·) consisting of three dierent bang arcs satisfying the terminalconditions x(0) = x0 and x3(tf ) = x3f , x4(tf ) = 0. These trajectories are of the formA+A−A+ or A−A+A−. Note that, for the trajectoires in Fx3 , according to the targetM1 dened by (2.2.4), only the nal condition (3.3.1) must be satised. Denote by x3

the maximal (resp., minimal) value of x3(t), t ∈ [0, tf ]. We have the following result.

Lemma 3.3.2. The optimal control u(t) and the trajectory x(t) of the form A+A−A+

(resp. A−A+A−) in Fx3 are given by

u(t) =

+1, t ∈ [0, τ1), (resp.,−1)

−1, t ∈ [τ1, τ2) ∪ [τ2, τ3), (resp.,+1)

+1, t ∈ [τ3, tf ], (resp.,−1)

and

x1(t) = v0 cos γ0 +

∫ t

0

a cosx3(s)ds,

x2(t) = v0 sin γ0 +

∫ t

0

a sinx3(s)− g0ds,

(3.3.2)


x3(t) =

x30 + x40t+ bt2/2, t ∈ [0, τ1), (resp., x30 − x40t− bt2/2, )x3(τ1) + (x40 + bτ1)(t− τ1)− b(t− τ1)2/2, t ∈ [τ1, τ2),

(resp., x3(τ1)− (x40 + bτ1)(t− τ1) + b(t− τ1)2/2, )

x3 − b(t− τ2)2/2, t ∈ [τ2, τ3), (resp., x3 + b(t− τ2)2/2, )

x3(τ3)− b(τ3 − τ2)(t− τ3) + b(t− τ3)2/2, t ∈ [τ3, tf ],

(resp., x3(τ3) + b(τ3 − τ2)(t− τ3)− b(t− τ3)2/2, )

(3.3.3)

x4(t) =

x40 + bt, t ∈ [0, τ1), (resp., x40 − bt, )x40 + bτ1 − b(t− τ1), t ∈ [τ1, τ2), (resp., x40 − bτ1 + b(t− τ1), )

−b(t− τ2), t ∈ [τ2, τ3), (resp. b(t− τ2), )

−b(τ3 − τ2) + b(t− τ3), t ∈ [τ3, tf ], (resp., b(τ3 − τ2)− b(t− τ3), )

with

τ1 = −x40

b+

√x2

40

2b2− x30 − x3

b, τ2 = 2τ1 +

x40

b,

τ3 = τ2 +

√−x3f − x3

b, tf = 2τ3 − τ2,

(resp., τ1 = −x40

b+

√x2

40

2b2+x30 − x3

b, τ2 = 2τ1 +

x40

b,

τ3 = τ2 +

√x3f − x3

b, tf = 2τ3 − τ2,

)Moreover,

p3(0) =ap2

(τ3 − τ1) cos γf

(∫ τ3

0

∫ s

0

cos(x3(τ)− γf ) dτ ds

−∫ τ1

0

∫ t

0

cos(x3(τ)− γf ) dτ ds), (3.3.4)

and

p4(0) =ap2

cos γf

( τ1

(τ3 − τ1)

∫ τ3

0

∫ s

0

cos(x3(τ)− γf ) dτ ds

− τ3

(τ3 − τ1)

∫ τ1

0

∫ s

0

cos(x3(τ)− γf ) dτ ds). (3.3.5)

Proof 3.3.2. By taking the switching times τi, i = 1, 2, 3 and the nal time tf asunknowns, and by solving the equations x4(τ2) = x4(tf ) = 0, x3(τ2) = x3, x3(tf ) = x3f ,we get u(t), x(t), τi, i = 1, 2, 3 and tf as functions of x3. Then, the expressions (3.3.4)-


(3.3.5) are obtained by integrating the extremal equations of p3 and p4, i.e.,

p3(t) =p3(0) +

∫ t

0

a(p1 sinx3(τ)− p2 cosx3(τ)) dτ

=p3(0)− ap2

cos γf

∫ t

0

cos(x3(τ)− γf ) dτ,

p4(t) =p4(0)− p3(0)t+ap2

cos γf

∫ t

0

∫ s

0

cos(x3(τ)− γf ) dτ ds.

(3.3.6)

and calculating p3(0) and p4(0) from (3.3.6) by requiring that p4(τ1) = p4(τ3) = 0.

Remark 3.3.2. From H(0) = 0 and using the transversality condition (3.2.2), weinfer p1 and p2 as functions of x3 provided that p0 6= 0. Actually, p0 is indeed nonzero,otherwise, using H(0) = 0, condition (3.2.2) and equations (3.3.4)-(3.3.5), we wouldinfer that p = 0, which is absurd.

Remark 3.3.3. For given terminal conditions x(0) = x0, x3(tf ) = x3f and x4(tf ) =0, Fx3 can be seen as a one-parameter family of trajectories with parameter x3. Thevalue x3 can be numerically derived from the condition (3.3.1), and then (x(t), p(t)) isobtained.

Recall that if condition (3.3.1) is satised, i.e., x2(tf ) = x1(tf ) tanx3f , then thetrajectories x(t) in Fx3 together with p(t) is an extremal of (MTTP).

Especially, according to Lemma 3.3.2 and Remark 3.3.3, the nal time tf is also afunction of x3. Hence, for any given x3 ∈ (max(x30, x3f ), x

∗3] (resp., x3 ∈ [x∗3,min(x30, x3f )))

with x30 = x30 − x240

2bsignx40, we have

γf (x3) := γ(tf (x3)) = arctan x2(tf (x3))/x1(tf (x3)). (3.3.7)

If we have

∂γf (x3)

∂x3

=1

vf (x3)

(∂x2(tf (x3))

∂x3

cos γf (x3)− ∂x1(tf (x3))

∂x3

sin γf (x3)

)=

1

vf (x3)tf (x3)

∫ tf (x3)

0

(a(T1(x3) sin(x3 − γf (x3)) + tf (x3)

cos(x3 − γf (x3)))− g0T1(x3) cos γf (x3)

)dt

=1

vf (x3)tf (x3)

∫ tf (x3)

0

(a√T 2

1 (x3) + t2f (x3) sin(x3 − γf (x3) + ϕ)

− g0T1(x3) cos γf (x3))dt > 0,

(3.3.8)


where

vf (x3) =√x1(tf (x3))2 + x2(tf (x3))2, ϕ = arctan

(tf (x3)

T1(x3)

),

T1(x3) =1√

(x3 − x30)/b+ x240/(2b

2)+

1√(x3 − γf (x3))/b

,

for all x(t) in Fx3 , then we have that γf (x3) is monotone with x3. Therefore, the valueof γf (x3) reaches its maximum (resp., minimum) when x3 = x∗3. In this sense, we havea reachable set of γ(tf ) as a function of x3, dened by

AccDx3(γ(tf )) = γf (x3)|x3 ∈ Dx3,

whereDx3 = (max(x30, x3f ), x

∗3] ∪ [x∗3,min(x30, x3f ).

Remark 3.3.4. In the anticlockwise case, the trajectories generally take the formof A+A−A+. However, if the condition (3.3.8) is valid, γf (x3) achieves a minimumextremal value over [max(x30, x3f ), x

∗3] when x3 = max(x30, x3f ). Then, if the expected

x3f = γf < γf (max(x30, x3f )), the trajectory takes the form A−A+A−. There exists ax3 = x3 ∈ [x∗3,min(x30, x3f )) such that γf (x3) = γf . Hence, x3 takes value in

Dacx3= (max(x30, x3f ), x

∗3] ∪ [x3,min(x30, x3f )),

for anticlockwise maneuvers. For the clockwise maneuvers, we have that x3 takes valuein

Dcx3= (max(x30, x3f ), x3] ∪ [x∗3,min(x30, x3f )),

where x3 ∈ (max(x30, x3f ), x∗3] being the extremal value such that γf (x3) = γf .

The positivity condition (3.3.8) is hard to check explicitly, however, numerically thiscondition can be veried easily for given terminal conditions. This is why we take itas an assumption. Accordingly, we make the following assumptions throughout thissection. The rst assumption is that τ1, τ2, τ3 and tf are nonnegative real numbers.The second assumption is that (3.3.8) holds. The third one is that the spacecraft wouldnot crash after the maneuver. The results of our numerical simulations are consistentwith these assumptions:

• the real numbers x30, x40, x3f are chosen such that τ1 ≥ 0, τ2 ≥ 0, τ3 ≥ 0 andtf > 0;

• for every x3 ∈ (max(x30, x3f ), x∗3] (resp., x3 ∈ [x∗3,min(x30, x3f ))) with x30 =

x30 − x240

2bsignx40, we have∫ tf

0

sin(x3 − γf + ϕ) dt >g0tf cos γf

a√

1 + tan2 ϕ;


• x1(tf ) > 0, x2(tf ) > 0.

Under these assumptions, we have the following chattering prediction result.

Theorem 3.3.1 (Chattering prediction). Let x(·) ∈ Fx3 be an optimal trajectoryof (MTTP) in the at-Earth case. In the anticlockwise case (resp., in the clockwisecase), if

SC ≥ 0 (resp., if SC ≤ 0), (3.3.9)

with SC dened bySC = x2(tf (x

∗3))− x1(tf (x

∗3)) tan γf , (3.3.10)

where x1(tf (x∗3)) = x10 +

∫ tf (x∗3)

0a cosx3(t) dt, x2(tf (x

∗3)) = x20 +

∫ tf (x∗3)

0(a sinx3(t) −

g0) dt, and x3(t) is calculated from (3.3.2)-(3.3.3) with x3 = x∗3, then x(·) does notinvolve any singular arc.

Our aim here is to check whether there exists a value of x3 such that the terminalcondition (3.3.1) is satised. Under our assumptions, condition (3.3.9) means that thereexists a value of x3, denoted by x3, such that the corresponding value γf (x3) belongsto the reachable set AccDx3

(γ(tf )).

Proof 3.3.3. In the counterclockwise case, if SC ≥ 0, then we get from (3.3.7) and(3.3.10) that tan γf (x

∗3) ≥ tan γf provided that x1(tf ) > 0 and that x2(tf ) > 0. Using

that γf = x3f ∈ (0, π/2), it follows that γf (x∗3) ≥ γf , and thus (γf (x

∗3)−γ0) ≥ (γf −γ0).

Since ∂γf (x3)/∂x3 > 0, we infer from the implicit function theorem that there exists ax3 := x3 = X (γf ) ∈ Dacx3

, where X (·) is a C1 function, such that γf (x∗3) ≥ γf (x3) = γf .

Hence, the corresponding trajectory x(t) is an optimal trajectory for (MTTP) withterminal value of γf (x3). The proof is similar in the clockwise case.

Remark 3.3.5. If (3.3.9) is not satised, then there are two possible types of solu-tions: one has more bang arcs, the other has a singular arc with chattering arcs aroundthe singular junctions. The points of S2 actually correspond to the zeros of the second-order time derivative of the switching function (z3 = 0 in the semi-canonical form),and so the zeros of S2 will not impose an immediate eect on the switching function,but ensure the switching function to have more possible switchings. The numerical re-sults show that the additional bang arcs lead to extremals that are closer to the singularsurface with an exponential speed.

3.3.2 Non-Flat Case c > 0

If c > 0 then the analysis of (MTTP) becomes more complicated, but we are able aswell to describe the set of terminal data for which optimal trajectories do not have anysingular arc, as we will see next.

We assume that the condition (3.3.8) still holds, i.e., ∂γf (x3)

∂x3> 0, and the real

numbers x30, x40, x3f are chosen such that τ1 ≥ 0, τ2 ≥ 0, τ3 ≥ 0 and tf > 0. Assume


moreover that the real numbers v0 and γ0 are chosen such that the two components ofthe velocity are positive along the whole trajectory, i.e., x1(t) > 0, x2(t) > 0, t ∈ [0, tf ].Using Table 2.1, we have

x1(t) ∈ [a cosx3 − cv2max, a cosx3],

x2(t) ∈ [a sinx3 − g0, a sinx3 − g0 + cv2max],

x3(t) ∈ [x4 − cvmax, x4],

where v2max ≈ (x10 + aT )2 + (x20 + aT + a2cT 3/3)2 < v2

m. It can be seen that the termsin c in the dynamics cause a decrease of x1 and x3, and an increase of x2. We considerthe auxiliary problem

min tf

x1 = a cosx3 − c1x1x2, x2 = a sinx3 − g0 + c1x21, x3 = x4 − cvmax, x4 = bu,

x(0) = x0, x3(tf ) = x3f , x4(tf ) = 0,(3.3.11)

where c, c1 ∈ [0, 10−6]. Similarly to Lemma 3.3.2 in the at-Earth case, the solutions ofthis problem, of the form A+A−A+ (resp., A−A+A−), can be obtained by integratingthe dynamical system, by using the control

u(t) =

+1, t ∈ [0, τ1), (resp.,−1)

−1, t ∈ [τ1, τ2) ∪ [τ2, τ3), (resp.,+1)

+1, t ∈ [τ3, tf ], (resp.,−1)

with

τ1 = −(x40 − cvmax)b

+

√(x40 − cvmax)2

2b2− x30 − x3

b, τ2 = 2τ1 +

(x40 − cvmax)b

,

τ3 = τ2 +

√(−cvmax)2

2b2− x3f − x3

b, tf = 2τ3 −

cvmaxb− τ2.

(resp., τ1 = −(x40 − cvmax)

b+

√(x40 − cvmax)2

2b2+x30 − x3

b,

τ2 = 2τ1 +(x40 − cvmax)

b, τ3 = τ2 +

√(−cvmax)2

2b2+x3f − x3

b,

tf = 2τ3 −cvmaxb− τ2

)As in Remark 3.3.3, the trajectories of the form A+A−A+ and A−A+A− for (MTTP)in the non-at case are also parametrized by x3. Let us denote the angle γ(tf ) ofthe auxiliary problem (3.3.11) by γ(tf (x3), c, c1). Let γf (x∗3, c, c1) = min

(γ(tf (x

∗3), c >

0, c1 = 0), γ(tf (x∗3), c > 0, c1 > 0)

). Based on the numerical results, we make the


following assumptions:

• γf (x∗3, c, c1) ≥ γf (x∗3) = γf (tf (x

∗3), c = 0, c1 = 0) in the anticlockwise maneuvers;

• γf (x∗3, c, c1) ≤ γf (x∗3) = γf (tf (x

∗3), c = 0, c1 = 0) in the clockwise maneuvers;

Under these assumptions, we have the following result.

Corollary 3.3.1. Let x(·) be an optimal trajectory of (MTTP) in the non-at case.Then:

• for an anticlockwise maneuver, if (3.3.9) holds true then x(·) does not have anysingular arc;

• for a clockwise maneuver, if

SC = x2(tf (x∗3), c, c1)− x1(tf (x

∗3), c, c1) tan γf ≤ 0,

where x2(tf (x∗3), c, c1)/x1(tf (x

∗3), c, c1) = tan γf (x

∗3, c, c1), then x(·) does not have

any singular arc.

Proof 3.3.4. For an anticlockwise maneuver (resp. a clockwise maneuver), wehave that if SC ≥ 0 (resp. SC ≤ 0), then 0 ≤ γf ≤ γf (x

∗3) ≤ γf (x

∗3, c, c1), (resp.,

0 ≥ γf ≥ γf (x∗3, c, c1)), and thus there exists a x3 = X (γf ) such that

(γf (x3, c > 0, c1 >

0)−γ0

)≤(γf (x

∗3, c, c1)−γ0

), (resp.,

(γf (x3, c > 0, c1 > 0)−γ0

)≥(γf (x

∗3, c, c1)−γ0

)),

and its associated trajectory is an optimal solution of (MTTP).

Remark 3.3.6. Similarly to Remark 3.3.5, in the non-at case, numerical resultsshow that if the conditions in Corollary 3.3.1 are not satised, then the trajectories willhave more bang arcs until the singular arc nally appear with chattering type junctions.

3.4 Numerical Results

In this section, we compute numerical optimal strategies, for dierent initial conditions,either by means of a direct method, or by means of an indirect one (shooting method).It is important to note that, if the optimal trajectory involves a singular arc and thushas chattering, then the shooting method fails in general. Indeed, the innite numberof switchings may cause a failure in the numerical integration of the dynamical system,and then direct methods may therefore be more appropriate to approach chattering.However, since they are based on a discretization, they can only provide a sub-optimalsolution of the problem, having a nite number of switchings.

In the rst subsection, we provide several numerical simulations, where the optimalsolutions are computed by means of a shooting method, in situations where the optimaltrajectory is known to be bang-bang, without any singular arc, and with a nite numberof switchings.

78 3.4. Numerical Results

In the second subsection, we describe in more details sub-optimal strategies, and weprovide evidence of their relevance in cases where we have chattering.

We consider the initial and nal conditions settled in Table 3.1. Here, we set γ(0) =

x30 x40 x3f x40 x2f − x1f tanx3f

Counter-clockwise 1.3 0.0 1.5 0.0 0.0Clockwise 1.5 0.0 1.3 0.0 0.0

v0 = (x210 + x2

20)1/2, x10 = v0 cosx30, x20 = v0 sinx30

Table 3.1: Initial and nal conditions.

γ0 = x30, meaning that before the maneuver the spacecraft was on a trajectory withangle of attack equal to zero.

Recall that when the optimal trajectory contains a singular arc, then the extremalis normal, i.e. p0 6= 0 (see Lemma 3.2.2). Moreover, in the at-Earth case, we have seenfrom the analysis in Section 3.3.1 that the bang-bang extremals are normal in case oftwo control switchings. The argument was based on Remark 3.3.2. Furthermore, it isnot dicult to see that if the control switches at least two times, then the extremals arenormal. Indeed, when the control switches more than two times, expressions of p3(0)and p4(0) similar to (3.3.4) and (3.3.5) can be obtained, and the fact that p0 6= 0 followsfrom the same argument as in Remark 3.3.2. Therefore, abnormal extremals may onlyoccur whenever the control switches at most one time.

In the non-at case, since c > 0 is very small, we can assume that p0 < 0 though theabnormal extremals may also exist with a few certain terminal conditions. Thus, theadjoint vector can be normalized by p0 = −1. The results of the numerical simulationsare consistent with this assumption.

3.4.1 Chattering Prediction

In practice, the terminal condition that can take very dierent values is the initialmodulus of velocity v0. Hence, we next investigate the inuence of v0 on the occurrenceof optimal singular arcs.

Flat-Earth case with two switchings. If we consider v0 as variable and if wetake c = 0, then, by solving SC = 0, we get vup = v0 = 1086.2m/s (resp., vdown =v0 = 1694.3m/s) for anticlockwise maneuvers (resp., for clockwise maneuvers). Whenv0 ≤ vup (resp., v0 ≤ vdown), we have SC ≥ 0 for anticlockwise maneuvers (resp., SC ≤ 0for clockwise maneuvers). In this case, according to Theorem 3.3.1, there is no singulararc in the optimal solution. Moreover, the maneuver times for both maneuvers are thesame, i.e., tf = 36.5437 s.

Using an indirect method (shooting method), we compute the optimal solutions of(MTTP), in the absence of a singular arc. Recall that the indirect method does notwork when there are chattering arcs. From the prediction above, we should therefore


be able to use successfully an indirect method when v0 ≤ vup. We will see in numericalsimulations that the indirect method works when the trajectory consists of three bangarcs, but fails otherwise due to chattering.

Figure 3.5 provides the solutions for two dierent values of the initial velocity mod-ulus v0 for the anticlockwise case, i.e., v0 = vup = 1086.2m/s (plotted in solid lines) andv0 = 1080m/s (plotted in dashed lines). Figure 3.6 shows the solutions of clockwisemaneuvers with v0 = vdown = 1694.3m/s (plotted in solid lines) and v0 = 1690m/s(plotted in dashed lines). The red star points represent the touching point of the tra-jectories with the surface S2 (where x3(t) touches x∗3). It is shown in Figure 3.5 thatthere is no singular arc in the trajectories when v0 < vup (resp., in Figure 3.6 whenv0 < vdown). The control switches two times and the x3 associated with the dashed lineis smaller than x∗3 = x3f + π/2 (resp. bigger than x∗3 = x3f − π/2). In Remark 3.3.5,

0 10 20 30

1.5

2

2.5

3

t: s

x 3: rad

0 10 20 30−0.2

−0.1

0

0.1

0.2

t: s

x 4: rad

/s

0 10 20 30−1

−0.5

0

0.5

1

t: s

u15 20

3.053.1

9.5 100.18

0.185

9 9.5100.460.480.5

Figure 3.5: Time history of x3, x4 and u when v0 = vup and v0 = 1080m/s

10 20 30

0

0.5

1

1.5

t: s

x 3: rad

0 10 20 30−0.2

−0.1

0

0.1

0.2

t: s

x 4: rad

/s

0 10 20 30−1

−0.5

0

0.5

1

t: s

u

16 18 20−0.26−0.24

9.49.59.6−0.188−0.186−0.184 9 9.5

−0.68−0.66−0.64

Figure 3.6: Time history of x3, x4 and u when v0 = vdown and v0 = 1690m/s

we mentioned that, when the condition (3.3.9) is not satised, there are more bangarcs until the appearance of a singular arc. We will show next the solutions with moreswitchings. As remarked, these results will show that the extremals will get closer tothe singular surface S when more bang arcs are present.


Flat-Earth case with more switchings. In fact, we can compute the correspondingvalue of v0 for optimal controls with dierent number of switchings, in the followingway. Let us assume that the optimal control u has 2m, m = 1, · · · , N switchings andu(0) = u0 being +1 or −1, i.e.,

u(t) =

u0, t ∈ [τ0, τ1],

−u0, t ∈ [τ4j−3, τ4j−1],

u0, t ∈ [τ4j−1, τ4j+1],

−u0, t ∈ [τ4m−3, τ4m−1],

u0, t ∈ [τ4m−1, τ4m],

with j = 1, · · · , (m−1), t0 = τ0, tf = τ4m, then we know that ϕ(τ2k+1) = p4(τ2k+1) = 0,k = 0, · · · , 2m − 1. Here we have additionally h1(τ2m) = p4(τ2m) = 0, because themaximum v0 corresponding to 2m switchings happens when u is about to have onemore switchings between τ2m−1 and τ2m+1.

Let q =(x3(τ2k)

)k=1,··· ,2m−1

be the variable vector (of dimension 2m − 1). OnFigure 3.7 are represented p4(t), p3(t), x4(t) and x3(t) for an anticlockwise maneuverwith m = 3, the variable q = (q1, · · · , q5) is of dimension 2m− 1.

Figure 3.7: Example of trajectory associated with optimal control of 6 switchings.

Using (3.3.6), we derive 2m− 1 constraints on q without the adjoint vector p, i.e.,

τk1 − τk2

τk3 − τk4

=

∫ τk10

∫ τ0

cos(x3(s)− γf ) ds dτ −∫ τk2

0

∫ τ0

cos(x3(s)− γf ) ds dτ∫ τk30

∫ τ0

cos(x3(s)− γf ) ds dτ −∫ τk4

0

∫ τ0

cos(x3(s)− γf ) ds dτ, (3.4.1)

where k1, k2, k3, k4 ∈ 2k + 1 | k = 0, · · · , 2m− 1 ∪ 2m and k1 6= k2, k3 6= k4. Notethat at least one of these equations must involve τ2m.

Since x3(τ2k), k = 1, · · · , 2m − 1 are local extrema, we must have x4(τ2k) = 0,


k = 1, · · · , 2m− 1. By integrating the system from x(0) = x0 and requiring that

x4(τ2k) = 0, k = 1, · · · , 2m− 1,

x3(τ2k) = q, k = 1, · · · , 2m− 1,

x3(tf ) = x3f , x4(tf ) = 0,

we can parametrize τk, k = 1, · · · , 4m by q, and hence as well the trajectories x3(t) andx4(t) which are parametrized by τk, k = 1, · · · , 4m. More precisely, we have

τ1 = −x40

b+

√x2

40

2b2+|q(1)− x30|

b, τ2 = τ1 +

√x2

40

2b2+|q(1)− x30|

b,

τ2k+1 = τ2k +

√|q(k)− q(k + 1)|

b, τ2k+2 = τ2k+1 +

√|q(k)− q(k + 1)|

b,

k = 1, · · · , 2m− 2,

τ4m−1 = τ4m−2 +

√|q(2m− 1)− x3f |

b, τ4m = τ4m−1 +

√|q(2m− 1)− x3f |

b.

Hence, we can get the value of q by solving (3.4.1). Then taking v0 as variable andγ(tf ) = γf as shooting function, we can derive the maximum v0 that can be used whenwe expect the control to have 2m− 1 switchings.

Using this method, we get that, when v0 ∈ (vup, 1183.4],m/s, in the anticlockwisecase, the control u(t) has two switchings. When v0 ∈ (1183.4, 1999.3]m/s, the controlu(t) has four switchings. Then when v0 ∈ (1999.3, 2132.1]m/s, the control u(t) hassix switchings. Figures 3.8 and 3.9 give the time history of the switching function

0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

t: s

h 1

22 24 26

05

1015

x 10−5

Figure 3.8: Switching function ϕ(t) when v0 = 1999.3m/s in the anticlockwise case.

ϕ(t) = h1(t) when v0 = 1999.3m/s and v0 = 2132.1m/s, respectively. Observingfrom the zoom-in windows of the gures, we see that the switching function is almostequal to zero when t ∈ [22, 26] s and t ∈ [22, 28] s. This implies that the associatedextremals are very close to the singular surface S along these time intervals. Thesegures also show that the additional bang arcs lead rapidly the extremals to get closer


to the singular surface S (see Remark 3.3.5). Note that when u(t) has 2m switchings,

0 5 10 15 20 25 30 35 40 450

0.2

0.4

0.6

0.8

1

t: s

h 1

22 24 26 280

5

10

15x 10−5

24.5 25 25.5−1

0

1x 10−6

Figure 3.9: Switching function ϕ(t) when v0 = 2132.1m/s in the anticlockwise case.

the trajectory of x3(t) has between max(0, 2m − 2) and 2m contact points with thesurface S2. Figure 3.10 shows the comparison of solutions with v0 = 1350m/s (solidline) and v0 = 1683m/s (dashed line). They both belong to the four switchings case,i.e., m = 2. We can see that the solid line touches the surface S2 two times, while thedashed line touches four times.

0 20 40

1.5

2

2.5

3

t: s

x 3: rad

0 20 40−0.2

−0.1

0

0.1

0.2

t: s

x 4: rad

/s

0 20 40−1

−0.5

0

0.5

1

t: s

u

Figure 3.10: Time history of x3, x4 and u when v0 = 1350m/s and v0 = 1683m/s

Non-at case. When c > 0, according to Corollary 3.3.1, there does not exist anysingular arc for anticlockwise maneuvers when v0 ≤ vup. For clockwise maneuvers, ifv0 ≤ vdown = 1624.3m/s, then there is no singular arc (this condition is obtained bysolving SC = 0 with c = 0 and c1 = 10−6). The assumptions are also veried.

In Figure 3.11, setting v0 = vup, we compare in anticlockwise case the solution withc > 0 (plotted with solid line) and the solution with c = 0 (plotted with dashed line).The trajectory x3(t) in the at-Earth case in fact reaches the surface S2 in smallertime than in the non-at case. Let v0 = vdown = 1624.3m/s. Figure 3.12 gives a


0 10 20 30

1.5

2

2.5

3

t: s

x 3: rad

0 10 20 30−0.2

−0.1

0

0.1

0.2

t: s

x 4: rad

/s

0 10 20 30−1

−0.5

0

0.5

1

t: s

u

15 203

3.05

8 9 100.18

0.1850.19

9 9.5100.550.6

Figure 3.11: Time history of x3, x4 and u when c = 0, c = 10−6 and v0 = vup

comparison in the clockwise cases of the solution with c > 0 (plotted with solid lines)and the solution with c = 0 (plotted with dashed lines). Both trajectories do not touchthe surface S2 and the trajectory in the non-at case gets closer" to S2. The controlswitches two times and there is no singular arc. In other simulations, we also observe

0 10 20 30−0.5

0

0.5

1

1.5

t: s

x 3: rad

0 10 20 30−0.2

−0.1

0

0.1

0.2

t: s

x 4: rad

/s

0 10 20 30−1

−0.5

0

0.5

1

t: s

u

Figure 3.12: Time history of x3, x4 and u when c = 0 and c = 10−6

that when v0 < vdown, the optimal control only has two switchings and x3 will not reachto x∗3. However, when v0 > vdown, new bang arcs appear and the trajectory tends tohave chattering arcs. These results illustrate Corollary 3.3.1 and Remark 3.3.6.

3.4.2 Sub-Optimal Strategies

Let N be a positive integer. We consider a subdivision 0 = t0 ≤ t1 ≤ · · · ≤ tN = tf ofthe interval [0, tf ] (where ti are unknown), and we consider piecewise constant controlsover this subdivision, thus enforcing the control to switch at most N times. We considerthe optimal control problem (MTTP) with this restricted class of controls, that wedenote by (MTTP)N .

Solving this problem provides what we call a sub-optimal strategy (with at most Nswitchings), because the optimal value of (MTTP)N must be less than or equal to the


optimal value of (MTTP).By the way, we expect that, (MTTP)N Γ-converges to (MTTP) as N → +∞,

meaning that, in particular, the optimal value of (MTTP)N converges to that of(MTTP). We will come back on this issue later.

As in classical direct methods in optimal control, we propose to solve numerically(MTTP)N , where the unknowns are the nodes ti of the subdivision, and the values uiof the control over each interval (ti, ti+1). More conveniently, instead of considering theswitching times ti as unknowns, we consider the durations ti+1 − ti as unknowns. Notethat these durations may be equal to 0.

The control is kept constant along each interval of the subdivision, but in orderto discretize the state in a ner way, we consider another (much) ner subdivision tocompute the discretized state.

We solve the resulting optimization problem by using IPOPT (see [108]) combinedwith the modeling language AMPL (see [42]).

Numerical results for non-at anticlockwise maneuvers. We consider rst thecase of anticlockwise maneuvers. Let v0 = 3000m/s. For N = 500, the numericaloptimal solution of (MTTP)N is provided on Figures 3.13 and 3.14. This simulationprovides numerical evidence of the fact that we have a singular arc for t ∈ [25.7, 28.1] s,with a chattering phenomenon at the junction points with the singular arc (see Figure3.13, on the right, where a zoom is made on those points). The singular control takesvalues in [−0.0016,−0.0013].

Moreover, the coordinates x3(t) and x4(t) oscillate around x3 = x∗3 and x4 = 0respectively, and the coordinates x1(t) and x2(t) oscillate around a straight line in thevicinity of the singular arc of the at-Earth case. This indicates that the singular arcof the non-at case does not vary much from that of the at-Earth case.

0 20 40 60−1

−0.5

0

0.5

1

t: s

u

25 30 35 40−1

−0.5

0

0.5

1

t: s

u

24.5 25 25.5−0.02

00.02

33.15 33.2 33.25−4−2

0x 10−3

Figure 3.13: Control u(t) in anticlockwise maneuver

Numerical results for non-at clockwise maneuvers. For clockwise maneuvers,still taking N = 500, the numerical optimal solution of (MTTP)N is provided on


200 400 600 8002400

2600

2800

3000

x1

x 2

0 20 40 60

0

1

2

3

t: sx 3: r

ad0 20 40 60

−0.2

−0.1

0

0.1

0.2

t: s

x 4: rad

/s

20 25 303

3.1

24 26 2805

10x 10−3

Figure 3.14: State variable x(t) in anticlockwise maneuver.

Figures 3.15 and 3.16. By comparing the clockwise maneuver in Figure 3.15 and 3.16and the anticlockwise maneuver in Figure 3.13 and 3.14, we see that, when x4(0) = 0,to realise the same |γf−γ0|, one need 62.43 s for the anticlockwise case and only 59.26 sfor the clockwise case.

10 20 30 40 50−1

−0.5

0

0.5

1

t: s

u

25 30 35−1

−0.5

0

0.5

1

t: s

u

25 26 27−0.04−0.02

00.020.04

30.9 310

1020

x 10−4

Figure 3.15: Optimal control in clockwise maneuver.

Γ-convergence of (MTTP)N towards (MTTP). It seems natural to expect that,if N → +∞, then the solution of (MTTP)N converges to the solution (if it is unique)of (MTTP). At least, Γ-convergence is expected. Such an analysis is beyond the scopeof the present chapter, however it is interesting to provide numerical simulations, foran anticlockwise maneuver, with several values of N :

N ∈ 6, 8, 10, 12, 14, 16, 18, 20, 30, 40, 50, 100, 200, 300, 400.

Figure 3.17 provides the numerical optimal control obtained for (MTTP)N . We observethat, when N becomes larger, then the optimal control seems to converge to its expectedlimit, that is the optimal control of (MTTP) with a singular arc and chattering. On


400 6002400

2600

2800

3000

3200

x1

x 2

10 20 30 40 50

0

0.5

1

1.5

t: sx 3: r

ad10 20 30 40 50

−0.2

−0.1

0

0.1

0.2

t: s

x 4: rad

/s24 25 26−0.32

−0.315−0.31

24 26 28−10

−505

x 10−3

Figure 3.16: State x(t) in clockwise maneuver.

Figure 3.18, we have reported the values of the maneuver time, in function of N . Weobserve that they seem to decrease exponentially with respect to N . This numericalobservation is important because, in practice, this means that it is not necessary totake N too large. Even with quite small values of N , the minimal time obtained for(MTTP)N seems to be very close to the minimal time for (MTTP). Hence the sub-optimal strategy seems to be a very good solution in practice, to bypass the problemsdue to chattering. We conclude with the following conjecture.

Conjecture. With obvious notations, we denote by (xN(·), uN(·), tNf ) the optimal so-lution of (MTTP)N , and by (x(·), u(·), tf ) the optimal solution of (MTTP) (assumingthat they are unique). Then tNf → tf exponentially, xN(·) → x(·) in C0-topology, anduN(·)→ u(·) in L1-topology, as N → +∞.

Remark 3.4.1. Such convergence properties have been established in [54, 98], butfor problems not involving any singular arc. Here, the diculty of establishing such aresult (in particular, for the control) is in the presence of an optimal singular arc.

Remark 3.4.2. These simulations were done by using hot-restart, that is, by usingthe solution of (MTTP)N to initialize the problem with a larger value of N .


0 20 40 60−1

0

1N=6

0 20 40 60−1

0

1N=8

0 20 40 60−1

0

1N=10

0 20 40 60−1

0

1N=12

0 20 40 60−1

0

1N=14

0 20 40 60−1

0

1N=16

0 20 40 60−1

0

1N=18

0 20 40 60−1

0

1N=20

0 20 40 60−1

0

1N=30

0 20 40 60−1

0

1N=40

0 20 40 60−1

0

1N=50

0 20 40 60−1

0

1N=100

0 20 40 60−1

0

1N=200

0 20 40 60−1

0

1N=300

0 20 40 60−1

0

1N=400

Figure 3.17: Control u(t) with dierent discretization step N .

0 100 200 300 40062

64

66

68

70

72

74

76

78

80

Discretization number

Man

euve

r tim

e (c

ost):

s

Figure 3.18: Maneuver time tf with respect to the discretization step N .

Chapter 4

Minimum Time Control Problem

(MTCP)

Contents

4.1 Some General Results for Bi-Input Control-Ane Systems 89

4.1.1 Application of the Pontryagin maximum principle . . . . . . . 90

4.1.2 Computation of Singular Arcs, and Necessary Conditions for

Optimality . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91

4.1.3 Chattering Phenomenon . . . . . . . . . . . . . . . . . . . . . 94

4.2 Geometric Analysis of the Extremals of (MTCP) . . . . . . 97

4.2.1 Regular Extremals . . . . . . . . . . . . . . . . . . . . . . . . 97

4.2.2 Singular and Chattering Extremals . . . . . . . . . . . . . . . 101

4.3 Numerical Approaches . . . . . . . . . . . . . . . . . . . . . . 105

4.3.1 Indirect Method and Numerical Continuation . . . . . . . . . 106

4.3.2 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . 111


4.4.1 Numerical Results without Chattering . . . . . . . . . . . . . 113

4.4.2 Numerical Results with Chattering Arcs . . . . . . . . . . . . 117

Introduction

As we have mentioned in the general introduction, the chattering phenomenon that mayoccur according to the terminal conditions under consideration, makes in particularthe problem quite dicult. Another important property in (MTCP) is due to thecoupling of the attitude movement with the trajectory dynamics. Indeed the system

88

Chapter 4: Minimum Time Control Problem (MTCP) 89

contains both slow (trajectory) and fast (attitude) dynamics. This observation will beparticularly important in order to design appropriate numerical approaches.

In order to analyze the extremals of the problem, we use geometric optimal controltheory (see [2, 93, 104]). The Pontryagin maximum principle and the geometric optimalcontrol, especially the concept of Lie bracket, will be used in this chapter in order toestablish an existence result of the chattering phenomenon. More precisely, based onthe Goh and generalized Legendre-Clebsch conditions, we prove that there exist optimalchattering arcs when trying to connect a regular arc with a singular arc of order two.

We will use numerical continuation, which has proved to be very powerful tool tobe combined with the PMP. However, due to the chattering phenomenon, numericalcontinuation combined with shooting cannot give an optimal solution to the problemfor certain terminal conditions for which the optimal trajectory contains a singulararc of higher-order. In that case, we propose sub-optimal strategies by using directmethods computing approximate piecewise constant controls. It is noticeable that ourindirect approach can also be adapted to generate sub-optimal solutions, by stoppingthe continuation procedure before its failure due to chattering. This approach happensto be faster than the direct approach, and appears as an interesting alternative forpractice.

From the engineer point of view, the theoretical analysis as well as the numericalstrategies and the way to design them (in particular, the design of the problem of orderzero) are strongly based on the fact that the trajectory movement is much slower thanthe attitude movement.

The chapter is organized as follows. In Section 4.1, we apply the PMP and recallsome higher necessary conditions of optimality (Goh and generalized Legendre-Clebschconditions) for bi-input control ane systems, where the two-dimensional control takesits values in the Euclidean unit disk. Based on these necessary conditions of optimality,we establish a result on the existence of the optimal chattering extremals. In Section 4.2,we analyze the regular and singular extremals of (MTCP). For the regular extremals,we classify the switching points and state some useful properties. For the singularextremals (which are of order two), we show that the chattering phenomenon occurs for(MTCP) by using the results given in the previous section. In Section 4.3, we proposea numerical approach to solve (MTCP) by implementing numerical PC continuationcombined with single shooting. Numerical results are given in Section 4.4.

4.1 Some General Results for Bi-Input Control-Ane

Systems

In this section, we focus on the chattering phenomenon for bi-input control-ane sys-tems with control constraints and with commuting controlled vector elds. The resultsthat we are going to give are general and will be used in the next section to analyze(MTCP).

90 4.1. Some General Results for Bi-Input Control-Ane Systems

We consider the following general framework. Let M be a smooth manifold ofdimension n, let x0 ∈M be arbitrary, and let M1 be a submanifold of M . We consideron M the minimal time control problem

min tf ,

x(t) = f(x(t)) + u1(t)g1(x(t)) + u2(t)g2(x(t)), u = (u1, u2)

‖u(t)‖2 = u1(t)2 + u2(t)2 ≤ 1,

x(0) = x0, x(tf ) ∈M1, tf ≥ 0 free,

(4.1.1)

where f , g1 and g2 are smooth vector elds on M .Assume that M1 is accessible from x0, and there exists a constant Btf such that for

every admissible control u, the corresponding trajectory xu(t) satises ‖xu(t)‖ ≤ Btf

for all t ∈ [0, tf ]. Then, according to classical results (see, e.g., [28, 105]), there existsat least one optimal solution (x(·), u(·)), dened on [0, tf ].

4.1.1 Application of the Pontryagin maximum principle

According to the PMP (see Theorem 1.1.1), there must exist an absolutely continuousmapping p(·) dened on [0, tf ], such that p(t) ∈ T ∗x(t)M (cotangent space) for everyt ∈ [0, tf ], and a real number p0 ≤ 0, with (p(·), p0) 6= 0, such that

x(t) =∂H

∂p(x(t), p(t), p0, u(t)), p(t) = −∂H

∂x(x(t), p(t), p0, u(t)),

almost everywhere on [0, tf ], where H(x, p, p0, u) = h0(x, p)+u1h1(x, p)+u2h2(x, p)+p0

is the Hamiltonian of the optimal control problem (4.1.1). Here, we have set h0(x, p) =〈p, f(x)〉, h1(x, p) = 〈p, g1(x)〉, and h2(x, p) = 〈p, g2(x)〉. The maximization conditionof the PMP yields, almost everywhere on [0, tf ],

u(t) =(h1(t), h2(t))√h1(t)2 + h2(t)2

=Φ(t)

‖Φ(t)‖, (4.1.2)

whenever Φ(t) = (h1(t), h2(t)) 6= (0, 0). We call Φ (as well as its components) theswitching function. Note that Φ is continuous. Here and throughout the chapter, wedenote by hi(t) = hi(x(t), p(t)), i = 0, 1, 2 with a slight abuse of notation.

Moreover, we have the transversality condition p(tf ) ⊥ Tx(tf )M1, where Tx(tf )M1 isthe tangent space to M1 at the point x(tf ), and, the nal time tf being free and thesystem being autonomous, we have also h0(x(t), p(t)) + ‖Φ(t)‖+ p0 = 0, ∀t ∈ [0, tf ].

We say that an arc (restriction of an extremal to a subinterval I) is regular if‖Φ(t)‖ 6= 0 along I. Otherwise, the arc is said to be singular. Note that a singularextremal may be both normal or abnormal. We will see in Section 4.2.2 that the singularextremals of (MTCP) must be normal.

A switching time is a time t at which Φ(t) = (0, 0), that is, both h1 and h2 vanish at


time t. Note that, at a switching time, the expression (4.1.2) is not well dened. If onlyone component of Φ(t) vanishes, then the optimal control is still regular. An arc thatis a concatenation of an innite number of regular arcs is said to be chattering. Thechattering arc is associated with a chattering control that switches an innite numberof times, over a compact time interval. A junction between a regular arc and a singulararc is said to be a singular junction.

4.1.2 Computation of Singular Arcs, and Necessary Conditions

for Optimality

We next dene the order of a singular control, since it is important to understand andexplain the occurence of chattering. This concept is related to the way singular controlsare computed, and since it is a bit technical to dene, we start with a preliminary quiteinformal discussion.

Preliminary informal discussion. In order to compute singular controls, the usualmethod is to dierentiate several times the switching function, until the control appearsin a nontrivial way. If ‖Φ(t)‖ = 0 for every t ∈ I, then h1(t) = h2(t) = 0, and,dierentiating in t, we get, using the Poisson bracket, h1 = h0, h1 + u2h2, h1 = 0and h2 = h0, h2 + u1h1, h2 = 0 along I. According to the Goh condition (see[50], see also Lemma 4.1.1), if the singular arc is optimal and the associated singularcontrol is not saturating, then the Goh condition h1, h2 = 〈p, [g1, g2](x)〉 = 0 mustbe satised along I. Therefore we get that h1 = h0, h1 = 〈p, [f, g1](x)〉 = 0 andh2 = h0, h2 = 〈p, [f, g2](x)〉 = 0 along I.

Let us now assume that the vector elds g1 and g2 commute, i.e., [g1, g2] = 0. Bydierentiating again, we get

h1 = h0, h0, h1+ u1h1, h0, h1+ u2h2, h0, h1 = 0,

h2 = h0, h0, h2+ u1h1, h0, h2+ u2h2, h0, h2 = 0.

If

det ∆1 = det

(h1, h0, h1 h2, h0, h1h1, h0, h2 h2, h0, h2

)6= 0

along I, thenu1 =

(− h0, h0, h1h2, h0, h2+ h0, h0, h2h2, h0, h1

)/ det ∆1,

u2 =(h0, h0, h1h1, h0, h2 − h0, h0, h2h1, h0, h1

)/ det ∆1,

(4.1.3)and we say that the control u = (u1, u2) is of order 1 (also called minimal order in[13, 30]). Note that u1 and u2 must moreover satisfy the constraint u2

1 + u22 ≤ 1. Note


also that, if moreover [g1, [f, g2]] = 0 and [g2, [f, g1]] = 0, then (4.1.3) yields

u1 = −h0, h0, h1/h1, h0, h1, u2 = −h0, h0, h2/h2, h0, h2.

Now, if h1, h0, h1 = 0 and h2, h0, h2 = 0 along I, then the singular controlis of higher order, and we must have hi, h0, hj = 0, i, j = 1, 2, i 6= j according tothe Goh condition (see [50, 64], see also Denition 4.1.1 and Lemma 4.1.1), and hencewe go on dierentiating. Assuming that [g1, [f, g1]] = 0 and [g2, [f, g2]] = 0, we have

[gi, ad2f.gi]] = [gi, [f, adf.gi]] = −[f, [adf.gi, gi]]− [adf.gi, [gi, f ]] = 0, i = 1, 2,

and we get

h(3)1 = h0, ad2h0.h1+ u2h2, ad2h0.h1 = 0,

h(3)2 = h0, ad2h0.h2+ u1h1, ad2h0.h2 = 0.

(4.1.4)

Using [g1, g2] = 0 and [gi, [f, gi]] = 0, i = 1, 2, it follows that [gk, [gi, [f, gj]] = 0,i, j, k = 1, 2 and

d

dth2, h0, h1 =

d

dth1, h0, h2 = h0, h1, h0, h2 = 0.

This is a new constraint along the singular arc. Note that the time derivative of thisconstraint is equal to zero and hence does not induce any additional constraint.

Due to higher-order necessary conditions for optimality (see Lemma 4.1.1), an opti-mal singular control cannot appear in a nontrivial way with an odd number of deriva-tives, therefore we must have h2, ad2h0.h1 = 0 and h1, ad2h0.h2 = 0 along I.Accordingly, h(3)

i = 0, i = 1, 2, gives three additional constraints along the singular arch0, ad2h0.h1 = 0, h0, ad2h0.h2 = 0, and h2, ad2h0.h1 = −h1, ad2h0.h2 = 0.Dierentiating the third constraint with respect to t, we have d

dth1, ad2h0.h2 =

h0, h1, ad2h0.h2 = 0, and thus

h2, ad3h0.h1 = ad2h0.h1, adh0.h2,h1, ad3h0.h2 = ad2h0.h2, adh0.h1.

Then, by dierentiating the rst two constraints with respect to t, we get

h(4)1 = ad4h0.h1 + u1h1, ad3h0.h1+ u2ad2h0.h1, adh0.h2 = 0,

h(4)2 = ad4h0.h2 + u1ad2h0.h2, adh0.h1+ u2h2, ad3h0.h2 = 0.

Assuming that hi, ad3h0.hi < 0, i = 1, 2 (generalized Legendre-Clebsch condition, see


below) and that

det ∆2 = det

(h1, ad3h0.h1 ad2h0.h1, adh0.h2

ad2h0.h2, adh0.h1 h2, ad3h0.h2

)6= 0

along I, the singular control is given byu1 =

(− (ad4h0.h1)h2, ad3h0.h2+ (ad4h0.h2)h2, ad3h0.h1

)/ det ∆2,

u2 =((ad4h0.h1)h1, ad3h0.h2 − (ad4h0.h2)h1, ad3h0.h1

)/ det ∆2.

We say, then, that the singular control u = (u1, u2) is of intrinsic order two.

Precise denitions. Now, following [45], let us give a precise denition of the orderof a singular control.

Denition 4.1.1. The singular control u = (u1, u2) dened on a subinterval I ⊂[0, tf ] is said to be of order q if

1. the rst (2q − 1)-th time derivative of hi, i = 0, 1, do not depend on u and

dk

dtk(hi) = 0, k = 0, 1, · · · , 2q − 1,

2. the 2q-th time derivative of hi, i = 0, 1, depends on u linearly and

∂

∂ui

d2q

dt2q(hi) 6= 0, det

(∂

∂u

d2q

dt2qΦ

)6= 0, i = 1, 2,

along I.

The control u is said to be of intrinsic order q if, moreover, the vector elds satisfy

[gi, adkf.gi] ≡ 0, k = 1, · · · , 2q − 2, i = 1, 2.

The condition of a nonzero determinant guarantees that the optimal control can becomputed from the 2q-th time derivative of the switching function. Note that, in thedenition, it is required that the two components of the control have the same order.

We next recall the Goh and generalized Legendre-Clebsch conditions (see [50, 58,64]). It is worth noting that in [64], the following higher-order necessary conditions aregiven even when the components of the control u have dierent orders.

Lemma 4.1.1. (higher-order necessary conditions) Assume that a singular controlu = (u1, u2) dened on I is of order q. Suppose that u is optimal and is not saturating,i.e. ‖u‖ < 1. Then the Goh condition

∂

∂uj

dk

dtk(hi) = 0, k = 0, 1, · · · , 2q − 1, i, j = 1, 2, i 6= j,


must be satised along I. Moreover, the matrix of which the (i, j)-th component

(−1)q∂

∂uj

d2q

dt2q(hi), i, j = 1, 2,

is symmetric and nonpositive denite along I (generalized Legendre-Clebsch Condition).

In (MTCP), as we will see, it happens that singular controls are of intrinsic order2, and that [g1, g2] = 0, [g1, [f, g2]] = 0, and [g2, [f, g1]] = 0, so that the conditionsgiven in the above denition yield [g1, [f, g1]] = 0, [g2, [f, g2]] = 0, [g1, ad2f.g1] = 0,[g2, ad2f.g2] = 0, 〈p, [g1, ad3f.g1](x)〉 6= 0, 〈p, [g2, ad3f.g2](x)〉 6= 0, and

〈p, [g1, ad3f.g1](x)〉〈p, [g2, ad3f.g2](x)〉 − 〈p, [g2, ad3f.g1](x)〉〈p, [g1, ad3f.g2](x)〉 6= 0,

and we have the following higher-order necessary conditions, that will be used in thestudy of (MTCP).

Corollary 4.1.1. We assume that the optimal trajectory x(·) contains a singulararc, dened on the subinterval I of [0, tf ], associated with a non saturating controlu = (u1, u2) of intrinsic order 2. If the vector elds satisfy [g1, g2] = 0, [gi, [f, gj]] = 0,for i, j = 1, 2, then the Goh condition

〈p(t), [g1, adf.g2](x(t))〉 = 0, 〈p(t), [g1, ad2f.g2](x(t))〉 = 〈p(t), [g2, ad2f.g1](x(t))〉 = 0,

and the generalized Legendre-Clebsch condition (in short, GLCC )

〈p(t), [gi, ad3f.gi](x(t))〉 ≤ 0, i = 1, 2,

〈p(t), [g1, ad3f.g2](x(t))〉 = 〈p(t), [g2, ad3f.g1](x(t))〉

must be satised along I. Moreover, we say that the strengthened GLCC is satised ifwe have a strict inequality above, that is, 〈p(t), [gi, ad3f.gi](x(t))〉 < 0.

Corollary 4.1.1 follows from Lemma 4.1.1 and from the arguments developed in theprevious informal discussion. It will be used in Section 4.2.2.

We next investigate the singular junctions for the problem (4.1.1), and the chatteringphenomenon.

4.1.3 Chattering Phenomenon

One can nd in [78] some results on the junction between an optimal regular arc andan optimal singular arc, for single-control ane systems, among which a result statingthat, if the singular arc is of even order and if the control is discontinuous at thejunction, then the junction must be nonanalytical (meaning that the control is notpiecewise analytic in any neighborhood of the junction). In [93, 112, 113], it is proved


that such a nonanalytical junction between a regular arc and a singular arc of intrinsicorder two causes chattering (see also chapter 3). When the control takes values in theunit disk, explicit analytic expressions for some optimal trajectories of linear-quadraticproblems were given, e.g., in [73, 112]. However these results cannot be applied to(MTCP) because the control system is bi-input and the cost functional is the time;they are anyway a good source of inspiration to establish the results of this section.The following result is valid for general bi-input control-ane systems.

Theorem 4.1.1. Consider the optimal control problem (4.1.1). Let (x(·), p(·), p0, u(·))be an optimal extremal lift on [0, tf ]. We assume that u is singular of order two alongan open interval I ⊂ [0, tf ], and we denote this control by us = (u1s, u2s). We as-sume that ‖us(t)‖ < 1 (i.e., the singular control does not saturate the constraint) andthat ∂

∂u1

d4

dt4h2(x(t), p(t)) = 0 along I. Then the optimal control u must switch innitely

many times at the junction with the singular arc. In other words, there is a chatteringphenomenon, which is due to the connection of a regular arc with a singular arc ofhigher-order.

Proof 4.1.1. Since the singular control is of order two, it follows from the denitionthat

∂

∂ui

dk

dtkhi(x(t), p(t)) = 0, k = 0, · · · , 3, i = 1, 2,

∂

∂ui

d4

dt4hi(x(t), p(t)) 6= 0.

Thus, we get from ∂∂ui

d4

dt4hi(x(t), p(t)) 6= 0 and Lemma 4.1.1 that

∂

∂uj

dk

dtkhi(x(t), p(t)) = 0, k = 0, · · · , 3, i, j = 1, 2, i 6= j,

and

∂

∂ui

d4

dt4hi(x(t), p(t)) < 0,

∂

∂u1

d4

dt4h2(x(t), p(t)) =

∂

∂u2

d4

dt4h1(x(t), p(t)),

along the singular arc I. By assumption, we have ∂∂u1

d4

dt4h2(x(t), p(t) = 0, and hence

we can write h(4)i (x(t), p(t)) = ai0(x(t), p(t)) + uisaii(x(t), p(t)) with aii(x(t), p(t)) =

∂∂ui

d4

dt4hi(x(t), p(t)) < 0.

Without loss of generality, we consider a concatenation of a singular arc with aregular arc at time τ ∈ I. Assume that for some ε > 0 the control u is singular along(−ε+τ, τ), and that, along (τ, τ+ε), the control u = (u1, u2) is given by ui = hi/‖Φ‖ ≥0, i = 1, 2. It can be easily seen from the assumption that ‖us‖ < 1 that there exists atleast one component of the singular control that is smaller than the same component of


the regular control, i.e., uks < uk for k = 1 or k = 2. Then, it follows that

h(4)k (τ) = ak0(x(τ), p(τ)) + uk(τ)akk(x(τ), p(τ))

≤ ak0(x(τ), p(τ)) + uks(τ−)akk(x(τ), p(τ)) = h

(4)k (τ−) = 0.

(4.1.5)

Hence the switching function hk has a local maximum at t = τ and is nonpositive alongthe interval (τ, τ+ε). It follows from the maximization property of the Hamiltonian thatuk ≤ 0. Thus, together with the assumption uk = 0. It follows that uj = 1 and ujs < ujfor j 6= k. Replacing k in (4.1.5) with j, we get uj ≤ 0, which gives a contradictionwith uj = 1.

If, instead, we assume ui = hi/‖Φ‖ ≤ 0, i = 1, 2 over (τ, τ + ε), then there mustexists a control component uks that is larger than uks, i.e., uks > uk, and then weobtain h

(4)k (τ) ≥ h

(4)k (τ−) = 0, which yields uk ≥ 0, and thus uk = 0. As in the

previous case, it follows that uj = −1 and ujs > uj for j 6= k, and thus we get from

h(4)k (τ) ≥ h

(4)k (τ−) = 0 that uj ≥ 0, which yields a contradiction.

Then, if we assume ui = hi/‖Φ‖ < 0 and uj = hj/‖Φ‖ > 0, i, j = 1, 2, i 6= j, wewill have either uis < ui which gives a contradiction with the sign of ui, or uis ≥ ui andujs < uj which gives a contradiction with the sign of uj. A similar reasoning can bedone for regular-singular type concatenations.

Recall that the extremal is said singular if ‖Φ(t)‖ =√h2

1(t) + h22(t) = 0, t ∈ I.

Thus, the obtained contradiction indicates that the concatenation of a singular arc witha regular arc violates the PMP and thus there exists a chattering arc when trying toconnect a regular arc with a singular arc.

Remark 4.1.1. Note that, in this result, we have assumed that ‖us‖ < 1. In the(nongeneric) case where the singular control saturates the constraint, in order to get thesame result we need to assume that the strengthened GLCC is satised at the junctionpoint, i.e., aii(x(τ), p(τ)) < 0, and the control is discontinuous at the singular junction.

In addition, we have assumed that ∂∂u1

d4

dt4h2(x(t), p(t)) = 0. Actually, if

∂

∂u1

d4

dt4h2(x(t), p(t)) 6= 0,

then singular and regular extremals can be connected without chattering. For example,(4.1.5) gives

h(4)k (τ) = ak0(x(τ), p(τ)) + uk(τ)akk(x(τ), p(τ)) + um(τ)akm(x(τ), p(τ))

≤ ak0(x(τ), p(τ)) + uks(τ−)akk(x(τ), p(τ)) + um(τ)akm(x(τ), p(τ))

= h(4)k (τ−) + akm(x(τ), p(τ))(um(τ)− ums(τ))

= akm(x(τ), p(τ))(um(τ)− ums(τ)).

where akm(x(τ), p(τ)) = ∂∂um

d4

dt4hk(x(t), p(t)), k,m = 1, 2, k 6= m. In contrast to the


previous reasoning, now the fact that akm(x(τ), p(τ))(um− ums) > 0 does not raise anymore a contradiction.

In the next section, we analyze the regular, singular and chattering extremals for(MTCP) by using the results presented previously.

4.2 Geometric Analysis of the Extremals of (MTCP)

In this section, we classify the switching points by their contact with the switchingsurface, and we establish that the optimal singular arcs of the (MTCP), if they exist,cause chattering.

4.2.1 Regular Extremals

Normal extremals. Here, we consider normal extremals and we take p0 = −1. Letus consider the system (2.2.6), with the vector elds f , g1 and g2 dened by (2.2.7). De-noting the adjoint vector by p = (pvx , pvy , pvz , pθ, pψ, pφ, pωx , pωy), the adjoint equationsgiven by the PMP are

pvx = 0, pvy = 0, pvz = 0,

pθ = −a cosψ(pvx cos θ − pvz sin θ),

pψ = a sinψ sin θpvx + a cosψpvy + a cos θ sinψpvz− sinψ(ωx sinφ+ ωy cosφ)/ cos2 ψpθ − (ωx sinφ+ ωy cosφ)/ cos2 ψpφ,

pφ = −(ωx cosφ− ωy sinφ)/ cosψpθ + (ωx sinφ+ ωy cosφ)pψ

− tanψ(ωx cosφ− ωy sinφ)pφ,

pωx = − sinφ/ cosψpθ − cosφpψ − sinψ sinφ/ cosψpφ,

pωy = − cosφ/ cosψpθ + sinφpψ − sinψ cosφ/ cosψpφ,

(4.2.1)

with the transversality condition

pvx(tf ) sin θf cosψf − pvy(tf ) sinψf + pvz(tf ) cos θf cosψf = 0. (4.2.2)

The switching function is Φ(t) = (h1(t), h2(t)) = (bpωy(t),−bpωx(t)) is of class C1. Theswitching manifold Γ is the submanifold of R16 of codimension two dened by

Γ = z = (x, p) ∈ R16 | pωx = pωy = 0.

Let us x an arbitrary reference regular extremal z(·) = (x(·), p(·)) of (MTCP).If z(·) never meets Γ, then the extremal control is given by (4.1.2) along the whole

extremal.If it meets Γ then there is a singularity to be analyzed. It is not even clear if the

extremal ow is well dened when crossing such a point (we could lose uniqueness).

98 4.2. Geometric Analysis of the Extremals of (MTCP)

Let us assume that the extremal z(·) meets Γ at some time t0, and we set z0 = z(t0) =(x0, p0). Following [14, 16, 65], we classify the regular extremals by their contact withthe switching surface, i.e., if Φ(k−1)(t0) = 0 and Φ(k)(t0) 6= 0 for some k ∈ N∗, then thepoint z0 is said to be a point of order k. Without loss of generality, we assume thatt0 = 0.

Let us analyze the singularity occurring at points of order 1, 2, 3, and 4 for (MTCP).

Points of order 1.

Lemma 4.2.1. We assume that z0 is of order 1. Then the reference extremal iswell dened in a neighborhood of t = 0, in the sense that there exists a unique extremalassociated with the control u = (u1, u2) passing through the point z0. The control turnswith an angle π when passing through the switching surface Γ, and is locally given by

u1(t) =a1√a2

1 + a22

t

|t|+ o(1), u2(t) =

a2√a2

1 + a22

t

|t|+ o(1),

with a1 = h0, h1(z0), a2 = h0, h2(z0).

Proof 4.2.1. In (MTCP), the vector elds g1 and g2 (dened by (2.2.7)) commute,i.e., [g1, g2] = 0. By dierentiating the switching function, we get h1 = h0, h1 andh2 = h0, h2 along the extremal, and then, since the point is of order 1, we have,locally, h1(t) = a1t + o(t) and h2(t) = a2t + o(t), with a2

1 + a22 6= 0, and we also have

pωx(t) = −a2t/b+o(t), pωy(t) = a1t/b+o(t),√pωx(t)

2 + pωy(t)2 =

√a2

1 + a22|t|/b+o(t).

The expression of the optimal control in the lemma follows. At the crossing point, bothcontrol components change their sign, i.e., ui(0

−) = −ui(0+), i = 1, 2, which meansthat the control direction turns with an angle π when crossing Γ.

Points of order 2.

Lemma 4.2.2. We assume that z0 is of order 2. Then the reference extremal iswell dened in a neighborhood of t = 0, in the sense that there exists a unique extremalassociated with the control u = (u1, u2) passing through the point z0. Moreover, theswitching function is of class C∞ in the neighborhood of t = 0, the control is of classC∞, and we have

u1(t) =α1√α2

1 + α22

+ o(1), u2(t) =α2√α2

1 + α22

+ o(1),

with α1 = h0, h0, h1(z0) and α2 = h0, h0, h2(z0).

Proof 4.2.2. The vector elds f , g1 and g2, dened by (2.2.7), are such that[gi, [f, gj]] = 0, for i, j = 1, 2 (see Lemma 4.2.4). Then, according to the calcula-tions done in Section 4.1.2, we have h1 = h0, h0, h1 and h2 = h0, h0, h2,


and hence the functions t 7→ hi(t), i = 1, 2 are of class C2 at 0. Locally, we haveh1(t) = 1

2α1t

2 + o(t2) and h2(t) = 12α2t

2 + o(t2). The expression of the optimal controlfollows and the control is continuous.

Dierentiating again the switching function, we have

h(3)1 (t) = ad3h0.h1 + u2h2, ad2h0.h1,

h(3)2 (t) = ad3h0.h2 + u1h1, ad2h0.h2,

because [gi, [f, [f, gi]]] = 0, for i = 1, 2 (see Lemma 4.2.4). Since the control iscontinuous, the switching function is at least of class C3 at 0. Hence, locally wecan write h1(t) = 1

2α1t

2 + 16β1t

3 + o(t3) and h2(t) = 12α2t

2 + 16β2t

3 + o(t3), where

βi = ad3h0.hi(z0) + αj/√α2

1 + α22hj, ad2h0.hi(z0) for i, j = 1, 2 and i 6= j. We infer

that the control is at least of class C1 at 0, with u1(0) = 16(α2

2β1−2α1α2β2)/(α21 +α2

1)3/2

and u2(0) = 16(α2

1β2 − 2α1α2β1)/(α21 + α2

1)3/2. We get the smoothness by an immediateinduction argument: for k > 2, assuming that the switching function is of class Ck andthat the control is of class Ck−2, then the (k + 1)-th time derivative of the switching

function can be written as h(k+1)i = u

(k−2)i term1 +term2, where termi, i = 1, 2 are terms

involving time derivatives of the control of order lower than k− 2. Hence the switchingfunction is of class Ck+1 since u is of class Ck−2, and hi =

∑k+1p=2 ai,pt

p + o(tk+1) whereall coecients can be computed explicitly. The (k − 1)-th time derivative of the controlcan be computed using the coecients ai,p, p = 2, · · · , k+ 1, and hence the control is ofclass Ck−1. The result follows.

Points of order 3.

Lemma 4.2.3. We assume that z0 is of order 3. Then the reference extremal is welldened in a neighborhood of t = 0, in the sense that there exists a unique extremal asso-ciated with the control u = (u1, u2) passing through the point z0. If bi = ad3h0.hi(z0) 6= 0and c = h2, ad2h0.h1(z0) 6= 0, then the switching function is of class C3 in the neigh-borhood of t = 0 and the control turns with an angle π when passing through the switch-ing surface Γ locally, and we have

ui(t) =βi√

β21 + β2

2

t

|t|+ o(1), i = 1, 2,

where, setting d =√

(−c2 + b21 + b2

2) and eij = −bic2 + bib2j + b3

i ,

βi =(−1)jbjcd+ eij

b21 + b2

2

, i, j = 1, 2, i 6= j.

Proof 4.2.3. Using (2.2.7) and (4.1.4), we have h(3)1 = ad3h0.h1 +u2h2, ad2h0.h1

and h(3)2 = ad3h0.h2−u1h2, ad2h0.h1 (see also Lemma 4.2.4). Assuming that we have


locally hi(t) = 16βit

3 + o(t3), we infer that ui(t) = βi√β2

1+β22

t|t| + o(1). By substituting the

control into the expression of h(3)i , we get β1 = b1 +c β2√

β21+β2

2

t|t| and β2 = b2−c β1√

β21+β2

2

t|t| .

The result follows by solving these two equations for t > 0 and t < 0.

Points of order 4. We assume that z0 is of order 4. If h2, ad2h0.h1(z0) 6= 0,then 0 = h

(3)1 = b1 + u2c and 0 = h

(3)2 = b2 − u1c, where bi = ad3h0.hi(z0), c =

h2, ad2h0.h1(z0). We have u1 = b2/c and u2 = −b1/c. If moreover b21 + b2

2 = c2, whichindicates that the control ui = αi/

√α2

1 + α22 can be a regular control according to the

value of hi(t) = 18αit

4 + o(t4), i = 1, 2, at time 0, we get that u2/u1 = α2/α1 = −b1/b2,sign(α1) = b2/c and sign(α2) = −b1/c. Then

u1(t) = sign(c)b2√b2

1 + b22

+ o(1), u2(t) = −sign(c)b1√b2

1 + b22

+ o(1).

If

α1 = h(4)1 (z0) = h0, b1(z0) + u1h1, b1(z0) + u2(h2, b1(z0) + h0, c(z0))

+u1u2h1, c(z0) + u22h2, c(z0),

α2 = h(4)2 (z0) = h0, b2(z0) + u1(h1, b2(z0)− h0, c(z0)) + u2h2, b2(z0)

−u21h1, c(z0)− u1u2h2, c(z0),

then the extremal is well dened in a neighborhood of t = 0 and the control is continuouswhen passing the switching surface Γ.

If h2, ad2h0.h1(z0) = 0, then h0, b1(z0) = h0, b2(z0) = 0, h1, b2(z0) = 0,h2, b1(z0) = 0 and h1, b1(z0) = h2, b2(z0) (see the proof of Lemma 4.2.5 further),and we have h(4)

1 (z0) = u1h1, b1(z0) and h(4)2 (z0) = u2h1, b1(z0). Assuming that we

have locally

Φ(t) = R0t4eiαln|t| + o(t4) = R0t

4(cos(αln|t|), sin(αln|t|)) + o(t4) (4.2.3)

(we identify C = R2 for convenience), with R0 > 0, we get u(t) = Φ(t)/‖Φ(t)‖ = eiαln|t|

and

Φ(4)(t) = R0(4 + iαt/|t|)(3 + iαt/|t|)(2 + iαt/|t|)(1 + iαt/|t|)eiαln|t| + o(1),

which leads to R0(α4 − 35α2 + 24) = h1, b1(z0) and R0(−10α3 + 50α)t/|t| = 0. Itfollows that, R0 = −h1, b1(z0)/126 with α ∈ −

√5,√

5 and h1, b1 < 0. It isclear that the uniqueness of the extremal when crossing the point z0 does not hold trueanymore. The switching function Φ(t) converges to (0, 0) when t→ 0, while the controlswitches innitely many times when t→ 0. Indeed, we will see in the following sectionthat this situation is related to the chattering phenomenon.


Abnormal extremals. Abnormal extremals correspond to p0 = 0 in the PMP. Wesuspect the existence of optimal abnormal extremals in (MTCP) for certain (non-generic) terminal conditions. In the planar version of (MTCP) studied in chapter 3,if the optimal control switches at least two times then there is no abnormal minimizer.We expect that the same property is still true here. We are able to prove that thesingular extremals of (MTCP) are normal (see section 4.2.2), however, we are not ableto establish a clear relationship between the number of switchings and the existence ofabnormal minimizers as in the previous chapter. Thus, in our numerical simulationsfurther, we will assume that there is at least one normal extremal for problem (MTCP)and compute it. Note moreover that Lemmas 4.2.1, 4.2.2 and 4.2.3 are also valid forabnormal extremals.

4.2.2 Singular and Chattering Extremals

Let us compute the singular arcs of (MTCP). According to [13], the singular trajecto-ries are feedback invariants since they correspond to the singularities of the end-pointmapping. This concept is related to the feedback group induced by the feedback trans-formation and the corresponding control systems are said to be feedback equivalent(see[13, Section 4] for details).

Since the singular trajectories are feedback invariants under feedback transforma-tions, we can replace the vector elds g1 and g2 with g1 = ∂

∂ωyand g2 = ∂

∂ωx. It can

be easily shown that the singular control computed with g1 and g2 coincides with theone computed with g1 and g2. Let us make precise the Lie bracket conguration of thecontrol system associated with the vector elds f , g1 and g2.

Lemma 4.2.4. We have

g1 =∂

∂ωy, g2 =

∂

∂ωx, [g1, g2] = 0,

adf.g1 = − cosφ/ cosψ∂

∂θ+ sinφ

∂

∂ψ− tanψ cosφ

∂

∂φ= 0,

adf.g2 = − sinφ/ cosψ∂

∂θ− cosφ

∂

∂ψ− tanψ sinφ

∂

∂φ= 0,

ad2f.g1 = −ωx∂

∂φ+ a(cos θ cosφ+ sin θ sinφ sinψ)

∂

∂vx+ a sinφ cosψ

∂

∂vy

− a(cosφ sin θ − sinφ cos θ sinψ)∂

∂vz,


ad2f.g2 = ωy∂

∂φ+ a(cos θ sinφ− sin θ cosφ sinψ)

∂

∂vx− a cosφ cosψ

∂

∂vy

− a(sinφ sin θ + cosφ cos θ sinψ)∂

∂vz,

[g1, adf.g1] = [g1, adf.g2] = [g2, adf.g1] = [g2, adf.g2] = 0,

ad3f.g1 = ωxΩ1/ cosψ∂

∂θ− ωxΩ2

∂

∂ψ+ ωx tanψΩ1

∂

∂φ

− aωy cosψ sin θ∂

∂vx+ aωy sinψ

∂

∂vy− aωy cosψ cos θ

∂

∂vz,

ad3f.g2 = −ωyΩ1/ cosψ∂

∂θ+ ωxΩ2

∂

∂ψ− ωx tanψΩ1

∂

∂φ

− aωx cosψ sin θ∂

∂vx+ aωx sinψ

∂

∂vy− aωx cosψ cos θ

∂

∂vz,

[g1, ad2f.g1] = [g2, ad2f.g2] = 0, [g1, ad2f.g2] = −[g2, ad2f.g1] =∂

∂φ,

ad4f.g1 = −a(ω2x + ω2

y)(cosφ cos θ + sinφ sinψ sin θ)∂

∂vx− a cosψ sinφ(ω2

x + ω2y)

∂

∂vy

+ a(ω2x + ω2

y)(cosφ sin θ − cos θ sinφ sinψ)∂

∂vz+ (ω3

x + ωxω2y)∂

∂φ,

ad4f.g2 = −a(ω2x + ω2

y)(cos θ sinφ− cosφ sinψ sin θ)∂

∂vx+ a cosφ cosψ(ω2

x + ω2y)

∂

∂vy

+ a(ω2x + ω2

y)(sinφ sin θ + cosφ cos θ sinψ)∂

∂vz+ (−ω2

xωy − ω3y)∂

∂φ,

[g1, ad3f.g1] = −a cosψ sin θ∂

∂vx+ a sinψ

∂

∂vy− a cosψ cos θ

∂

∂vz− (ωx sinφ)/ cosψ

∂

∂θ

− ωx cosφ∂

∂ψ− ωx sinφ tanψ

∂

∂φ,


[g1, ad3f.g2] = −2aωy(cos θ sinφ− cosφ sinψ sin θ)∂

∂vx+ 2aωy cosφ cosψ

∂

∂vy

+ 2aωy(sinφ sin θ + cosφ cos θ sinψ)∂

∂vz+ (−ω2

x − 3ω2y)∂

∂φ,

[g2, ad3f.g1] = −2aωx(cosφ cos θ + sinφ sinψ sin θ)∂

∂vx− 2aωx cosψ sinφ

∂

∂vy

+ 2aωx(cosφ sin θ − cos θ sinφ sinψ)∂

∂vz+ (3ω2

x + ω2y)∂

∂φ,

[g2, ad3f.g2] = −a cosψ sin θ∂

∂vx+ a sinψ

∂

∂vy− a cosψ cos θ

∂

∂vz− (ωy cosφ)/ cosψ

∂

∂θ

+ ωy sinφ∂

∂ψ− ωy cosφ tanψ

∂

∂φ,

where Ω1 = ωx cosφ− ωy sinφ and Ω2 = ωx sinφ+ ωy cosφ. Moreover, we have

dim Span(g1, g2, adf.g1, adf.g2, ad2f.g1, ad2f.g2

)= 6.

The proof of Lemma 4.2.4 was done with symbolic computations.

Lemma 4.2.5. In (MTCP), let us assume that (x(·), p(·), p0, u(·)) is a singulararc along the subinterval I, which is locally optimal in C0 topology. Then we have u =(u1, u2) = (0, 0) along I, and u is a singular control of intrinsic order two. Moreover,the extremal must be normal, i.e., p0 6= 0, and the GLCC

a+ gx sin θ cosψ − gy sinψ + gz cos θ cosψ ≥ 0, (4.2.4)

must hold along I.

Proof 4.2.4. Using Lemma 4.2.4, we infer from Φ = 0 and Φ = 0 that

〈p, g1(x)〉 = pωy = 0, 〈p, g2(x)〉 = pωx = 0,

〈p, adf.g1(x)〉 = −pθ cosφ/ cosψ + pψ sinφ− pφ tanψ cosφ = 0,

〈p, adf.g2(x)〉 = −pθ sinφ/ cosψ − pψ cosφ− pφ tanψ sinφ = 0,

(4.2.5)

and from Φ = 0, that

〈p, ad2f.g1(x)〉 =− ωxpφ + a(cos θ cosφ+ sin θ sinφ sinψ)pvx + a sinφ cosψpvy− a(cosφ sin θ − sinφ cos θ sinψ)pvz = 0,

〈p, ad2f.g2(x)〉 = ωypφ + a(cos θ sinφ− sin θ cosφ sinψ)pvx − a cosφ cosψpvy− a(sinφ sin θ + cosφ cos θ sinψ)pvz = 0,

(4.2.6)


along the interval I. Since dim Span(g1, g2, adf.g1, adf.g2, ad2f.g1, ad2f.g2

)= 6, the six

equations in (4.2.5)-(4.2.6) are independent constraints along the singular arc. There-fore, writing Φ(3) = 0, we get from Corollary 4.1.1 that

〈p, [g1, ad2f.g2(x)]〉 =pφ = 0, 〈p, [g2, ad2f.g1(x)]〉 = −pφ = 0,

〈p, ad3f.g1(x)〉 =pθωxΩ1/ cosψ − pψωxΩ2 + pφωx tanψΩ1 − pvxaωy cosψ sin θ

+ pvyaωy sinψ − pvzaωy cosψ cos θ = 0,

〈p, ad3f.g2(x)〉 =− pθωyΩ1/ cosψ + pψωxΩ2 − pφωx tanψΩ1 − pvxaωx cosψ sin θ

+ pvyaωx sinψ − pvzaωx cosψ cos θ = 0,

(4.2.7)

These four constraints are dependent: they reduce to two functionally independent con-straints. Hence, with (4.2.5)-(4.2.6)-(4.2.7), we have 8 independent constraints alongI. Now, we infer from (4.2.5)-(4.2.6)-(4.2.7) that pωx = pωy = 0 and pθ = pψ = pφ = 0and

pvx = tan θpvz , pvy = − tanψ/ cos θpvz , (4.2.8)

along I. Dierentiating again, we get that θ = ψ = 0. It follows that ωx = ωy = 0.Using that H = 0 along any extremal, we get

pvz =−p0 cos θ cosψ

a+ gx sin θ cosψ − gy sinψ + gz cos θ cosψ, (4.2.9)

Substituting (4.2.8) and (4.2.9) into 〈p, ad4f.g1〉 and 〈p, ad4f.g2〉, we get

〈p, ad4f.g1〉 = 〈p, ad4f.g2〉 = 0, 〈p, [g1, ad3f.g2](x)〉 = 〈p, [g2, ad3f.g1](x)〉 = 0,

〈p, [g1, ad3f.g1](x)〉 = 〈p, [g2, ad3f.g2](x)〉 = − apvzcosψ cos θ

.

According to Denition 4.1.1, to prove that u is of intrinsic order two, it suces to provethat 〈p, [gi, ad3f.gi](x)〉 6= 0 along I. We prove it by contradiction. If 〈p, [gi, ad3f.gi](x)〉 =0, then necessarily pvz = 0 and this would lead to pvx = pvy = 0. It follows then fromH = 0 that p0 = 0. We have obtained that (p, p0) = 0, which is a contradiction.

The fact that u = (u1, u2) = (0, 0) simply follows from the fact that

u1 = −h1, ad3h0.h1/ad4h0.h1, u2 = −h2, ad3h0.h2/ad4h0.h2.

Besides, if p0 = 0, then pvz = 0 and pvx = pvy = 0, which leads to (p, p0) = 0 andthus to a contradiction as well. Therefore, p0 < 0 (i.e., the singular arc is normal), andthen (4.2.4) follows by applying the GLCC of Corollary 4.1.1.

We dene the singular surface S, which is lled by singular extremals of (MTCP),


by

S =

(x, p) | ωx = ωy = 0, pθ = pψ = pφ = pωx = pωy = 0, pvx = tan θpvz ,

pvz =−p0 cos θ cosψ

a+ gx sin θ cosψ − gy sinψ + gz cos θ cosψ, pvy = − tanψ/ cos θpvz

. (4.2.10)

We will see, in the next section, that the solutions of the problem of order zero (denedin Section 4.3.1.1) live in this singular surface S.

The following result, establishing chattering for (MTCP), is a consequence of The-orem 4.1.1, Lemma 4.2.4 and Lemma 4.2.5.

Corollary 4.2.1. For (MTCP), any optimal singular arc cannot be connected witha nontrivial bang arc. There is a chattering arc when trying to connect a regular arcwith an optimal singular arc. More precisely, let u be an optimal control, solution of(MTCP), and assume that u is singular on the sub-interval (t1, t2) ⊂ [0, tf ] and isregular elsewhere. If t1 > 0 (resp., if t2 < tf) then, for every ε > 0, the controlu switches an innite number of times over the time interval [t1 − ε, t1] (resp., on[t2, t2 + ε]).

As mentioned in Section 4.2.1, points of order 4 are related to the chattering phe-nomenon. Indeed, from (4.2.3), the switching function for (MTCP) has the form

Φ(t) =

R0(t1 − t)4eiαln(t1−t) + o((t1 − t)4) if t < t1,

R0(t− t2)4eiαln(t−t2) + o((t− t2)4) if t > t2,

in the neighborhood of the singular junction. The extremal converges to the switchingsurface when t → t1, t < t1, and escapes from the singular surface when t > t2. In-between, the control switches an innite number of times if t→ t1, t > t1, or if t→ t2,t < t2.

This result is important for solving (MTCP) in practice. Indeed, when using numer-ical methods to solve the problem, the chattering control is an obstacle to convergence,especially when using an indirect approach (shooting). The existence of the chatteringphenomenon in (MTCP) explains well why the indirect methods may fail for certainterminal conditions.

Note that, in the planar version of (MTCP), one can give sucient conditions onthe initial conditions under which the chattering phenomenon does not occur. Unfor-tunately, we are not able to derive such conditions in the general problem (MTCP).

4.3 Numerical Approaches

In this section, we design two dierent numerical strategies for solving (MTCP): oneis based on combining indirect methods with numerical continuation, and the other

106 4.3. Numerical Approaches

is based on a direct transcription approach. The rst one may be successfully imple-mented when dealing with solutions without chattering arcs, and the second one is moreappropriate to compute solutions involving chattering arcs. However, both approachesare dicult to initialize successfully because (MTCP) is of quite high dimension, ishighly nonlinear, and moreover, as a main reason, the system consists of fast (Eulerangles and angular velocity) and slow (trajectory velocity) dynamics at the same time.

The occurrence of chattering arcs is an obstacle to convergence. Especially for in-direct methods, the chattering phenomenon raises an important diculty due to thenumerical integration of the discontinuous Hamiltonian system. Direct transcriptionapproaches provide a sub-optimal solution of the problem that has a nite number ofswitchings based on a (possibly rough) discretization. Actually, in case of chattering,we are also able to provide a sub-optimal solution with our indirect approach, by stop-ping the continuation before it would fail due to chattering. Though the sub-optimalsolutions provided in this way may be less optimal" compared with those given by adirect approach, in practice they can be computed in a much faster way and also muchmore accurately.

4.3.1 Indirect Method and Numerical Continuation

The idea of this continuation procedure is to use the (easily computable) solution ofa simpler problem, that we call hereafter the problem of order zero, in order then toinitialize an indirect method for the more complicated problem (MTCP). Then weare going to plug this simple, low-dimensional problem in higher dimension, and thencome back to the initial problem by using appropriate continuations.

This method actually gives an optimal solution with high accuracy. The problemof order zero dened below is used as the starting problem because the trajectorymovement is much slower compared with the attitude movement and it is easy to solveexplicitly. As well, it is worth noting that the solution of the problem of order zero iscontained in the singular surface S lled by the singular solutions for (MTCP), denedby (4.2.10).

4.3.1.1 Two Auxiliary Problems

Problem of order zero. We dene the problem of order zero, denoted by (POZ), asa subproblem of the complete problem (MTCP), in the sense that we consider onlythe trajectory dynamics and that we assume that the attitude angles (Euler angles) canbe driven to the target values instantaneously. Thus, the attitude angles are consideredas control inputs in that simpler problem. Denoting the LV longitudinal axis as ~e andconsidering it as the control vector (which is consistent with the attitude angles θ, ψ),we formulate the problem as follows:

~v = a~e+ ~g, ~v(0) = ~v0, ~v(tf )//~w, ‖~w‖ = 1, min tf ,


where ~w is a given vector that refers to the desired target velocity direction. Thisproblem is easy to solve, and the solution is the following.

Lemma 4.3.1. The optimal solution of (POZ) is given by

~e∗ =1

a

(k ~w − ~v0

tf− ~g), tf =

−a2 +√a2

2 − 4a1a3

2a1

, ~pv =−p0

a+ 〈~e∗, ~g〉~e∗.

with k = 〈~v0, ~w〉 + 〈~g, ~w〉tf , a1 = a2 − ‖〈~g, ~w〉~w − ~g‖2, a2 = 2(〈~v0, ~w〉〈~g, ~w〉 − 〈~v0, ~g〉),and a3 = −‖〈~v0, ~w〉~w − ~v0‖2.

Proof 4.3.1. The Hamiltonian is H = p0 + ~pv(a~e + ~g), and we have ~pv = ~0,with ~pv = (pvx , pvy , pvz)

>, and H = 0 along any extremal. It follows that ~pv 6= ~0(indeed otherwise we would get also p0 = 0, and thus a contradiction). Hence there areno singular controls for this problem. The maximization condition of the PMP yields~e∗ = ~pv/‖~pv‖, and hence the optimal control is a constant vector. Moreover, according tothe nal condition ~v(tf )//~w, the transversality condition is ~pv ⊥ ~w, hence 〈~e∗, ~w〉 = 0,and using ~v(tf ) = ~v0 + (a~e + ~g)tf = k ~w we get that ~e∗ = 1

a(k ~w−~v0

tf− ~g). It follows from

the transversality condition that k = 〈~v0, ~w〉 + 〈~g, ~w〉tf . The expression of tf follows,

using that ‖~e∗‖2 = 1. Using that H = 0, we get ~pv = −p0

a+〈~e∗,~g〉~e∗.

Since the vector ~e is expressed in the launch frame as

(~e)R = (sin θ cosψ,− sinψ, cos θ sinψ)>,

the Euler angles θ∗ ∈ (−π, π) and ψ∗ ∈ (−π/2, π/2) are given by

θ∗ =

arctan(e∗1/e

∗3), e∗3 > 0,

π/2 sign(e∗1), e∗3 = 0

(π − arctan(|e∗1|/|e∗3|))sign(e∗1), e∗3 < 0,

, ψ∗ = −arcsin(e∗2), (4.3.1)

where e∗i is the i-th component of ~e∗, for i = 1, 2, 3.Using the denition (4.2.10) of the singular surface S, we check that the optimal

solution of (POZ) is contained in S with θ = θ∗, ψ = ψ∗ and φ = φ∗ (φ∗ is any realnumber). Therefore, the relationship between (POZ) and (MTCP) is the following.

Lemma 4.3.2. The optimal solution of (POZ) actually corresponds to a singularsolution of (MTCP) with the terminal conditions given by

vx(0) = vx0 , vy(0) = vy0 , vz(0) = vz0 ,

θ(0) = θ∗, ψ(0) = ψ∗, , φ(0) = φ∗, ωx(0) = 0, ωy(0) = 0,(4.3.2)

vz(tf ) sinψf + vy(tf ) cos θf cosψf = 0, vz(tf ) sin θf − vx(tf ) cos θf = 0, (4.3.3)

θ(tf ) = θ∗, ψ(tf ) = ψ∗, , φ(tf ) = φ∗, ωx(tf ) = 0, ωy(tf ) = 0. (4.3.4)


Due to this result, a natural idea of numerical continuation strategy consists ofdeforming continuously (step by step) the terminal conditions given in Lemma 4.3.2,to the terminal conditions (2.2.8)-(2.2.9) of (MTCP).

However, because of the chattering phenomenon, we cannot make converge theshooting method in such a strategy. More precisely, when the terminal conditionsare in the neighborhood of the singular surface S, the optimal extremals are likely tocontain a singular arc (and thus chattering arcs). In that case, the shooting method willcertainly fail due to the diculty of numerical integration of discontinuous Hamiltoniansystem. Hence, we introduce hereafter an additional numerical trick and we dene thefollowing regularized problem, in which we modify the cost functional with a parameterK, so as to overcome the problem caused by chattering.

Regularized problem. LetK > 0 be arbitrary. The regularized problem (RMTCP)Kconsists of minimizing the cost functional

CK = tf +K

∫ tf

0

(u21 + u2

2) dt, (4.3.5)

for the bi-input control-ane system (2.2.6), under the control constraints −1 ≤ ui ≤1, i = 1, 2, with terminal conditions (2.2.8)-(2.2.9). Here, we replace the constraintu2

1 + u22 ≤ 1 (i.e., u takes its values in the unit Euclidean disk) with the constraint that

u takes its values in the unit Euclidean square. Note that we use the Euclidean square(and not the disk) because we observed that our numerical simulations worked betterin this case. The advantage, for this intermediate optimal control problem with thecost (4.3.5), is that the extremal controls are then continuous.

The Hamiltonian is

HK = 〈p, f(x)〉+ u1〈p, g1(x)〉+ u2〈p, g2(x)〉+ p0(1 +Ku21 +Ku2

2), (4.3.6)

and according to the PMP, the optimal controls are

u1(t) = sat(−1,−bpωy(t)/(2Kp0), 1), u2(t) = sat(−1, bpωx(t)/(2Kp0), 1), (4.3.7)

where the saturation operator sat is dened by sat(−1, f(t), 1) = −1 if f(t) ≤ −1; 1 iff(t) ≥ 1; and f(t) if −1 ≤ f(t) ≤ 1.

As mentioned previously, one of the motivations for considering the intermediateproblem (RMTCP)K is that the solution of (POZ) is a singular trajectory of thefull problem (MTCP), and hence, passing directly from (POZ) to (MTCP) causesdiculties due to chattering (see Corollary 4.2.1). The following result shows that whenwe embed the solutions of (POZ) into the (RMTCP)K , they are not singular.

Lemma 4.3.3. An extremal of (POZ) can be embedded into (RMTCP)K by set-


ting

u(t) = (0, 0), θ(t) = θ∗, ψ(t) = ψ∗, φ(t) = φ∗, ωx(t) = 0, ωy(t) = 0,

pθ(t) = 0, pψ(t) = 0, pφ(t) = 0, pωx(t) = 0, pωy(t) = 0,

where θ∗ and ψ∗ are given by (4.3.1), with terminal conditions given by (4.3.2) and(4.3.3)-(4.3.4). Moreover, it is not a singular extremal for (RMTCP)K. The extremalequations for (RMTCP)K are the same than for (MTCP), as well as the transver-sality conditions.

Proof 4.3.2. It is easy to verify that the embedded extremal is an extremal of(RMTCP)K and that the transversality conditions are the same. The control is com-puted from (4.3.7) which maximizes the Hamiltonian HK, and we have HK = 0 withp0 = −1. It follows from the PMP that the extremal equations are the same than for(MTCP). Then, for (RMTCP)K, we have ∂2HK

∂u2i

= Kp0. Note that, in this case,

the control ui, i = 1, 2 is singular if ∂2HK∂u2

i= 0. Hence there is no normal singular

extremal for (RMTCP)K. From Lemma 4.3.1, it is easy to see that p0 6= 0 and thusthe extremals of (POZ) are not singular extremals of (RMTCP)K.

4.3.1.2 Strategy for Solving (MTCP)

Continuation procedure. The ultimate objective is to compute the optimal solutionof (MTCP), starting from the explicit, simple to compute, solution of (POZ). Weproceed as follows:

• First, according to Lemma 4.3.3, we embed the solution of (POZ) into (RMTCP)K .For convenience, we still denote by (POZ) the (POZ) seen in high dimension.

• Then, we pass from (POZ) to (MTCP) by means of a numerical continuationprocedure, involving three continuation parameters: the rst two parameters λ1

and λ2 are used to pass continuously from the optimal solution of (POZ) tothe optimal solution of the regularized problem (RMTCP)K , for some xedK > 0, and the third parameter λ3 is then used to pass to the optimal solutionof (MTCP) (see Figure 4.1).

(POZ)start (RMTCP)K (MTCP)λ1, λ2 = 0 → 1 λ3 = 0 → 1

Figure 4.1: Continuation procedure.

The parameter λ1 is used to act, by continuation, on the initial conditions, accordingto

θ(0) = θ∗(1− λ1) + θ0λ1, ψ(0) = ψ∗(1− λ1) + ψ0λ1, φ(0) = φ∗(1− λ1) + φ0λ1,


ωx(0) = ω∗x(1− λ1) + ωx0λ1, ωy(0) = ω∗y(1− λ1) + ωy0λ1,

where ω∗x = ω∗y = 0, φ∗ = 0, and θ∗, ψ∗ are calculated through equation (4.3.1).Using the transversality condition (4.2.2) and the extremal equations pvx = 0, pvy =

0 and pvz = 0, the unknown pvy can be expressed in terms of pvx and pvz as

pvy = (pvx sin θf cosψf + pvz cos θf cosψf )/ sinψf ,

and hence the unknowns of the shooting problem are reduced to pvx , pvz , pθ(0), pψ(0),pφ(0), pωx(0), pωy(0) and tf . The shooting function Sλ1 for the λ1-continuation is denedby

Sλ1 =(pωx(tf ), pωy(tf ), pθ(tf ), pψ(tf ), pφ(tf ), HK(tf ),

vz(tf ) sinψf + vy(tf ) cos θf cosψf , vz(tf ) sin θf − vx(tf ) cos θf),

whereHK(tf ) with p0 = −1 is calculated from (4.3.6) and u1 and u2 are given by (4.3.7).In fact, from Lemma 4.2.5, we know that a singular extremal of problem (MTCP) mustbe normal, and since we are starting to solve the problem from a singular extremal,here we assume that p0 = −1.

Note that we can use Sλ1 as shooting function thanks for (RMTCP)K . For problem(MTCP), if Sλ1 = 0, then together with ωx(tf ) = 0 and ωy(tf ) = 0, the nal point(x(tf ), p(tf )) of the extremal is then lying on the singular surface S dened by (4.2.10)and this will cause the fail of the shooting. However, for problem (RMTCP)K , evenwhen x(tf ) ∈ S, the shooting problem can still be solved.

Initializing with the solution of (POZ), we can solve this shooting problem withλ1 = 0, and we get a solution of (RMTCP)K with the terminal conditions (4.3.2)-(4.3.3) (the other states at tf being free). Then, by continuation, we make λ1 varyfrom 0 to 1, and in this way we get the solution of (RMTCP)K for λ1 = 1. With thissolution, we can integrate extremal equations (??) and (4.2.1) to get the values of thestate variable at tf . Then denote θe := θ(tf ), ψe := ψ(tf ), φe := φ(tf ), ωxe := ωx(tf )and ωye := ωy(tf ).

In a second step, we use the continuation parameter λ2 to act on the nal conditions,in order to make them pass from the values θe, ψe, φe, ωxe and ωye, to the desired targetvalues θf , ψf , φf , ωxf and ωyf . The shooting function is

Sλ2 =(ωx(tf )− (1− λ2)ωxe − λ2ωxf , ωy(tf )− (1− λ2)ωye − λ2ωyf ,

θ(tf )− (1− λ2)θe − λ2θf , ψ(tf )− (1− λ2)ψe − λ2ψf ,

φ(tf )− (1− λ2)φe − λ2φf , vz(tf ) sinψf + vy(tf ) cos θf cosψf ,

vz(tf ) sin θf − vx(tf ) cos θf , HK(tf )).

Solving this problem by making vary λ2 from 0 to 1, we obtain the solution of (RMTCP)Kwith the terminal conditions (2.2.8)-(2.2.9).


Finally, in order to compute the solution of (MTCP), we use the continuationparameter λ3 to pass from (RMTCP)K to (MTCP). We add the parameter λ3 tothe Hamiltonian HK and to the cost functional (4.3.5) as follows:

CK = tf +K

∫ tf

0

(u21 + u2

2)(1− λ3) dt,

H(tf , λ3) = 〈p, f〉+ 〈p, g1〉u1 + 〈p, g2〉u2 + p0 + p0K(u21 + u2

2)(1− λ3).

Then, according to the PMP, the extremal controls are given by ui = sat(−1, uie, 1),i = 1, 2, where

u1e =bpωy

−2p0K(1− λ3) + bλ3

√p2ωx + p2

ωy

, u2e =−bpωx

−2p0K(1− λ3) + bλ3

√p2ωx + p2

ωy

.

The shooting function Sλ3 is dened as Sλ2 , replacing HK(tf ) with HK(tf , λ3). Thesolution of (MTCP) is then obtained by making vary λ3 continuously from 0 to 1.

Remark 4.3.1. Note that the above continuation procedure fails in case of chat-tering (see Corollary 4.2.1), and thus cannot be successful for any possible choice ofterminal conditions. In particular, if chattering occurs then the λ3-continuation is ex-pected to fail for some value λ3 = λ∗3 < 1. But in that case, with this value of λ3, wehave generated a sub-optimal solution of (MTCP), which appears to be acceptable andvery interesting for practice. Moreover, the overall procedure is very fast and accurate.Note that the resulting sub-optimal control is continuous.

4.3.2 Direct Method

We now propose a direct approach for solving (MTCP), where the control is approx-imated by a piecewise constant control over a given time subdivision. The solutionsderived from such a method are therefore sub-optimal, in particular when the controlis chattering (and in such a case the number of switchings is limited by the time step).Note that this approach is much more computationally demanding than the indirectone.

Since the initialization of a direct method may also raise some diculties, we proposethe following strategy. The idea is to start from the solution of (MTCP) with lessterminal requirements, which is easy to obtain with a direct method, and then weintroduce step by step the nal conditions (2.2.9) of (MTCP). We implement thisdirect approach with the software BOCOP and its batch optimization option (see [12]).

• Step 1: we solve (MTCP) with initial conditions (2.2.8) and nal conditions

ωy(tf ) = 0, θ(tf ) = θf , vz(tf ) sin θf − vx(tf ) cos θf = 0.


These nal conditions are the ones of the planar version of (MTCP) in which themotion of the spacecraft is 2D (see chapter 3 for details). Numerical simulationsshow that, with such terminal conditions, (MTCP) is easy and fast to solve bymeans of a direct method (a constant initial guess for the discretized variablessuces to ensure convergence).

• Then, in Steps 2, 3, 4 and 5, we add successively (and step by step) the nalconditions vz(tf ) sinψf + vy(tf ) cos θf cosψf = 0, ψ(tf ) = ψf , φ(tf ) = φf , andωx(tf ) = ωxf , and for each new step we use the solution of the previous one as aninitial guess.

At the end of this process, we have obtained the solution of (MTCP). Note again thatthis direct approach is much slower than the indirect one, and that the resulting controlhas many numerical oscillations (see numerical results in Section 4.4.2).


The structure of the LV is presented in Figure 2.2 (b). We assume that the thrust I isexible, i.e., it can turn ±6 in all directions, and its thrust is around Tatt = 1400 kN.The other thrusts are xed with a total thrust Ttot = 1× 105 kN. The LV mass is 800t, the length of the LV is lr = 50 m and its radius is rr = 2.5 m. Considering the LV asa cylinder, we have Ix = Iy = m(3r2

r + l2r)/12 and Iz = mr2r/2. The parameters a and

b in (2.1.12) and (2.1.6) are therefore a = Ttot/m ≈ 12 and b = Tattlr2Ix

µmax ≈ 0.02.During the atmospheric ascent phase, the velocity of the LV remains between several

hundreds m/s and around 1000 m/s. Let v =√v2x + v2

y + v2z be the modulus of the

velocity, and let ψv and φv be the ight path angles that we use to calculate thecomponents of the velocity in SR frame, i.e., vx = v sin θv cosψv, vy = −v sinψv andvz = v cos θv cosψv. In this section, the initial values of the angles θv and ψv are chosenequal to the initial values of the angles θ and ψ. This means that, before the maneuver,the LV is on a trajectory with angle of attack equal to zero.

In the numerical simulations, we set vx0 = v0 sin θ0 cosψ0, vy0 = −v0 sinψ0, vz0 =v0 cos θ0 cosψ0. The terminal values are given in Table 4.1.

(TC1): ωx0 = ωy0 = 0, θ0 = 75, ψ0 = 0.5, φ0 = 0

ωxf = ωyf = 0, θf = 85, ψf = 5, φf = 0

(TC2): ωx0 = ωy0 = 0, θ0 = 70, ψ0 = 0.5, φ0 = 0

ωxf = ωyf = 0, θf = 85, ψf = 5, φf = 0

(TC3): ωx0 = ωy0 = 0, θ0 = 85, ψ0 = 0.5, φ0 = 0

ωxf = ωyf = 0, θf = 75, ψf = 5, φf = 0

Table 4.1: Terminal conditions

Note that v0 is the module of velocity at time 0. In the next two subsections,we will choose dierent values of v0, and so here we do not assign to it a specic


value. Moreover, we set K = 50 as the weight of the L2-norm control term in the costfunctional of (RMTCP)K .

4.4.1 Numerical Results without Chattering

In this section, we solve (MTCP) using the indirect method combined with the nu-merical continuation described in Section 4.3.1. This method is implemented using apredictor-corrector (PC) continuation method (see [1]). Note that the PC continuationmethod is discrete, in contrast with dierential methods used in [25], for which the Ja-cobian of the homotopy method must be computed. The Fortran routines hybrd.f (see[80]) and dop853.f (see [53]) are used, respectively, for solving the shooting problem(Newton method) and for integrating the ordinary dierential equations (with predic-tion).

The Euler angle θ is usually called the pitch angle, and a pitching up maneuverdesignates a maneuver with terminal condition θf > θ0, while a pitching down maneuverdesignates a maneuver with terminal condition θf < θ0.

Pitching up maneuvers. We set v0 = 1000 m/s and we use the numerical valuesdenoted by (TC1) in Table 4.1. The components of the state variable are reportedon Figure 4.2. The optimal control, the adjoint variables pωx(t) and pωy(t) and themodulus of the switching function Φ(t) = b(pωy ,−pωx) are reported on Figure 4.3. We

0 10 20 30

−10

−5

0

5

10

t

angular velocity-unit degree/s

ωxωy

0 10 20 30

0

50

100

150

t

Euler angles-unit degree

θψφ

0 10 20 30

0

200

400

600

800

1,000

t

velocity-unit m/s

vxvyvz

Figure 4.2: State variable and the optimal control with (TC1) and v0 = 1000m/s.

observe that the optimal control switches twice, at times 8.8 s and 25.8 s. These twoswitching points are of order 1 (i.e., Φ(t) = 0 and Φ(t) 6= 0). Accordingly with Lemma4.2.1, the control turns with an angle π at those points.

Let us give another numerical example, taking the same terminal conditions aspreviously except for v0, and we take v0 = 1500 m/s. The time history of the state, ofthe optimal control and of the switching function are reported on Figures 4.4 and 4.5.One can see on Figure 4.5 that the optimal control turns two more times with an angleπ due to two switching points of order one.


0 10 20 30 40 50

−1

−0.5

0

0.5

1

t

u1u2

‖u‖

0 10 20 30

0

20

40

60

t

pωxpωy

0 10 20 30

0

5

10

t

‖Φ‖

Figure 4.3: Adjoint variable and the switching function with (TC1) and v0 = 1000m/s.

0 10 20 30 40

−10

−5

0

5

10

t


ωxωy

0 10 20 30 40

0

50

100

150

200

t


θψφ

0 10 20 30 40

0

500

1,000

1,500

t

velocity-unit m/s

vxvyvz

Figure 4.4: State with (TC1) and v0 = 1500m/s.


0 10 20 30 40

−1

−0.5

0

0.5

1

t

u1u2

‖u‖

0 10 20 30 40

0

20

40

60

t

pωxpωy

0 10 20 30 40

0

5

10

t

‖Φ‖

Figure 4.5: Control and switching function with (TC1) and v0 = 1500m/s.

Pitching down maneuvers. We set v0 = 1500 m/s and we use the numerical valuesdenoted by TC3 in Table 4.1. The optimal solution is drawn on Figures 4.6 and 4.7.The shorter maneuver time tf indicates that it is easier to turn clockwise the axis of

0 10 20 30−10

−5

0

5

t


ωxωy

0 10 20 30

−20

0

20

40

60

80

t


θψφ

0 10 20 30

0

500

1,000

1,500

t

velocity-unit m/s

vxvyvz

Figure 4.6: State with TC3 and v0 = 1500m/s.

the velocity vector than to turn it anti-clockwise. This corresponds to the intuition.The reason is that the total force induced by the gravity force tends to reduce vx, i.e.,it helps the velocity to turn clockwise, and so together with the LV thrust force, themaneuver time is less than that of the anti-clockwise case.


Note that the derived time history of the adjoint variable (for both pitching up andpitching down maneuvers) do not have the same order of magnitude, i.e., pωx and pωyare ten times larger than pθ and pψ, and are thousand times larger than pvx , pvy andpvz . This indicates again that the shooting method is dicult to initialize successfully.

We note that the indirect strategy proposed in Section 4.3.1.2 is ecient also becausethe smallest adjoint variables pvx , pvy and pvz are already quite accurately estimatedthanks to the problem of order zero.

0 10 20 30

−1

−0.5

0

0.5

1

t

u1u2

‖u‖

0 10 20 30−60

−40

−20

0

20

t

pωxpωy

0 10 20 30

0

2

4

6

8

10

t

‖Φ‖

Figure 4.7: Control and switching function with TC3 and v0 = 1500m/s.

0 20 40 60−10

−5

0

5

10

t


ωxωy

0 20 40 60

0

50

100

150

t


θψφ

0 20 40 60

0

500

1,000

1,500

2,000

t

velocity-unit m/s

vxvyvz

Figure 4.8: State with (TC2) and v0 = 2000m/s.


4.4.2 Numerical Results with Chattering Arcs

Sub-optimal solution by the indirect approach. On Figures 4.8 and 4.9 is givena sub-optimal solution of (MTCP) with the terminal conditions (TC2) of Table 4.1and v0 = 2000 m/s. Due to chattering, the continuation parameter λ3 stops at valueλ∗3 = 0.98 (see Remark 4.3.1).

0 20 40 60 80 100

−0.5

0

0.5

1

t

control

u1u2

‖u‖0 20 40 60

0

20

40

60

t

time history of pωx and pωy

pωxpωy

0 20 40 60

0

5

10

t

module of the switching function

‖Φ‖

25 30 35 40

−0.2

−0.1

0

0.1

t

zoom of pωx and pωy

pωxpωy

Figure 4.9: Control and switching function with (TC2) and v0 = 2000m/s.

Observing from Figure 4.9, the switching function passes four times the switchingsurface Γ and remains small between times 26.5 and 40.3. The control, instead of bang-bang or singular, is continuous. The cost of this trajectory is 69.3 and the nal timetf = 66.0 s.

Sub-optimal solution by the direct approach. With the same terminal condi-tions as above, we now use the direct method described in Section 4.3.2. Numericalsimulations show that the initialization step for the direct method procedure is quiterobust (a constant initial guess is enough). The results are reported on Figures 4.10,4.11 and 4.12.

We observe that, when t ∈ [23, 44], the control oscillates much with a modulus lessthan 1: this indicates that there is a singular arc in the true" optimal trajectory, andtherefore chattering according to Corollary 4.2.1.


0 20 40 60

−10

−5

0

5

10

t


ωxωy

0 20 40 60

0

50

100

150

t


θψφ

0 20 40 60

0

500

1,000

1,500

2,000

t

velocity-unit m/s

vxvyvz

25 30 35 40 45

−5

0

5

·10−3

t

zoom of ωx and ωy (degree/s)

ωxωy

Figure 4.10: State variable x(t) with (TC2) and v0 = 2000m/s obtained by BOCOP.

0 20 40 60

−1

−0.5

0

0.5

1

t

control u1

0 20 40 60

−1

−0.5

0

0.5

1

t

control u2

0 20 40 60

0

0.5

1

1.5

t

‖u‖‖Φ‖/max(‖Φ‖)

Figure 4.11: Control u(t) with (TC2) and v0 = 2000m/s obtained by BOCOP.


0 20 40 60

0

20

40

60

t

pωxpωy

0 20 40 60

−10

−5

0

5

10

t

pθpψpφ

0 20 40 60

−8

−6

−4

−2

0

·10−2

t

pvxpvypvz

25 30 35 40 45

0

0.2

0.4

0.6

0.8

1

·10−3

t

zoom of pωx and pωy

pωxpωy

Figure 4.12: Adjoint vector p(t) with (TC2) and v0 = 2000m/s obtained by BOCOP.

Note that, along the singular arc, the variables ωx, ωy, pωx , pωy , pθ, pψ and pφ arealmost equal to 0, and we check that this arc indeed lives on the singular surface Sdened by (4.2.10). Therefore, it turns out that there is a singular arc in the optimaltrajectory, causing chattering at the junction with regular arcs.

The maneuver time is tf = 65.4 s. Compared with that of the sub-optimal solutionderived from the indirect strategy, only 0.6 s are gained with the direct method. Thedirect approach is hundreds of times slower than the indirect approach and the obtainedcontrol presents many oscillations, which is not much appropriate for a practical use.

On Figure 4.2, 4.4, 4.6, 4.8 and 4.10, we note that the attitude angles rst tend toreach the values θ∗ deg, ψ∗ (i.e., θ∗ = 176.9 deg and ψ∗ = 18.5 deg for Figures 4.2 and4.4; θ∗ = −17.4 deg and ψ∗ = 25.2 deg for Figure 4.6; θ∗ = 176.1 deg and ψ∗ = 11.2 degfor Figures 4.8 and 4.10), and then turn back to reach their nal values. Actually, doingmore numerical simulations with dierent terminal conditions (note reported here), weobserve that the extremals have a trend to rst go towards the singular surface andthen to get back to the target submanifold. We suspect that this is due to a turnpikephenomenon as described in [107], at least when the required transfer time is quitelarge.


Conclusion

We have studied the time optimal control of the LV attitude motion combined with thetrajectory dynamics. (MTCP) is of interest because of the coupling of guidance andnavigation systems. However, this problem is dicult to solve because of the occurrenceof the chattering phenomenon for certain terminal conditions. Using geometric control,we have established a chattering result for bi-input control-ane systems. We havealso classied the switching points for the extremals of (MTCP), according to theorder of vanishing of the switching function, showing the behavior of the control at thesingularities.

In order to compute numerically the solutions of (MTCP), we have implementedtwo approaches. The indirect approach, combining shooting and numerical continua-tion, is time-ecient when the solution does not contain any singular arcs. For certainterminal conditions, the optimal solution of (MTCP) involves a singular arc that is oforder two, and the connection with regular arcs can only be done by means of chat-tering. The occurrence of chattering causes the failure of the indirect approach. Forsuch cases, we have proposed two possible numerical alternatives. Since our indirectapproach involves three continuations, one of them being concerned with a continuationon the cost function (and thus on the Hamiltonian and the control), we have proposed,as a rst alternative, to stop this last continuation before its failure: in such a way,we obtain a sub-optimal solution, which seems to be very acceptable for a practicaluse. The second alternative is based on a direct approach, and then we obtain as wella sub-optimal solution having a nite number of switchings, this nite number beinglimited by the chosen step of the subdivision in the discretization scheme. In any case,the direct strategy is much more time consuming than the indirect approach. Notethat, in both cases, it is not required to know a priori the structure of the optimalsolution (in particular, the number of switchings).

Chapter 5

Application to the Airborne Launchers

Contents

5.1 Pull-Up Maneuver Problem . . . . . . . . . . . . . . . . . . . 123

5.2 Application of the PMP . . . . . . . . . . . . . . . . . . . . . 124

5.3 Resolution algorithm . . . . . . . . . . . . . . . . . . . . . . . 127

5.3.1 Comparison with (MTCP) . . . . . . . . . . . . . . . . . . . 127

5.3.2 Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133


5.4.1 Pull-Up Maneuvers of the AVL . . . . . . . . . . . . . . . . . 133

5.4.2 Rapid Attitude Maneuver of the LVs . . . . . . . . . . . . . . 138

Introduction

Since the rst successful ight of Pegasus vehicle in April 1990, the airborne launchvehicles (ALVs) have always been a potentially interesting technique for small andmedium-sized space transportation systems. The mobility and deployment of the ALVsprovide increased performance and reduced velocity requirements due to non-zero initialvelocity and altitude (see, e.g., [57, 59, 68, 90, 92]).

ALVs consist of a carrier aircraft and a rocket-powered LV, and they are typicallystarted several seconds after they are dropped almost horizontally from the carrieraircraft for the safety of the carrier aircraft. In order to benet from the airbornelaunch [91, 92], a pull-up maneuver is required to rotate the ALVs in order to attain theoptimal release ight path angle (30±15 for subsonic release velocities). Consider forexample the Pegasus vehicle [5, 33, 81, 89]. It is released horizontally with an altitudeof 12.65 km. Its rst stage is ignited with an altitude of 12.54 km and a velocity of236.8m/s (0.8 Mach). Then it has to fulll a pull-up maneuver until having a ight

121

122

path angle of 13.8 (for the ignition of the second stage) subject to a maximal loadfactor of 2.5 g and a maximal dynamic pressure of 47.6 kPa.

To tackle the pull-up maneuver problem for ALVs, the vehicle cannot be regardedas a single mass point, since the pull-up maneuver consists of performing an attitudemaneuver such that the ight path angle increases up to its expected value, while beingsubject to state constraints. In this chapter, we address the minimum time-energypull-up maneuver problem for ALVs with a focus on the numerical resolution of theproblem.

The problem consists of minimizing the cost functional

C(tf , u) = tf +K

∫ tf

0

‖u(t)‖2 dt, (5.0.1)

for the six-degree-of-freedom (6DOF) dynamical system (2.1.6)-(2.1.7)-(2.1.8)-(2.1.12)

rx = vx, ry = vy, rz = vz,

vx = a sin θ cosψ + gx + (Dx + Lx)/m,

vy = −a sinψ + gy + (Dy + Ly)/m,

vz = a cos θ cosψ + gz + (Dz + Lz)/m,



φ = (ωx sinφ+ ωy cosφ) tanψ,

ωx = −bu2, ωy = bu1,

(5.0.2)

with prescribed initial conditions, nal conditions, and state constraints on the loadfactor n and on the dynamic pressure q.

Compared to (MTCP), an additional diculty is due to the state constraint. Itis well known that state constraints are dicult to treat by using maximum principle,and thus we use a penalty method (also called soft constraint method) to treat stateconstraints.

This chapter is organized as follows. In Section 5.1, we formulate the optimalcontrol problem (PUMP). Then, in Section 5.2, the PMP and the penalty method areapplied to (PUMP). Next, in Section 5.3, the numerical strategy for solving (PUMP)is detailed. Finally, in Section 5.4, numerical results are given, including an example ofALV minimum time-energy pull-up maneuver, statistical results for pull-up maneuversof an ALV, and an example of rapid maneuver for a LV.

Chapter 5: Application to the Airborne Launchers 123

5.1 Pull-Up Maneuver Problem

Dening the state variable x = (rx, ry, rz, vx, vy, vz, θ, ψ, φ, ωx, ωy), we write the system(5.0.2) as a bi-input control-ane system

x = f(x) + u1g1(x) + u2g2(x), (5.1.1)

where the controls u1 and u2 satisfy the constraint u21 + u2

2 ≤ 1, and the vector elds f ,g1 and g2 are smooth vector elds dened by

f =vx∂

∂rx+ vy

∂

∂ry+ vz

∂

∂rz+ (a sin θ cosψ + gx + (Dx + Lx)/m)

∂

∂vx

+ (−a sinψ + gy + (Dy + Ly)/m)∂

∂vy+ (a cos θ cosψ + gz + (Dz + Lz)/m)

∂

∂vz

+ (ωx sinφ+ ωy cosφ)/ cosψ∂

∂θ+ (ωx cosφ− ωy sinφ)

∂

∂ψ

+ tanψ(ωx sinφ+ ωy cosφ)∂

∂φ,

g1 =b∂

∂ωy, g2 = −b ∂

∂ωx.

Terminal conditions. Let rx0, ry0, rz0, vx0 , vy0 , vz0 , θ0, ψ0, φ0, ωx0 , ωy0 , θf , ψf , φf ,ωxf and ωyf be real numbers. The initial conditions are given by

rx(0) = rx0, ry(0) = ry0, rz(0) = rz0, vx(0) = vx0 , vy(0) = vy0 , vz(0) = vz0 ,

θ(0) = θ0, ψ(0) = ψ0, φ(0) = φ0, ωx(0) = ωx0 , ωy(0) = ωy0 .

(5.1.2)

Let the nal position to be free, then the nal conditions are the same with that of(MTCP) given by (2.2.9), i.e.,

vzf sinψf + vyf cos θf cosψf = 0, vzf sin θf − vxf cos θf = 0,

θ(tf ) = θf , ψ(tf ) = ψf , φ(tf ) = φf , ωx(tf ) = ωxf , ωy(tf ) = ωyf .(5.1.3)

State constraint. During the atmospheric ight, the lateral load factor due to aero-dynamic forces must be limited according to

n =qSCNmg0

=ρ‖v‖2SCN

2mg0

≤ nmax,

124 5.2. Application of the PMP

where q = ρ‖v‖22

is the dynamic pressure and nmax is the maximal admissible load factor.The coecient CN is approximated by CN = CN0 + CNαα. Note that(

CzCx

)=

(cosα − sinαsinα cosα

)(CNCA

),

where the angle of attack α is dened by α = (vx sin θ cosψ−vy sinψ+vz cos θ cosψ)/v.We also require that the dynamic pressure be bounded above by a maximal value qmax,i.e., q ≤ qmax.

These constraints can be formulated as state constraints for our optimal controlproblem, in the form

c(x) = (c1(x), c2(x))> = (n− nmax, q − qmax)> ≤ 0. (5.1.4)

Pull-up maneuver problem (PUMP). We set

x0 = (rx0, ry0, rz0, vx0 , vy0 , vz0 , θ0, ψ0, φ0, ωx0 , ωy0) ∈ R11,

and we dene the target set (submanifold of R11)

M1 = (rx, ry, rz, vx, vy, vz, θ, ψ, φ, ωx, ωy) ∈ R11 | vz sinψf + vy cos θf cosψf = 0,

vz sinψf + vy cos θf cosψf = 0, θ = θf , ψ = ψf , φ = φf ,

ωx = ωxf , ωy = ωyf.

The pull-up maneuver problem denoted by (PUMP) consists of minimizing the costfunctional (5.0.1) such that the bi-input control-ane system (5.1.1) goes from x(0) =x0 to the nal target M1 with controls satisfying the constraint u2

1 + u22 ≤ 1. Moreover,

the state constraint (5.1.4) is also satised.

5.2 Application of the PMP

Hard constraint formulation. First we recall the order of a state constraint.

Denition 5.2.1. A state constraint c(x) ≤ 0 is of order m if gi.c = gif.c = · · · =gif

m−2.c = 0 and gifm.c 6= 0, i = 1, 2.

A boundary arc is an arc (not reduced to a point) solution of the system satisfying

c(x(t)) = c(1)(x(t)) = · · · = c(m−1)(x(t)) = 0,

and the control along the boundary arc is a feedback control calculated by solving

c(m) = fm.c+ u1 g1f(m−1).c+ u2 g2f

(m−1).c = 0.


Here, by some manipulations, we nd that the constraint on the load factor n is oforder 2 and the constraint on the dynamic pressure q is of order 3.

Then, according to the maximum principle with state constraints (see e.g. [55]),there exists a nontrivial triple of Lagrangian multipliers (p, p0, η), with p0 ≤ 0, p ∈BV (0, tf )

11 and η = (η1, η2) ∈ BV (0, tf )2, where BV (0, tf ) is the set of functions of

bounded variation over [0, tf ], such that

x =∂H(x, p, u, p0, η)

∂p, dp = −∂H(x, p, u, p0, η)

∂xdt−

2∑i=1

∂ci(x)

∂xdηi,

almost everywhere on [0, tf ], where the Hamiltonian of the problem is H(x, p, u, p0, η) =〈p, f(x)+u1g1(x)+u2g2(x)〉+

∑2i=1 ηici(x)+p0(1+K‖u‖2), and we have the maximiza-

tion condition u(t) ∈ argmaxwH(x(t), p(t), w, p0, η(t)) for almost every t. In addition,we have dηi ≥ 0 and

∫ tf0ci(x) dηi = 0 for i = 1, 2.

Along a boundary arc, we must have hi = 〈p, gi(x)〉 = 0, i = 1, 2. Assume thatonly the rst constraint (which is of order 2) is active along this boundary arc. Thenby dierentiating two times the switching functions hi, i = 1, 2, we have d2hi =〈p, ad2f.gi(x)〉dt2 − dη1 · (adf.gi).c1dt. Moreover, at an entry point, letting t = τ ,we have dhi(τ+) = dhi(τ

−) − dη1 · (adf.gi).c1 = 0. Hence we can calculate dη1. Asimilar result is obtained at an exit point.

The main drawback of this formulation is that the adjoint vector p is no longerabsolutely continuous: a jump dη may occur at the entry or at the exit point of aboundary arc. Then, in order to design a robust algorithm for solving (PUMP), weuse another approach.

An alternative to treat the dynamic pressure state constraint, used in [35, 39, 71],is to design a feedback law that reduces the commanded throttle based on an errorsignal. According to [39], this approach works well when the trajectory does not violatetoo much the maximal dynamic pressure constraint. In contrast, if the constraint isviolated signicantly, this approach may cause instability in the ight. Moreover, thederived solutions are suboptimal.

Another alternative is the penalty function method (also called soft constraint method). The soft constraint is implemented using a penalty function to discard solutionsentering the constrained region [36, 76].

For (PUMP), we adopt the soft constraint method, because unconstrained solutionsfor ALV ights generally violate signicantly the state constraint, and the continuationprocedure that we will use is actually starting from a solution lying in the constrainedregion.

Soft constraint formulation. Following the penalty method, the constraint (PUMP)is recast as an unconstrained optimal control problem by adding a penalty function to

126 5.2. Application of the PMP

the cost functional, i.e.,

C(tf , u,Kp) = tf +K

∫ tf

0

‖u‖2dt+Kp

∫ tf

0

P (x(t))dt, (5.2.1)

where the penalty function P (·) for the state constraint (5.1.4) is given by

P (x) = (max(0, n− nmax))2 + (max(0, q − qmax))2.

Tuning the parameter Kp allows one to control the constraint violation. Then, we applythe PMP to this unconstrained problem. We still denote this unconstrained problemby (PUMP).

Application of the PMP without state constraint. The Hamiltonian is nowgiven by

H(x, p, p0, u) = 〈p, f(x)〉+ u1〈p, g1(x)〉+ u2〈p, g2(x)〉+ p0(1 +K‖u‖2 +KpP (x)).

Here we assume p0 = −1 (normal case, see the chapter 4). The adjoint equation is

p(t) = −∂H∂x

(x(t), p(t), p0, u(t)), (5.2.2)

where we have set p = (prx , pry , prz , pvx , pvy , pvz , pθ, pψ, pφ, pωx , pωy). Let h = (h1, h2) bethe switching function and let

h1(t) = 〈p(t), g1(x(t))〉 = bpωy(t), h2(t) = 〈p(t), g2(x(t))〉 = −bpωx(t).

The maximization condition of the PMP gives

u =

(h1, h2)/(2K) if ‖h‖ ≤ 2K,

(h1, h2)/‖h‖ if ‖h‖ > 2K.(5.2.3)

Moreover, the transversality condition p(tf ) ⊥ Tx(tf )M1, where Tx(tf )M1 is the tan-gent space to M1 at the point x(tf ), yields the additional conditions pvy(tf ) sinψf =pvx(tf ) sin θf cosψf + pvz(tf ) cos θf cosψf and prx(tf ) = pry(tf ) = prz(tf ) = 0. Whenthe nal time tf being free and the system being autonomous, we have in addition thatH(x(t), p(t),−1, u(t)) = 0 almost everywhere on [0, tf ].

Recall that the optimal control given by (5.2.3) is regular. When K = 0 and‖h(t)‖ = 0, the control is said singular. Note that the term K

∫ tf0‖u(t)‖2dt in the

cost functional (5.0.1) is used to avoid chattering [73, 43, 88, 112, 113], and the exactminimum time solution can be approached by decreasing step by step the value ofK ≥ 0.


Remark 5.2.1 (Choice of the penalty parameter). According to the cost functional(5.2.1), we see that if Kp is small compared with the other two terms, minimizingC(tf , u,Kp) may not lead a feasible solution. Therefore, the penalty parameter Kp

should be chosen large enough. However, large values of Kp may create steep valleysat the constraint boundaries, which raise diculties for search methods. In order tochoose an adequate value for Kp, a simple strategy [44, 96] is to start with a quite smallvalue of Kp = Kp0 and then to increase Kp while solving a sequence of problems untilreaching a quite large Kp = Kp1. We stop increasing Kp when ‖c(x(t))‖ < εc, for everyt ∈ [0, tf ], for some given tolerance εc > 0.

5.3 Resolution algorithm

5.3.1 Comparison with (MTCP)

We can extend our numerical method for solving (MTCP) to a more complex strat-egy for solving (PUMP). By comparing (PUMP) with (MTCP), we see that in(PUMP):

• (a) the cost functional is the same with that of (RMTCP)K ;• (b) the position of the launcher is considered;• (c) the gravity acceleration vector ~g is no longer constant and the aerodynamicforces (lift force ~L and the drag force ~D) are considered;• (d) state constraints are considered;

As for the point (a), it suces to ignore the continuation of parameter λ3 introducedto the resolution approach for solving (MTCP).

Then, for the point (b), we need to plug the solution of (POZ) to higher orderproblem of zero with the adjoint variable of the position ~pr = (prx, pry, prz)

> beingzero. More precisely, consider the following problem, denoted by (HPOZ), in whichthe position and the velocity are considered

min tf , s.t. ~r = ~v, ~v = a~e+ ~g0, ~r(0) = ~r0, ~v(0) = ~v0, ~v(tf )//~w, ‖~w‖ = 1.

Lemma 5.3.1. The solution to (HPOZ) is given by

tf =−a2 +

√a2

2 − 4a1a3

2a1

, ~pr = ~0, ~pv =−p0

a+ 〈~e∗, ~g〉~e∗ ,

and the optimal control ~e = ~e∗ = 1a

(k ~w−~v0

tf− ~g0

), with k = 〈~v0, ~w〉 + 〈~g0, ~w〉tf , a1 =

a2 − ‖〈~g0, ~w〉~w − ~g0‖2, a2 = 2(〈~v0, ~w〉〈~g0, ~w〉 − 〈~v0, ~g0〉), and a3 = −‖〈~v0, ~w〉~w − ~v0‖2.

128 5.3. Resolution algorithm

Proof 5.3.1. According to the PMP, we have ~pr = ~0 and ~pr(tf ) = ~0 (transversalitycondition), and thus we get that ~pr(t) = ~0 for t ∈ [0, tf ]. The rest of this Lemma isproved using the same arguments for Lemma 4.3.1.

Similar to Lemma 4.3.2, this solution can be embedded into the solution of (PUMP).We use this solution as the initialization for solving (PUMP).

Next, the point (c) can be easily tackled by adding one more continuation to re-introduce the variational gravity acceleration and the aerodynamics forces. Introducea new continuation parameter λ4 to act on the gravity acceleration vector (2.1.1), theaerodynamic forces (2.1.11) and the air density ρ, according to

vx = a sin θ cosψ + g0x(1− λ4) + λ4gx + λ4Dx + Lx

m,

vy = −a sinψ + g0y(1− λ4) + λ4gy + λ4Dy + Ly

m,

vz = a cos θ cosψ + g0z(1− λ4) + λ4gz + λ4Dz + Lz

m,

and

ρ(t) = ρ0

((1− λ4) exp(−(‖~rd(0)‖ −RE)/hs) + λ4 exp(−(‖~rd(t)‖ −RE)/hs)

),

where gx, gy and gz are given by (2.1.1), (~g0)R = (gx0, gy0, gz0)>, and hs = 7143,ρ0 = 1.225. Applying the PMP, the equations of the adjoint variable p also involve λ4.This step is a continuation on the dynamics of the system, and the parameter λ4 doesnot explicitly appear in the shooting function.

Finally, for the point (d), as we have mentioned, we use the penalty method. Recallthat to use this method, we only need to add a penalty function P (x(t)) to the costfunctional (4.3.5) with a positive penalisation parameter Kp. See details in section 5.2.

For convenience, we dene the exo-atmospheric pull-up maneuver problem (ERMP)as (PUMP) without state constraints and without aerodynamic forces. Moreover, wedene the unconstrained pull-up maneuver problem (URMP) as (PUMP) withoutstate constraints.

We proceed as follows:

• First, we embed the solution of (HPOZ) into the higher dimensional problem(PUMP). For convenience, we still denote by (HPOZ) the problem (HPOZ)seen in higher dimension.• Then, we pass from (HPOZ) to (PUMP) by using a numerical continuationprocedure, involving four continuation parameters: two parameters λ1 and λ2 areused to introduce the terminal conditions (5.1.2)-(5.1.3) in (ERMP); λ4 is usedto introduce aerodynamic forces and variational gravity acceleration in (URMP);λ5 is used to introduce soft constraints in (PUMP).


The overall continuation procedure is pictured in Fig. 5.1. We note that the pro-cedure of increasing λ3 (decreasing K) is ignored, but it is somehow important whendealing with the maneuver problem of the upper stage of LVs, since in that case amaneuver within smaller time is to be expected.

(HPOZ)start

(ERMP) (URMP)

(PUMP)(RMTCP)K=0

λ4 = 0 → 1

λ3 = 0 → 1

(K → 0)

λ1 = 0 → 1

λ2 = 0 → 1

λ5 = 0 → 1

Kp = Kp0 → Kp1

K → 0

Figure 5.1: Continuation procedure.

More precisely, we are to solve the following problem with continuation parametersλi, i = 1, 2, 4, 5,

min tf + (1− λ3)

∫ tf

0

‖u‖2dt+ λ5Kp

∫ tf

0

P (x(t))dt,

such thatrx = vx, ry = vy, rz = vz,

vx = a sin θ cosψ + g0x(1− λ4) + λ4gx + λ4Dx + Lx

m,

vy = −a sinψ + g0y(1− λ4) + λ4gy + λ4Dy + Ly

m,

vz = a cos θ cosψ + g0z(1− λ4) + λ4gz + λ4Dz + Lz

m,



φ = (ωx sinφ+ ωy cosφ) tanψ,

ωx = −bu2, ωy = bu1.

and the initial conditions

~r(0) = (rx0, ry0, rz0)>, ~v(0) = (vx0, vy0, vz0)>,

θ(0) = θ∗(1− λ1) + θ0λ1, ψ(0) = ψ∗(1− λ1) + ψ0λ1, φ(0) = φ0λ1,


ωx(0) = ωx0λ1, ωy(0) = ωy0λ1,

and the nal conditions~r(tf ) free, ~v(tf ) ⊥ zb,

θ(tf ) = θe(1− λ2) + θfλ2, ψ(tf ) = ψe(1− λ2) + ψfλ2, φ(tf ) = φe(1− λ2) + φfλ2,

ωx(tf ) = ωxe(1− λ2) + ωxfλ2, ωy(tf ) = ωye(1− λ2) + ωxfλ2,

where θe, ψe, φe, ωxe, and ωye are the values of the state variable at tf by integratingthe extremal system derived from the PMP, and θ∗, ψ∗ are calculated through (4.3.1).

To implement eciently the above multi-parameter continuation, we use the PCcontinuation combined with the multiple shooting method. Moreover, we use two ad-ditional numerical tricks in order to improve the robustness of the algorithm and totackle Euler singularities (see Remark 2.1.1).

Multiple shooting. The unknowns of this shooting problem are p(0) ∈ R11, tf ∈ R,and zi = (xi, pi) ∈ R22, i = 1, · · · , N − 1, where zi are the node points of the multipleshooting method (see section 1.2.1). We set Z = (p(0), tf , zi), and let E = (θ, ψ, φ),ω = (ωx, ωy), pr = (prx, pry, prz), pE = (pθ, pψ, pφ), and pω = (pωx, pωy). Then, theshooting function with continuation parameter λ1 is given by

Gλ1 =(vz(tf ) sinψf + vy(tf ) cos θf cosψf , vz(tf ) sin θf − vx(tf ) cos θf

pvy(tf ) sinψf − (pvx(tf ) sin θf cosψf + pvz(tf ) cos θf cosψf ),

pr(tf ), pω(tf ), pE(tf ), H(tf ), zi(t−i ) = zi(ti)+, i = 1, · · · , N − 1

),

where H(tf ) is given by

H(tf ) = 〈p, f(x)〉+ u1〈p, g1(x)〉+ u2〈p, g2(x)〉+ p0(1 + (1− λ3)K‖u‖2 + λ5KpP (x)).

The shooting function with λ2 is

Gλ2 =(vz(tf ) sinψf + vy(tf ) cos θf cosψf , vz(tf ) sin θf − vx(tf ) cos θf ,

pvy(tf ) sinψf − (pvx(tf ) sin θf cosψf + pvz(tf ) cos θf cosψf ),

E(tf )− (1− λ2)Ee − λ2Ef , ω(tf )− (1− λ2)ωe − λ2ωf ,

pr(tf ), H(tf ), zi(t−i ) = zi(ti)+, i = 1, · · · , N − 1

),

and the shooting functions Gλ4 and Gλ5 are the same with Gλ2 .

PC continuation. As introduced in section 1.3.2, one needs to calculate the Jacobianmatrix JG when using a classical predictor-corrector continuation, and this is compu-tationally heavy for our problem. Therefore, we use an approximation to acceleratethe solving process. According to [26], the rst turning point of λ(s) (where dλ

ds(s) = 0


and d2λds2

(s) 6= 0) corresponds to a conjugate point (the rst point where extremalslose local optimality) at time tf . By contraposition, if we assume the absence of theconjugate point, then there is no turning point in λ(s), and then λ increases monoton-ically along the zero path. Assume that we know three zeros (Zi−2, λi−2), (Zi−1, λi−1)and (Zi, λi), and let s1 = ‖(Zi−1, λi−1) − (Zi−2, λi−2)‖, s2 = ‖(Zi, λi) − (Zi−2, λi−2)‖,s3 = ‖(Zi, λi)− (Zi−1, λi−1)‖, then we approximate t(JG) by

t(JG) =(Zi, λi)− (Zi−1, λi−1)

s2 − s1

|s2 − s1||s3|

. (5.3.1)

We note that when the step length hs is small enough, the predicted point (1.3.2) withthis approximation is very close to the true zero.

Change of Frame Changing the reference frame can improve the problem condi-tioning and enhance the numerical solution process. The new frame S ′R is designed byadequately rotating the initial frame SR. This is a nonlinear state transformation, lead-ing to a preconditionner that makes the proposed continuation procedure more robust.More precisely, we dene the new coordinate SR′ by two single-axis rotations from theframe SR, given by

SRRy(β1)−−−−→ Rx(β2)−−−−→ SR′ ,

and the transition matrix from SR to S ′R is LR′R = Rx(β2)Ry(β1). Denoting the Eulerangles of Sb with respect to the new frame SR′ as θ′, ψ′, and φ′, the transition matrixfrom S ′R to Sb is LbR′ = Rz(φ

′)Rx(ψ′)Ry(θ

′).Using that LbRLRR′ = Rz(φ)Rx(ψ)Ry(θ)L

>R′R = LbR′ , the angles θ′, ψ′, φ′ are func-

tions of θ, ψ, φ, β1, and β2. Moreover, the velocity vector ~v in the S ′R can also beobtained by (~v)′R = LR′R(~v)R. The angular velocity vector ~ω is expressed in the bodyframe Sb and it is therefore not altered by this coordinate change.

Given xed β1 and β2, the change of frame corresponds to the nonlinear invertiblechange of state variable x′ = diag(LR′R(~r)R, LR′R(~v)R, ϕatt(x), Id), where ϕatt(·) mapsEuler angles in SR to Euler angles in S ′R, and Id is the 3-dimensional identity matrix.

This state transformation can be seen as a preconditionner for our numerical method.Indeed, we use a Newton-like method to solve the boundary value problem in the shoot-ing method. Let us for instance consider the simplest Newton method and let us denoteby Z the variables of the shooting method. Given an initial guess of Z = Z0, the equa-tion G(Z) = 0 is solved iteratively according to J(Zk)Zk+1 = J(Zk)Zk −G(Zk), whereJ = ∂G/∂Z. Dene a dieomorphism ϕ(·) such that Z = ϕ(y). Then the originalproblem consists of solving G(y) = 0 and the Newton method iterative step becomesJ(yk)yk+1 = J(yk)yk − G(yk), where J(yk) = J(Zk)

∂Z∂y

(yk). The matrix M = (∂Z∂y

)−1

actually acts as a preconditionner in the shooting method and it can be used to reducethe condition number of the Jacobian matrix J . In numerical experiments, we use theFortran subroutine hybrd.f (see [80]) which uses a modication of the Powell hybridmethod: the choice of the correction is a convex combination of the Newton and scaled


gradient directions, and the updating of the Jacobian by the rank-1 method of Broyden.Note in addition that the dierential equations for the new variable x′ keep the same

form as for the old variable x, and by using the PMP, the adjoint vector p′ to the newstate x′ can also be derived from (x, p), according to

~p′r = LR′R~pr, ~p′v = LR′R~pv, ~p′E =

(dϕatt(x)

dx

)−1>

~pE, ~p′ω = ~pω,

where ~pE = (pθ, pψ, pφ)>, ~p′E = (p′θ, p′ψ, p

′φ)>, ~pv = (pvx, pvy, pvz)

>, ~p′v = (p′vx, p′vy, p

′vz)>,

~pω = (pωx, pωy)>, and ~p′ω = (p′ωx, p

′ωy)>.

To sum up, the new reference frame S ′R can be chosen such that (PUMP) is easierto solve numerically. However, a priori, we do not know what values (β1, β2) are themost suitable. We propose to choose the pair such that ψ′f = −ψ′0 and |ψ′f |+ |ψ′0| beingminimal. By doing this, the terminal values on the yaw angle are closer to the originand hence farther from Euler singularities. We observe from numerical experimentsthat this choice enhances robustness of the algorithm.

Singularities of the Euler Angles Smoothing the vector elds at singular pointsof Euler angles also assists to tackle singularities. Assuming that θ is bounded, wehave ωx sinφ + ωy cosφ → 0 when ψ → π/2 + kπ. Since θφ = limψ→π/2+kπ(ωx sinφ +

ωy cosφ)2 sinψ → 0 and θ/φ → 1 as ψ → π/2 + kπ, it follows that θ = φ = 0 asψ → π/2 + kπ. Assuming that −pθ+pφ sinψ

cosψ→ A <∞ as ψ → π/2 + kπ, it follows from

the fact that

A = limψ→π/2+kπ

−pθ + pφ sinψ

cosψ= lim

ψ→π/2+kπ

pθ + pφ sinψ + pφ cosψψ

sinψψ= −A

that A = 0. Hence, we get pθ = 0, pφ = 0, pψ = a sin θpvx + a cos θpvz, pωx = −pψ cosφ,pωy = pψ sinφ. Summing up, at points ψ → π/2+kπ, (2.1.6)-(2.1.7) and (5.2.2) become

θ = 0, ψ = ωx cosφ− ωy sinφ, φ = 0, ωx = −bu2, ωy = bu1,

pθ = 0, pψ = a sin θpvx + a cos θpvz , pφ = 0,

pωx = −pψ cosφ, pωy = pψ sinφ.

(5.3.2)

When we are close to a singularity, we rather use (5.3.2).


5.3.2 Algorithm

We describe the whole numerical strategy in the following algorithm.

Result: The solution of the problem (PUMP)· Change of frame: compute (β1, β2) and the new initial condition x(0) = x′0;· Solve (HPOZ) to get a solution Z0;· Initialize the multiple shooting method with Z0 and λi = 0, i = 1, · · · 4;for i = 1, · · · , 4 do

while λi ≤ 1 and 4λimin ≤ 4λi ≤ 4λimax do(Predictor) Predict a point (Z, λi) according to (1.3.2) and (5.3.1);(Corrector) Find the solution (Z, λi) of Gλi(Z, λi) = 0;if successful then

(Z, λi) = (Z, λi);else

Choose a new step-length hs;end


The λi-continuation is successful;else

The λi-continuation failed;end

endAlgorithm 3: Prediction-Corrector continuation


In this section, we solve (PUMP) with the algorithm proposed in Section 5.3.2. Werst present a pull-up maneuver of an ALV just after its release from the airplane andwe present some statistical results showing robustness of our algorithm. Then we applythe algorithm to the three-dimensional reorientation maneuver of the upper stage of aLV after a stage separation.

5.4.1 Pull-Up Maneuvers of the AVL

We consider a pull-up maneuver of an ALV. The data used in (5.0.2) approximate aPegasus-like airborne launcher: a = 15.8, b = 0.2, S = 14m2, Cx0 = 0.06, Cxα = 0,Cz0 = 0, and Czα = 4.7. Let nmax = 2.2g and qmax = 47 kPa. The initial conditions


(5.1.2) are

rx0 = 11.9 km, ry0 = rz0 = 0, v0 = 235m/s, θv0 = −10,

ψv0 = 0, θ0 = −10, ψ0 = φ0 = 0, ωx0 = ωy0 = −1/s,

and the nal conditions (5.1.3) are

θf = 42, ψf = 10, φf = 0, ωxf = ωyf = 0.

Note that generally the pull-up maneuvers are planar (ψf = 0). Here we set ψf = 10

with the aim of showing that the algorithm can also deal eciently with the non-planarpull-up maneuvers (ψf 6= 0).

The multiple shooting method is applied with three node points. The components ofthe state variable x and the control u are reported on Figs 5.2 and 5.3. The componentsof the adjoint variable p are given on Fig. 5.4. The time histories of the load factor nand of the dynamic pressure q are reported on Fig. 5.5.

0 5 10 15 20

0

5

10

15

t

position-unit km

rx (altitude)ryrz

0 5 10 15 20

0

200

400

600

800

t

velocity-unit m/s

vθvψv

0 5 10 15 20

0

20

40

60

t


θψφ

0 5 10 15 20

−5

0

5

10

15

t


ωxωy

Figure 5.2: Time history of state variable x(t) for an ALV.

From Fig. 5.5, we see that there is a boundary arc of the load factor constraint.


0 5 10 15 20

−0.1

0

0.1

0.2

t

u1u2

Figure 5.3: Time history of control variable u(t) for an ALV.

According to the pωy(t) in Fig. 5.4, we see that, over the boundary arc, the switchingfunction h(t) = b(pωy ,−pωx) indeed stays close to zero. Comparing Figs. 5.3 and 5.4,we see that the control follows the form of the switching function. Moreover, the stateconstraint of the dynamic pressure is not active.

As mentioned in Remark 5.2.1, the large value of Kp often causes numerical dicul-ties. In this example, we observe on Fig. 5.4 that, at t = 5.86 s, the curve pθ(t) is notas smooth as the other parts. Indeed, at t = 5.86 s, the penalty function P (x) starts tobe positive and thus provides nonzero terms in the adjoint dierential equation.

Running this example requires 24.6 s to compute the optimal solution, with CPU:Intel(R) Core(TM) i5-2500 CPU 3.30GHz; Memory: 3.8 Gio; Compiler: gcc version4.8.4 (Ubuntu 14.04 LTS). If the multiple shooting method is applied with four nodepoints, the computing time is 31.2 s.

In the following, we present some statistical results done with the same computersettings.

Statistical results We solve (PUMP) with dierent terminal conditions. Initialand nal conditions are swept in the range given in Table 5.1. The last cell of the tabledenes the restriction applied to the terminal conditions in order to exclude unrealisticcases. For each variable, we choose a discretization step and we solve all possible

Table 5.1: Parameter ranges.v0 θv0 ψv0 θ0 ψ0

xed 0.8Ma [−10, 0] xed 0 [−10, 10] xed 0

θf ψf ωx0 ωy0 θ0 − θv0

[20, 80] [−10, 10] [−2, 2]/s [−2, 2]/s [0, 10]

combinations of this discretization (factorial experiment).Statistical results are reported in Table 5.3-5.4. Set Kp0 = 0.1, Kp1 = 100, and

let the maximum number of iteration steps in each continuation be 200 (i.e., in Table


0 5 10 15 20

−2

−1.5

−1

−0.5

0

·10−3

t

adjoint position

prxpryprz

0 5 10 15 20

−0.15

−0.1

−5 · 10−2

0

5 · 10−2

0.1

t

adjoint velocity

pvxpvypvz

0 5 10 15 20

−10

0

10

20

t

adjoint Euler angles

pθpψpφ

0 5 10 15 20

0

20

40

t

adjoint angular velocity

pωxpωy

Figure 5.4: Time history of the adjoint variable p(t) for an ALV.

5.3-5.4, number of simulations for each continuation parameter are limited to 200). Ineach table, there are 1701 cases.

We see from the results that the algorithm is robust with respect to terminal con-ditions and that it is rather fast in comparison with a simple direct method.

On one hand, from Table 5.3-5.2, we see that the choice of the regularization param-eter K aects the resolution results: (i) the rate of success increases (resp. decreases) inthe non-planar case (resp. planar case) when K increases from K = 800 to K = 1000;(ii) in term of the execution time, we see that in both cases, it is faster to get a resultin planar case than in non-planar case, and most time is devoted to deal with the stateconstraints.

We suggest that systematical numerical experiments could be done to nd out thebest K for both planar and non-planar cases. For example, we have tested the planarcases with dierent values ofK. The rate of success and the execution time with respectto K is illustrated in Fig. 5.4.1. We see that the value of K should not be too large,neither too small. In fact, when K is large, the control may be too small to steer thesystem to the aimed conguration, and thus it leads to smaller success rate and largerexecution time. On contrast, if K is too small, it is also problematic. Not only causes


0 5 10 15 20

0

0.5

1

1.5

2

t

load factor n

0 5 10 15 20

10

20

30

t

α-unit degree, q-unit kPa

Figure 5.5: Time history of the constraints c(x(t)) for an ALV.

Table 5.2: Statistical results (N = 2 and K = 8× 102).planar non-planar

Number of cases 567 1134Rate of success (%) 89.07 80.04

Number of failure cases- In λ1-continuation 0 14- In λ2-continuation 21 172- In λ4-continuation 41 26

- In λ5+Kp-continuation 0 10Average execution time (s)

- Total 26.94 44.05- In λ1-continuation 0.49 0.48- In λ2-continuation 2.07 2.37- In λ4-continuation 2.54 2.99

- In λ5+Kp-continuation 23.16 37.35

it more failures in the rst two continuations, but also the introduction of the stateconstraint could be more tricky.

Moreover, from Table 5.3-5.4, we observe that in the non-planar case, it is λ2-continuation that causes more failure. We suggest that this is due to the choice ofK.

On the other hand, from Table 5.2 and Table 5.4, we nd that, compared to singleshooting method used in chapter 4 (N = 0), the multiple shooting method (N =2) improves the robustness of the algorithm. Though intuitively when N = 0, theexecution time should be smaller, this is not the case. We observe that the executiontime does not vary much when we set N = 0 and N = 2.

It is worth noting that a compromise between the numerical stability and the domainof convergence should be made. As shown in Fig. 5.4.1 where K is set to 5.5 × 103,








0 2 4 6 8

·104

0.5

0.6

0.7

0.8

0.9

K

Success rate

0 2 4 6 8

·104

0.2

0.4

0.6

0.8

1

1.2

1.4

K

Execution time: × 100 s

Figure 5.6: Rate of success with respect to K

the rate of success (for planar maneuvers) does not increase monotonically with respectto the number of node points, and the execution time does not change too much whenN = 0, · · · , 5. When N ≥ 6, the success rate decreases quickly and equals to zero whenN = 7. Actually, when there are too much node points, the number of unknowns for theshooting method becomes too large and the domain of convergence of a Newton-typemethod becomes extremely small, which nally leads to lower rate of success.

5.4.2 Rapid Attitude Maneuver of the LVs

We note that, when solving multi-burn ascent problems for LVs, it is possible to nd acontrol (Euler angles) that contains a jump between dierent stages (see for example[72, Fig. 3]). In this case, a rapid attitude maneuver has to be done such that the








0 2 4 6

0

0.2

0.4

0.6

0.8

N

Success rate

0 2 4 6

0.2

0.4

0.6

0.8

N

Execution time: × 100 s

Figure 5.7: Rate of success with respect to N

LV can follow the optimal trajectory of the next stage. For this reason, we apply thepresented algorithm as well to the maneuver problem of the upper stages of the LVs.

In contrast to the ALV's pull-up maneuver, these attitude maneuvers are in generalthree-dimensional and of lower magnitude. They occur at high altitudes (typicallyhigher than 50 km) since a suciently low dynamic pressure is required to ensure theseparation safety. In addition, compared to ALV, the velocity of in this case is in generalmuch larger.

In this case, the state constraints are not active due to the low dynamic pressure,and we are rather interested in obtaining the fastest possible maneuver. To this aim,we need to vary the parameter λ3 from 0 to 1, as shown in Fig. 5.1.

In the example, we set the system parameters in (5.0.2) to a = 20, b = 0.2, which


approximate an Ariane-like launcher. The initial conditions (5.1.2) are

rx0 = 100 km, ry0 = rz0 = 0, v0 = 5000m/s, θv0 = 30,

ψv0 = 0, θ0 = 40, ψ0 = φ0 = 0, ωx0 = ωy0 = 0,

and the nal conditions (5.1.3) are

θf = 60, ψf = 10, φf = 0, ωxf = ωyf = 0.

0 50 100 150

0

200

400

600

t

position, unit km

rxryrz

0 50 100 150−1,000

0

1,000

2,000

3,000

4,000

t

velocity, unit m/s

vxvyvz

0 50 100 150

0

50

100

150

t

Euler angles, unit degree

θψφ

0 50 100 150

−5

0

5

10

t

angular velocity, unit degree/s

ωxωy

Figure 5.8: Time history of state variable x(t) for a LV.

The multiple shooting method is applied with four node points. On Figs 5.8 and5.9, we report the components of state and control variables. We observe that, when t ∈[32, 145] s, the control is very small, and the state variable θ = 151.5 ≈ θ∗ = 151.57,ψ = 8.6 ≈ ψ∗ = 8.85 with θ∗ and ψ∗ calculated by (4.3.1). Indeed, the control in Fig.5.9 is continuous thanks to the regularization term K

∫ tf0‖u‖2dt in the cost functional.

It helps to generate a solution and to avoid chattering. In the presented result, wehave started with (1 − λ3)K = 8 × 104 (λ = 0) and stopped with (1 − λ3)K = 240(λ3 = 0.997) and the computing time is about 110 s. This maneuver time tf is about


0 50 100 150

−0.1

0

0.1

0.2

0.3

t

u1u2

Figure 5.9: Time history of control variable u(t) for a LV.

175 s, due to the fact that we require as well the direction of the trajectory velocityto change as much as the Euler angles. While for real LVs, only minor adjustmentof velocity direction is needed while doing a large attitude maneuver. Therefore, inpractice, the maneuver time is in general only several seconds. Our aim of presentingthis non-realistic case is indeed to show that the proposed algorithm is also robust toa large range of system congurations and quite extreme terminal conditions.

Conclusion

In this chapter, we have addressed the problem of minimum time-energy pull-up ma-neuver problem for ALVs. The dynamics couple attitude and trajectory motion. Weadapted the algorithm constructed for (MTCP) for solving (PUMP). The mainidea does not change: starting from the explicit solution of a simplied problem oflower dimension, the successive continuations consist of retrieving the true dynamicsand terminal conditions; with two continuation parameters, the terminal conditionsare successively retrieved; aerodynamic forces and variable gravity are then introducedwith a third continuation parameter. Finally, the state constraints are applied withother continuations. The multiple shooting method, the PC continuation method, thechange of reference frame and the smoothing of vector elds are employed to improvenumerical stability and robustness of the algorithm. An example of the pull-up maneu-ver for an ALV is given and statistical experiments show that our approach is fast androbust. In addition, the application for the rapid maneuver of an upper stage for a LVis exemplied numerically. Indeed, with this algorithm, the solution of similar classesof problems can be obtained from scratch in a quite short time, whatever the launcherdata and the terminal conditions are.

Conclusion

Contribution. In this work, we have studied the time minimum attitude reorienta-tion problem coupled with the trajectory dynamics for launch vehicles. The objectivewas to design an automatic software capable of computing the optimal solution fromscratch within short time, whatever the launcher data and the terminal conditions are.Diculties are twofold: 1) the system consists of slow dynamics (trajectory) combinedwith rapid dynamics (attitude), 2) chattering may occur, depending on the terminalconditions.

We have established the physical model of the coupled movement, and we haveformulated optimal control problems for planar maneuvers and three-dimensional ma-neuvers.

We have rst studied the planar maneuver problem (minimum time tilting prob-lem, (MTTP)) with a focus on the chattering phenomenon. We have investigated theextremals of (MTTP) near the singular surface, and proved that there are optimalchattering arcs when connecting a singular arc with regular arcs (in this case, bangarcs). This analysis is based on the Pontryagin Maximum Principle and on resultsby M.I. Zelikin and V.F. Borisov. Then, we have derived sucient conditions on theterminal values under which the optimal solutions do not contain any singular arc (andthus do not contain any chattering arcs), and are bang-bang with a nite number ofswitchings. In the cases where chattering occurs, we have provided sub-optimal strate-gies by replacing the chattering control with a piecewise constant control. Numericalsimulations have been carried out to illustrate our results.

Then, we have studied the three-dimensional maneuver problem (minimum timecontrol problem, (MTCP)). We have established a chattering result for bi-inputcontrol-ane systems by using geometric control theory, in particular the concept ofLie bracket, and we have applied this result to (MTCP). Meanwhile, we have clas-sied the switching points for the extremals of (MTCP), according to the order ofvanishing of the switching function, and we have described the qualitative behavior ofoptimal trajectories near the singularities. For the numerical resolution, we have de-signed a time-ecient indirect approach which is a combination of the shooting methodand of numerical continuation. For certain terminal conditions, the optimal solution of(MTCP) involves chattering arcs. For such cases, we have designed two sub-optimal

142

Chapter 5: Conclusion 143

strategies. One of them consists of stopping the last continuation (on the cost function)in the indirect approach before its failure: in such a way, we obtain a sub-optimal solu-tion, which proves to be relevant for practical use. The other one is a direct approachwhich generates a sub-optimal solution having a nite number of switchings. Notethat, in both methods, it is not required to know a priori the structure of the optimalsolution. The direct strategy is however much more time consuming than the indirectapproach.

Finally, we have applied our indirect approach to the minimum time-energy pull-upmaneuver problem for airborne launch vehicles by adding several more continuationsteps. In (PUMP), the position of the launcher, the aerodynamic forces, and thestate constraints are all considered, thus making the problem very close to real-life.Moreover, multiple shooting (instead of simple shooting), a more complex Predictor-Corrector continuation method, change of reference frame and smoothing of vector eldsare used to improve numerical stability and robustness of our approach. More precisely,starting from the explicit solution of a simplied problem of lower dimension, the suc-cessive continuations consist of retrieving the true dynamics and terminal conditions.With two continuation parameters, the terminal conditions are successively retrieved.Aerodynamic forces and variable gravity are then introduced with a third continuationparameter. Finally, the state constraints are recovered thanks to additional contin-uations. Specic examples and statistical numerical results illustrate eciency androbustness of our method.

Perspectives. Various aspects of this work raise issues that are open or interestingto study. We list some of them hereafter.

The rst aspect is related to the system property, i.e., the system consists of slowand fast dynamics. Indeed, this property suggests that the Tychonov theorem and thesingular perturbation method may be useful in our context. They were applied to thelaunch ascent guidance problem (see e.g., [21, 40, 47, 56]) to separate the dynamics intotwo time scales describing fast and slow dynamics respectively, and then to developa feedback guidance law (see e.g., [24, 34, 70, 82]). However, to use the Tychonovtheorem, the assumption of negative eigenvalue should be satised, which is not thecase in our context. However, by observing the numerical solutions, we suspect thatthe optimal solutions of our problem have a turnpike property as described in [107],at least when the required maneuver time is large enough. Recall that turnpike (resp.exact turnpike) means that the optimal trajectory stays close to (resp. stays exactlyat) a specic steady state during the major part of the time. In our problem, thesingular surface can be seen as the steady state", on which the middle part of theoptimal trajectories lie on when the maneuver time is large enough. Therefore, as anopen issue, it is interesting to be able to prove theoretically that the exact turnpike isa property of all optimal solutions with large maneuver times. This would greatly helpin obtaining the global optimal synthesis for our problem, and this would be a concretetheoretical support when designing sub-optimal strategies.

144

A second issue concerns chattering.(1) In the third chapter, we have used the approach designed by Zelikin and Borisov

to describe the phaseportrait of extremals near the singular surface for our single-inputcontrol ane system. We could try to extend the techniques used by Zelikin andBorisov (including semi-canonical systems, Poincaré mapping, and invariant manifoldtheorem) to establish the phaseportrait of optimal extremals near the singular surfacefor a general bi-input control ane system with the control taking values in the Eulideanunit disk.

Moreover, we think that writing the original system as a semi-canonical system maybe a possible way to establish the turnpike property of optimal trajectories, since sucha system reveals more directly the qualitative behavior of extremals near the singularsurface.

(2) At the end of the third chapter, a conjecture has been done on the convergence forthe designed direct approaches (suboptimal strategies, in which the chattering controlis approximated with a piecewise constant control, with a xed number of switchings).It would be interesting to establish rigorously this convergence, and to estimate andcompare the convergence rates of our suboptimal methods.

(3) When chattering occurs, it would also be interesting to quantify precisely the lossof optimality of such suboptimal strategies. Note again that the diculty of establishingsuch a result is in the presence of an optimal singular arc and the chattering arcs. Inaddition, we think that the results obtained in the third and fourth chapter could beapplied to other classes of problems, such as the minimum time control of underwatervehicles (see e.g. [31, 32]) and the optimal control of biological problems (see e.g. [69]).

A next concern refers to geometric control theory.(1) Following [14, 16, 65], we have classied regular extremals by their contact

with the switching surface for (MTCP) in the fourth chapter. In [14] and [16], theclassication is also done for specic problems. It would be interesting to study thegeometric classication of optimal trajectories for general bi-input control systems.

(2) As mentioned in the last chapter, using the maximum principle to treat theoptimal control problem with state constraints is very dicult, since the adjoint vectorbecomes a Radon measure. In the case of single-input control ane systems, the cal-culation of the boundary control (control during the boundary arc) and of the adjointvector with jumps, and the classication of optimal trajectories by their contact withthe boundary arc (on which the state constraint equals to zero) have been studied (seee.g. [13, 16, 18]). However, when dealing with a multi-input control system, even howto calculate the boundary control is not very clear. Thus, deriving similar results formulti-input control systems is an open issue.

(3) We have studied only necessary conditions for (MTCP), and thus the optimal-ity of the obtained solutions remains to be proved, though the numerical results areconsistent (direct and indirect approaches). For treating this problem, the conjugatepoint theory available for the continuous and discontinuous problems (see e.g. [3, 15])

Chapter 5: Conclusion 145

could be applied.

Finally, as suggested in the last chapter, systematic tests of our method remain to bedone, in order to obtain the best possible combination of the regularization parameterand number of node points. The choice of the new reference frame also remains to beinvestigated.

List of Figures

1.1 Possible zero paths (left) and impossible zero paths (right). . . . . . . . 28

2.1 Coordinate systems and relations. . . . . . . . . . . . . . . . . . . . . . 342.2 Thrust in the body Frame . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 An illustration of chattering phenomenon. . . . . . . . . . . . . . . . . 432.4 Optimal synthesis for the Fuller's problem. . . . . . . . . . . . . . . . . 44

3.1 Phase portrait of optimal extremals near the singular surface. . . . . . 563.2 (a) Illustration of sucient optimality condition; (b)-(c) Optimal syn-

thesis of the Fuller problem. . . . . . . . . . . . . . . . . . . . . . . . . 583.3 Illustration of Proposition 3.1.2. . . . . . . . . . . . . . . . . . . . . . . 583.4 Illustration of M∗

1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.5 Time history of x3, x4 and u when v0 = vup and v0 = 1080m/s . . . . . 793.6 Time history of x3, x4 and u when v0 = vdown and v0 = 1690m/s . . . . 793.7 Example of trajectory associated with optimal control of 6 switchings. . 803.8 Switching function ϕ(t) when v0 = 1999.3m/s in the anticlockwise case. 813.9 Switching function ϕ(t) when v0 = 2132.1m/s in the anticlockwise case. 823.10 Time history of x3, x4 and u when v0 = 1350m/s and v0 = 1683m/s . 823.11 Time history of x3, x4 and u when c = 0, c = 10−6 and v0 = vup . . . . 833.12 Time history of x3, x4 and u when c = 0 and c = 10−6 . . . . . . . . . . 833.13 Control u(t) in anticlockwise maneuver . . . . . . . . . . . . . . . . . . 843.14 State variable x(t) in anticlockwise maneuver. . . . . . . . . . . . . . . 853.15 Optimal control in clockwise maneuver. . . . . . . . . . . . . . . . . . . 853.16 State x(t) in clockwise maneuver. . . . . . . . . . . . . . . . . . . . . . 863.17 Control u(t) with dierent discretization step N . . . . . . . . . . . . . . 873.18 Maneuver time tf with respect to the discretization step N . . . . . . . 87

4.1 Continuation procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . 1094.2 State variable and the optimal control with (TC1) and v0 = 1000m/s. . 1134.3 Adjoint variable and the switching function with (TC1) and v0 = 1000m/s.1144.4 State with (TC1) and v0 = 1500m/s. . . . . . . . . . . . . . . . . . . . 1144.5 Control and switching function with (TC1) and v0 = 1500m/s. . . . . . 1154.6 State with TC3 and v0 = 1500m/s. . . . . . . . . . . . . . . . . . . . . 115

146

List of Figures 147

4.7 Control and switching function with TC3 and v0 = 1500m/s. . . . . . . 1164.8 State with (TC2) and v0 = 2000m/s. . . . . . . . . . . . . . . . . . . . 1164.9 Control and switching function with (TC2) and v0 = 2000m/s. . . . . . 1174.10 State variable x(t) with (TC2) and v0 = 2000m/s obtained by BOCOP. 1184.11 Control u(t) with (TC2) and v0 = 2000m/s obtained by BOCOP. . . . 1184.12 Adjoint vector p(t) with (TC2) and v0 = 2000m/s obtained by BOCOP. 119

5.1 Continuation procedure. . . . . . . . . . . . . . . . . . . . . . . . . . . 1295.2 Time history of state variable x(t) for an ALV. . . . . . . . . . . . . . . 1345.3 Time history of control variable u(t) for an ALV. . . . . . . . . . . . . 1355.4 Time history of the adjoint variable p(t) for an ALV. . . . . . . . . . . 1365.5 Time history of the constraints c(x(t)) for an ALV. . . . . . . . . . . . 1375.6 Rate of success with respect to K . . . . . . . . . . . . . . . . . . . . . 1385.7 Rate of success with respect to N . . . . . . . . . . . . . . . . . . . . . 1395.8 Time history of state variable x(t) for a LV. . . . . . . . . . . . . . . . 1405.9 Time history of control variable u(t) for a LV. . . . . . . . . . . . . . . 141

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Abstract

In this thesis, we investigate the minimum time control problem for the controland guidance of a launch vehicle, whose motion is described by its attitude kinemat-ics and dynamics but also by its trajectory dynamics. The diculty of this problemis essentially due to the chattering phenomenon and to the coupling of dynamics ofdierent time scales. With a rened geometric study of the extremals coming fromthe application of the Pontryagin Maximum Principle, we establish a general result forbi-input control-ane systems, providing sucient conditions under which the chat-tering phenomenon occurs. We show how this result can be applied to our problem.Based on this preliminary theoretical analysis, we implement an ecient indirect nu-merical method, combined with numerical Predictor-Corrector continuation, in orderto compute numerically the optimal solutions of the problem. In case of chattering,two sub-optimal strategies are designed: one is a direct method in which the controlis approximated by a piecewise constant control, and the other consists of stoppingthe continuation procedure before its failure due to chattering. With several additionalnumerical continuation steps, we apply nally the developed indirect approach to theminimum time-energy pull-up maneuver problem, in which state constraints are alsoconsidered, for airborne launchers. Numerical simulations illustrate the eciency androbustness of our method.

Key words. minimum time, optimal control, coupled problem, attitude, trajec-tory, Pontryagin maximum principle, singular control, chattering arcs, single shooting,multiple shooting, numerical continuation, Predictor-Corrector method, airborne, stateconstraint.

AMS subject classication. 49K15, 49M05, 49M37, 65H20

Résumé

Cette thèse porte sur un problème couplé des lanceurs, à savoir une man÷uvre del'attitude couplée avec la trajectoire minimisant le temps de man÷uvre. La dicultéde ce problème vient essentiellement du phénomène de chattering et du couplage desdynamiques n'ayant pas la même échelle de temps. Avec une analyse géométrique desextrémales venant de l'application du Principe du Maximum de Pontryagin, nous don-nons des conditions susantes sous lesquelles le phénomène de chattering se produit,pour des systèmes anes bi-entrée. Nons appliquons ensuite ce résultat à notre prob-lème, et montrons que le phénomène de chattering arrive pour les trajectoires optimales,pour certaines données terminales. A l'aide de cette analyse théorique préliminaire,nous mettons en ÷uvre une méthode de résolution indirecte ecace, combinée à uneméthode de continuation Prédicteur-Correcteur. En cas de chattering, deux stratégiessous-optimales sont proposées: soit une méthode directe dont le contrôle est approchépar un contrôle constant par morceaux, soit en stoppant la continuation avant l'échecdû au chattering. Avec le tir multiple et plusieurs paramètres de continuations sup-plémentaires, cette méthode de résolution est appliquée à chercher une man÷uvre depull-up avec des contraintes sur l'état en minimisant le temps-énergie pour des lanceursaéroportés. Les résultats numériques permettent de mettre en évidence l'ecacité et larobustesse de notre méthode de résolution.

Key words. temps minimal, contôle optimal, problème couplé, attitude, trajec-toire, Principe du maximum de Pontryagin, contrôle singulier, arc de chattering, tirsimple, tir multiple, continuation numérique, méthode Prédicteur-Correcteur, lanceuraéroporté, contrainte sur l'état.

Classication. 49K15, 49M05, 49M37, 65H20

Documents

Contrôle optimal de l'attitude d'un lanceur