ÉCOLE POLYTECHNIQUEDÉPARTEMENT DE MATHÉMATIQUES APPLIQUÉES

Master M2 Mathématiques de la modélisationAnalyse numérique et EDP

Rapport de stage de recherche

La condition d’excitation persistante dansles systèmes avec retard et dans une

équation de transport avec amortissement

Élève : Guilherme MAZANTITuteurs : Yacine CHITOUR

Mario SIGALOTTIPériode de stage : 01/03/2013 à 31/08/2013Lieu de stage : Inria Saclay - Équipe GECO

Bâtiment Alan Turing1 rue Honoré d’Estienne d’Orves91120 PALAISEAU

AbstractWe consider in this document two problems involving the condition of persistence of excitation. The

first one deals with the stabilization to the origin of a persistently excited linear system by means of alinear state feedback, where we suppose that the feedback law is not applied instantaneously, but after acertain positive delay (not necessarily constant). The main result is that, under certain spectral hypotheseson the linear system, stabilization by means of a linear delayed feedback is indeed possible, generalizing aprevious result already known for non-delayed feedback laws. The second problem consists of a transportequation defined in two tangent circles with a certain transmission condition at the intersection point ofthe circles and with a persistently excited damping term in a part of one of the circles. The motivationfor this problem is the study of the wave equation on networks of strings. We present here a study of thestability of the undamped system and of the system with an always active damping, not submitted to acondition of persistent excitation, before turning to some preliminary results on the persistently exciteddamped case.

RésuméCe document présente deux problèmes faisant intervenir la condition d’excitation persistante. Le pre-

mier d’entre eux concerne la stabilisation à l’origine d’un système linéaire à excitation persistante par unretour d’état linéaire, où l’on suppose que le retour d’état n’est pas appliqué instantanément, mais aprèsun certain retard positif (pas forcément constant). Le résultat principal est que, sous certaines hypothèsesspectrales sur le système linéaire, la stabilisation par retour d’état avec retard est possible, généralisant unrésultat précédent établi pour les retours d’état sans retard. Le deuxième problème concerne l’équationdu transport posée sur deux cercles tangents avec une certaine condition de transmission dans le pointd’intersection des cercles et avec un terme d’amortissement à excitation persistante dans une partie d’undes cercles. La motivation pour ce problème vient de l’étude de l’équation d’onde dans des réseauxde cordes. Nous présentons ici une étude de la stabilité du système non-amorti et du système avec unamortissement toujours actif, sans l’hypothèse d’excitation persistante, avant de passer à des résultatspréliminaires concernant le système amorti à excitation persistante.

ResumoEste documento apresenta dois problemas envolvendo a condição de excitação persistente. O pri-

meiro considera a estabilização à origem de um sistema linear à excitação persistente através de umretorno de estado linear, em que se supõe que o retorno de estado não é aplicado instantaneamente, masapenas após um certo atraso positivo (não necessariamente constante). O resultado principal é que, sobcertas hipóteses espectrais do sistema linear, a estabilização por um retorno de estado linear atrasado épossível, generalisando um resultado anterior sobre retornos de estado sem atraso. O segundo problemaconsiste em uma equação de transporte definida em duas circunferências tangentes com uma certa con-dição de transmissão no ponto de interseção das circunferências e com um amortecimento à excitaçãopersistente em uma parte de uma das circunferências. A motivação para este problema vem do estudoda equação de onda em redes de cordas. Apresentamos aqui um estudo da estabilidade do sistema não-amortecido e do sistema com amortecimento sempre ativo, sem a condição de excitação persistente, antesde apresentarmos alguns resultados preliminares no caso de amortecimento à excitação persistente.

i

ii

Contents

1 Introduction 11.1 Hybrid systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Switched systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.3 Persistently excited systems . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.4 Problems studied in this project . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Stabilization of persistently excited linear systems by delayed feedback laws 92.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92.2 Notations, definitions and previous results . . . . . . . . . . . . . . . . . . . . 102.3 The d-integrator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.4 Main result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152.5 Further discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.A Appendix: A continuity result for delayed systems . . . . . . . . . . . . . . . 212.B Appendix: On the proof of Theorem 2.5 . . . . . . . . . . . . . . . . . . . . . 24

3 Transport equation on circles 313.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 313.2 Notations and definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323.3 From the wave equation on a segment to the transport equation on a circle . . . 333.4 The undamped transport equation on circles . . . . . . . . . . . . . . . . . . . 36

3.4.1 Well-posedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373.4.2 Asymptotic behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . 403.4.3 Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 433.4.4 Uniformity of the convergence . . . . . . . . . . . . . . . . . . . . . . 443.4.5 Explicit solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

3.5 The damped transport equation with an always active damping . . . . . . . . . 523.5.1 Explicit solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533.5.2 Uniform exponential decay of the coefficients . . . . . . . . . . . . . . 563.5.3 Exponential convergence of the solutions . . . . . . . . . . . . . . . . 63

3.6 Developments on the persistently excited damped case . . . . . . . . . . . . . 653.6.1 Explicit solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 663.6.2 The case of rationally dependent lengths L1 and L2 . . . . . . . . . . . 703.6.3 Persistently exciting signals and the flow of the transport equation . . . 723.6.4 Perspectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

3.A Appendix: Lyapunov functions in Banach spaces . . . . . . . . . . . . . . . . 773.B Appendix: Well-posedness of a class of time-dependent differential equations

in Banach spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

iii

References 80

iv

Chapter 1

Introduction

1.1 Hybrid systems

Hybrid systems are systems whose behavior is determined by the interaction between both con-tinuous and discrete dynamics [6, 7, 22, 34, 35, 41, 53]. This is the case, for instance, of anautomatic control of the temperature of a room, where the continuous dynamics of the temper-ature depends on and influences the on/off state of the heating system. An automotive internalcombustion engine is another example of a hybrid system [10], with four discrete states, eachone corresponding to one cycle of the engine, and continuous variables such as the temperatureand the pressure, whose dynamics depend on the cycle of the engine at a given time and alsoprovoke the switches between the cycles.

The study of hybrid systems has attracted much research effort recently due to its appli-cations in the control of mechanical systems, industrial processes, the automotive industry,electrical power systems, air traffic control, chemical processes, transport systems, among sev-eral other applications involving fields such as control engineering, mathematics and computerscience [6, 7, 35, 41]. Even though hybrid models may appear naturally in some contexts, theirstudy is in general intricate due to the interactions between the discrete and the continuousmodels.

1.2 Switched systems

In several applications of hybrid systems, the main goal is to study the continuous dynamics andtheir properties of stability and stabilization, and the discrete dynamics play only a secondaryrole, being seen as modes or subsystems which determine the continuous dynamics. Thus,instead of studying the details of the discrete dynamics, one may more simply consider thatthe system is determined by several continuous dynamics and that a certain switching logicor switching signal determines which of these dynamics is active at each time. This is theframework of the switched systems [34, 41], where we focus not on the specific evolution ofthe discrete variable in time, but only on the effects of this evolution on the dynamics of thecontinuous variable, and we typically consider a family of possible switching signals in orderto obtain robust properties with respect to signals in this family.

Mathematically, a switched control system in Rd can be described by a family of applica-tions fk : Rd×Rm→ Rd , k ∈ I, where I is a certain set of indices, and by a piecewise constant

1

function α : R+→ I, the switching signal, as

x(t) = fα(t)(x(t),u(t)), t ∈ R+. (1.1)

The continuous state x is a vector of Rd or, more generally, belongs to a manifold M or a Banachspace X, and u is the control of the system. The switching signal α determines, at each time,which of the dynamics fk is active. In general, α is not completely known in advance and weonly dispose of partial information on α . The interest is thus to study (1.1) not for a fixed α butfor a whole class G of signals α in order to obtain robust properties of the system with respectto G.

In general, one may model the signal α in several ways, each model being more adaptedto a certain problem and to a certain kind of analysis. We may consider, for instance, that α

depends only on t, but also possibly on the current state x(t) or on the past values α(τ) orx(τ) for τ < t. In some situations, α can also be a project parameter, to be chosen in order toachieve a prescribed behavior for the continuous variable x, whereas in other cases α comesfrom a natural condition from the system and cannot be changed. One may also consider α tobe deterministic or probabilistic. In this document, we shall consider only deterministic signalsα depending only on the current time t and belonging to a given class G.

An important particular case of (1.1) is when all the functions fk are linear and given byfk(x) =Akx+Bku for certain matrices A∈Md(R) and B∈Md,m(R). In this case, (1.1) becomes

x(t) = Aα(t)x(t)+Bα(t)u(t), t ∈ R+. (1.2)

Even though (1.2) is only a particular case of (1.1), we highlight this case here due both toits importance and to the difficulties that may arise from switched systems even in the linearcase. Much research effort on switched systems has been dedicated to the linear case, studyingsubjects such as controllability, observability, stability and stabilizability [8, 16, 17, 19, 24, 35,43, 49, 53].

The stability analysis of a switched system is not a trivial problem, since the switching signalmay introduce in a system a behavior which does not occur on its subsystems when studiedseparately. In particular, a switched system whose subsystems are all stable may actually beunstable, as one may see in the following example, adapted from [24].

Example 1.1. Let us consider the linear switched system

x(t) = Aα(t)x(t) (1.3)

with x(t) ∈ R2, α(t) ∈ I= 1,2 and

A1 =

(−1 −91 −1

), A2 =

(−1 1−9 −1

).

The matrices A1 and A2 are both Hurwitz and have the same eigenvalues λ1,2 = −1± 3i.Let us consider the piecewise constant switching signal α , right-continuous, switching from1 to 2 when x2(t) = 0 and the current dynamics are α(t−) = 1, and switching from 2 to 1when x1(t) = 0 and the current dynamics are α(t−) = 2; that is,

α(t) =

2 if α(t−) = 1 and x2(t) = 0,1 if α(t−) = 2 and x1(t) = 0,α(t−) otherwise.

2

We denote respectively by Φ1(t) and Φ2(t) the fundamental matrices of the linear systemsx = A1x and x = A2x; these matrices are

Φ1(t) = e−t(

cos3t −3sin3t13 sin3t cos3t

), Φ2(t) = e−t

(cos3t 1

3 sin3t−3sin3t cos3t

).

We claim that every non-zero solution of this system tends exponentially to the infinityas t → +∞. We first consider the trajectory corresponding to the initial condition x(0) =(0 1

)T and such that α(0) = 1. Since α is piecewise constant, one can find an interval[0, t1) such that α(t) = 1 for t ∈ [0, t1), and thus, in this interval, the solution is

x(t) = e−t(−3sin3t

cos3t

).

The solution thus turns in the counterclockwise sense around the origin. By the definitionof α , the system switches to α(t) = 2 when this solution attains the x1 axis, that is, at timet = π/6. Hence α(t) = 1 for t ∈ [0,π/6) and α(π/6) = 2. A straightforward computationshows that

‖x(t)‖ ≥ e−π/6 for t ∈ [0,π/6) (1.4)

and

x(π/6) =

(−3e−π/6

0

).

Similarly, α is constant and equal to 2 on a certain interval starting from π/6, and, in thisinterval, the solution is

x(t) =−3e−t(

cos(3(t− π/6))3sin(3(t− π/6))

),

so that the solution turns in the clockwise sense around the origin. By the definition ofα , the system remains in α(t) = 2 until the solution reaches the x2 axis, which happens att = π/3, when the system switches back to α(t) = 1. Hence α(t) = 2 for t ∈ [π/6,π/3) andα(π/3) = 1. Again, a straightforward computation from the explicit formula of the solutiongives

‖x(t)‖ ≥ 3e−π/3 for t ∈ [π/6,π/3) (1.5)

and we have

x(π/3) =

(0

9e−π/3

).

The solution thus returns to the axis x2 after a time t = π/3 with α(t) = 1, and its forwardbehavior can be deduced from the previous study by homogeneity. In particular, by (1.4)and (1.5), we obtain

‖x(t)‖ ≥ 3ne−(n+1) π

6 for t ∈[nπ

6 ,(n+1)π

6

), n ∈ N,

and, since 3e−π/6 > 1, we conclude that ‖x(t)‖ t→+∞−−−−→+∞ exponentially. We represent thissolution in Figure 1.1.

This particular solution allows us to determine the behavior of all the other solutionsof the system. Indeed, by homogeneity, we obtain that ‖x(t)‖ t→+∞−−−−→+∞ exponentially forevery solution x(t) of (1.3) with initial condition x(0) 6= 0 in the x2 axis and with α(0) = 1.

3

−7 −6 −5 −4 −3 −2 −1 00

1

2

3

4

5

6

7

8

9

10Solution x(t)

x1

x 2

FIGURE 1.1: Behavior of the solution x(t) of the switched system (1.3) with x(0) = (0,1)T andα(0) = 1.

Similarly, if the initial condition x(0) 6= 0 is in the x1 axis and α(0) = 2, then the corre-sponding solution x(t) coincides with x(t+π/6) for a certain solution x with initial conditionx(0) 6= 0 in the x2 axis and α(0) = 1, and thus ‖x(t)‖ t→+∞−−−−→+∞ exponentially. If finally x(t)is a solution with initial condition lying outside the axes and with a certain value of α(0),then x rotates around the origin, counterclockwise if α(0) = 1 and clockwise if α(0) = 2,until reaching one of the axes, from where its behavior coincides with a trajectory previ-ously described, and so ‖x(t)‖ t→+∞−−−−→ +∞ exponentially. Our previous analysis also holdswhen the non-zero initial condition x(0) is in the x1 axis and α(0) = 1 or when x(0) is inthe x2 axis and α(0) = 2.

Hence every non-zero solution of the switched system (1.3) diverges exponentially ast→+∞, even though each subsystem x = A1x and x = A2x is exponentially stable.

It is important to note that Example 1.1 is not a purely academic example: [24] remarks thatsuch a phenomenon of switching between two asymptotically stable systems may appear in thecontrol of the longitudinal dynamics of an aircraft with restricted attack angle, which highlightsthe practical importance of the study of switched systems.

1.3 Persistently excited systemsA particular case of switching phenomenon in control systems is when the switching influencesonly the action of the control on the system, which is the case for instance when the controllermight be disconnected from the system at certain unknown times. Mathematically, this can bedescribed by fixing a certain application f : Rd×Rm→ Rd and writing (1.1) as

x(t) = f (x(t),α(t)u(t)), t ∈ R+, (1.6)

where α(t) takes its values in 0,1. In particular, (1.6) can be written in the linear case as

x(t) = Ax(t)+α(t)Bu(t), (1.7)

4

where we consider α(t) ∈ 0,1, or, more generally, the convexified case α(t) ∈ [0,1]. System(1.7) corresponds to a modification of the autonomous linear control system

x(t) = Ax(t)+Bu(t)

where the control u(t) is not always active and, when α takes its values on 0,1, the systemactually switches between the controlled and the non-controlled dynamics, x = Ax+Bu andx = Ax.

The control problem for (1.6) consists on designing a robust control which should not beaffected by the incertitudes on α . We thus consider that α belongs to a certain class G ⊂L∞(R+, [0,1]) which contains all the known information on α , and in particular the informationconcerning the instants where α may or must be active, and our aim is to obtain a robust controlstrategy with respect to α ∈ G.

The problem of controlling (1.6) by a suitable choice of u is obviously not interesting whenα ≡ 0, or when α is zero for a large amount of time, since in this case the control u hasa very limited effect on (1.6). The class G should thus ensure that the control input has asufficient amount of action on the system. Among the possible choices for G, the class of(T,µ)-persistently exciting signals (or simply (T,µ)-PE signals) has attracted much interestrecently (see, for instance, [15, 16, 18, 19, 31, 37, 43, 45], and also [38] for a similar condition)and, for T ≥ µ > 0, it consists on the signals α ∈ L∞(R+, [0,1]) such that, for every t ∈ R+,

w t+T

tα(s)ds≥ µ. (1.8)

The class of these signals α is noted G(T,µ). This condition of persistence of excitation (alsocalled PE condition) guarantees that, on every time window of length T , the control u acts onthe system, and we also have a certain measure on the action of u on the system by the boundµ . We say that (1.6) with the condition α ∈ G(T,µ) is a persistently excited system or simply aPE system.

Several different phenomena may be modeled by signal α in (1.6), such as a failure in thetransmission of the control u to the plant, leading to an intermittent action of u; a time-varyingparameter affecting the control efficiency, leading to the effective application of a rescaled con-trol α(t)u(t); the allocation of control resources, activating the control only up to a certainfraction of its designed value or only on certain time intervals; among other possible phenom-ena. A more concrete example presented in [37, 38] is the control of spacecrafts with magneticactuators, described by the system

ω = S(ω)ω +g(t)u

with ω ∈R3 the state variable representing the orientation of the spacecraft, u the control, S(ω)a matrix depending on ω and g(t) a time-dependent matrix with rank(g(t)) < 3 for every timet and satisfying a persistent excitation condition similar to (1.8).

The condition of persistence of excitation (1.8) arises naturally in identification and adaptivecontrol problems (see, e.g., [3–5,14,40]). In this context, we are led to study systems of the kindx =−P(t)x, x ∈ Rd , where P(t) is a symmetric non-negative definite matrix for every t. If P isbounded and has bounded derivative, it has been shown in [45] that the persistence of excitationof P, in the sense that α(t) = ξ TP(t)ξ is (T,µ)-persistently exciting for all unitary vectors ξ ∈Rd and for certain constants T ≥ µ > 0 independent of ξ , is a necessary and sufficient conditionfor the global exponential stability of x = −P(t)x. This is what motivates the assumption thatα is persistently exciting in (1.6). Further examples of systems similar to (1.6) where thepersistent excitation condition appears are given in [16,18,37], where the motivation for the useof persistently exciting signals is also more deeply discussed.

5

1.4 Problems studied in this project

Even though much research effort has been spent recently in the study of systems of the form(1.7) submitted to persistent excitation conditions [15, 16, 19, 37, 43], several important ques-tions remain unanswered, the general phenomena related to the presence of the signal α nothaving been yet completely understood. Our goal in this project is to further develop the the-ory of persistently excited systems in two directions: in Chapter 2, we consider the problemof stabilization of a persistently excited system by means of a delayed linear feedback law,and, in Chapter 3, we study the stability of a transport equation on two circles submitted to apersistently excited damping in a part of one of the circles.

Chapter 2 corresponds almost entirely to an article recently submitted [42]. It deals with theproblem of stabilization of (1.7) for α ∈ G(T,µ) by means of a delayed linear state feedbacku(t) =−Kx(t−τ(t)), where τ(t)≥ 0 is a time-varying bounded time-delay. Our stabilizabilityproblem is to find, for given matrices A and B, for constants T ≥ µ > 0 and for a certain classof time-dependent delays L∞(R+,T), a feedback matrix K such that the closed-loop system

x(t) = Ax(t)−α(t)BKx(t− τ(t)),α ∈ G(T,µ), τ ∈ L∞(R+,T)

is exponentially stable, uniformly with respect to α and τ . We show in Chapter 2 that this ispossible under certain hypotheses on A, B and T. This generalizes [19, Theorem 3.2], where thesame result is given in the case of the non-delayed feedback u(t) =−Kx(t), corresponding thusto the particular case T = 0.

Chapter 3 considers a transport equation defined on a pair of tangent circles, with a certaintransmission coefficient in the intersection point between the circles and with a persistentlyexcited damping in a part of one of the circles. More precisely, we are interested in the problem

∂tu1(t,x)+∂xu1(t,x) = 0, t ∈ R+, x ∈ [0,L1],

∂tu2(t,x)+∂xu2(t,x)+α(t)χ(x)u2(t,x) = 0, t ∈ R+, x ∈ [0,L2],

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, t ∈ R+,

u j(0,x) = u j,0(x), x ∈ [0,L j], j ∈ 1,2,α ∈ G(T,µ),

(1.9)

where L1,L2 > 0 and T ≥ µ > 0. This system can be seen as the transport equation with unitaryvelocity on two circles, C1 and C2, as in Figure 1.2, of respective lengths L1 and L2, whichintersect in one point, chosen to be the origin of the measure of length along the circles andwhere we have a transmission condition stating that the arriving mass is split in equal partsgoing on each circle. The equation in circle C2 is damped at the support of a certain functionχ , which we take here as the characteristic function of a certain subinterval of [0,L2], and thisdamping is submitted to a persistently exciting signal α .

This “toy model” of transport equations on circles is actually a preliminary study aiming atthe study of the wave equation on networks of strings, a research subject much studied recently[23,58,60], where we use the D’Alembert decomposition of the solutions of the wave equationinto traveling waves [50, 51] in order to obtain a correspondence between the solutions of thewave equation on a segment and the transport equation on a circle. Indeed, consider the wave

6

0 0L1 L2C1 C2

FIGURE 1.2: Representation of (1.9) as a transport equation on two tangent circles of lengths L1 andL2. The arrows indicate the sense of the transport along the corresponding circle, and the highlighted

interval in C2 represents the support of the damping term χ .

equation on a string of length L with Dirichlet boundary conditions,∂

2tt u(t,x) = ∂

2xxu(t,x), t ∈ R+, x ∈ [0,L],

u(0,x) = u0(x), x ∈ [0,L],∂tu(0,x) = u1(x), x ∈ [0,L],

u(t,x) = 0, t ∈ R+, x ∈ 0,L.

(1.10)

It is a classical result, from the works of D’Alembert on the wave equation [50, 51], that thesolution of (1.10) can be decomposed in two traveling waves, solutions of the two transportequations on [0,L],

∂t f (t,x)+∂x f (t,x) = 0, t ∈ R+, x ∈ [0,L],

f (0,x) =12

[u0(x)−

w x

0u1(ξ )dξ

], x ∈ [0,L],

f (t,0) =−g(t,0), t ∈ R+,

(1.11a)

∂tg(t,x)−∂xg(t,x) = 0, t ∈ R+, x ∈ [0,L],

g(0,x) =12

[u0(x)+

w x

0u1(ξ )dξ

], x ∈ [0,L],

g(t,L) =− f (t,L), t ∈ R+.

(1.11b)

If f and g are (classical regular) solutions of (1.11), it is easy to verify that u = f + g is asolution of (1.10), and, conversely, a (classical regular) solution u of (1.10) can be decomposedas u = f +g with f and g solutions of (1.11). We can now define v on R+× [0,2L] as

v(t,x) =

f (t,x), if x ∈ [0,L],−g(t,2L− x), if x ∈ (L,2L].

Then v satisfies

∂tv(t,x)+∂xv(t,x) = 0, t ∈ R+, x ∈ [0,2L],

v(0,x) =12

[u0(x)−

w x

0u1(ξ )dξ

], x ∈ [0,L],

v(0,x) =−12

[u0(2L− x)+

w 2L−x

0u1(ξ )dξ

], x ∈ (L,2L],

v(t,0) = v(t,2L), t ∈ R+,

7

which is a transport equation on [0,2L] that can be seen as a transport equation on a circle thanksto the periodicity condition v(t,0)= v(t,2L). The solution u of the original wave equation (1.10)can be obtained from v by u(t,x) = v(t,x)− v(t,2L− x). This reduction of the wave equationon a segment to the transport equation on a circle is more precisely justified in Section 3.3, andis our key motivation in the study of (1.9).

The main interest when considering the problem of stability of a persistently excited system,submitted to a persistently exciting signal α , is to give hypotheses under which one may retrieve,for the PE system, the same stability properties of the non-PE system, i.e., the system withα ≡ 1. It is thus important to understand the behavior of the non-PE system in order to knowwhich properties to expect from the corresponding PE system. For this reason, we study, inChapter 3, equation (1.9) in the undamped case, corresponding to χ ≡ 0, and in the case ofan always active damping, corresponding to α ≡ 1. This preliminary study highlights the maincharacteristics of the behavior of (1.9) without the PE condition on the damping, which we shalluse as a guide in the study of the corresponding PE system. We could not go as far as proving astability result for (1.9), but we present in Chapter 3 several advances in the study of (1.9) withthe PE condition, which we hope will ultimately lead to a stability result for (1.9).

8

Chapter 2

Stabilization of persistently excited linearsystems by delayed feedback laws

2.1 IntroductionConsider a control system of the form

x(t) = Ax(t)+α(t)Bu(t), x(t) ∈ Rd, u(t) ∈ Rm, α ∈ G(T,µ), (2.1)

where x is the state variable, u is a control input, A and B are matrices of appropriate dimensions,and α belongs to the class G(T,µ) of (T,µ)-persistently exciting signals, which, for givenT ≥ µ > 0, consists on the signals α ∈ L∞(R+, [0,1]) satisfying

w t+T

tα(s)ds≥ µ (2.2)

for every t ≥ 0. System (2.1) corresponds to the introduction on the linear control systemx = Ax+Bu of a certain signal α that determines when and how much the control u is active.Note that, when α takes its values on 0,1, (2.1) is actually a switched system between thedynamics of the uncontrolled system x = Ax and the controlled one x = Ax+Bu.

We consider the problem of stabilization of system (2.1) to the origin by means of a linearstate feedback u = −Kx, where we require the choice of the gain matrix K not to depend on aparticular signal α but instead on the class G(T,µ). In many practical situations, this feedbackcannot be done instantaneously, for a certain state x(t) may not be available for measure beforea certain delay τ has elapsed, and so the state measured in time t is actually x(t− τ(t)). Thischapter considers the problem of stabilization of (2.1) by a delayed feedback u(t) = −Kx(t−τ(t)), where the delay τ(t) may depend on t, and the closed-loop system becomes

x(t) = Ax(t)−α(t)BKx(t− τ(t)),α ∈ G(T,µ), τ ∈ L∞(R+,T)

(2.3)

where T ⊂R+ is the set where the delay τ takes its values. The goal of this chapter is to presenta stabilization result for system (2.3), showing that, under certain hypotheses on A and B, givenT ≥ µ > 0 and τ0 ≥ 0, there exist a neighborhood T of τ0 in R+ and K ∈Mm,d(R) such that,for any α ∈ G(T,µ) and any delay function τ ∈ L∞(R+,T), system (2.3) is exponentially stable,uniformly with respect to α and τ . This generalizes [19, Theorem 3.2], where the same resultis given in the case of the non-delayed feedback u(t) =−Kx(t), corresponding thus to T = 0.

9

Let us comment briefly on the technique used in [19] to prove this result in the non-delayedcase. The main problem when dealing with the class G(T,µ) is that a signal α ∈ G(T,µ)may be zero on certain time intervals, and so the system follows its uncontrolled dynamicsx = Ax. On the other hand, for every ρ > 0, it is known by a result from [27] that one canchoose a linear feedback u(t) = −Kx(t) that stabilizes (2.1) uniformly with respect to α ∈L∞(R+, [ρ,1]). The main idea in [19] is to perform a change of variables corresponding to a timecontraction by a factor ν > 0, which transforms a (T,µ)-signal α into a (T/ν ,µ/ν)-signal αν

with αν(t) = α(νt). It is possible to show that the family (αν)ν>0 admits a weak-? convergentsubsequence (ανn)n∈N∗ in L∞(R+, [0,1]) with νn→+∞ and that any weak-? subsequential limitα? of (αν)ν>0 as ν → +∞ satisfies α?(t) ≥ µ/T almost everywhere. The idea is thus to studya certain limit system obtained as ν → +∞, for which stabilization can be obtained using theresult from [27] mentioned above. It can then be shown by a limit procedure that the samefeedback gain K also stabilizes a time-contracted system for a certain ν > 0 large enough, andone may finally adapt such a feedback gain K in order to obtain a stabilizer for the originalsystem.

This time-contraction technique used in [19] is well-adapted to deal with delays in the feed-back, since a delay τ(t) in the original system will correspond to a delay τ(νt)

νin the time-

contracted system. We may thus expect to obtain a non-delayed limit system as ν → +∞ sim-ilar to the one obtained in [19] and to conclude the stabilizability of the original system by asimilar argument. This intuition is actually true, as proved in Theorem 2.5 below, where weprove our stabilizability result by following the same time-contraction argument of the proofof [19, Theorem 3.2].

In their article [19], the authors first prove their stabilization result in the particular casewhere the dynamics are given by the Jordan block Jd (see (2.7) below), since it is a representa-tive example containing most of the difficulties of the proof of the general case. We also treatthe case of the Jordan block separately in this article (see Theorem 2.6), but in this particularcase we have a stronger result, showing that stabilizability is possible for any bounded intervalT ⊂ R+ where the delay τ ∈ L∞(R+,T) may take its values, whereas in the general case wemay only guarantee stabilizability for delays τ which are perturbations around a certain con-stant prescribed value τ0. This difference between the statements of our result in the generalcase and in the particular case of the Jordan block is more deeply discussed in Section 2.5.

The plan of the chapter is the following. In Section 2.2, we present the notations and def-initions used throughout this chapter and recall the previous result of [19]. We then proceedto prove, in Section 2.3, the main theorem of this chapter in the particular case of the Jordanblock, which allows us to highlight the main ideas of the proof in a setting where the notationsare much clearer than in the general case, and also leads to a stronger result than in the generalcase. The proof of our main theorem is presented in Section 2.4, and Section 2.5 discusses theresults we obtained, and specially the difference in the statements of Theorems 2.6 and 2.5. Theproofs of some technical lemmas used in this chapter are given in the Appendices 2.A and 2.B.

2.2 Notations, definitions and previous results

In this chapter, Md,m(R) denotes the set of d ×m matrices with real coefficients, which isdenoted simply by Md(R) when d = m. As usual, we identify column matrices in Md,1(R)with vectors in Rd . The identity matrix in Md(R) is denoted by Idd and 0d×m ∈Md,m(R)denotes the matrix whose entries are all zero, the dimensions being possibly omitted if they are

10

implicit. The block-diagonal matrix whose diagonal blocks are the square matrices a1, . . . ,adis denoted by diag(a1, . . . ,ad). The notation ‖x‖ indicates both the Euclidean norm of a vectorx ∈ Rd and the associated matrix norm. The real and imaginary parts of a complex number zare denoted by ℜ(z) and ℑ(z) respectively. The sets R+ and N∗ denote, respectively, the sets ofthe non-negative real numbers R+ = [0,+∞) and the positive integers N∗ = 1,2,3,4, . . .. Fortwo topological spaces X and Y , we denote by C0(X ,Y ) the set of all continuous functions fromX to Y .

Throughout this chapter, we consider the system

x(t) = Ax(t)+α(t)Bu(t), x(t) ∈ Rd, u(t) ∈ Rm, α ∈ G(T,µ), (2.4)

where A∈Md(R), B∈Md,m(R), and we take persistently exciting signals α in the class G(T,µ)defined as follows.

Definition 2.1. Let T , µ be two positive constants with T ≥ µ . We say that a measurablefunction α : R+→ [0,1] is a (T,µ)-signal if, for every t ∈ R+, one has

w t+T

tα(s)ds≥ µ.

The set of (T,µ)-signals is denoted by G(T,µ). System (2.4) with α ∈ G(T,µ) is called apersistently excited system (PE system for short).

We shall consider the problem of stabilization of system (2.4) by means of a delayed linearstate feedback u(t) =−Kx(t− τ(t)), where the delay τ is a function in L∞(R+,T) for a certainbounded set T ⊂ R+ and K ∈Mm,d(R). With this feedback, system (2.4) takes the form

x(t) = Ax(t)−α(t)BKx(t− τ(t)),α ∈ G(T,µ), τ ∈ L∞(R+,T).

(2.5)

Note that, for T ≥ µ > 0 and T ⊂ R+ bounded, for every α ∈ L∞(R+, [0,1]) and every τ ∈L∞(R+,T), (2.5) satisfies the Carathéodory conditions for delayed equations (see, for instance,[30, Section 2.6 and Theorem 6.1.1]), and so, noting r = supT, for any given initial condi-tion x0 ∈ C0([−r,0],Rd), (2.5) admits a unique continuous solution x defined on [−r,+∞),which is absolutely continuous on R+, coincides with x0 on [−r,0], and satisfies x(t) = Ax(t)−α(t)BKx(t − τ(t)) for almost every t ∈ R+. In order to make explicit the dependence of thesolution x on τ , x0, α and K, we denote x(t) = x(t;τ,x0,α,K).

In the context of delayed systems, stability is defined in terms of the uniform norm of theinitial condition (see, for instance, [30, Chapter 5]), which motivates the following definition.

Definition 2.2. Let T ≥ µ > 0 and T be a bounded subset of R+, and denote r = supT.We say that K ∈ Mm,d(R) is a (T,µ,T)-stabilizer for (2.5) if there exist constants C ≥ 1and γ > 0 such that, for every α ∈ G(T,µ), every τ ∈ L∞(R+,T), and every initial conditionx0 ∈ C0([−r,0],Rd), the solution x(t;τ,x0,α,K) of (2.5) satisfies

‖x(t;τ,x0,α,K)‖ ≤Ce−γt sups∈[−r,0]

‖x0(s)‖ , ∀t ≥ 0.

Remark 2.3. If K is a (T,µ,T)-stabilizer for (2.5), then, for every constant α? ∈ [µ/T ,1] andevery constant delay τ? ∈ T, the linear delayed system

x(t) = Ax(t)−α?BKx(t− τ?) (2.6)

11

is exponentially stable. This is an important remark, since the stability and stabilization ofsystems with a constant delay of the form (2.6) can be more easily studied (see, for instance,[44,47]), giving rise to necessary conditions for K to be a (T,µ,T)-stabilizer. We shall use thisapproach later in Example 2.9.

Let us recall that a pair of matrices (A,B) ∈Md(R)×Md,m(R) is said to be stabilizable ifthere exists a matrix K ∈Mm,d(R) such that A−BK is Hurwitz. This is equivalent to sayingthat there exists an invertible matrix P ∈Md(R) such that

PAP−1 =

(A1 A30 A2

), PB =

(B10

),

where A2 is Hurwitz and (A1,B1) is controllable. Stabilizability of a pair (A,B) means that thelinear control system x = Ax+Bu admits a linear state feedback u =−Kx such that the closed-loop system x = (A−BK)x is exponentially stable, and thus, in order to achieve the requiredstabilizability property for system (2.5), the stabilizability of (A,B) is a necessary conditionwhen 0∈ T. This is what motivates us to consider only stabilizable pairs (A,B) in what follows.

The stabilizability of (2.5) by means of a non-delayed feedback law has been studied in [19]in the case of a single-input system, i.e., when m = 1, and it has been generalized to the multi-input case in [18]. In terms of Definition 2.2, this result can be stated as follows.

Theorem 2.4. Let (A,B)∈Md(R)×Md,m(R) be a stabilizable pair and assume that the eigen-values of A have non-positive real part. Then, for every T ≥ µ > 0, there exists a (T,µ,0)-stabilizer for (2.5).

The hypothesis that the eigenvalues of A have non-positive real part may seem restrictive,but it was shown in [19] that Theorem 2.4 is not true for certain stabilizable pairs (A,B) andcertain values of T,µ when A admits an eigenvalue with positive real part. This is actually aneffect of the signal α in the dynamics of the system; note that, when α(t) ∈ 0,1, the closed-loop system actually switches between the dynamics given by x = Ax and x = (A−BK)x, andthe phenomena related to this switch, such as the overshooting phenomenon, may lead to thenon-stabilizability of the switched system when A has an eigenvalue with positive real part, asdetailed in [19]. For more general information on the behavior of switched systems, we referto [9, 12, 34, 35, 41, 53].

The main result of this chapter is the following generalization of Theorem 2.4.

Theorem 2.5. Let (A,B)∈Md(R)×Md,m(R) be a stabilizable pair and assume that the eigen-values of A have non-positive real part. Then, for every T ≥ µ > 0 and every τ0 ≥ 0, thereexists a neighborhood T of τ0 in R+ and a (T,µ,T)-stabilizer for (2.5).

We prove this theorem here by generalizing the proof given in [19] in the non-delayed case.The main point is that the time-contraction argument given in [19], when applied to a delayedsystem, reduces the effects of the delay in the system, in such a way that the limit systemobtained by making the time-contraction parameter tend to infinity is essentially the same in thedelayed and the non-delayed cases. In order to highlight these main ideas, we first consider aparticular case of Theorem 2.5.

12

2.3 The d-integratorBefore turning to the proof of Theorem 2.5, let us first consider the particular case where thedynamics of the system are given by the d-integrator, defined by the Jordan block

Jd =

0 1 0 0 · · · 0 00 0 1 0 · · · 0 00 0 0 1 · · · 0 00 0 0 0 · · · 0 0...

......

... . . . ......

0 0 0 0 · · · 0 10 0 0 0 · · · 0 0

(2.7)

and by taking m = 1 and B =(0 · · · 0 1

)T ∈Md,1(R). This particular case will allow us tohighlight the main ideas of the proof of Theorem 2.5, since it contains most of the difficulties ofthe general case. Furthermore, we can give in this case a stronger result, showing the existenceof a (T,µ,T)-stabilizer for any bounded interval T ⊂R+, and not only for perturbations arounda certain value as in the general case of Theorem 2.5.

Theorem 2.6. Let A = Jd , B =(0 · · · 0 1

)T ∈ Rd , and let T ≥ µ > 0 and r > 0 be given.Then there exists a (T,µ, [0,r])-stabilizer K ∈M1,d(R) for (2.5).

Proof. The proof follows the same idea of the proof of [19, Theorem 3.1]: we first performa change of variables corresponding to a time contraction in order to relate (T,µ, [0,r])-stabilizers to (T/ν ,µ/ν , [0,r/ν ])-stabilizers for ν > 0. We then study the stabilizability ofa certain limit system, and this allows us to conclude the stabilizability of the original systemfor a certain ν > 0 large enough, thanks to the continuity result presented in the Appendix2.A.

Step 1. Time contraction

The system we consider is

x(t) = Jdx(t)−α(t)BKx(t− τ(t)),α ∈ G(T,µ), τ ∈ L∞(R+, [0,r]).

(2.8)

For ν > 0, we defineDd,ν = diag(νd−1, . . . ,ν ,1), (2.9)

which satisfies the relations

νD−1d,νJdDd,ν = Jd, Dd,νB = B. (2.10)

Noting, for simplicity, x(t) = x(t;τ,x0,α,K), and defining

xν(t) = D−1d,νx(νt), (2.11)

xν satisfiesddt

xν(t) = Jdxν(t)−α(νt)νBKDd,νxν

(t− τ(νt)

ν

)(2.12)

13

and hence

xν(t) = x(

t;τ(ν ·)

ν,D−1

d,νx0(ν ·),αν ,νKDd,ν

)with αν(t) = α(νt), which is a (T/ν ,µ/ν)-signal. Thus K is a (T,µ, [0,r])-stabilizer for(2.8) if and only if νKDd,ν is a (T/ν ,µ/ν , [0,r/ν ])-stabilizer. This equivalence is crucialin what follows: instead of looking for a (T,µ, [0,r])-stabilizer for (2.8), we look for a(T/ν ,µ/ν , [0,r/ν ])-stabilizer for a certain ν > 0 large enough. The technique is thus tostudy a certain limit system obtained as ν → +∞, obtain a stabilizer for this non-delayedsystem and then show that this stabilizer is actually a (T/ν ,µ/ν , [0,r/ν ])-stabilizer for acertain ν > 0 large enough.

Step 2. Limit system

We turn to the systemx(t) = Jdx(t)−α?(t)BKx(t),α? ∈ L∞(R+, [µ/T,1]).

(2.13)

It has been proved in [19, Theorem 3.1], using a result from [27], that one can find K ∈M1,d(R) and a positive definite matrix S ∈Md(R), both independent of the particular sig-nal α? ∈ L∞(R+, [µ/T,1]), such that (2.13) is globally uniformly exponentially stable andV (x) = xTSx decreases along all trajectories of (2.13), uniformly with respect to α?. Inparticular, there exists a time σ such that every trajectory of (2.13) starting in BV

2 = x ∈Rd |V (x)≤ 2 at time 0 lies in BV

1 = x ∈ Rd |V (x)≤ 1 for every time larger than σ .

Step 3. Study of (2.12) through the limit system.

We wish to deduce from the conclusion obtained in the previous step that (2.8) admitsa (T/ν ,µ/ν , [0,r/ν ])-stabilizer for a certain ν > 0 large enough. We claim that, for someν > 0 large enough, every trajectory of

x(t) = Jdx(t)−α(t)BKx(t− τ(t)),α ∈ G(T/ν ,µ/ν), τ ∈ L∞(R+, [0,r/ν ]),

with initial condition x0 ∈ C0([−r/ν ,0],BV2 ) stays in BV

1 for every time larger than 2σ . Inparticular, by homogeneity, this will imply that K is a (T/ν ,µ/ν , [0,r/ν ])-stabilizer of(2.8) and thus ν−1KD−1

d,ν is a (T,µ, [0,r])-stabilizer, concluding the proof. To prove this,

assume, by contradiction, that for every n ∈ N∗ there exist τn ∈ L∞(R+, [0,r/n]), x(n)0 ∈C0([−r/n,0],BV

2 ), αn ∈ G(T/n,µ/n), and tn ∈ [2σ ,4σ ] such that, for every n ∈ N∗,

x(

tn;τn,x(n)0 ,αn,K

)/∈ BV

1 . (2.14)

Up to the extraction of a subsequence, we can suppose that, as n→+∞, tn→ t? ∈ [2σ ,4σ ],x(n)0 (0)→ x?0 ∈ BV

2 , and αn α? ∈ L∞(R+, [0,1]) weakly-?; we also note that τn(t)→ 0 asn→ +∞ uniformly on t ∈ R+. Then, applying Lemma 2.10 proved in the Appendix 2.A,we obtain that x

(tn;τn,x

(n)0 ,αn,K

)converges to x(t?;0,x?0,α?,K) as n→+∞. We also note

that, by [19, Lemma 2.5], α?(t) ≥ µ/T almost everywhere in R+, and so, by our previousstudy of (2.13), since t? ≥ 2σ , by homogeneity, we have

V (x(t?;0,x?0,α?,K))≤ 12.

This contradicts (2.14), establishing the desired result.

14

2.4 Main result

We now turn to the proof of our main result, Theorem 2.5. For a given stabilizable pair ofmatrices (A,B) ∈Md(R)×Md,m(R) and for given T ≥ µ > 0 and τ0 ≥ 0, we wish to find aninterval T ⊂ R+ of admissible perturbations around τ0 and a (T,µ,T)-stabilizer for (2.5).

Proof of Theorem 2.5.

Step 1. Reduction to a canonical form

Notice that we may reduce the theorem to the case where (A,B) is controllable, m = 1,and all the eigenvalues of A lie on the imaginary axis; this is detailed in Lemmas 2.11, 2.12,and 2.13 in the Appendix 2.B. We thus suppose from now on that (A,B) is controllable,m = 1, and ℜ(λ ) = 0 for every eigenvalue λ of A. We also reduce (A,B) to a normal formwith which it shall be easier to work.

Lemma 2.7. Suppose (A,B) ∈Md(R)×Rd is a controllable pair and ℜ(λ ) = 0 for everyeigenvalue λ of A. Then, up to a linear transformation of coordinates, (2.4) can be writtenas

x0(t) = Jr0x0(t)+α(t)b0u(t), x0(t) ∈ Rr0,

x j(t) = (ω jA( j)+ JCr j)x j(t)+α(t)b ju(t), x j(t) ∈ R2r j , j = 1, . . . ,h,

(2.15)

where the spectrum of A is σ(A) = ±iω j, j = j0, j0+1, . . . ,h with all the ω j ≥ 0 distinct,j0 = 1 if 0 /∈ σ(A), j0 = 0 and ω0 = 0 otherwise; r j is the algebraic multiplicity of theeigenvalue iω j (with r0 = 0 if 0 /∈ σ(A)); Jr0 is the real Jordan block defined in (2.7);JC

n ∈M2n(R) is the Jordan block for complex eigenvalues,

JCn =

02×2 Id2 02×2 02×2 · · · 02×2 02×202×2 02×2 Id2 02×2 · · · 02×2 02×202×2 02×2 02×2 Id2 · · · 02×2 02×202×2 02×2 02×2 02×2 · · · 02×2 02×2

......

...... . . . ...

...02×2 02×2 02×2 02×2 · · · 02×2 Id202×2 02×2 02×2 02×2 · · · 02×2 02×2

,

that is, JCn = Jn⊗ Id2 in terms of the Kronecker product; A( j) = diag(A0, . . . ,A0) ∈M2r j(R)

with

A0 =

(0 1−1 0

);

and b0 and b j are respectively the vectors of Rr0 and R2r j with all the coordinates equal tozero except the last one that is equal to one.

This lemma was proved in [19] during the proof of Theorem 3.2 therein; for the sake ofcompleteness, we present briefly its proof in the Appendix 2.B.

Step 2. Time contraction

15

We work from now on with system (2.15). Given K ∈M1,d(Rd), we decompose K inblocks as K =

(K0 K1 · · · Kh

)with K0 ∈M1,r0(R), K j ∈M1,2r j(R), j = 1, . . . ,h, so that

the feedback law u(t) =−Kx(t−τ(t)) is written as u(t) =−K0x0(t−τ(t))−∑hj=1 K jx j(t−

τ(t)). As in the proof of Theorem 2.6, we perform a change of time-space variables in theclosed-loop system corresponding to a time contraction. Define

y0(t) = D−1r0,ν

x0(νt),

y j(t) = (DCr j,ν)

−1e−νtω jA( j)x j(νt), j = 1, . . . ,h,

with Dn,ν as in (2.9), satisfying (2.10), and

DCn,ν = Dn,ν ⊗ Id2 = diag(νn−1,νn−1, . . . ,ν ,ν ,1,1) ∈M2n(R),

which satisfies

ν(DCr j,ν)

−1JCr j

DCr j,ν = JC

r j, DC

r j,νb j = b j, j = 1, . . . ,h.

Then y0,y1, . . . ,yh satisfy

y0(t) = Jr0y0(t)−αν(t)b0

[K0,νy0

(t− τ(νt)

ν

)+

h

∑`=1

K`,νe(νt−τ(νt))ω`A(`)y`(

t− τ(νt)ν

)],

y j(t) = JCr j

y j(t)−αν(t)e−νtω jA( j)b j[K0,νy0

(t− τ(νt)

ν

)+

+h

∑`=1

K`,νe(νt−τ(νt))ω`A(`)y`(

t− τ(νt)ν

)], j = 1, . . . ,h,

(2.16)with αν(t) = α(νt), K0,ν = νK0Dr0,ν , K`,ν = νK`DC

r`,ν for ` = 1, . . . ,h, and where we usethat A( j)DC

r j,ν = DCr j,νA( j) and A( j)JC

r j= JC

r jA( j) for j = 1, . . . ,h. This shows that the gain

K =(K0 K1 · · · Kh

)is a (T,µ,T)-stabilizer for (2.15) if and only if the gain Kν =(

K0,ν K1,ν · · · Kh,ν)

is a (T/ν ,µ/ν ,T/ν)-stabilizer for (2.16), where T/ν = t/ν | t ∈T.

Step 3. Choice of the feedback family

We turn to the problem of finding a neighborhood T of τ0 in R+ and a (T/ν ,µ/ν ,T/ν)-stabilizer for (2.16) for a certain ν > 0, which will imply the theorem. We shall look forsuch a stabilizer Kν under a particular form. We write b0 =

(0 1

)T and we take Kν =(K0,ν K1,ν · · · Kh,ν

)with

K0,ν =K0, K0 =(k0

1 · · · k0r0

)∈M1,r0(R)

K j,ν =K j⊗bT0 eτ0ω jA0, K j =

(k j

1 · · · k jr j

)∈M1,r j(R), j = 1, . . . ,h.

(2.17)

Now, since A(`) = Idr`⊗A0, we have, for `= 1, . . . ,h, that

K`,νe(νt−τ(νt))ω`A(`)=K`⊗bT

0 e(νt−τ(νt)+τ0)ω`A0 =

=K`⊗bT0 eνtω`A0 +K`⊗

[bT

0 eνtω`A0(

e−(τ(νt)−τ0)ω`A0− Id2

)].

16

Noting b j ∈Rr j the vector with all coordinates equal to zero except the last one that is equalto one, we have b j = b j⊗b0, and thus e−νtω jA( j)

b j = b j⊗ e−νtω jA0b0. We finally write, forj, ` ∈ 1, . . . ,h,

C(ν)00 (t) = αν(t),

C(ν)0 j (t) = αν(t)bT

0 eνtω jA0,

C(ν)j0 (t) = αν(t)e−νtω jA0b0,

C(ν)j` (t) = αν(t)e−νtω jA0b0bT

0 eνtω jA0,

P(ν)00 (t) = P(ν)

j0 (t) = 0,

P(ν)0 j (t) = αν(t)bT

0 eνtω jA0[e−(τ(νt)−τ0)ω jA0− Id2

],

P(ν)j` (t) = αν(t)e−νtω jA0b0bT

0 eνtω`A0[e−(τ(νt)−τ0)ω`A0− Id2

],

(2.18)

and thus system (2.16) can be written under the formy0(t) = Jr0y0(t)−

h

∑`=0

[b0K`⊗ (C(ν)0` (t)+P(ν)

0` (t))]y`(

t− τ(νt)ν

),

y j(t) = JCr j

y j(t)−h

∑`=0

[b jK`⊗ (C(ν)j` (t)+P(ν)

j` (t))]y`(

t− τ(νt)ν

), j = 1, . . . ,h.

(2.19)

We can arrange all the matrices C(ν)j` in a (2h+1− j0)×(2h+1− j0) symmetric matrix and

all the matrices P(ν)j` in a (2h+1− j0)× (2h+1− j0) matrix respectively as

C(ν)(t) =(

C(ν)j` (t)

)j0≤ j,`≤h

, P(ν)(t) =(

P(ν)j` (t)

)j0≤ j,`≤h

. (2.20)

We take from now on T under the form T = [τ0− r,τ0 + r]∩R+ for a certain r > 0 to bechosen, and so∥∥∥P(ν)

j` (t)∥∥∥≤ ∥∥∥e−(τ(νt)−τ0)ω jA0− Id2

∥∥∥==√

2[1− cos((τ(νt)− τ0)ω j)

]≤∣∣(τ(νt)− τ0)ω j

∣∣≤ rΩ

with Ω = maxω j | j = j0, . . . ,h.

Step 4. Limit system

We wish to study (2.19) through a limit system, as we did with (2.12) in Theorem 2.6.The stability result for the limit system is given in the following lemma, proved later on inAppendix 2.B.

Lemma 2.8. Consider the systemy0(t) = Jr0y0(t)−

h

∑`=0

[b0K`⊗ (C0`(t)+P0`(t))]y`(t),

y j(t) = JCr j

y j(t)−h

∑`=0

[b jK`⊗ (C j`(t)+Pj`(t))]y`(t), j = 1, . . . ,h,

(2.21)

17

where y0 ∈ Rr0 , y j ∈ R2r j , Jn and JCn are the Jordan blocks defined above, b0 and b j are

the vectors defined above, K j ∈ M1,r j(R) are constant matrices, j = j0, . . . ,h, C?,P? ∈L∞(R+,M2h+1− j0(R)) and the 2×2 time-dependent matrices C j`,Pj`, 1≤ j, `≤ h, the (1−j0)× 2 time-dependent matrices C0`,P0`, the 2× (1− j0) time-dependent matrices C j0,Pj0and the signals C00,P00 are defined by the relations

C?(t) =(C j`(t)

)j0≤ j,`≤h , P?(t) =

(Pj`(t)

)j0≤ j,`≤h , (2.22)

and we also assume that∥∥Pj`(t)∥∥≤ rΩ, for almost every t ∈ R+, ∀ j, ` ∈ j0, . . . ,h. (2.23)

We write y =(yT

0 yT1 · · · yT

h

)T.Let ξ > 0. Then there exist C ≥ 1, γ > 0, r > 0, and K j ∈M1,r j(R), j = j0, . . . ,h, such

that, for every symmetric matrix C? ∈ L∞(R+,M2h+1− j0(R)) satisfying C?(t)≥ ξ Id2h+1− j0almost everywhere, every P? ∈ L∞(R+,M2h+1− j0(R)) satisfying (2.23) and every solution yof (2.21), we have

‖y(t)‖ ≤Ce−γt ‖y(0)‖ , ∀t ≥ 0.

Step 5. Study of (2.19) through the limit system

To conclude the proof, we deduce the stability of (2.19) from that of (2.21) in the sameway as we did in the proof of Theorem 2.6. Take T ≥ µ > 0 and τ0 ≥ 0. By [19, Lemma2.5], there exists ξ > 0 depending only on T,µ and ω j, j = j0, . . . ,h, such that, for anyα ∈ G(T,µ) and any ν > 0, the time-dependent matrix C(ν) constructed from α as in (2.18)and (2.20) is in L∞(R+,M2h+1− j0(R)) and satisfies

w t+ Tν

tC(ν)(s)ds≥ ξ

Tν

Id2h+1− j0 .

For this ξ > 0, take C ≥ 1, γ > 0, r > 0, and K j ∈M1,r j(R) as in Lemma 2.8. Set T =

[τ0−r,τ0+r]∩R+ and construct K =(K0 · · · Kh

)from the K j, j = j0, . . . ,h as in (2.17).

We want to show that, for ν > 0 large enough, K is a (T/ν ,µ/ν ,T/ν)-stabilizer for (2.16),and this will conclude the proof by the conclusion of Step 2.

Note that, by Lemma 2.8, there exists a time σ > 0 depending only on C and γ suchthat, for every trajectory y of (2.21) starting in B2 = x ∈ Rd | ‖x‖ ≤ 2 at time 0 lies inB1 = x ∈ Rd | ‖x‖ ≤ 1 for every time larger than σ . We claim that, for some ν > 0 largeenough, for every α ∈ G(T/ν ,µ/ν), every τ ∈ L∞(R+,T/ν) and every initial conditiony0 ∈ C0([−R/ν ,0],B2), with R = supT, the solution y of (2.19), with C(ν) and P(ν) given by(2.18) and (2.20), stays in B1 for every time larger than 2σ . This will show, by homogeneity,that K is a (T/ν ,µ/ν ,T/ν)-stabilizer for (2.16).

Assume, by contradiction, that for every n ∈ N∗ there exist τn ∈ L∞(R+,T/n), y0n ∈

C0([−R/n,0],B2), αn ∈ G(T/n,µ/n), and tn ∈ [2σ ,4σ ] such that, for every n ∈ N∗, thesolution yn of (2.19), with C(n) and P(n) given by (2.18) and (2.20), satisfies

yn(tn) /∈ B1. (2.24)

18

Up to the extraction of a subsequence, we can suppose that

limn→∞

tn = t? ∈ [2σ ,4σ ],

limn→∞

y0n(0) = y0

? ∈ B2,

limn→∞

C(n) =C? ∈ L∞(R+,M2h+1− j0(R)) weakly-?,

limn→∞

P(n) = P? ∈ L∞(R+,M2h+1− j0(R)) weakly-?,

and we also note that τn(t)→ 0 uniformly on t ∈ R+ as n→ +∞. Then, by Lemma 2.10,yn converges to the solution y? of (2.21) associated to C?, P? and with initial condition y0

?,uniformly on compact time intervals, and in particular yn(tn)→ y?(t?). By [19, Lemma2.5], we have C?(t) ≥ ξ Id2h+1− j0 for almost every t and, since

∥∥∥P(n)j` (t)

∥∥∥ ≤ rΩ for everyj, ` ∈ j0, . . . ,h and almost every t ∈ R+, we have, by the lower semicontinuity of thenorm of L∞(R+,M2h+1− j0(R)), that

∥∥Pj`(t)∥∥ ≤ rΩ for every j, ` ∈ j0, . . . ,h and almost

every t ∈ R+, where Pj` is obtained from P? by (2.22). Thus we are under the hypothesesof Lemma 2.8, and so our previous discussion shows us that y? remains in B1 for everytime larger than σ ; by homogeneity, ‖y?(t?)‖ ≤ 1/2 since t? ≥ 2σ . This contradicts (2.24),establishing the desired result.

2.5 Further discussion

We proved that persistently excited linear systems can be stabilized by a delayed feedback lawwhen the uncontrolled dynamics of the system is given by a matrix A whose eigenvalues haveall non-positive real part and when the delay varies in an interval around a constant value τ0,with the feedback matrix K depending on the matrices A, B, on the constants T and µ of thecondition of persistence of excitation and on the reference delay τ0. This is a generalizationof [19, Theorem 3.2], originally proved for the non-delayed case.

The technique of the proof consists on adapting the time-contraction argument of [19, The-orem 3.2] to the delayed case. Indeed, the time contraction also contracts the delay, reducing itseffect, and the limit system obtained in the time-contraction procedure is the same as in [19],except for the new terms Pj`, which are treated as perturbations of the limit system of [19].

It is actually by treating these terms Pj` as perturbations that we arrive to the construction ofthe delay neighborhood T around τ0 where we can guarantee stabilizability. Note that the termsPj` do not appear in the limit system obtained when A = Jd in Theorem 2.6, since they dependon the eigenvalues iω j, and this is the reason why we can obtain a (T,µ,T)-stabilizer for anybounded T ⊂ R+ when A = Jd in Theorem 2.6.

This is a fundamental difference between Theorems 2.6 and 2.5 which we would like tohighlight: in Theorem 2.6, stabilization can be achieved for any bounded set T ⊂ R+ wherethe delay takes its values, whereas in Theorem 2.5 T is chosen as T = [τ0− r,τ0 + r]∩R+, aperturbation around the constant value τ0.

A natural question is then to study if Theorem 2.5 might not be generalized for any boundedset T instead of considering only perturbations around τ0. This is actually not possible, asshown in the following example, where we take α identically equal to one, i.e., the control iscompletely active the whole time.

19

Example 2.9. Consider the control system

x = Ax+Bu (2.25)

with

A =

(0 1−1 0

), B =

(01

)and submitted to the feedback law

u(t) =−Kx(t− τ(t)) (2.26)

This control system does not depend on a persistently exciting signal α , but, in order tokeep the notations we used previously, we shall consider it as a persistently excited systemwith constants T = µ , so that G(T,µ) = G(T,T ) reduces to the class containing only theconstant signal identically equal to one. We want to prove that the conclusion of Theorem2.6 does not hold for (2.25), that is, we want to show that there exists a bounded interval Tfor which (2.25) with the feedback (2.26) does not admit a (T,T,T)-stabilizer. Obviously,this also implies the non-existence of a (T,µ,T)-stabilizer for every µ ∈ (0,T ] since such astabilizer would be in particular a (T,T,T)-stabilizer.

We claim that (2.25) with the feedback (2.26) does not admit a (T,T, [0,2π])-stabilizer.In order to simplify our analysis, we shall consider only constant-in-time delays in the inter-val [0,2π], which allow us to apply the techniques of stability analysis for delayed systemspresented in [44].

The closed-loop system obtained from (2.25) with the feedback (2.26) and a constantdelay τ ∈ [0,2π] is

x(t) = Ax(t)−BKx(t− τ). (2.27)

According to [44, Proposition 1.6], the stability of (2.27) can be studied through the complexroots λ of the characteristic equation

det(

λ Id2−A+BKe−λτ

)= 0; (2.28)

the origin of (2.27) is exponentially stable if and only if all the roots λ of (2.28) satisfyℜ(λ )< 0, and exponential stability and asymptotic stability are also equivalent in this case.

Writing K =(k1 k2

), the characteristic equation (2.28) is

λ2 + k2λe−λτ +1+ k1e−λτ = 0. (2.29)

We now want to show that, for every K ∈M1,2(R), there exists τ ∈ [0,2π] such that (2.29)admits a root λ with ℜ(λ ) ≥ 0. As remarked in [44, Theorem 1.15], by the continuity ofthe real part of the largest eigenvalue with respect to the delay, this study is reduced to theproblem of finding a delay τ ∈ [0,2π] such that (2.29) admits a root λ with ℜ(λ ) = 0.

The feedback K = 0 obviously does not stabilize the system to the origin, and so wesuppose from now on that k1 and k2 are not simultaneously zero. We look for a certainτ ∈ [0,2π] and a root λ = iω of (2.29) with ω ∈ R. We thus want ω to satisfy

1−ω2 + k1 cos(τω)+ k2ω sin(τω) = 0,−k1 sin(τω)+ k2ω cos(τω) = 0.

20

This is equivalent to

sinθ =k2ω(ω2−1)

k22ω2 + k2

1,

cosθ =k1(ω

2−1)k2

2ω2 + k21,

θ = τω

(2.30)

and such a system can only have a solution if sin2θ + cos2 θ = 1, which is the case if and

only if (ω2−1)2 = k22ω2 + k2

1. This last equation is a polynomial in ω2 of degree 2, whosesolutions can be computed explicitly as

ω2 =

12

[2+ k2

2±√

(2+ k22)

2−4(1− k21)

].

We consider from now on the solution

ω =

√√√√2+ k22 +√

(2+ k22)

2−4(1− k21)

2.

Note that ω is well-defined in R since (2+ k22)

2 > 4(1− k21) for any K ∈M1,2(R) \ 0,

and that ω ≥ 1. With this ω , we can thus find θ ∈ [0,2π] such that (2.30) is satisfied, andso τ = θ/ω ∈ [0,2π] since ω ≥ 1. Since the constructed (θ ,τ,ω) satisfies (2.30), (2.29) ishence satisfied for τ and λ = iω , and thus (2.27) is not asymptotically stable. Hence (2.25)admits no (T,T, [0,2π])-stabilizer.

Note that we could replace [0,2π] in Example 2.9 for any other interval T ⊂R+ with lengthgreater than or equal 2π , and so we conclude that (2.25) does not admit a (T,µ,T)-stabilizer ifT contains an interval with length greater than or equal 2π .

The value 2π obtained in these computations comes from the fact that the dynamics given bythe matrix A we chose correspond to rotations around the origin with unitary angular velocity,and 2π is the total time that a solution of x = Ax takes to make a complete turn around theorigin. If we choose A as

A =

(0 ω0−ω0 0

)for ω0 6= 0, then the same computations as in Example 2.9 show that no (T,µ,T)-stabilizercan exist for (2.25) if T contains an interval of length at least 2π

ω0. In particular, this gives a

link between an upper bound on the maximal length of an interval contained in T for which a(T,µ,T)-stabilizer exists and the eigenvalues of A on the imaginary axis.

This example shows that the fundamental difference in the statement of Theorems 2.6 and2.5 concerning the choice of the set T actually comes from the dynamics of the system itself,and that no improvement of Theorem 2.5 as good as Theorem 2.6 can be obtained.

2.A Appendix: A continuity result for delayed systemsWe show here a continuity result of the solution of a delayed system with respect to its param-eters, in the spirit of [16, Proposition 21], which is used in the proof of Theorems 2.6 and 2.5.We place ourselves in a more general setting than (2.5), considering the system

x(t) = Ax(t)+B(t)x(t− τ(t)), (2.31)

21

where τ ∈ L∞(R+, [0,r]), and B ∈ L∞(R+,Md(R)) is a time-dependent matrix. We remark that(2.31) satisfies the Carathéodory conditions for delayed equations, and so, for fixed τ and B andfor any given initial condition x0 ∈ C0([−r,0],Rd), (2.31) admits a unique continuous solutionx defined on [−r,+∞), which we denote by x(t) = x(t;τ,x0,B); this solution is absolutely con-tinuous on R+, coincides with x0 on [−r,0], and satisfies (2.31) for almost every t ∈ R+. Ourcontinuity result can then be stated as follows.

Lemma 2.10. Let (τn)n∈N∗ be a sequence on L∞(R+, [0,r]) such that τn(t)→ 0 as n→ +∞

uniformly on R+. Suppose (x(n)0 )n∈N∗ is a sequence of functions in C0([−r,0],Rd) and (Bn)n∈N∗a bounded sequence on L∞(R+,Md(R)) satisfying

1. limn→+∞

x(n)0 (0) = x?0 for a certain x?0 ∈ Rd;

2. there exists Λ > 0 such that∥∥∥x(n)0 (t)

∥∥∥≤ Λ for all n ∈ N∗ and all t ∈ [−r,0];

3. Bn −−−−n→+∞

B? weakly-? for a certain B? ∈ L∞(R+,Md(R)).

Then x(t;τn,x(n)0 ,Bn)−−−−→

n→+∞x(t;0,x?0,B?), uniformly on compact time intervals in R+.

Proof. We can extend B? outside R+ to the whole real line in such a way that this extensionis an element of L∞(R,Md(R)). We fix such an extension, so that x(·;0,x?0,B?) is abso-lutely continuous in R and satisfies (2.31) for almost every t ∈ R; note that this is possiblesince x(·;0,x?0,B?) is the solution of a non-delayed system. For simplicity, we shall notexn(t) = x(t;τn,x

(n)0 ,Bn) and x?(t) = x(t;0,x?0,B?). We also note by M an upper bound on

‖Bn‖L∞(R+,Md(R)) and rn = supt∈R+τn(t), and, by the uniform convergence of τn to 0, we

have that rn→ 0 as n→+∞.Define en(t) = xn(t)− x?(t) for t ≥−r. Then, for t ≥ 0, en satisfies

en(t) = Aen(t)+Bn(t)en(t− τn(t))+ fn(t) (2.32)

with fn given by fn(t) = Bn(t)(x?(t− τn(t))− x?(t))+(Bn(t)−B?(t))x?(t).Since x? is continuous, it follows from Lebesgue’s Dominated Convergence Theorem

thatlim

n→+∞

w t

0Bn(s)(x?(s− τn(s))− x?(s))ds = 0

for every t ≥ 0. By the weak-? convergence of (Bn), we have that

limn→+∞

w t

0(Bn(s)−B?(s))x?(s)ds = 0,

and so fn satisfieslim

n→+∞

w t

0fn(s)ds = 0

for every t ≥ 0. Letting Fn(t) =r t

0 fn(s)ds, this shows that Fn(t) −−−−→n→+∞

0 for every t ≥0. This limit is uniform on compact time intervals in R+. Indeed, let T > 0 and X? =supt∈[−r,T ] ‖x?(t)‖; we thus see that ‖ fn(t)‖ ≤ 2MX? and so ‖Fn(t)‖ ≤ 2MX?T for everyt ∈ [0,T ]. Furthermore, for 0≤ t1 < t2 ≤ T , we have

‖Fn(t2)−Fn(t1)‖ ≤w t2

t1‖ fn(s)‖ds≤ 2MX? (t2− t1) ,

22

and hence (Fn) is equicontinuous. Thus, by Arzelà-Ascoli Theorem, the closure of Fn |n ∈N∗ is a compact subset of C0([0,T ],Rd) with the topology of the uniform convergence,and so this set has at least one limit point; it has exactly one, for, if it had two distinctlimit points, this would contradict the fact that (Fn(t))n∈N∗ tends pointwise to 0, and so thesequence (Fn)n∈N∗ converges uniformly to 0 in [0,T ].

Integrating (2.32) from 0 to t ≥ 0, we obtain

en(t) = en(0)+Fn(t)+w t

0Aen(s)ds+

w t

0Bn(s)en(s− τn(s))ds,

which gives us the estimate

‖en(t)‖ ≤ ‖en(0)‖+‖Fn(t)‖+w t

0‖A‖‖en(s)‖ds+M

w t

0‖en(s− τn(s))‖ds. (2.33)

DefineXn,t = s ∈ [0, t] | s− τn(s)< 0.

This set is measurable and, since 0≤ τn(t)≤ rn for all t ∈R+, we have that Xn,t ⊂ [0,rn], sothat λ (Xn,t)≤ rn for all t ∈ R+, where λ denotes the Lebesgue measure. Define also

En(t) = sups∈[t−rn,t]∩[0,t]

‖en(s)‖

and M′ = ‖A‖+M. From (2.33), we obtain

‖en(t)‖ ≤ ‖en(0)‖+‖Fn(t)‖+Mw

Xn,t‖en(s− τn(s))‖ds+M′

w t

0En(s)ds,

so that, for t ≥ 0,En(t)≤ ϕn(t)+M′

w t

0En(s)ds,

with ϕn given by ϕn(t)= ‖en(0)‖+supσ∈[t−rn,t]∩[0,t]

[‖Fn(σ)‖+M

rXn,σ‖en(s− τn(s))‖ds

].

Applying Gronwall’s Lemma, we get

En(t)≤ ϕn(t)+M′w t

0ϕn(s)eM′(t−s)ds (2.34)

for t ≥ 0.Fix T > 0. Since limn→+∞ Fn(t) = 0 uniformly on [0,T ], we have that

limn→+∞

[sup

σ∈[t−rn,t]∩[0,t]‖Fn(σ)‖

]= 0 uniformly on t ∈ [0,T ].

Moreover, for s ∈ Xn,σ , we have that

‖en(s− τn(s))‖= ‖xn(s− τn(s))− x?(s− τn(s))‖ ≤C,

where C = Λ+ supt∈[−r,0] ‖x?(t)‖, and so

supσ∈[t−rn,t]∩[0,t]

w

Xn,σ‖en(s− τn(s))‖ds≤Crn −−−−→

n→+∞0

uniformly on t ∈ [0,T ]. Hence ϕn(t) −−−−→n→+∞

0 uniformly on [0,T ], from where we get,

together with (2.34), that En(t) −−−−→n→+∞

0 uniformly on [0,T ]. So en(t) −−−−→n→+∞

0 uniformly

on [0,T ], and, since T > 0 is arbitrary, this gives the desired result.

23

2.B Appendix: On the proof of Theorem 2.5We prove here some of the results that were used in the proof of Theorem 2.5. The first threeresults, Lemmas 2.11, 2.12 and 2.13, deal with the reduction of Theorem 2.5 to the case where(A,B) is controllable, m = 1 and all the eigenvalues of A lie on the imaginary axis. We begin byreducing the theorem to the case where (A,B) is controllable.

Lemma 2.11. It suffices to prove Theorem 2.5 in the case where (A,B) is controllable.

Proof. Up to a linear change of variables, A and B can be decomposed on the controllableand uncontrollable parts according to Kalman decomposition as

A =

(A1 A30 A2

), B =

(B10

)with A1 ∈Md′(R), A2 ∈Md−d′(R), B1 ∈Md′,m(R), the other matrices having appropriatedimensions, and where (A1,B1) is controllable (see, for instance, [52, Theorem 13.1]); since(A,B) is stabilizable, A2 is Hurwitz. The open-loop system (2.4) can thus be written afterthe change of variables as

x1(t) = A1x1(t)+A3x2(t)+α(t)B1u(t),x2(t) = A2x2(t),

(2.35)

with x1(t) ∈Rd′ , x2(t) ∈Rd−d′ , and x(t) =(x1(t)T x2(t)T)T. Now, suppose the theorem is

proved for the controllable case and K′ ∈Mm,d′(R) is a (T,µ,T)-stabilizer for (A1,B1) fora certain neighborhood T of τ0 in R+, associated with certain constants C1 ≥ 1, γ1 > 0 asin Definition 2.2. Take K =

(K′ 0

)∈Mm,d(R), so that, with the feedback u(t) =−Kx(t−

τ(t)), (2.35) becomesx1(t) = A1x1(t)−α(t)B1K′x1(t− τ(t))+A3x2(t),x2(t) = A2x2(t).

(2.36)

Let us note r = supT. Take α ∈ G(T,µ), τ ∈ L∞(R+,T), and an initial condition x0 ∈C0([−r,0],Rd), written as x0(t) =

(x0,1(t)T x0,2(t)T)T. Note by y(t) ∈ Rd the solution of

y(t) = A1y(t)−α(t)B1K′y(t− τ(t)), t > 0,y(t) = x0,1(t), t ∈ [−r,0].

Then, by the hypothesis on K′, we have that

‖y(t)‖ ≤C1e−γ1t sups∈[−r,0]

∥∥x0,1(s)∥∥ . (2.37)

The result on [30, Section 6.2] allows us to write the solution x(t) =(x1(t)T x2(t)T)T of

(2.36) associated with α and τ and with initial condition x0 asx1(t) = y(t)+w t

0X(t,s)A3x2(s)ds,

x2(t) = eA2tx0,2(0),(2.38)

24

where X(t,s) ∈Md′(R) is the fundamental matrix solution associated with the delayed sys-tem z(t) = A1z(t)−α(t)B1K′z(t−τ(t)) (see [30, Section 6.1]). Our choice of K′ guaranteesthat this last system is exponentially stable, uniformly with respect to α ∈ G(T,µ) andτ ∈ L∞(R+,T), and so, by [30, Lemma 6.5.3], there exist constants C0 ≥ 1, γ0 > 0 indepen-dent of α and τ such that

‖X(t,s)‖ ≤C0e−γ0(t−s) for all t ≥ s≥ 0; (2.39)

note that [30, Lemma 6.5.3] is proved only for the case of uniformity with respect to theinitial time, but the same proof also applies for the case of uniformity with respect to otherparameters. Note also that we do not need to consider uniformity with respect to the initialtime since the classes G(T,µ) and L∞(R+,T) are invariant with respect to positive timetranslations and a non-zero initial time may be translated into terms of a different choice ofα and τ .

Since A2 is Hurwitz, there exist C2 ≥ 1, γ2 > 0 such that∥∥∥eA2t∥∥∥≤C2e−γ2t . (2.40)

Using the estimates (2.37), (2.39) and (2.40) in (2.38), we can find C ≥ 1 and γ > 0, de-pending only on C0, C1, C2, γ0, γ1, γ2, and thus independent of α and τ , such that

‖x(t)‖ ≤Ce−γt sups∈[−r,0]

‖x0(s)‖ ,

which proves that K is a (T,µ,T)-stabilizer for (A,B), as desired.

The following lemma shows that we may further reduce Theorem 2.5 to the single-inputcase. Its proof follows the same idea of [18, Chapter 4, Theorem 4], where the original stabi-lization result for single-input systems of [19, Theorem 3.2] is generalized to the multi-inputcase by a recurrence on the number of inputs.

Lemma 2.12. It suffices to prove Theorem 2.5 in the case where (A,B) is controllable andm = 1.

Proof. We may suppose (A,B) controllable by Lemma 2.11. We suppose the theorem tobe proved in the case m = 1 and we prove the general case by induction on m. Supposethe theorem has been proved for m− 1, that is, for every d ∈ N∗, for every A ∈Md(R)and B ∈Md,m−1(R) such that (A,B) is a controllable pair and the eigenvalues of A havenon-positive real part, for every T,µ with T ≥ µ > 0, and for every τ0 ≥ 0, there exists aneighborhood T of τ0 in R+ and a (T,µ,T)-stabilizer for (2.5).

Take A ∈Md(R) and B ∈Md,m(R) such that (A,B) is a controllable pair and the eigen-values of A have non-positive real part and fix T ≥ µ > 0 and τ0≥ 0. Note by b∈Rd the firstcolumn of B; we may suppose, without loss of generality, that b 6= 0, for otherwise the firstinput does not influence the system and it may thus be excluded, reducing the system to thecase with m−1 inputs. We consider the pair (A,b), which may not be controllable, but canbe decomposed according to Kalman decomposition: there exists an invertible P ∈Md(R)such that

PAP−1 =

(A1 A30 A2

), Pb =

(b10

),

25

with A1 ∈Md′(R), b1 ∈Rd′ , all the other matrices have appropriate dimensions, and (A1,b1)is controllable. Now, performing the change of variables z = Px in (2.4), the open-loopsystem becomes

z =(

A1 A30 A2

)z+α(t)

(b1 B30 B2

)u (2.41)

with B2 ∈Md−d′,m−1(R) and B3 ∈Md′,m−1(R).By the controllability of (A,B) and (A1,b1), it follows that (A2,B2) is also controllable.

Now B2 ∈Md−d′,m−1(R), and so, by the induction hypothesis, (A2,B2) admits a (T,µ,T2)-stabilizer K2 ∈Mm−1,d−d′(R) for a certain neighborhood T2 of τ0 in R+. If Theorem 2.5is proved in the controllable case with m = 1, then we can take a (T,µ,T1)-stabilizer K1 ∈M1,d′(R) for (A1,b1) for a certain neighborhood T1 of τ0 in R+. We claim that K ∈Mm,d(R)given by

K =

(K1 00 K2

)is a (T,µ,T)-stabilizer for (A,B) for the neighborhood T = T1∩T2. Indeed, with this feed-back, system (2.41) becomes

z(t) =(

A1 A30 A2

)z(t)−α(t)

(b1K1 B3K2

0 B2K2

)z(t− τ(t)).

Noting z =(zT

1 zT2)T with z1 ∈ Rd′ and z2 ∈ Rd−d′ , we can thus write

z1(t) = A1z1(t)−α(t)b1K1z1(t− τ(t))+A3z2(t)−α(t)B3K2z2(t− τ(t)),z2(t) = A2z2(t)−α(t)B2K2z2(t− τ(t)).

(2.42)

We denote by X(t,s) the fundamental matrix solution of x(t) = A1x(t)−α(t)b1K1x(t −τ(t)); by construction of K1 and by [30, Lemma 6.5.3], we can find C0 ≥ 1 and γ0 > 0, bothindependent of α ∈ G(T,µ) and τ ∈ L∞(R+,T), such that

‖X(t,s)‖ ≤C0e−γ0(t−s), ∀t ≥ s≥ 0.

Note r = supT. Given an initial condition(zT

0,1 zT0,2)T ∈ C0([−r,0],Rd), note by y1 and y2

the solutions to

y1(t) = A1y1(t)−α(t)b1K1y1(t− τ(t)), y1(t) = z0,1(t) for t ∈ [−r,0],y2(t) = A2y2(t)−α(t)B2K2y2(t− τ(t)), y2(t) = z0,2(t) for t ∈ [−r,0].

(2.43)

By construction of K1 and K2, there exist C1,C2 ≥ 1 and γ1,γ2 > 0 such that∥∥y j(t)∥∥≤C je−γ jt sup

s∈[−r,0]

∥∥z0, j(s)∥∥ , j = 1,2.

We can now write the solution of (2.42) in terms of the initial condition(zT

0,1 zT0,2)T ∈

C0([−r,0],Rd) using the variation-of-constants formula in [30, Section 6.2] asz1(t) = y1(t)+w t

0X(t,s)(A3z2(s)−α(s)B3K2z2(s− τ(s)))ds,

z2(t) = y2(t).

26

It is thus easy to see that‖z1(t)‖ ≤C1e−γ1t sup

s∈[−r,0]

∥∥z0,1(s)∥∥+C′e−γ ′t sup

s∈[−r,0]

∥∥z0,2(s)∥∥ ,

‖z2(t)‖ ≤C2e−γ2t sups∈[−r,0]

∥∥z0,2(s)∥∥ ,

for certain constants C′≥ 1, γ ′> 0, and so K is a (T,µ,T)-stabilizer for (2.41), as we wantedto prove. The result is thus established by induction.

We further reduce our proof of Theorem 2.5 to the case where all the eigenvalues of A lieon the imaginary axis.

Lemma 2.13. It suffices to prove Theorem 2.5 in the case where (A,B) is controllable, m = 1,and ℜ(λ ) = 0 for every eigenvalue λ of A.

Proof. We may suppose (A,B) controllable and m = 1 by Lemma 2.12. Up to a linearchange of variables, A and B can be written as

A =

(A1 A30 A2

), B =

(B1B2

)with A1 ∈Md′(R), A2 ∈Md−d′(R), B1 ∈Rd′ , the other matrices having appropriate dimen-sions, and where A1 is Hurwitz and all the eigenvalues of A2 have real part 0. Since (A,B)is controllable, (A2,B2) is also controllable. The open-loop system (2.4) can thus be writtenafter the change of variables as

x1(t) = A1x1(t)+A3x2(t)+α(t)B1u(t),x2(t) = A2x2(t)+α(t)B2u(t),

(2.44)

with x1(t) ∈Rd′ , x2(t) ∈Rd−d′ , and x(t) =(x1(t)T x2(t)T)T. Now, suppose the theorem is

proved for the case stated above and take K′ ∈M1,d−d′(R) a (T,µ,T)-stabilizer for (A2,B2)for a certain neighborhood T of τ0 in R+, associated with certain constants C2 ≥ 1, γ2 > 0 asin Definition 2.2. Take K =

(0 K′

)∈M1,d(R), so that, with the feedback u(t) =−Kx(t−

τ(t)), (2.44) becomesx1(t) = A1x1(t)+A3x2(t)−α(t)B1K′x2(t− τ(t)),x2(t) = A2x2(t)−α(t)B2K′x2(t− τ(t)).

(2.45)

Let us note r = supT. Take α ∈ G(T,µ), τ ∈ L∞(R+,T), and an initial condition x0 ∈C0([−r,0],Rd), written as x0(t) =

(x0,1(t)T x0,2(t)T)T. By the hypothesis on K′, we have

that the solution x(t) =(x1(t)T x2(t)T)T of (2.45) associated with α and τ and with initial

condition x0 satisfies‖x2(t)‖ ≤C2e−γ2t sup

s∈[−r,0]‖x2(s)‖ .

Applying the variation-of-constants formula to (2.45) and using an exponential estimate on∥∥eA1t∥∥, it is immediate to verify that K is a (T,µ,T)-stabilizer for (A,B).

Let us now present a proof of Lemma 2.7, which was originally done in [19] and that werecall here for the sake of completeness.

27

Proof of Lemma 2.7. Up to a linear change of variables in (2.4), we may suppose thatA is in its real Jordan normal form. A has a unique Jordan block associated with each−iω j, iω j, j = j0, . . . ,h, for, otherwise, the rank of the matrix

(A− iω j Idd B

)would be

strictly smaller than d, contradicting the Hautus test for controllability. Thus, up to a permu-tation of variables on Rd , we can write A = diag(Jr0,ω1A(1)+JC

r1, . . . ,ωhA(h)+JC

rh), and B∈

Rd is such that (A,B) is controllable. Now, take b ∈Rd as b =((b0)T (b1)T · · · (bh)T

)T

with b0 and b j, j = 1, . . . ,h, as defined in the statement of the lemma. It follows from Hautustest for controllability that (A, b) is controllable. But all controllable linear control systemsassociated with a pair (A,B) that have in common the eigenvalues of A, counted accordingto their multiplicity, are state-equivalent, since they can be transformed by a linear transfor-mation of coordinates into the same system under controller form (see, e.g., [55]), and so(A,B) can be transformed into (A, b) by a linear transformation of coordinates, leading tothe desired result.

Finally, to complete the proof of Theorem 2.5, we prove Lemma 2.8, which gives the uni-form exponential stability of the limit system considered in the proof of Theorem 2.5.

Proof of Lemma 2.8. We consider the matrices Pj` as a perturbations in (2.21), and so weconsider first the non-perturbed system

y0(t) = Jr0y0(t)−h

∑`=0

[b0K`⊗C0`(t)]y`(t),

y j(t) = JCr j

y j(t)−h

∑`=0

[b jK`⊗C j`(t)]y`(t), j = 1, . . . ,h.

(2.46)

Let ξ > 0. It has been proved in [19, Theorem 3.2] that, for a given ξ > 0, one can finda gain K=

(K0 K1 · · · Kh

)and a positive definite matrix S ∈Md(R) such that, for ev-

ery symmetric C? ∈ L∞(R+,M2h+1− j0(R)) satisfying C?(t) ≥ ξ Id2h+1− j0 for almost everyt ≥ 0, (2.46) is globally uniformly exponentially stable and V (y) = yTSy decreases exponen-tially along all trajectories of (2.46), uniformly with respect to C? ∈ L∞(R+,M2h+1− j0(R))satisfying C?(t) ≥ ξ Id2h+1− j0 almost everywhere; i.e., there exist C ≥ 1 and γ > 0 suchthat, for every symmetric C? ∈ L∞(R+,M2h+1− j0(R)) satisfying C?(t)≥ ξ Id2h+1− j0 almosteverywhere and every solution y of (2.46), we have

‖y(t)‖ ≤Ce−2γt ‖y(0)‖ .

We denote by X(t,s) the fundamental matrix solution of (2.46), i.e., for any y0 ∈Rd , y(t) =X(t,s)y0 is the unique solution to (2.46) with y(s) = y0. Hence we have the estimate

‖X(t,s)‖ ≤Ce−2γ(t−s). (2.47)

We now turn to the perturbed system (2.21). For a given ξ > 0, we take C≥ 1, γ > 0 andK j as before. For every symmetric matrix C? ∈ L∞(R+,M2h+1− j0(R)) satisfying C?(t) ≥ξ Id2h+1− j0 almost everywhere, and every P? ∈ L∞(R+,M2h+1− j0(R)) satisfying (2.23), weset A= diag(Jr0,J

Cr1, . . . ,JC

rh) ∈Md(R),

B(t) =(b jK`⊗C j`(t)

)j0≤ j,`≤h , P(t) =

(b jK`⊗Pj`(t)

)j0≤ j,`≤h

with b0 = b0. System (2.21) can thus be written under the form

y(t) =Ay(t)−B(t)y(t)−P(t)y(t)

28

and, using the fundamental matrix X of (2.46), we can write its solution for a given initialcondition y0 as

y(t) = X(t,0)y0−w t

0X(t,s)P(s)y(s)ds.

By (2.23), we can write ‖P(t)‖ ≤C′rΩ for a certain constant C′ > 0, and thus, up to increas-ing C, we have, by (2.47),

‖y(t)‖ ≤Ce−2γt ∥∥y0∥∥+CrΩ

w t

0e−2γ(t−s) ‖y(s)‖ds.

Applying Gronwall’s Lemma to e2γt ‖y(t)‖, we thus obtain

‖y(t)‖ ≤Ce−(2γ−CrΩ)t ∥∥y0∥∥ .We choose r > 0 small enough so that 2γ−CrΩ≥ γ , and so

‖y(t)‖ ≤Ce−γt ∥∥y0∥∥ ,which gives us the desired result.

29

30

Chapter 3

Transport equation on circles

3.1 IntroductionIn this chapter, we study the following persistently excited damped transport equation,

∂tu1(t,x)+∂xu1(t,x) = 0, t ∈ R+, x ∈ [0,L1],

∂tu2(t,x)+∂xu2(t,x)+α(t)χ(x)u2(t,x) = 0, t ∈ R+, x ∈ [0,L2],

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, t ∈ R+,

u j(0,x) = u j,0(x), x ∈ [0,L j], j ∈ 1,2,α ∈ G(T,µ).

(3.1)

System (3.1) can be seen as the transport equation on two tangent circles C1 and C2 with re-spective lengths L1,L2 > 0. The origin of the measure of length along these circles is definedto be their intersection point, at which we have a transmission condition stating that the massu1(t,L1)+u2(t,L2) arriving at the intersection point at time t splits in two equal parts going oneach circle. The function χ determines the damping on a certain part of the circle C2 and will betaken here as the characteristic function of a subinterval [a,b] of [0,L2]. The signal α belongsto the class G(T,µ) of (T,µ)-persistently exciting signals, which, for given T ≥ µ > 0, consistson the signals α ∈ L∞(R+, [0,1]) satisfying

w t+T

tα(s)ds≥ µ (3.2)

for every t ≥ 0. Such a signal α determines when and how much of the damping is active.As we have mentioned in Section 1.4, and as we shall detail in Section 3.3, (3.1) is a “toy

model” for the study of the wave equation on networks of strings. Networks of strings, or,more generally, partial differential equations defined on several coupled domains, is an activeresearch subject that attracts much interest due to both its applications and the complexity of itsanalysis [2, 11, 23, 28, 39, 46, 57, 58, 60]. Several practical problems, particularly in mechanics,involve a structure composed of a number of elements, such as strings, membranes, plates,which are coupled by junctions at some points or surfaces, and the interest is on some physicalphenomenon on this structure, such as the propagation of waves or of heat. These systems,usually called multi-link or multi-body structures, are of a high practical relevance, and theiranalysis is difficult due to the phenomena arising from the coupling between the parts of thesystem. In this context, the study of the networks of strings, i.e. of systems where the constituent

31

elements are one-dimensional strings distributed along a planar graph, is of much importance,since this simple model highlights the several phenomena arising from the coupling of thestrings.

As remarked in [60], the study of networks of strings submitted to switching controls is aninteresting open problem, with some results being presented, for instance, in [28]. Our moti-vation to consider (3.1) is to understand the effects of a persistently excited internal dampingon the dynamics of our toy model before turning to more complicated problems involving apersistently excited damping in one or several strings of a network.

The plan of this chapter is the following. Section 3.2 gives the main notations and definitionsused throughout this chapter. In Section 3.3, we motivate the study of our toy model (3.1) ofthe transport equation on circles by establishing the correspondence between the wave equationon a segment and the transport equation on a circle. Section 3.4 studies (3.1) in the undampedcase, i.e., when χ ≡ 0, obtaining its asymptotic properties by using the L2 norm as a Lyapunovfunction and applying LaSalle’s Principle, both of these being recalled in Appendix 3.A. Inparticular, we will see that the asymptotic properties of (3.1) depend on the rationality of theratio L1

L2: when this quantity is rational, (3.1) admits non-constant periodic solutions, whereas

every solution of (3.1) converges to a constant when L1L2

/∈ Q. We also derive, in Section 3.4,an explicit formula for the solution of (3.1) with χ ≡ 0, by following the flow of the transportequation in order to determine which points of the initial condition influence the value of thesolution u j(t,x) at a certain time t in a certain position x∈ [0,L j], and we obtain a combinatorialformula giving the weight that each of these points has in the expression of u j(t,x).

Section 3.5 studies (3.1) in the case of an always active damping, i.e., with α ≡ 1. Theinterest of this study is to understand the behavior of (3.1) when α ≡ 1 in order to know thebehavior we might expect to obtain in the case of a persistently excited damping, since, as usual,the goal of the study of a persistently excited system is to understand if the properties of thenon-PE system still hold in the PE case with a certain robustness with respect to the PE signalα . The technique used in Section 3.5 is to modify the explicit formula of the solution derived inSection 3.4.5 to take into account the damping and show the exponential stability of the systemdirectly from this explicit formula. The choice of this technique is done bearing in mind thepossible extension of our method to a persistently excited damping, which is what we describein Section 3.6. Even though we still do not have a stability result for the PE damped case, wepresent several advances in this direction, showing how the PE condition can be translated interms of the coefficients of the explicit formula of the solution and giving some preliminaryestimates on these coefficients.

3.2 Notations and definitions

In this chapter, we denote by Z the set of all integers, N= 0,1,2,3, . . . the set of nonnegativeintegers, N∗ = 1,2,3, . . . the set of positive integers, Q the set of rational numbers, R the setof real numbers, R+ = [0,+∞) the set of nonnegative real numbers and R∗+ = (0,+∞) the setof positive real numbers.

All Banach and Hilbert spaces considered are supposed real, but this is done only to fixthe ideas when doing computations, and all our results still hold true with the same proof inthe case of complex vector spaces. Accordingly, we also suppose that the elements of theLebesgue spaces Lp and the Sobolev spaces Hs and Hs

0 are real-valued functions. The scalarproduct between two elements u,v on a Hilbert space X is denoted by 〈u,v〉X, and the norm of

32

an element u on a Banach space X is denoted by ‖u‖X, this norm being ‖u‖X =√〈u,u〉X if X

is a Hilbert space. The indices X will be abandoned from the previous notations when the spaceconsidered is clear from the context. Strong convergence of a sequence (un)n∈N in a Banach

space X to u ∈ X is denoted by unX−−−→

n→∞u, and the weak convergence of (un)n∈N in a Hilbert

space X to u∈X is denoted by unX−−−

n→∞u, the indices X and n→∞ being possibly omitted when

clear from the context.We shall refer to linear operators in a Banach space X simply as operators. The domain of

an operator T on X is denoted by D(T ). The set of all bounded operators from a Banach spaceX to a Banach space Y is denoted by L(X,Y) and is provided with its usual norm ‖·‖L(X,Y),which gives to L(X,Y) the structure of a Banach space. The notation L(X) is used to denoteL(X,X).

For two topological spaces X and Y , C(X ,Y ) = C0(X ,Y ) denotes the set of continuousfunctions from X to Y , with the convention C0(X) = C0(X ,R). For an interval I ⊂ R andk ∈N, Ck(I) denotes the set of the k times differentiable real-valued functions defined on I, andCk

c(I) is the subset of Ck(I) of the compactly supported functions.For a real number x, bxc ∈ Z denotes the greatest integer k ∈ Z such that k ≤ x, and dxe ∈ Z

denotes the smallest integer k ∈ Z such that k≥ x. Clearly, x−1 < bxc ≤ x and x≤ dxe< x+1for every x ∈ R. For y > 0, we denote by xy the number xy = x−

⌊xy

⌋y, which satisfies

0 ≤ xy < y; when y = 1, x1 is simply the fractional part of x. If n ∈ N, its factorial isdenoted as usual by n!, and, for k ∈ Z, we denote the binomial coefficient of n and k as

(nk

),

which is(n

k

)= n!

k!(n−k)! if 0≤ k≤ n and(n

k

)= 0 otherwise. For a finite set F , we shall denote its

cardinality by #F . These notations will be useful when considering the explicit formula for thesolutions of (3.1) in Sections 3.4.5, 3.5.1 and 3.6.1.

The interest of this chapter is the study of System (3.1). The signal α is a persistentlyexciting signal, whose definition we now recall.

Definition 3.1. Let T , µ be two positive constants with T ≥ µ > 0. We say that a measurablefunction α : R+→ [0,1] is a (T,µ)-signal if, for every t ∈ R+, one has

w t+T

tα(s)ds≥ µ.

The set of (T,µ)-signals is denoted by G(T,µ). If α ∈ G(T,µ) for certain constants T ≥ µ > 0,we say that α is a persistently exciting signal, or PE signal for short.

We shall refer to System (3.1) as being undamped in the particular case where χ ≡ 0, ashaving an always active damping or as being a non-PE system when α ≡ 1, and as a persistentlyexcited damped system or with a persistently excited damping in the general case.

3.3 From the wave equation on a segment to the transportequation on a circle

Consider the wave equation on a segment [0,L] with Dirichlet boundary conditions,∂

2tt u(t,x) = ∂

2xxu(t,x), t ∈ R+, x ∈ [0,L],

u(0,x) = u0(x), x ∈ [0,L],∂tu(0,x) = u1(x), x ∈ [0,L],

u(t,x) = 0, t ∈ R+, x ∈ 0,L.

(3.3)

33

We introduce the Hilbert space H = H10 (0,L)× L2(0,L) with its usual scalar product, given

by 〈(u1,u2),(v1,v2)〉H = 〈u1,v1〉H10 (0,L)

+〈u2,v2〉L2(0,L) = 〈u′1,v′1〉L2(0,L)+〈u2,v2〉L2(0,L), and wedefine the operator T : D(T )⊂ H→ H by

D(T ) = (H2(0,L)∩H10 (0,L))×H1

0 (0,L),

T(

u1u2

)=

(u2u′′1

).

This operator allows us to write (3.3) as a differential equation in the Hilbert space H asz = T z,

z(0) = (u0,u1),(3.4)

where the state z ∈ H is z = (u,∂tu). Clearly, any regular solution of (3.4) is a solution of (3.3)in the classical sense. It is also easy to see that T is a closed densely defined operator, and astraightforward computation shows that T ∗ =−T . Furthermore, for every u = (u1,u2) ∈D(T ),

〈Tu,u〉H =⟨u′2,u

′1⟩

L2(0,L)+⟨u′′1,u2

⟩L2(0,L) = 0,

and thus both T and T ∗ =−T are dissipative. Hence T is the generator of a strongly continuousgroup of isometries etTt∈R on H (see, e.g., [48, Chapter 1, Corollary 4.4]). For every u ∈D(T ), (3.4) admits thus a unique solution etT u continuously differentiable on R+. For u ∈ H,we shall also say that the continuous function t 7→ etT u is a solution of (3.4).

We now wish to define a unitary transformation on H corresponding to the D’Alembertdecomposition of (3.3) in two traveling waves [50, 51]. To do so, we first consider the space

G0 = (u1,u2) ∈ H1(0,L)×H1(0,L) |u1(x)+u2(x) = 0 for x ∈ 0,L.

We provide this space with the equivalence relation ∼ defined by

(u1,u2)∼ (v1,v2) ⇐⇒ ∃c ∈ R |u1− v1 = v2−u2 = c

and we consider the quotient space G=G0/∼. Elements of G are classes of equivalences of pairsof functions, but, in order to simplify the notations, we shall treat elements of G themselves aspairs of functions whenever this will not cause confusion. We provide the vector space G withthe scalar product 〈(u1,u2),(v1,v2)〉G = 〈u′1,v′1〉L2(0,L)+ 〈u

′2,v′2〉L2(0,L), which is clearly well-

defined since it depends on u1, v1, u2 and v2 only through its derivatives, and it is easy to seethat this scalar product provides G with the structure of a Hilbert space.

Note that the quotient G0/∼ is taken because the D’Alembert decomposition is not unique:if u1 and u2 are traveling waves in opposite directions such that u = u1 +u2 is a solution to thewave equation (3.3), then, for every c∈R, u1+c and u2−c are also traveling waves representingthe same solution u of (3.3).

We define the operator U : H→ G by

U(

u1u2

)(x) =

(1√2

(u1(x)−

r x0 u2(ξ )dξ

)1√2

(u1(x)+

r x0 u2(ξ )dξ

)) , (3.5)

which is clearly well-defined; furthermore, one easily sees that 〈Uu,Uv〉G = 〈u,v〉H for everyu,v ∈ H, so that U is an isometry. It is also easy to check that U is invertible, its inverse beinggiven by

U−1(

u1u2

)=

(u1+u2√2

u′2−u′1√2

). (3.6)

34

Hence U is a unitary transformation from H to G. The operator S =UTU−1 is thus the generatorof a strongly continuous group of isometries etSt∈R on G with etS =UetTU−1.

Let us describe this operator S. Its domain is given by

D(S) = u ∈ G |U−1u ∈ D(T ),

so that any u = (u1,u2) ∈ D(S) can be written under the form u =Uv for a certain v ∈ D(T ) =(H2(0,L)∩H1

0 (0,L))×H10 (0,L). By the explicit formula (3.5) for U , one obtains

(u1,u2) ∈ H2(0,L)×H2(0,L), u′2(x) = u′1(x) for x ∈ 0,L. (3.7)

Conversely, if u = (u1,u2) ∈ G satisfies (3.7), then the explicit formula for U−1 shows easilythat U−1u ∈ D(T ), and thus

D(S) = (u1,u2) ∈ G∩ (H2(0,L)×H2(0,L)) |u′2(x) = u′1(x) for x ∈ 0,L.

Now, for u = (u1,u2) ∈ D(S), we have

Su =UTU−1u =UT

(u1+u2√2

u′2−u′1√2

)=U

u′2−u′1√2

u′′1+u′′2√2

=

=

(12

[u′2−u′1−

r x0 (u′′1(ξ )+u′′2(ξ ))dξ

]12

[u′2−u′1 +

r x0 (u′′1(ξ )+u′′2(ξ ))dξ

])=

(−u′1 +

u′1(0)+u′2(0)2

u′2−u′1(0)+u′2(0)

2

)=

(−u′1u′2

),

where the last equality holds as an equality in G.If z(t) = (u1(t),u2(t)) is a solution of z = Sz in R+, i.e., if z(t) ∈ D(S) is such that z(t) =

etSz(0), then we see that u1 and u2 satisfy the system of transport equations∂tu1(t,x)+∂xu1(t,x) = 0, t ∈ R+, x ∈ [0,L],∂tu2(t,x)−∂xu2(t,x) = 0, t ∈ R+, x ∈ [0,L],u1(t,x) =−u2(t,x), t ∈ R+, x ∈ 0,L.

(3.8)

Note that we do not include the condition ∂xu2(t,x) = ∂xu1(t,x) for x ∈ 0,L since it can bededuced from the previous ones. System (3.8) is thus composed of two transport equations,one corresponding to a transport to the right, the other corresponding to a transport to the left,and with a transmission condition at the extremities that couples the two equations. The unitarytransformation U thus corresponds to the decomposition of the solution of the wave equationinto traveling waves.

We now want to perform another transformation in G in order to obtain the transport equa-tion on a circle. We define the space F0 by

F0 = u ∈ H1(0,2L) |u(2L) = u(0),

where we define the equivalence relation ≈ by

u≈ v ⇐⇒ ∃c ∈ R |u− v = c.

As before, the quotient space F = F0/≈ can be provided with the scalar product 〈u,v〉F =〈u′,v′〉L2(0,2L), with which F becomes a Hilbert space.

35

Consider the operator V : G0→ F0 defined by

V(

u1u2

)(x) =

u1(x) if x ∈ [0,L],−u2(2L− x) if x ∈ (L,2L].

(3.9)

V is well-defined, since u1(L) =−u2(L) and u1(0) =−u2(0), and V is also invertible, with

V−1u(x) =(

u(x)−u(2L− x)

). (3.10)

Note also that, for u,v ∈ G0, u ∼ v ⇐⇒ Vu ≈ V v, and thus V and V−1 define operators (stillnoted the same way) V : G→ F and V−1 : F→ G. It is also easy to see, by the definitions ofthe scalar products in G and F, that 〈Vu,V v〉F = 〈u,v〉G, so that V is an isometry and thus aunitary operator. The operator R =V SV−1 is thus the generator of a strongly continuous groupof isometries etRt∈R on F with etR =VetSV−1.

Proceeding in a similar way as before, we can see that the domain of R is

D(R) = u ∈ H2(0,2L) |u′(0) = u′(2L)

and thatRu =−u′.

Thus the differential equation z = Rz corresponds to the transport equation∂tu(t,x)+∂xu(t,x) = 0, t ∈ R+, x ∈ [0,2L],u(t,0) = u(t,2L), t ∈ R+,

which can be seen as a transport equation on a circle thanks to the periodicity condition u(t,0) =u(t,2L) for every t ≥ 0.

We thus conclude that the unitary operators U and V transform a wave equation on a segmentinto a transport equation on a circle. They give a correspondence between solutions of thistransport equation and solutions of the original wave equation, this correspondence being givenby (3.5), (3.6), (3.9) and (3.10), and it is this correspondence that we use to justify and motivateour study of transport equations on circles as a preliminary study for the study of the waveequation on networks of strings. With this in mind, we will restrict ourselves in the sequel tothe study of the transport equation on circles (3.1).

3.4 The undamped transport equation on circlesWe consider here the system of undamped transport equations

∂tu j(t,x)+∂xu j(t,x) = 0, t ∈ R+, x ∈ [0,L j], j ∈ 1,2,

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, t ∈ R+,

u j(0,x) = u j,0(x), x ∈ [0,L j], j ∈ 1,2,

(3.11)

which corresponds to (3.1) with χ ≡ 0. This system can be seen as the transport equation withunitary velocity on two circles, C1 and C2, as in Figure 3.1, of respective lengths L1 and L2,which intersect in one point, chosen to be the origin of the measure of length along the circles

36

0 0L1 L2C1 C2

FIGURE 3.1: Interpretation of (3.11) as a transport equation on two circles C1 and C2 of respectivelengths L1 and L2, the arrows giving the sense of the transport on each circle.

and where we have a transmission condition stating that the arriving mass u1(t,L1)+ u2(t,L2)at time t is split in equal parts going on each circle.

We wish to write (3.11) as a differential equation in an appropriate Hilbert space. To doso, we consider the Hilbert space X = L2(0,L1)×L2(0,L2) and we introduce the operator A :D(A)⊂ X→ X defined on its domain D(A) by

D(A) =(u1,u2) ∈ H1(0,L1)×H1(0,L2) |u1(0) = u2(0) =

u1(L1)+u2(L2)

2

,

A(u1,u2) =

(−du1

dx,−du2

dx

).

Note that A is well-defined: the Sobolev embedding H1(0,L j) → C0([0,L j]) (see, for instance,[1, Theorem 5.4]) justifies that the punctual values u j(0),u j(L j), j ∈ 1,2, are well-defined

for every (u1,u2) ∈ D(A), and is it clear that(−du1

dx ,−du2dx

)∈ X for every (u1,u2) ∈ D(A).

Setting z = (u1,u2), (3.11) can be written in an abstract manner asz = Az,

z(0) = z0(3.12)

with z0 = (u1,0,u2,0).

3.4.1 Well-posednessWe wish to study the well-posedness of the equation (3.12), that is, we wish to investigate if, forevery z0 ∈D(A), (3.12) has a unique solution z(t) depending continuously on z0. As remarked in[48, Chapter 4, Theorem 1.3], in the case where A is densely defined with a nonempty resolventset, this is equivalent to A being the generator of a strongly continuous semigroup etAt≥0 onX, which is what we show in the sequel.

Proposition 3.2. A is closed and densely defined. With the norm of the graph, D(A) is a Hilbertspace compactly embedded in X.

Proof. Clearly, A is a densely defined operator since C∞c ((0,L1))×C∞

c ((0,L2)) is a subsetof D(A).

By the definition of A, the norm of the graph in D(A) is the usual norm in H1(0,L1)×H1(0,L2), that is,

‖z‖2D(A) = ‖u1‖2

L2(0,L1)+‖u2‖2

L2(0,L2)+∥∥u′1∥∥2

L2(0,L1)+∥∥u′2∥∥2

L2(0,L2)

37

for z = (u1,u2) ∈ D(A). But we also have that D(A) is closed in H1(0,L1)×H1(0,L2),thanks to the Sobolev embedding H1(0,L j) → C0([0,L j]), and so D(A) is a Hilbert space.Note that this also shows that A is closed. Furthermore, the embedding H1(0,L j) →L2(0,L j) is compact by Rellich-Kondrachov Theorem (see, e.g., [13, Théorème IX.16]),and thus D(A) is compactly embedded in X.

Proposition 3.3. The adjoint operator A∗ of A is given by

D(A∗) =(u1,u2) ∈ H1(0,L1)×H1(0,L2) |u1(L1) = u2(L2) =

u1(0)+u2(0)2

,

A∗(u1,u2) =

(du1

dx,du2

dx

).

Proof. Note that, since A is densely defined, A∗ is well-defined and closed; furthermore, A∗

is also densely defined since A is closed.Let z = (u1,u2) ∈ D(A∗), y = (v1,v2) ∈ D(A), and note A∗z = (w1,w2). We have

〈A∗z,y〉= 〈z,Ay〉=−w L1

0u1v′1−

w L2

0u2v′2. (3.13)

Taking in particular v1 ∈ C∞c ((0,L1)) and v2 = 0, we obtain that

w L1

0u1v′1 =−〈A∗z,y〉=−

w L1

0w1v1, ∀v1 ∈ C∞

c ((0,L1)).

This shows that u′1 = w1 and thus u1 ∈ H1(0,L1). Similarly, u′2 = w2 and u2 ∈ H1(0,L2),and thus A∗(u1,u2) = (u′1,u

′2).

For any y ∈ D(A), we can integrate (3.13) by parts to obtain

〈A∗z,y〉=w L1

0u′1v1 +

w L2

0u′2v2−u1(L1)v1(L1)+u1(0)v1(0)−u2(L2)v2(L2)+u2(0)v2(0).

On the other hand, since A∗z = (w1,w2) = (u′1,u′2), we have

〈A∗z,y〉=w L1

0u′1v1 +

w L2

0u′2v2,

and so

−u1(L1)v1(L1)+u1(0)v1(0)−u2(L2)v2(L2)+u2(0)v2(0) = 0, ∀(v1,v2) ∈ D(A).

By the definition of D(A), we obtain

v1(L1)

[u1(0)+u2(0)

2−u1(L1)

]+

+ v2(L2)

[u1(0)+u2(0)

2−u2(L2)

]= 0, ∀(v1,v2) ∈ D(A),

and thus

u1(L1) = u2(L2) =u1(0)+u2(0)

2.

38

This shows that

D(A∗)⊂(u1,u2) ∈ H1(0,L1)×H1(0,L2) |u1(L1) = u2(L2) =

u1(0)+u2(0)2

.

Conversely, suppose that z = (u1,u2) ∈ H1(0,L1)×H1(0,L2) is such that u1(L1) =

u2(L2) =u1(0)+u2(0)

2 . Then, for y = (v1,v2) ∈ D(A), we have

〈z,Ay〉=−w L1

0u1v′1−

w L2

0u2v′2 =

=w L1

0u′1v1 +

w L2

0u′2v2−u1(L1)v1(L1)+u1(0)v1(0)−u2(L2)v2(L2)+u2(0)v2(0) =

=w L1

0u′1v1 +

w L2

0u′2v2

since−u1(L1)v1(L1)+u1(0)v1(0)−u2(L2)v2(L2)+u2(0)v2(0) = 0 thanks to the hypothesison z and to the fact that y ∈ D(A). This shows that y 7→ 〈z,Ay〉 can be extended to a linearform on X, and so z ∈ D(A∗), from where we get the desired result.

Proposition 3.4. The operators A and A∗ are both dissipative.

Proof. Take z = (u1,u2) ∈ D(A). We have

〈z,Az〉=−w L1

0u1u′1−

w L2

0u2u′2 =

=w L1

0u1u′1 +

w L2

0u2u′2−u1(L1)

2 +u1(0)2−u2(L2)2 +u2(0)2 =

=−〈z,Az〉−u1(L1)2−u2(L2)

2 +(u1(L1)+u2(L2))

2

2=−〈z,Az〉− (u1(L1)−u2(L2))

2

2,

and so

〈z,Az〉=−(u1(L1)−u2(L2))2

4≤ 0.

Thus A is dissipative.The computations are similar for A∗: taking z = (u1,u2) ∈ D(A∗), we have

〈z,A∗z〉=w L1

0u1u′1 +

w L2

0u2u′2 =

=−w L1

0u1u′1−

w L2

0u2u′2 +u1(L1)

2−u1(0)2 +u2(L2)2−u2(0)2 =

=−〈z,A∗z〉−u1(0)2−u2(0)2 +(u1(0)+u2(0))2

2=−〈z,A∗z〉− (u1(0)−u2(0))2

2,

and so

〈z,A∗z〉=−(u1(0)−u2(0))2

4≤ 0.

Thus A∗ is dissipative.

Propositions 3.2, 3.3, and 3.4 show that A is a closed densely defined operator such thatboth A and A∗ are dissipative, and thus A is the generator of a strongly continuous semigroupof contractions etAt≥0 on X (see, e.g., [48, Chapter 1, Corollary 4.4]). For every z0 ∈ D(A),(3.12) admits thus a unique solution etAz0 continuously differentiable on R+. For z0 ∈ X, weshall also say that the continuous function t 7→ etAz0 is a solution of (3.12).

39

3.4.2 Asymptotic behaviorWe now propose to study the asymptotic behavior of the solutions of (3.12) in the case wherethe ratio L1

L2is irrational. We will show that, in this case, every solution of (3.12) converges to

a constant, whose value can be determined explicitly. Our technique consists on using a non-strict Lyapunov function and applying LaSalle Principle in order to conclude the convergence,the value of the constant being computed thanks to a conservation law. We recall the frameworkfor Lyapunov functions in Banach spaces, as presented in [29, 32, 54], in Appendix 3.A, wherewe also provide the definitions of Lyapunov function and ω-limit set used in this section.

Let us consider the asymptotic behavior of the system (3.12). We take V : X→ R as thefunction V (z) = ‖z‖2

X.

Lemma 3.5. V is a Lyapunov function for etAt≥0. If z = (u1,u2) ∈ D(A), we have

V (z) = 2〈z,Az〉=−(u1(L1)−u2(L2))2

2.

Proof. This comes from the fact that A is dissipative. Indeed, take z∈D(A), so that t 7→ etAzis continuously differentiable in R+; thus t 7→ V (etAz) is continuously differentiable in R+

withddt

V (etAz) = 2⟨

etAz,AetAz⟩≤ 0

since A is dissipative. Thus V (z) = 2〈z,Az〉 ≤ 0 for every z ∈D(A), and the explicit compu-tation of 〈z,Az〉 has been done in Proposition 3.4. This also shows that∥∥∥etAz

∥∥∥X≤ ‖z‖X , ∀z ∈ D(A), ∀t ≥ 0,

and, by the density of D(A) in X, we obtain that∥∥∥etAz∥∥∥X≤ ‖z‖X , ∀z ∈ X, ∀t ≥ 0.

Thus V (z)≤ 0 for every z ∈ X, and so V is a Lyapunov function for etAt≥0.

Lemma 3.6. For every z ∈ D(A), etAz | t ≥ 0 is precompact in X.

Proof. Since etAt≥0 is a contraction semigroup in the Hilbert space X, etAt≥0 is also acontraction semigroup in the Hilbert space D(A), and so, for every z ∈ D(A), etAz | t ≥ 0is a bounded set in D(A), which is thus precompact in X thanks to the compact embeddingD(A) → X.

In the next lemma, for z0 ∈ X, ω(z0) denotes its ω-limit set, whose definition is recalled inAppendix 3.A, Definition 3.39.

Lemma 3.7. If z0 ∈ D(A), then ω(z0)⊂ D(A).

Proof. Take z ∈ ω(z0) and (tn)n∈N a nondecreasing sequence in R+ with tn −−−→n→∞

+∞ such

that etnAz0X−−−→

n→∞z. Since etAt≥0 is a contraction semigroup in X, it also defines a contrac-

tion semigroup in the Hilbert space D(A) with its graph norm, and so∥∥∥etnAz0

∥∥∥D(A)≤ ‖z0‖D(A) .

40

Thus (etnAz0)n∈N is a bounded sequence in D(A), and thus, up to the extraction of a sub-sequence of (tn)n∈N, which we still note by (tn)n∈N for the sake of simplicity, (etnAz0)n∈Nconverges weakly to a certain w ∈ D(A). Thanks to the compact embedding D(A) → X, we

obtain that etnAz0X−−−→

n→∞w, and so w = z. Since w ∈ D(A), we thus get that z ∈ D(A) and so

ω(z0)⊂ D(A).

We now set, as in LaSalle Principle (Appendix 3.A, Theorem 3.43), E = z ∈ X |V (z) = 0,and we denote by M the maximal invariant subset of E.

Lemma 3.8. Suppose L1L2

/∈Q. Then

D(A)∩M = (λ ,λ ) ∈ L2(0,L1)×L2(0,L2) |λ ∈ R,

i.e., D(A)∩M is the set of constant functions on L2(0,L1)×L2(0,L2).

Proof. Take z0 = (u1,0,u2,0) ∈ D(A)∩M. By the invariance of M, there exists a continuousfunction z : R→ M such that z(0) = z0 and etAz(s) = z(t + s) for every s ∈ R, t ≥ 0. Inparticular, z(t) = etAz0 for t ≥ 0, and so z(t) ∈ D(A)∩M for every t ≥ 0.

Let us note z(t) = (u1(t),u2(t)), which is a solution of (3.11) with initial condition z0.Since z(t) ∈ M, we have V (z(t)) = 0 for every t ≥ 0, which means, by Lemma 3.5, thatu1(t,L1) = u2(t,L2) for every t ≥ 0. Then we have that

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2= u1(t,L1) = u2(t,L2), ∀t ≥ 0.

We suppose that L1 < L2 to fix the ideas. For t ≥ L1 and x ∈ [0,L1], we have that

u1(t,x) = u1(t− x,0) = u2(t− x,0) = u2(t,x),

and so u1(t,x) = u2(t,x). Now, for t ≥ L1 and x ∈ [0,L1], we have that

u1(t +L1,x) = u1(t +L1− x,0) = u1(t +L1− x,L1) = u1(t− x,0) = u1(t,x)

and thus [L1,+∞)3 t 7→ u1(t,x) is a L1-periodic function for every x ∈ [0,L1]. Similarly, fort ≥ L2 and x ∈ [0,L2], we have that

u2(t +L2,x) = u2(t +L2− x,0) = u2(t +L2− x,L2) = u2(t− x,0) = u2(t,x)

and thus [L2,+∞) 3 t 7→ u2(t,x) is a L2-periodic function for every x ∈ [0,L2]. Sinceu1(t,x) = u2(t,x) for t ≥ L1, x ∈ [0,L1], we obtain that [L2,+∞) 3 t 7→ u1(t,x) is both L1-periodic and L2-periodic for every x ∈ [0,L1], and the fact that L1

L2/∈ Q thus implies that

[L2,+∞) 3 t 7→ u1(t,x) = u2(t,x) is constant for every x ∈ [0,L1]; let us note this constantvalue by λ (x). Clearly, λ (x) does not depend on x; indeed,

λ (x) = u1(t,x) = u1(t− x,0) = λ (0), ∀t ≥ L2 +L1, ∀x ∈ [0,L1],

and so we shall note this constant value simply by λ . We thus have that

u1(t,x) = u2(t,x) = λ , ∀t ≥ L2, ∀x ∈ [0,L1].

Note now that, since [L1,+∞) 3 t 7→ u1(t,x) is L1-periodic for every x ∈ [0,L1], we havethat

u1(t,x) = λ , ∀t ≥ L1, ∀x ∈ [0,L1].

41

We thus have that, for x ∈ [0,L1],

u1,0(x) = u1(0,x) = u1(L1− x,L1) = u1(L1− x,0) = u1(L1,x) = λ

and so u1,0 is constantly equal to λ .Note also that u2(t,x) = u2(t − x,0) = λ for every x ∈ [0,L2] and every t ≥ 2L2 and,

since [L2,+∞) 3 t 7→ u2(t,x) is L2-periodic for every x ∈ [0,L2], we obtain that

u2(t,x) = λ , ∀t ≥ L2, ∀x ∈ [0,L2].

We thus have that, for x ∈ [0,L2],

u2,0(x) = u2(0,x) = u2(L2− x,L2) = u2(L2− x,0) = u2(L2,x) = λ

and so u2,0 is constantly equal to λ . Thus z0 = (λ ,λ ), and we have thus shown that

D(A)∩M ⊂ (λ ,λ ) ∈ L2(0,L1)×L2(0,L2) |λ ∈ R.

The converse inclusion is trivial and this concludes the proof of our result.

Suppose now that z0 ∈ D(A). By Lemma 3.7, we have ω(z0) ⊂ D(A) and, by Lemma3.6 and Theorem 3.43, we have ω(z0) ⊂ M, so that ω(z0) ⊂ D(A)∩M and thus, by Lemma3.8, if L1

L2/∈ Q, we get that every function in ω(z0) is constant. We now wish to show that

ω(z0) contains exactly one function, which will imply that etAz0 converges to this function ast→+∞, since d(etAz0,ω(z0))→ 0 as t→+∞ by definition of ω(z0), where d denotes the usualpoint-to-set distance in X. To do so, we study a conservation law for (3.12).

We define U : X→ R by

U(u1,u2) =1

L1 +L2

(w L1

0u1(x)dx+

w L2

0u2(x)dx

).

Notice that U is well-defined and continuous in X since we have the continuous embeddingX → L1(0,L1)×L1(0,L2).

Lemma 3.9. For every z ∈ X and every t ≥ 0, we have U(etAz) =U(z).

Proof. By the density of D(A) in X and by the continuity of U , it suffices to show thisfor z ∈ D(A). In this case, the function t 7→ U(etAz) is differentiable in R+, and, notingetAz = (u1(t),u2(t)), we have

ddt

U(etAz) =1

L1 +L2

(w L1

0∂tu1(t,x)dx+

w L2

0∂tu2(t,x)dx

)=

=− 1L1 +L2

(w L1

0∂xu1(t,x)dx+

w L2

0∂xu2(t,x)dx

)=

=− 1L1 +L2

(u1(t,L1)−u1(t,0)+u2(t,L2)−u2(t,0)) = 0

since u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2 . We thus have the desired result.

Notice that U is actually a bounded linear operator. We denote by L : X→ X the boundedlinear operator defined by Lz = (U(z),U(z)). The main result of this section can thus be statedas follows.

42

Theorem 3.10. Suppose L1L2

/∈Q. Then, for every z0 ∈ X,

limt→+∞

etAz0 = Lz0.

Proof. Since L is a bounded operator and the semigroup etAt≥0 is uniformly bounded, itsuffices by density to show this result for z0 ∈ D(A). By Lemmas 3.6 and 3.7 and Theorem3.43, we have ω(z0) ⊂ D(A)∩M and thus, by Lemma 3.8, every function in ω(z0) is con-stant. Let z = (λ ,λ ) ∈ ω(z0) with λ ∈ R and take (tn)n∈N a nondecreasing sequence in R+

with tn→+∞ as n→ ∞ such that etnAz0→ z in X as n→ ∞. By the continuity of U and byLemma 3.9, we obtain that

λ =U(z) = limn→∞

U(etnAz0) =U(z0)

and thus z = Lz0. Hence ω(z0) = Lz0 and, by definition of ω(z0), this shows that etAz0→Lz0 as t→+∞, which gives the desired result.

3.4.3 Periodic solutionsIn Section 3.4.2, we established that every solution of (3.12) converges to a constant as t→+∞

if L1L2

/∈ Q. The situation, however, is quite different if L1L2∈ Q, and we now show that, in this

case, we have non-constant periodic solutions of (3.12).

Proposition 3.11. If L1L2∈Q, then there exists a non-constant periodic solution of (3.12).

Proof. Take p,q∈N∗ such that L1L2

= pq . We shall construct explicitly a non-constant periodic

solution of (3.12) in this case.Let us note `= L1

p = L2q . Take ϕ ∈ C∞

c (R) with support included in (0, `). For x ∈ [0,L1],we define u1,0 by

u1,0(x) =+∞

∑k=−∞

ϕ(x− k`);

note that, for each x ∈ R, there exists at most one k ∈ Z such that ϕ(x− k`) 6= 0, and sothis sum is actually reduced to at most one single term. In particular, u1,0 ∈ C∞([0,L1]).Similarly, for x ∈ [0,L2], we define u2,0 by the same expression,

u2,0(x) =+∞

∑k=−∞

ϕ(x− k`),

and u2,0 ∈ C∞([0,L2]). Define

u1(t,x) =+∞

∑k=−∞

ϕ(x− t− k`), u2(t,x) =+∞

∑k=−∞

ϕ(x− t− k`).

We clearly have u1(0,x)= u1,0(x) for every x∈ [0,L1], u2(0,x)= u2,0(x) for every x∈ [0,L2],and ∂tu j(t,x)+∂xu j(t,x) = 0 for every t ∈R+ and every x∈ [0,L j], j ∈ 1,2. We also have

u1(t,L1) = u2(t,L2) = u1(t,0) = u2(t,0) =+∞

∑k=−∞

ϕ(−t− k`)

43

since L1 = p`, L2 = q`. Thus

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, ∀t ≥ 0.

Hence (u1,u2) is the unique solution of (3.12) with initial data z0 = (u1,0,u2,0). It is periodicin time, and non-constant if ϕ is chosen to be non-constant.

3.4.4 Uniformity of the convergenceWe have shown in Theorem 3.10 the strong convergence of etA to L as t→+∞ in the case whereL1L2

/∈Q. We now show that this convergence is actually uniform with respect to the norm of theinitial data in D(A).

Theorem 3.12. If L1L2

/∈Q, then etA −−−−→t→+∞

L in L(D(A),X).

Proof. Let S = z ∈ D(A) | ‖z‖D(A) ≤ 1. Note that S is a closed subset of X: indeed, if

z is in the closure of S in X, then there exists a sequence (zn)n∈N in S such that znX−−−→

n→∞z.

Since zn ∈ S, the sequence (zn)n∈N is bounded in D(A), and so, since D(A) is a Hilbertspace, (zn)n∈N admits a weakly convergent subsequence, which we still denote by (zn)n∈N

for the sake of simplicity, and so there exists w∈D(A) such that znD(A)−−−n→∞

w. By the compact

embedding D(A) → X, we conclude that znX−−−→

n→∞w and thus w = z, from where we obtain

that z ∈ D(A) and znD(A)−−−n→∞

z. By the lower semicontinuity of the norm with respect to the

weak convergence, we obtain that ‖z‖D(A) ≤ 1, and thus z ∈ S, which shows that S is closedin X. Since S is bounded in D(A), the compact embedding D(A) → X shows that S is acompact subset of X. We endow S with the topology of X, and S is thus a compact metricspace.

For every t ≥ 0, we define the function ft : S→ X by

ft(z) = etAz.

We endow the set C(S,X) of continuous functions from S to X with the uniform norm

‖ f‖C(S,X) = supz∈S‖ f (z)‖X ; (3.14)

since S is compact, this norm is well-defined and C(S,X) is a Banach space. Clearly, ft ∈C(S,X) for every t ≥ 0.

Let F be the closure of ft | t ≥ 0 in C(S,X). We wish to apply Arzelà-Ascoli Theorem(see, for instance, [33]) to F in order to conclude its compactness on C(S,X).

Let us first note that, for every z∈ S, ft(z) |t ≥ 0= etAz |t ≥ 0 is precompact in X byLemma 3.6. By the definition of F , f (z) | f ∈ F is a subset of the closure of ft(z) | t ≥ 0in X, and thus f (z) | f ∈ F is precompact in X for every z ∈ S.

For every t ≥ 0 and for z1,z2 ∈ S, we have

‖ ft(z1)− ft(z2)‖X =∥∥∥etA(z1− z2)

∥∥∥X≤ ‖z1− z2‖X

44

since etAt≥0 is a contraction semigroup, and so the family ft | t ≥ 0 is equicontinuousin C(S,X); thus F is also equicontinuous in C(S,X).

By Arzelà-Ascoli Theorem, we conclude that F is a compact subset of C(S,X). Notenow that, by Theorem 3.10, ft(z)→ g(z) as t → +∞ for every z ∈ S, where g : S→ X isdefined by g(z) = Lz and thus g ∈ C(S,X). Since F is a compact subset of C(S,X), forevery nondecreasing sequence (tn)n∈N, ( ftn)n∈N admits a convergent subsequence in thetopology of C(S,X), and, due to the fact that ft(z)→ g(z) as t→+∞ for every z ∈ S, everyconvergent subsequence of ( ftn)n∈N in C(S,X) converges to g. This shows that ft → g inC(S,X) as t→+∞, which means that

limt→+∞

supz∈S‖ ft(z)−g(z)‖X = 0,

that is,

limt→+∞

supz∈D(A)‖z‖D(A)≤1

∥∥∥etAz−Lz∥∥∥X= 0,

which is the desired result.

Theorem 3.12 relies on the Arzelà-Ascoli Theorem to obtain the uniformity of the con-vergence of etA to L in L(D(A),X) as t → +∞. It is important here to consider etA and L asoperators in L(D(A),X) instead of L(X), since the set S = z ∈ X | ‖z‖X ≤ 1 is not compact inX, and compactness of S is essential for (3.14) to define a norm on C(S,X) and to apply Arzelà-Ascoli Theorem. We also remark that, even though our technique does give the uniformity ofthe convergence etA → L as t → +∞ in L(D(A),X), we cannot use this method to obtain therate at which this convergence occurs.

3.4.5 Explicit solution

We finish the study of the undamped transport equation (3.11) by giving an explicit formula forits solution. Let us first recall some notations introduced in Section 3.2. For x ∈ R, we denoteby bxc the integer part of x, i.e., bxc ∈ Z is the largest integer such that bxc ≤ x. For y > 0, wedenote by xy the number xy = x−

⌊xy

⌋y. Clearly, we have x−1 < bxc ≤ x and 0≤xy < y.

Theorem 3.13. Let z0 = (u1,0,u2,0) ∈ D(A). Then the solution of (3.11) is given by

u1(t,x) =

u1,0(x− t) if 0≤ t ≤ x,⌊t−xL2

⌋∑n=0

(n+⌊

t−x−nL2L1

⌋n

)2n+

⌊t−x−nL2

L1

⌋+1

u1,0(L1−t− x−nL2L1)+

+

⌊t−xL1

⌋∑

m=0

(m+⌊

t−x−mL1L2

⌋m

)2m+

⌊t−x−mL1

L2

⌋+1

u2,0(L2−t− x−mL1L2) if t > x,

(3.15a)

45

u2(t,x) =

u2,0(x− t) if 0≤ t ≤ x,⌊t−xL2

⌋∑n=0

(n+⌊

t−x−nL2L1

⌋n

)2n+

⌊t−x−nL2

L1

⌋+1

u1,0(L1−t− x−nL2L1)+

+

⌊t−xL1

⌋∑

m=0

(m+⌊

t−x−mL1L2

⌋m

)2m+

⌊t−x−mL1

L2

⌋+1

u2,0(L2−t− x−mL1L2) if t > x.

(3.15b)

Let us explain how to obtain these formulas using the flow of the transport equation and thetransmission condition at the contact point between the circles. The formula for 0 ≤ t ≤ x isclear since, in this case, we follow the transport flow simply from the initial condition. Whent ≥ x, also by following the transport flow, we see that u j(t,x) = u j(t− x,0) for j = 1,2, andthus it suffices to consider u j(t,0) for t ≥ 0 and j = 1,2. Thanks to the transmission conditionat the contact point between the circles, u1(t,0) = u2(t,0) for every t ≥ 0, and thus we consideronly u1(t,0) for t ≥ 0.

Note thatu1(t,0) =

12

u1(t,L1)+12

u2(t,L2). (3.16)

If t ≤minL1,L2, this can be written as

u1(t,0) =12

u1,0(L1− t)+12

u2,0(L2− t). (3.17)

Let us suppose for the moment, to fix the ideas, that L1 < L2 and suppose that L1≤ t ≤ L2. Thenwe write (3.16) as

u1(t,0) =12

u1(t−L1,0)+12

u2,0(L2− t) (3.18)

and, by considering (3.16) at time t−L1 and supposing that t ≤ 2L1, we obtain that

u1(t,0) =14

u1(t−L1,L1)+14

u2(t−L1,L2)+12

u2,0(L2− t) =

=14

u1,0(2L1− t)+14

u2,0(L2 +L1− t)+12

u2,0(L2− t)(3.19)

for L1 ≤ t ≤min2L1,L2. Note that both (3.17) and (3.19) give the same values as (3.15).The idea to obtain (3.17) and (3.19) can be summarized in Figure 3.2. The idea is to start

from the point O, representing the intersection point between the circles C1 and C2 at time t,and go backwards in time to find out which values of the initial condition will influence u1(t,0)and with which weight.

O

P

L1L2

t

`ab

t t

FIGURE 3.2: Graphical representation of the idea to obtain (3.17) and (3.19).

We start fromu1(t,0) =

12

u1(t,L1)+12

u2(t,L2)

46

and we follow the transport flow backwards in time in each of the terms u1(t,L1), u2(t,L2),which gives

u1(t,0) =12

u1(t− s,L1− s)+12

u2(t− s,L2− s), 0≤ s≤mint,L1,L2. (3.20)

Graphically, we represent this by two segments starting from O, at an angle of 45 degrees withthe vertical direction, perpendicular to each other, going upwards, one of length L1 and the otherof length L2, each one representing the solution in the circle with the corresponding length. Eachpoint of the segment of length L j represents a position x ∈ [0,L j] for j = 1,2 in the followingway: the intersection point O represents the maximal length x = L j, the upper extremity of thesegment represents the point x = 0 and, by noting s the distance from a point of the segment toO, the corresponding point represents x = L j− s.

We draw an horizontal line a which intersects both segments. The intersection points bothhave the same distance to O, which we note by s, and so these points represent the coordinatex = L j− s on the circle C j, j = 1,2. Hence, all the necessary information to (3.20) is given bythe line a through its intersection with the segments. Going backwards in time following thetransport flow corresponds to increasing s in (3.20) and thus to moving the line a upwards.

This description is sufficient for 0 ≤ t ≤ L1 (we recall that we suppose L1 < L2 to fix theideas), and it allows us to graphically obtain (3.20) by considering the intersection of a withthe segments. When t = L1, (3.20) gives (3.18), which means that we arrive at the point x = 0and, to move further backwards in time, we need to consider again the transmission condition(3.16). This corresponds to drawing again a pair of segments following the same constructionas before, but starting from the upper endpoint of the segment corresponding to C1, as we alsorepresent in Figure 3.2. When L1 ≤ t ≤ minL2,2L1, the horizontal line b intersects threesegments, corresponding to the three terms in (3.19): it intersects two segments correspondingto C2, and thus to the initial condition u2,0, and one segment corresponding to C1, and thus tothe initial condition u1,0. The distance ` between each intersection point and the lower end ofthe corresponding segment gives the point L j− ` where we evaluate the initial condition u j,0;this distance is ` = t−L1 for the intersection point with the segment corresponding to C1 and` = t − L1 and ` = t for the two intersection points with the segments corresponding to C2,which means that u1(t,0) is a linear combination of u1,0(L1− (t−L1)), u2,0(L2− (t−L1)) andu2,0(L2− t),

u1(t,0) = α1u1,0(L1− (t−L1))+α2u2,0(L2− (t−L1))+α3u2,0(L2− t). (3.21)

The coefficients α1, α2 and α3 can be computed thanks to the transmission condition at theintersection of the circles: for each intersection P between b and a segment (one such point isrepresented in Figure 3.2), we consider a path from P to O along the segments going strictlydownwards and we count the number n of intersections of segments that occur in this path,including O itself, which corresponds to the number of times that the initial condition u j,0(L j−`) must pass through the intersection between the circles in order to arrive at this intersectionin time t. We also count the number m of possible paths from P to O along the segmentsgoing strictly downwards (which is 1 for all the three intersection points in Figure 3.2), and thecoefficient α j corresponding to P is thus α j =

m2n . In our case, α1 =

122 , α2 =

122 and α3 =

121 ,

and thus (3.21) is simply (3.19).If we increase further t, thus increasing the distance from b to O, we arrive at another

endpoint of a segment, where we construct another pair of segments of lengths L1 and L2 asbefore. This construction can be continued at every upper endpoint of an interval, thus givingrise to a situation as in Figure 3.3, which is constructed as follows.

47

O

P

Q3,1(1,4,0) (2,1,2)

L1L2

bt

R0 R1 R2 R3 R4 S0S1S2S3S4S5S6S7

x

y

FIGURE 3.3: General method used to obtain (3.15), whose construction is given in Definition 3.14. Inorder to illustrate this definition, we represent here an horizontal line bt corresponding to a certain time

t, the point P of intersection between bt and R2, the point Q3,1 and the segments (1,4,0) and (2,1,2).

Definition 3.14. We consider the euclidean plan R2 with its canonical system of coordinates(x,y) and we place a point O at its origin. We place half-lines R0,R1,R2, . . . and S0,S1,S2, . . .defined by the equations

Rn : y =−x+n√

2L2, x≤ nL2√

2, n ∈ N,

Sm : y = x+m√

2L1, x≥−mL1√

2, m ∈ N.

The intersection between Rn and Sm occurs at the point Qm,n =(−m L1√

2+n L2√

2,m L1√

2+n L2√

2

).

We denote by R the rectangular grid obtained by the union of all the half-lines Rn and Sm,n,m ∈ N.

For a given point P ∈ R, a path connecting P to the origin O is a curve γ : [0, t]→ R,γ(s) = (γx(s),γy(s)), parametrized by arc length, with γ(0) =P, γ(t) =O and such that γy(s)< 0for every s ∈ [0, t]. Given P = (x,y) ∈R, it is easy to see that there is only a finite number ofsuch paths, that γt(s) = − 1√

2for every path γ = (γx,γy) connecting P to O, and that the total

length t of a path from P to O is t =√

2y, depending thus only on the y-coordinate of P.Segments of the type Qm,nQm+1,n have length L1 and represent a turn in the circle C1,

whereas segments of the type Qm,nQm,n+1 have length L2 and represent a turn in the circleC2. We represent each of these segments by a triple n = (n0,n1,n2) ∈ 1,2×N×N in thefollowing way: a segment of the kind Qm,nQm+1,n is represented by (1,m,n) and a segmentof the kind Qm,nQm,n+1 is represented by (2,m,n). We shall from now identify segments withelements of N = 1,2×N×N; segments of the kind Qm,nQm+1,n are thus the elements ofN1 = 1×N×N and segments of the kind Qm,nQm,n+1 are the elements of N2 = 2×N×N.Note that, for a given segment n= (n0,n1,n2) ∈N, n1 and n2 give the number of times that anypath connecting Qn1,n2 to the origin passes through segments of N1 and N2, respectively.

A given time t ≥ 0 is represented in this plan by the horizontal line bt of equation y = t√2.

48

The intersections between bt and the half-lines R j, S j allow us thus to obtain the expressionof u1(t,0), as we did before in the simpler case of (3.21), and we get

u1(t,0) =

⌊t

L2

⌋∑n=0

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

u1,0(L1−t−nL2L1)+

+

⌊t

L1

⌋∑

m=0

(m+⌊

t−mL1L2

⌋m

)2m+

⌊t−mL1

L2

⌋+1

u2,0(L2−t−mL1L2), ∀t ≥ 0. (3.22)

Indeed, let us consider the intersections between bt and the half-lines Rn, which give rise to theterm ⌊

tL2

⌋∑n=0

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

u1,0(L1−t−nL2L1) (3.23)

in (3.22). For n ∈ N, the half-line Rn has equation y = −x+ n√

2L2, x ≤ n L2√2, and thus an

intersection between Rn and bt occurs if and only if −n L2√2+ n√

2L2 ≤ t√2, that is, if and only

if n ≤ tL2

, and thus we only consider the intersections for n = 0, . . . ,⌊

tL2

⌋. Consider such an

intersection point P. Note that any path connecting P to O has total length t and passes exactlyn times on a segment of N2, which means that it passes

⌊t−nL2

L1

⌋times on a segment of N1,

that the lower endpoint of the segment containing P is Q⌊ t−nL2L1

⌋,n

, and that the distance from

P to Q⌊ t−nL2L1

⌋,n

is t − nL2L1 . Thus the corresponding term in the expression of u1(t,0) is

u1,0(L1−t−nL2L1), which is multiplied by a certain coefficient α . Since every path from Pto O is uniquely determined by the order in which it passes n times through the segments of N2

and⌊

t−nL2L1

⌋times through the segments of N1, the total number of such paths is the binomial

coefficient(n+

⌊t−nL2

L1

⌋n

), and every such path passes through n+

⌊t−nL2

L1

⌋+1 intersection points

Q j,k, which shows that

α =

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

and we thus get (3.23). The similar geometric argument gives the expression for the inter-sections between bt and the lines Sm and justifies the other term in (3.22). Since u1(t,x) =u1(t− x,0) for t ≥ x, this gives (3.15).

In order to conclude a rigorous proof of Theorem 3.13, we use the following lemma.

Lemma 3.15. Let z0 = (u1,0,u2,0) ∈ D(A) and define φ : R+→ R by

φ(t) =

⌊t

L2

⌋∑n=0

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

u1,0(L1−t−nL2L1)+

⌊t

L1

⌋∑

m=0

(m+⌊

t−mL1L2

⌋m

)2m+

⌊t−mL1

L2

⌋+1

u2,0(L2−t−mL1L2).

Then φ ∈ H1loc(R+).

49

Proof. We prove that, for every T > 0, φ ∈H1(0,T ), and we will deeply exploit the previousgeometric construction of φ .

Note that, except for a finite number of times 0 < t1 < t2 < · · · < tr ≤ T , the quantities⌊t

L1

⌋,⌊

tL2

⌋,⌊

t−mL1L2

⌋and

⌊t−nL2

L1

⌋are all locally constant in t, for every n ∈

0, . . . ,

⌊t

L2

⌋and every m ∈

0, . . . ,

⌊t

L1

⌋, and all the functions t 7→ t−mL1L2 and t 7→ t− nL2L1

are locally affine. This means that, for t /∈ t1, . . . , tr, φ can be written in a neighborhoodVt of t as a linear combination of H1 functions, which shows that φ ∈ H1(Vt). Since the sett1, . . . , tr is finite, to conclude that φ ∈ H1(0,T ), it suffices to show that φ is continuouson the points t1, . . . , tr.

The instants t1, . . . , tr correspond to the times for which the line bt from our previousconstruction intersects one of the half-lines Rn and one of the half-lines Sm at the samepoint Qm,n. Such an instant t j thus satisfies t j√

2= −x+ n

√2L2 = x+m

√2L1, where x is

the horizontal coordinate of Qm,n with respect to the coordinate system of Figure 3.3; thust j = nL2 +mL1. Let us now prove that

t 7→ ϕn,m(t) =

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

u1,0(L1−t−nL2L1)+

(m+⌊

t−mL1L2

⌋m

)2m+

⌊t−mL1

L2

⌋+1

u2,0(L2−t−mL1L2)

is continuous on t j. We have

limt→t j+

ϕn,m(t) = ϕn,m(t j) =

(n+mn

)2n+m+1 u1,0(L1)+

(m+nm

)2m+n+1 u2,0(L2) (3.24)

and

limt→t j−

ϕn,m(t) =

(n+m−1n

)2n+m u1,0(0)+

(m+n−1m

)2m+n u2,0(0). (3.25)

Since z0 ∈ D(A), we have u1,0(0) = u2,0(0) =u1,0(L1)+u2,0(L2)

2 , and thus

limt→t j−

ϕn,m(t) =1

2n+m+1

((n+m−1

n

)+

(m+n−1

m

))(u1,0(L1)+u2,0(L2)).

But(n+m−1

n

)+

(m+n−1

m

)=

(n+m−1

n

)+

(n+m−1

n−1

)=

(n+m

n

)=

(n+m

m

),

which gives the equality between (3.24) and (3.25) and thus the continuity of ϕn,m at t j.This holds for every n and m such that t j = nL2 +mL1; for the other values of n and m,⌊

t−mL1L2

⌋and

⌊t−nL2

L1

⌋are locally constant on t j and t 7→ t−mL1L2 and t 7→ t− nL2L1

are locally affine on t j, so that φ is continuous on t j. Thus φ is continuous on t1, . . . , trand this concludes the proof of the lemma.

Thanks to Lemma 3.15, we can prove Theorem 3.13.Proof of Theorem 3.13. Using the function φ introduced in Lemma 3.15, we can rewrite(3.15) as

u1(t,x) =

u1,0(x− t) if 0≤ t ≤ x,φ(t− x) if t > x,

u2(t,x) =

u2,0(x− t) if 0≤ t ≤ x,φ(t− x) if t > x.

50

Let us prove that (u1,u2) satisfies (3.11). Note that, for j = 1,2, u j(t,x) depends only

on the difference t − x and, since φ(0) = u1,0(L1)+u2,0(L2)2 = u1,0(0) = u2,0(0), we see that

u j ∈ H1loc(R+× [0,L j]). Moreover, since u j(t,x) depends only on the difference t − x, u j

satisfies the transport equation on R+× [0,L j]. It is also clear that u j(0,x) = u j,0(x) forevery x ∈ [0,L j], and thus it is left to prove that u1(t,0) = u2(t,0) =

u1(t,L1)+u2(t,L2)2 for every

t ≥ 0.Suppose, to fix the ideas, that L1 ≤ L2. For 0≤ t < L1, we have

u1(t,0) = u2(t,0) = φ(t) =12

u1,0(L1− t)+12

u2,0(L2− t) =u1(t,L1)+u2(t,L2)

2.

For L1 ≤ t < L2, we have

u1(t,0) = u2(t,0) = φ(t) =1

2⌊

tL1

⌋+1

u1,0(L1−tL1)+

⌊t

L1

⌋∑

m=0

12m+1 u2,0(L2− (t−mL1)),

whereas

u1(t,L1)+u2(t,L2)

2=

φ(t−L1)+u2,0(L2− t)2

=

=12

1

2⌊

tL1

⌋u1,0(L1−tL1)+

⌊t

L1

⌋−1

∑m=0

12m+1 u2,0(L2− (t− (m+1)L1))

+12

u2,0(L2− t) =

=1

2⌊

tL1

⌋+1

u1,0(L1−tL1)+

⌊t

L1

⌋∑

m=1

12m+1 u2,0(L2− (t−mL1))+

12

u2,0(L2− t)

and thus

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, ∀t ∈ [L1,L2) .

Finally, for t ≥ L2, we have

u1(t,L1)+u2(t,L2)

2=

φ(t−L1)+φ(t−L2)

2=

=12

⌊t−L1

L2

⌋∑n=0

(n+⌊

t−nL2L1

⌋−1

n

)2n+

⌊t−nL2

L1

⌋ u1,0(L1−t−nL2L1)+

+12

⌊t

L1

⌋−1

∑m=0

(m+⌊

t−(m+1)L1L2

⌋m

)2m+

⌊t−(m+1)L1

L2

⌋+1

u2,0(L2−t− (m+1)L1L2)+

+12

⌊t

L2

⌋−1

∑n=0

(n+⌊

t−(n+1)L2L1

⌋n

)2n+

⌊t−(n+1)L2

L1

⌋+1

u1,0(L1−t− (n+1)L2L1)+

+12

⌊t−L2

L1

⌋∑

m=0

(m+⌊

t−mL1L2

⌋−1

m

)2m+

⌊t−mL1

L2

⌋ u2,0(L2−t−mL1L2) =

51

=

⌊t−L1

L2

⌋∑n=0

(n+⌊

t−nL2L1

⌋−1

n

)2n+

⌊t−nL2

L1

⌋+1

u1,0(L1−t−nL2L1)+

⌊t

L1

⌋∑

m=1

(m−1+⌊

t−mL1L2

⌋m−1

)2m+

⌊t−mL1

L2

⌋+1

u2,0(L2−t−mL1L2)+

+

⌊t

L2

⌋∑n=1

(n−1+⌊

t−nL2L1

⌋n−1

)2n+

⌊t−nL2

L1

⌋+1

u1,0(L1−t−nL2L1)+

⌊t−L2

L1

⌋∑

m=0

(m+⌊

t−mL1L2

⌋−1

m

)2m+

⌊t−mL1

L2

⌋+1

u2,0(L2−t−mL1L2) =

=

⌊t

L2

⌋∑n=0

(n+⌊

t−nL2L1

⌋−1

n

)+(n−1+

⌊t−nL2

L1

⌋n−1

)2n+

⌊t−nL2

L1

⌋+1

u1,0(L1−t−nL2L1)+

+

⌊t

L1

⌋∑

m=0

(m−1+⌊

t−mL1L2

⌋m−1

)+(m+

⌊t−mL1

L2

⌋−1

m

)2m+

⌊t−mL1

L2

⌋+1

u2,0(L2−t−mL1L2) =

= φ(t) = u1(t,0) = u2(t,0)

where we use that, for m ∈⌊

t−L2L1

⌋+1, . . . ,

⌊t

L1

⌋, we have

m+

⌊t−mL1

L2

⌋−1≤ m+

t−mL1

L2−1≤ m+

tL2−(⌊

t−L2

L1

⌋+1)

L1

L2−1 <

< m+t

L2− t−L2

L2−1 = m,

so that(m+

⌊t−mL1

L2

⌋−1

m

)= 0, and similarly for n ∈

⌊t−L1

L2

⌋+1, . . . ,

⌊t

L2

⌋and the coefficient(n+

⌊t−nL2

L1

⌋−1

n

). This shows that

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, ∀t ≥ L2

and concludes the proof of the theorem.

3.5 The damped transport equation with an always activedamping

We now propose to study the system

∂tu1(t,x)+∂xu1(t,x) = 0, t ∈ R+, x ∈ [0,L1],

∂tu2(t,x)+∂xu2(t,x)+χ(x)u2(t,x) = 0, t ∈ R+, x ∈ [0,L2],

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, t ∈ R+,

u j(0,x) = u j,0(x), x ∈ [0,L j], j ∈ 1,2,

(3.26)

which corresponds to (3.1) with α ≡ 1, that is, with an always active damping. We supposehere that the function χ is the characteristic function of an interval [a,b]⊂ [0,L2]; (3.26) can beinterpreted as a transport equation in two tangent circles C1 and C2 with a damping term in thecircle C2, as we did in Figure 1.2 in Chapter 1.

52

Let us write (3.26) as a differential equation in the Hilbert space X= L2(0,L1)×L2(0,L2).We consider the same operator A from Section 3.4 and we defined the bounded linear operatorB on X by

B(

u1u2

)=

(0−χu2

).

Setting z = (u1,u2), (3.26) can be written as a differential equation in X asz = (A+B)z,

z(0) = z0(3.27)

with z0 = (u1,0,u2,0).Since B is bounded, the operator A+B in X is defined in the domain D(A), and so it is

closed and densely defined. A simple computation shows that B is self-adjoint and that, forevery z = (u1,u2) ∈ X,

〈Bz,z〉X =−w L2

0χ(x)u2(x)2dx≤ 0.

Combining this with Proposition 3.4, we obtain that both A+B and (A+B)∗ = A∗+B aredissipative, and thus A+B is the generator of a strongly continuous semigroup of contractionset(A+B)t≥0 on X. As in Section 3.4.1, this means that, for every z0 ∈ D(A), (3.27) admits aunique solution et(A+B)z0 continuously differentiable on R+, and, for z0 ∈ X, we also say thatthe continuous function t 7→ et(A+B)z0 is a solution of (3.27).

3.5.1 Explicit solution

We wish to study the stability of (3.26) by establishing an explicit formula for its solutions,similar to the one given in Theorem 3.13. Note that other techniques might be proposed to studythe stability of (3.26). For instance, one might look for an observability inequality, which is aclassical approach in Control Theory (see, for instance, [20,36,56]) that might actually be easierthan the study of the explicit formula of the solution in our case. However, we are studyingthe stability of (3.26) bearing in mind that our technique should also work for systems with apersistently excited damping, and the problem of finding a generalization of the observabilityinequality with the PE condition for our system seems quite difficult. We find that the use of theexplicit formula for the solution is valuable since this straightforward technique may prove tobe useful to study also the persistently excited damped case, as we shall later discuss in Section3.6.

In Section 3.4.5, the first step to obtain the explicit solution was to realize that the quantitiesu j(t,0), j = 1,2, suffice to describe the solution of (3.11). We will show that a similar resultalso holds in the damped case.

Lemma 3.16. Suppose (u1,0,u2,0) ∈ D(A). Then the corresponding solution (u1,u2) of (3.26)satisfies

u1(t,x) =

u1,0(x− t), if 0≤ t ≤ x,u1(t− x,0), if t ≥ x,

(3.28a)

53

u2(t,x) =

u2,0(x− t), if 0≤ t ≤ x and x≤ a,u2(t− x,0), if t ≥ x and x≤ a,

u2,0(x− t)e−t , if 0≤ t ≤ x−a and a≤ x≤ b,

u2,0(x− t)e−(x−a), if x−a≤ t ≤ x and a≤ x≤ b,

u2(t− x,0)e−(x−a), if t ≥ x and a≤ x≤ b,u2,0(x− t), if 0≤ t ≤ x−b and b≤ x≤ L2,

u2,0(x− t)e−t+x−b, if x−b≤ t ≤ x−a and b≤ x≤ L2,

u2,0(x− t)e−(b−a), if x−a≤ t ≤ x and b≤ x≤ L2,

u2(t− x,0)e−(b−a), if t ≥ x and b≤ x≤ L2.

(3.28b)

Proof. The undamped transport flow in the circle C1 gives u1(t,x) = u1,0(x− t) if 0≤ t ≤ xand u1(t,x) = u1(t− x,0) if t ≥ x, and so we have (3.28a).

For the second circle, we have the same situation if 0≤ x ≤ a, that is, before the actionof the damping, with u2(t,x) = u2,0(x− t) if 0≤ t ≤ x≤ a and u2(t,x) = u2(t−x,0) if t ≥ xand x≤ a. Now, if a≤ x≤ b, on can easily verify that

u2(t,x) =

u2,0(x− t)e−t , if 0≤ t ≤ x−a and a≤ x≤ b,

u2(t− x+a,a)e−(x−a), if t ≥ x−a and a≤ x≤ b

by inserting this formula in (3.26). Since u2(t−x+a,a) = u2(t−x,0) if t ≥ x or u2(t−x+a,a) = u2,0(x− t) if x− a ≤ t ≤ x, we obtain that u2(t,x) can be computed from u2,0 andu2(t− x,0) for 0 ≤ x ≤ b as in (3.28b). Finally, in the case b ≤ x ≤ L2, one can follow thetransport flow to obtain that u2(t,x) = u2(t− x+b,b) if t ≥ x−b and u2(t,x) = u2,0(x− t)otherwise, and thus the previous formulas applied for u2(t − x+ b,b) give the remainingcases of (3.28b).

Lemma 3.16 thus shows that, as in Section 3.4.5, it suffices to study u j(t,0), j = 1,2, fort ≥ 0 in order to obtain the solution (u1,u2) of (3.26). Thanks to the fact that all the exponentialdecays appearing in (3.28) are upper bounded by 1, one also obtains trivially the followingcorollary.

Corollary 3.17. The solution u2 of (3.26) satisfies the estimate

|u2(t,x)| ≤

∣∣u2,0(x− t)∣∣ , if 0≤ t ≤ x,

|u2(t− x,0)| , if t ≥ x.

For any p ∈ [1,∞], we have, for t ≥ L1,

‖u1(t, ·)‖Lp(0,L1)= ‖u1(·,0)‖Lp(t−L1,t) ,

and, for t ≥ L2,‖u2(t, ·)‖Lp(0,L2)

≤ ‖u2(·,0)‖Lp(t−L2,t) .

This corollary allows us to replace the spatial Lp norm of the solutions by the Lp norm intime of the solution at the point 0.

We define the function φ : R+→ R by φ(t) = u1(t,0) = u2(t,0), and thus our study of theexplicit formula for the solution can be concentrated on the study of φ . In order to simplify the

54

O

P

L1L2

bt

R0 R1 R2 R3 R4 S0S1S2S3S4S5S6S7

x

y

FIGURE 3.4: Method used to obtain the solution of (3.26) as a modification of the method of Section3.4.5.

notations, we define η = e−(b−a); note that η < 1 corresponds, by (3.28b), to the decay of asolution of (3.26) in the circle C2 after having passed through the decay interval [a,b].

In order to obtain the expression for φ , let us consider the same diagram as in Section 3.4.5,described by Definition 3.14, which we represent in Figure 3.4. We add in this diagram seg-ments corresponding to the interval [a,b] in [0,L2] where the solution is damped. We recall thatthe segments representing the circle C2 in Figure 3.4 are those in N2, of the kind Qm,nQm,n+1.If γ : [0,L2] → Qm,nQm,n+1 is a curve parametrized by arc length with γ(0) = Qm,n+1 andγ(L2) = Qm,n, then the segment corresponding to the interval [a,b] is γ(a)γ(b); these segmentsare highlighted in Figure 3.4.

As before, φ(t) can be written as a linear combination of the initial conditions u1,0 and u2,0 at

the respective points L1−t−nL2L1 , n = 0, . . . ,⌊

tL2

⌋, and L2−t−mL1L2 , m = 0, . . . ,

⌊t

L1

⌋,

as

φ(t) =

⌊t

L2

⌋∑n=0

αn,tu1,0(L1−t−nL2L1)+

⌊t

L1

⌋∑

m=0βm,tu2,0(L2−t−mL1L2) (3.29)

for certain coefficients αn,t and βm,t . Let us now determine these coefficients.The coefficient αn,t can be obtained from the intersection point P between bt and Rn. Indeed,

any path connecting P to O has total length t and passes exactly n times on a segment of N2 and⌊t−nL2

L1

⌋times on a segment of N1. Since every such path is uniquely determined by the order

in which it passes n times through the segments of N2 and⌊

t−nL2L1

⌋times through the segments

of N1, the total number of such paths is the binomial coefficient(n+

⌊t−nL2

L1

⌋n

), and every such

path passes through n+⌊

t−nL2L1

⌋+1 intersection points Q j,k. Each complete passage through a

segment of N2 corresponds to a decay of the solution by a factor η = e−(b−a) thanks to (3.28b),and thus we obtain that

αn,t =

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

ηn. (3.30)

55

The similar geometric argument can be applied to the intersection P between b and one half-line Sm to obtain βm,t . In this case, any path connecting P to O passes m times on a segment

of N1 and⌊

t−mL1L2

⌋times on a segment of N2, and thus there are

(m+⌊

t−mL1L2

⌋m

)such paths, each

one passing through m+⌊

t−mL1L2

⌋+1 intersection points Q j,k. Each complete passage through

an interval of length L2 corresponds to a decay of the solution by a factor η , and thus we have

a multiplicative factor of η

⌊t−mL1

L2

⌋in βm,t . Furthermore, we may also have a decay δm,t in this

case in the segment of N2 containing P, given by

δm,t =

1 if L2−t−mL1L2 ≥ b,

eb−L2−t−mL1L2 if a≤ L2−t−mL1L2 ≤ b,η if L2−t−mL1L2 ≤ a.

We thus obtain the coefficient

βm,t =

(m+⌊

t−mL1L2

⌋m

)2m+

⌊t−mL1

L2

⌋+1

η

⌊t−mL1

L2

⌋δm,t . (3.31)

Inserting (3.30) and (3.31) into (3.29) gives

φ(t) =

⌊t

L2

⌋∑n=0

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

ηnu1,0(L1−t−nL2L1)+

+

⌊t

L1

⌋∑

m=0

(m+⌊

t−mL1L2

⌋m

)2m+

⌊t−mL1

L2

⌋+1

η

⌊t−mL1

L2

⌋δm,tu2,0(L2−t−mL1L2). (3.32)

We thus obtain the following result.

Theorem 3.18. Suppose (u1,0,u2,0) ∈ D(A). The solution of (3.26) with this initial conditionis (3.28) with u1(t,0) = u2(t,0) being given for t ≥ 0 by u1(t,0) = u2(t,0) = φ(t) and φ beinggiven by (3.32).

The previous presentation is not a rigorous proof of Theorem 3.18, but it is clear that, byfollowing the same steps as in the proof of Theorem 3.13 in Section 3.4.5, we obtain a proof forTheorem 3.18.

3.5.2 Uniform exponential decay of the coefficientsWe now wish to study the asymptotic behavior of the coefficients αn,t and βm,t as t, n and mtend to +∞. Notice that these coefficients contain both a normalized binomial coefficient andan exponential decay, and thus, in order to be able to give fine estimates on them, we first givesome estimates on these normalized binomial coefficients.

The normalized binomial coefficient

1

2n+⌊

t−nL2L1

⌋(n+⌊

t−nL2L1

⌋n

)(3.33)

56

has a natural interpretation as a probability: given N = n+⌊

t−nL2L1

⌋independent and identically

distributed random variables X1, . . . ,XN taking values in 0,1 with a Bernoulli distribution ofparameter p = 1

2 , (3.33) gives the probability that n of these variables take the value 1 and

the other⌊

t−nL2L1

⌋take the value 0. This probabilistic interpretation allows us to give bounds

on (3.33) thanks to the Central Limit Theorem or to its specialized version to the binomialdistribution, the de Moivre-Laplace Theorem (see, for instance, [26, Chapter 8] for the CentralLimit Theorem and [25, Chapter 7] for the de Moivre-Laplace Theorem). The statement of thelatter can be written as follows.

Theorem 3.19 (de Moivre-Laplace, [25]). Let −∞ < a < b <+∞ and p ∈ (0,1). Then

limn→+∞

⌊np+b√

np(1−p)⌋

∑k=⌈

np+a√

np(1−p)⌉(

nk

)pk(1− p)n−k =

1√2π

w b

ae−

x22 dx.

and the convergence is uniform in a and b.

This theorem is given in its classical proof for the sum of the binomial coefficients, but itsproof actually estimates each binomial coefficient as follows.

Lemma 3.20 (de Moivre-Laplace punctual limit, [25]). Let p ∈ (0,1) and a > 0. Then(nk

)pk(1− p)n−k =

1√2πnp(1− p)

e−(k−np)2

2np(1−p) (1+ εn(k)) (3.34)

withlim

n→+∞max

k=dnp−a√

ne,...,bnp+a√

nc|εn(k)|= 0.

This is a good estimate to apply to (3.33), but it is established only for k on a neighborhoodof np with size

√n. As we will see in the sequel, we need an estimate for a neighborhood of

np whose size increases linearly with n, i.e., an estimate for k ∈ [µn,νn] for certain constants0 < µ < ν < 1. On the other hand, the conclusion (3.34) of Lemma 3.20 is much stronger thanwhat we need, since an upper bound on the binomial coefficient suffices for us. We can actuallyget such an estimate by following the same steps of the proof of Lemma 3.20, which we do inthe next lemma.

Before proving the next lemma, let us recall Stirling’s approximation for the factorial, whichstates that, for every n ∈ N∗,

√2πnn+1/2e−n < n! <

√2πnn+1/2e−n

(1+

14n

); (3.35)

see, for instance, [21, Chapter 6] for the formula given here. We remark that one may obtainsharper estimates, but the error term 1

4n given here will be sufficient for what we need.

Lemma 3.21. Suppose that p ∈ (0,1) and 0 < µ < ν < 1. Then there exist constants C,λ > 0such that, for every n ∈ N∗ and every k ∈ N with µn≤ k ≤ νn, we have(

nk

)pk(1− p)n−k ≤ C√

2πne−λ

(k−np)2n . (3.36)

57

Proof. By (3.35), we have(nk

)pk(1− p)n−k =

1√2π

√n

k(n− k)

(npk

)k(

n(1− p)n− k

)n−k 1+ 14n(

1+ 14k

)(1+ 1

4(n−k)

) .(3.37)

We start by remarking that a rough estimate gives, for every n ∈ N∗,

1+ 14n(

1+ 14k

)(1+ 1

4(n−k)

) ≤ 2. (3.38)

Consider now the term √n

k(n− k).

Since µn≤ k ≤ νn, we have that

k(n− k)≥minµ(1−µ)n2,ν(1−ν)n2= γn2

with γ = minµ(1−µ),ν(1−ν)> 0. Thus√n

k(n− k)≤ 1√

γn. (3.39)

We finally treat the term (npk

)k(

n(1− p)n− k

)n−k

.

By taking the natural logarithm of this term, we obtain

log

[(npk

)k(

n(1− p)n− k

)n−k]= k log

(1− k−np

k

)+(n− k) log

(1+

k−npn− k

). (3.40)

Let c > 2 and consider the function f : (−1,+∞)→ R given by

f (x) = x− x2

c− log(1+ x).

This function satisfies

limx→−1

f (x) = +∞, limx→+∞

f (x) =−∞, f (0) = 0. (3.41)

Its derivative is

f ′(x) = 1− 2xc− 1

1+ x= x

1− 2c (1+ x)1+ x

,

and so f ′(x)= 0 if and only if x= 0 or x= c2−1> 0. We can see that f ′(x)< 0 if−1< x< 0,

f ′(x) > 0 if 0 < x < c2 − 1 and f ′(x) < 0 if x > c

2 − 1, and thus f is strictly decreasing in(−1,0), strictly increasing in (0, c

2 − 1) and strictly decreasing in ( c2 − 1,+∞). By (3.41),

we obtain in particular that f (x) ≥ 0 for x ∈ (−1, c2 − 1]. Hence, for every M > 0, there

exists c > 2 such that

log(1+ x)≤ x− x2

c, ∀x ∈ (−1,M].

58

Notice now that, since µn≤ k ≤ νn, we have

µ− pν≤ k−np

k≤ ν− p

µ,

µ− p1−µ

≤ k−npn− k

≤ ν− p1−ν

.

The right-hand side of (3.40) uses the function log(1+ x) in two points, both of which areupper bounded by M = max p−µ

ν, ν−p

1−ν. For this M > 0, there exists a constant c > 2 such

that

log(1+ x)≤ x− x2

c, ∀x ∈ (1,M],

and thus one can estimate (3.40) as

log

[(npk

)k(

n(1− p)n− k

)n−k]≤

≤ k

(−k−np

k− 1

c

(k−np

k

)2)+(n− k)

(k−npn− k

− 1c

(k−npn− k

)2)

=

=−1c(k−np)2

(1k+

1n− k

)=−1

c(k−np)2 n

k(n− k)≤−4

c(k−np)2

n. (3.42)

We finally obtain (3.36) from (3.37), (3.38), (3.39) and (3.42) with C = 2√γ

and λ = 4c .

We recall here a simple result concerning the monotonicity of(n

k

)pk(1− p)n−k.

Lemma 3.22. Let p ∈ (0,1) and n ∈ N∗. The function f : 0,1, . . . ,n→ (0,1) given by

f (k) =(

nk

)pk(1− p)n−k

is strictly increasing for k < np+ p−1 and strictly decreasing for k > np+ p−1, in the sensethat

f (k+1)> f (k) if k < np+ p−1, f (k+1)< f (k) if k > np+ p−1.

Proof. Clearly, f (k) ∈ (0,1) for every k ∈ 0,1, . . . ,n. We have

f (k) =n!pk(1− p)n−k

k!(n− k)!.

For k ∈ 0,1, . . . ,n−1, we have

f (k+1)f (k)

=p(n− k)

(1− p)(k+1)

and a simple computation shows that

p(n− k)(1− p)(k+1)

> 1 ⇐⇒ k < np+ p−1,p(n− k)

(1− p)(k+1)< 1 ⇐⇒ k > np+ p−1.

59

Notice that Lemma 3.21 estimates the binomial term(n

k

)pk(1− p)n−k for k in an interval

of the kind [µn,νn] for 0 < µ < ν < 1. Lemma 3.22 allows us to study these terms also forintervals of the kind [0,µn] and [νn,n]. The following corollary will be useful in what follows.

Corollary 3.23. Let p∈ (0,1) and 0 < µ < p < ν < 1. Then there exist constants C,λ > 0 suchthat, for every n ∈ N, we have(

nk

)pk(1− p)n−k ≤Ce−λn, ∀k ∈ ([0,µn]∪ [νn,n])∩N. (3.43)

Proof. Let N0 =max⌈

2µ

⌉,⌈ 2

1−ν

⌉, so that, if n≥N0, both

[µ

2 n,µn]∩N and

[νn, ν+1

2 n]∩

N are non-empty. It suffices to prove (3.43) for n ≥ N0, since there is only a finite numberof cases when n < N0 and these can be incorporated in the estimate (3.43) by possiblyincreasing the constant C. We thus suppose from now on that n≥ N0.

Consider the interval[

µ

2 n,µn]

and take C1 > 0 and λ1 > 0 as in Lemma 3.21 such that,for every k ∈

[µ

2 n,µn]∩N, we have(

nk

)pk(1− p)n−k ≤C1e−λ1

(k−np)2n . (3.44)

The right-hand side of (3.44) is an increasing function of k for k ∈ [0,np], and thus, sinceµ < p, (

nk

)pk(1− p)n−k ≤C1e−λ1(µ−p)2n (3.45)

for every k ∈[

µ

2 n,µn]∩N. By Lemma (3.22), the left-hand side of (3.45) is an increasing

function of k for k < np+ p−1 and, since µn < np+ p−1, we conclude that (3.45) holdsfor every k ∈ [0,µn]∩N.

We proceed similarly for the interval[νn, ν+1

2 n], obtaining from Lemma 3.21 constants

C2 > 0 and λ2 > 0 such that, for every k ∈[νn, ν+1

2 n]∩N, we have(

nk

)pk(1− p)n−k ≤C2e−λ2

(k−np)2n ≤C2e−λ2(ν−p)2n. (3.46)

By Lemma 3.22, the left-hand side of (3.46) decreases for k ∈ [νn,n]∩N since νn > np+p−1, and thus (

nk

)pk(1− p)n−k ≤C2e−λ2(ν−p)2n

for every k ∈ [νn,n]∩N.The conclusion follows by taking C = maxC1,C2 and λ = minλ1(µ − p)2,λ2(ν −

p)2.

Corollary 3.23 can now be used to give an exponential estimate for the coefficients αn,t andβm,t . We define

αn,t =

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

, β m,t =

(m+⌊

t−mL1L2

⌋m

)2m+

⌊t−mL1

L2

⌋+1

, (3.47)

60

so that αn,t = αn,tηn and βm,t = β m,tη

⌊t−mL1

L2

⌋δm,t . Notice that, to study αn,t and β m,t , it suffices

to study

γn,τ =

(n+bτ−n`cn

)2n+bτ−n`c+1 (3.48)

and thus αn,t can be obtained from γn,τ by taking τ = tL1

and ` = L2L1

and β m,t can be obtainedfrom γn,τ by taking n = m, τ = t

L2and `= L1

L2.

Lemma 3.24. There exist constants C,λ > 0, τ0 > 0 and 0 < µ0 < ν0 <1` such that, for every

τ ≥ τ0 and every n ∈ ([0,µ0τ]∪ [ν0τ,τ/`])∩N, we have

γn,τ ≤Ce−λτ .

Proof. We have

γn,τ =12

(n+ bτ−n`c

n

)(12

)n(12

)bτ−n`c

and thus we wish to apply Corollary 3.23 with p = 12 . Notice first that

n+ bτ−n`c ≥ τ +n(1− `)−1

and that, for n ∈ [0,τ/`]∩N, if ` ≤ 1, we have τ + n(1− `)− 1 ≥ τ − 1 and, if ` > 1, wehave τ−n(`−1)−1≥ τ− τ

` (`−1)−1 = τ

` −1, so that, in any case,

n+ bτ−n`c ≥ τ

`max−1 (3.49)

with `max = max1, `.Take µ0 =

1`+4 . We remark that, for τ ≥ `+4, we have τ

`+4 ≤τ−1`+3 . Hence, for 0≤ n≤

µ0τ , we have

0≤ n≤ τ−1`+3

,

which implies that`+3

4n≤ τ−1

4,

from where we get that

n≤ 14

n+τ

4− n`

4− 1

4=

14(n+ τ−n`−1) ,

and so0≤ n≤ 1

4(n+ bτ−n`c) . (3.50)

Take ν0 =3

1+3` . Clearly, 0 < µ0 < ν0 <1` . If ν0τ ≤ n≤ τ/`, we have

n≥ 3τ

1+3`,

which implies that1+3`

4n≥ 3

4τ,

61

from where we get that

n≥ 34

n+34

τ− 34

n`≥ 34(n+ τ−n`) .

On the other hand, bτ−n`c ≥ 0 for n≤ τ/`, and thus

34(n+ bτ−n`c)≤ n≤ n+ bτ−n`c . (3.51)

Hence, if n ∈ ([0,µ0τ]∪ [ν0τ,τ/`])∩N, we have (3.50) and (3.51) and, applying Corol-lary 3.23 with µ = 1

4 , p = 12 and ν = 3

4 , we obtain constants C0 > 0 and λ0 > 0 such that,for every τ ≥ `+4 and every n satisfying (3.50) and (3.51), we have(

n+ bτ−n`cn

)(12

)n(12

)bτ−n`c≤C0e−λ0(n+bτ−n`c).

So, for n ∈ ([0,µ0τ]∪ [ν0τ,τ/`])∩N, we have, by (3.49)

γn,τ ≤C0

2e−λ0( τ

`max−1).

We have thus the required result with C = C02 eλ0 , λ = λ0

`maxand τ0 = `+4.

Remark 3.25. Since γn,τ ≤ 12 for every τ ≥ 0 and every n ∈ [0,τ/`]∩N, we can, by possibly

increasing C, suppose that τ0 = 0 in Lemma 3.24.

Lemma 3.24 allows us to obtain an exponential estimate on the coefficients αn,t and βm,t .

Theorem 3.26. There exist C,λ > 0 such that, for every t ≥ 0, we have

αn,t ≤Ce−λ t , ∀n ∈ [0, t/L2]∩N,βm,t ≤Ce−λ t , ∀m ∈ [0, t/L1]∩N.

(3.52)

Proof. Recall that

αn,t = αn,tηn, βm,t = β m,tη

⌊t−mL1

L2

⌋δm,t

and that δm,t ≤ 1 for every t ≥ 0 and every m ∈ [0, t/L1]∩N.Taking τ = t

L1and `= L2

L1, Lemma 3.24 gives constants C0,γ0 > 0 and 0 < µ0 <

L1L2

suchthat, for every t ≥ 0 and every n ∈ [0,µ0t/L1]∩N, we have

αn,t ≤C0e−γ0t

L1 .

Thus, for n ∈ [0,µ0t/L1]∩N, we have

αn,t ≤C0e−γ0t

L1 . (3.53)

If n ∈ [µ0t/L1, t/L2]∩N, we have

αn,t = αn,tηn ≤ 1

2e

µ0 logη

L1t (3.54)

62

since η < 1 and αn,t ≤ 12 .

Now, taking τ = tL2

and `= L1L2

, Lemma 3.24 gives constants C1,γ1 > 0 and 0 < ν1 <L2L1

such that, for every t ≥ 0 and every m ∈ [ν1t/L2, t/L1]∩N, we have

β m,t ≤C1e−γ1t

L2 .

Thus, for m ∈ [ν1t/L2, t/L1]∩N, we have

βm,t ≤C1e−γ1t

L2 . (3.55)

If m ∈ [0,ν1t/L2]∩N, we have

βm,t = β m,tη

⌊t−mL1

L2

⌋δm,t ≤

12η

e(L2−ν1L1) logη

L22

t. (3.56)

since η < 1, β m,t ≤ 12 , δm,t ≤ 1 and

⌊t−mL1

L2

⌋≥ 1−ν1`

L2t−1 = L2−ν1L1

L22

t−1 for m≤ ν1L2

t.

Combining (3.53), (3.54), (3.55) and (3.56), we obtain (3.52) with C =max

C0,C1,1

2η

and λ = min

γ0L1,−µ0 logη

L1, γ1

L2,− (L2−ν1L1) logη

L22

> 0.

3.5.3 Exponential convergence of the solutionsWe now want to show that every solution of (3.26) converges exponentially to the origin, whichcan be translated, in terms of the semigroup et(A+B), by the exponential decay of

∥∥∥et(A+B)∥∥∥L(X)

.

Theorem 3.27. There exist constants C > 0 and λ > 0 such that, for every t ≥ 0,∥∥∥et(A+B)∥∥∥L(X)≤Ce−λ t .

Proof. It suffices to show that there exist C,λ > 0 such that, for every z0 ∈ D(A) and everyt ≥ 0, we have ∥∥∥et(A+B)z0

∥∥∥X≤Ce−λ t ‖z0‖X , (3.57)

and the conclusion of the theorem will follow by the density of D(A) in X.Let C0 > 0 and λ0 > 0 be the constants given by Theorem 3.26. Set Lmax = maxL1,L2

and Lmin = minL1,L2. Take z0 = (u1,0,u2,0) ∈ D(A) and denote the solution et(A+B)z0 of(3.27) by et(A+B)z0 = z(t) = (u1(t),u2(t)). By Theorem 3.18 and Corollary 3.17, we have,for t ≥ Lmax,∥∥∥et(A+B)z0

∥∥∥2

X= ‖u1(t)‖2

L2(0,L1)+‖u2(t)‖2

L2(0,L2)≤

≤ ‖u1(·,0)‖2L2(t−L1,t)+‖u2(·,0)‖2

L2(t−L2,t) ≤ 2‖φ‖2L2(t−Lmax,t) (3.58)

with φ given by (3.29) as in Theorem 3.18. We have

‖φ‖2L2(t−Lmax,t) =

w t

t−Lmax|φ(s)|2 ds≤

63

≤ 2

w t

t−Lmax

∣∣∣∣∣∣∣⌊

sL2

⌋∑n=0

αn,su1,0(L1−s−nL2L1)

∣∣∣∣∣∣∣2

ds+

+w t

t−Lmax

∣∣∣∣∣∣∣⌊

sL1

⌋∑

m=0βm,su2,0(L2−s−mL1L2)

∣∣∣∣∣∣∣2

ds

≤

≤ 2

w t

t−Lmax

⌊s

L2

⌋⌊ sL2

⌋∑n=0

∣∣αn,su1,0(L1−s−nL2L1)∣∣2 ds+

+w t

t−Lmax

⌊s

L1

⌋⌊ sL1

⌋∑

m=0

∣∣βm,su2,0(L2−s−mL1L2)∣∣2 ds

≤≤

2C20

Lmint

⌊

tL2

⌋∑n=0

w t

t−Lmaxe−2λ0s ∣∣u1,0(L1−s−nL2L1)

∣∣2 ds+

+

⌊t

L1

⌋∑

m=0

w t

t−Lmaxe−2λ0s ∣∣u2,0(L2−s−mL1L2)

∣∣2 ds

≤≤

2C20e2λ0Lmax

Lminte−2λ0t

⌊

tL2

⌋∑n=0

w t

t−Lmax

∣∣u1,0(L1−s−nL2L1)∣∣2 ds+

+

⌊t

L1

⌋∑

m=0

w t

t−Lmax

∣∣u2,0(L2−s−mL1L2)∣∣2 ds

. (3.59)

Consider now the integralw t

t−Lmax

∣∣u j,0(L j−s−nLiL j)∣∣2 ds (3.60)

with j ∈ 1,2 and i = 3− j. Notice that, for k ∈ Z, we have s−nLiL j = s−nLi+kL j fors ∈[nLi + kL j,nLi +(k+1)L j

). We can thus take

kmin = maxk ∈ Z |nLi + kL j ≤ t−Lmax=⌊

t−Lmax−nLi

L j

⌋,

kmax = mink ∈ Z |nLi +(k+1)L j ≥ t=⌈

t−nLi

L j

⌉−1,

and so (3.60) can be estimated as

w t

t−Lmax

∣∣u j,0(L j−s−nLiL j)∣∣2 ds≤

64

≤kmax

∑k=kmin

w nLi+(k+1)L j

nLi+kL j

∣∣u j,0(L j− s+nLi− kL j)∣∣2 ds =

=kmax

∑k=kmin

w L j

0

∣∣u j,0(σ)∣∣2 dσ = (kmax− kmin +1)

∥∥u j,0∥∥2

L2(0,L j).

We have

kmax− kmin +1 =

⌈t−nLi

L j

⌉−⌊

t−Lmax−nLi

L j

⌋≤ Lmax

L j+2,

and thusw t

t−Lmax

∣∣u j,0(L j−s−nLiL j)∣∣2 ds≤

(Lmax

L j+2)∥∥u j,0

∥∥2L2(0,L j)

.

Inserting this into (3.59) gives

‖φ‖2L2(t−Lmax,t) ≤

≤2C2

0e2λ0Lmax

Lmint2e−2λ0t

(1L2

(Lmax

L1+2)∥∥u1,0

∥∥2L2(0,L1)

+1L1

(Lmax

L2+2)∥∥u2,0

∥∥2L2(0,L2)

)≤

≤2C2

0e2λ0Lmax

L2min

(Lmax

Lmin+2)

t2e−2λ0t ‖z0‖2X . (3.61)

We finally get (3.57) from (3.58) and (3.61).

3.6 Developments on the persistently excited damped caseThe study done on Section 3.5 is rather complicated, and, as we remarked before, one might pro-pose other methods in order to establish the exponential decay of the semigroup et(A+B)t≥0,for instance by establishing an observability inequality. However, we chose to take this ap-proach since it seems well-adapted to be generalized to the PE damped case, as we show in thissection.

We study in this section the following transport equation in two circles with a persistentlyexcited damping in one of them,

∂tu1(t,x)+∂xu1(t,x) = 0, t ∈ R+, x ∈ [0,L1],

∂tu2(t,x)+∂xu2(t,x)+α(t)χ(x)u2(t,x) = 0, t ∈ R+, x ∈ [0,L2],

u1(t,0) = u2(t,0) =u1(t,L1)+u2(t,L2)

2, t ∈ R+,

u j(0,x) = u j,0(x), x ∈ [0,L j], j ∈ 1,2,α ∈ G(T,µ),

(3.62)

where we suppose that χ is the characteristic function of an interval [a,b]⊂ [0,L2].We can write (3.62) as a differential equation in the Hilbert space X= L2(0,L1)×L2(0,L2)

using the operators A : D(A)⊂ X→ X and B : X→ X from Section 3.5 asz(t) = Az(t)+α(t)Bz(t),z(0) = z0,

α ∈ G(T,µ), (3.63)

65

with z0 = (u1,0,u2,0). Note that the operator we now consider is A+α(t)B, which is time-dependent. Since B ∈ L(X), we can apply a technique similar to [48, Chapter 3, Proposition1.2], which we detail in Appendix 3.B, in order to conclude the existence and the uniqueness ofa family T (t,s)t≥s≥0 of bounded operators in X such that, for every z0 ∈D(A), t 7→ T (t,s)z0 isthe unique absolutely continuous function satisfying z(t) = Az(t)+α(t)Bz(t) for almost everyt ≥ s and z(s) = z0. As usual, for z0 ∈ X, we shall say that t 7→ T (t,s)z0 is the solution ofz(t) = Az(t)+α(t)Bz(t) with z(s) = z0.

3.6.1 Explicit solutionWe now follow the same steps of Section 3.5.1 in order to obtain an explicit solution for (3.62).We first notice that, as in Lemma 3.16, it suffices to study u1(t,0) = u2(t,0) for every t ≥ 0 inorder to obtain the solution (u1,u2) at every point.

Lemma 3.28. Suppose (u1,0,u2,0) ∈ D(A). Then the corresponding solution (u1,u2) of (3.62)satisfies

u1(t,x) =

u1,0(x− t), if 0≤ t ≤ x,u1(t− x,0), if t ≥ x,

(3.64a)

u2(t,x) =

u2,0(x− t), if 0≤ t ≤ x and x≤ a,u2(t− x,0), if t ≥ x and x≤ a,

u2,0(x− t)e−r t

0 α(s)ds, if 0≤ t ≤ x−a and a≤ x≤ b,

u2,0(x− t)e−r t

t−x+a α(s)ds, if x−a≤ t ≤ x and a≤ x≤ b,

u2(t− x,0)e−r t

t−x+a α(s)ds, if t ≥ x and a≤ x≤ b,u2,0(x− t), if 0≤ t ≤ x−b and b≤ x≤ L2,

u2,0(x− t)e−r t−x+b

0 α(s)ds, if x−b≤ t ≤ x−a and b≤ x≤ L2,

u2,0(x− t)e−r t−x+b

t−x+a α(s)ds, if x−a≤ t ≤ x and b≤ x≤ L2,

u2(t− x,0)e−r t−x+b

t−x+a α(s)ds, if t ≥ x and b≤ x≤ L2.

(3.64b)

Proof. Equation (3.64a) is obtained trivially from the undamped transport flow in [0,L1],and the two first cases of (3.64b) are obtained similarly since the damping is not active in[0,L2] for 0≤ x≤ a.

If a≤ x≤ b, we have

u2(t,x) =

u2,0(x− t)e−

r t0 α(s)ds, if 0≤ t ≤ x−a and a≤ x≤ b,

u2(t− x+a,a)e−r t

t−x+a α(s)ds, if t ≥ x−a and a≤ x≤ b,(3.65)

as we can easily verify by inserting these expressions in (3.62). The term u2(t− x+ a,a)can be computed from the first two cases of (3.64b), and hence the third, fourth and fifthcases of (3.64b) can be obtained from (3.65). Finally, by following the transport flow, onecan see that, if b≤ x≤ L2,

u2(t,x) =

u2,0(x− t), if 0≤ t ≤ x−b and b≤ x≤ L2,u2(t− x+b,b), if t ≥ x−b and b≤ x≤ L2,

and thus the remaining cases of (3.64b) follow from the five first.

66

Note that the difference between (3.64b) and (3.28b) lie on the exponential decays: insteadof having a simple decay e−t or e−(b−a), for instance, the decay now depends on the integralof α on an appropriate time interval. In particular, this integral may be zero for certain timeintervals, which corresponds to an absence of decay. We expect, however, that, with the per-sistent excitation of the damping, and under the addition hypothesis that L1

L2is irrational, one

might obtain a sufficient number of intervals where the decay is large enough in order to ensurea stability result similar to Theorem 3.27, as we develop in the sequel. We shall see the originof the hypothesis that L1

L2is irrational later on, in Section 3.6.2.

It follows trivially from Lemma 3.28 that Corollary 3.17, which allows us to replace thespatial Lp norm of the solutions by the Lp norm in time of the solution at the point 0, also holdsin this case; we restate it here for the sake of completeness.

Corollary 3.29. The solution u2 of (3.62) satisfies the estimate

|u2(t,x)| ≤

∣∣u2,0(x− t)∣∣ , if 0≤ t ≤ x,

|u2(t− x,0)| , if t ≥ x.

For any p ∈ [1,∞], we have, for t ≥ L1,

‖u1(t, ·)‖Lp(0,L1)= ‖u1(·,0)‖Lp(t−L1,t) ,

and, for t ≥ L2,‖u2(t, ·)‖Lp(0,L2)

≤ ‖u2(·,0)‖Lp(t−L2,t) .

Let us now obtain the explicit formula for the solutions of (3.62), which, by Lemma 3.28,reduces to obtaining an explicit formula for the function φ : R+→R given by φ(t) = u1(t,0) =u2(t,0). We construct the same diagram of Section 3.5.1, described by Definition 3.14, whichwe represent in Figure 3.5 using the same notations as previously. The decay intervals high-lighted in Figure 3.5 represent no longer a constant decay η < 1 but rather a decay dependingon the PE signal α in the corresponding time interval. As in Section 3.5.1, φ is of the form

φ(t) =

⌊t

L2

⌋∑n=0

αn,tu1,0(L1−t−nL2L1)+

⌊t

L1

⌋∑

m=0βm,tu2,0(L2−t−mL1L2) (3.66)

for certain coefficients αn,t and βm,t which we now want to determine.The coefficient αn,t can be computed from the intersection point P between the line bt rep-

resenting the time and the half-line Rn. Any path connecting P to O has total length t and passesthrough n+

⌊t−nL2

L1

⌋+1 intersection points Q j,k, which gives a term 1

2n+⌊

t−nL2L1

⌋+1

on the expres-

sion of αn,t . We let P′ be the lower endpoint of the segment where P lies, and we note that everypath connecting P to O passes through P′. Each such path has total length t and passes exactly ntimes on a segment of N2 and

⌊t−nL2

L1

⌋times on a segment of N1, but, differently from the case

of Sections 3.4.5 and 3.5.1, it is not sufficient to count the number of such paths, since now eachpath will correspond to a different decay, depending on the values of α at the decay intervals.In order to simplify the notations, we define n1 =

⌊t−nL2

L1

⌋and n2 = n, which are the number of

times that a path passes through segments of N1 and N2, respectively. We define

Vn1,n2 = v = (v1, . . . ,vn1+n2) ∈ 1,2n1+n2 |# j | v j = 1= n1,

67

O

P

P′

L1L2

bt

R0 R1 R2 R3 R4 S0S1S2S3S4S5S6S7

x

y

FIGURE 3.5: Method used to obtain the solution of (3.62). We construct segments of lengths L1 andL2 representing the two circles. Line bt represents the time t by its equation y = t/

√2. The intersections

between bt and the half-lines Rn give the coefficients αn,t , and the intersections between bt and thehalf-lines Sm give the coefficients βm,t . In the figure, we represent the intersection P between bt and R2

used to obtain α2,t . The point P lies on a segment of N1 whose lower endpoint is P′ = Q3,2, and eachpath from P to O passes through P′. For this point, we have n1 = 3, n2 = 2, and the element of V3,2

corresponding to the path from P′ to O represented in blue in the figure is v = (1,2,2,1,1).

where we recall that, for a set F , #F denotes its cardinality. Clearly, #Vn1,n2 =(n1+n2

n1

). Each

v ∈V defines a path from P to O in the following way: starting from O, the path correspondingto v goes upwards first in the segment (v1,0,0) ∈Nv1 , arriving either at the point Q1,0 if v1 = 1or at the point Q0,1 if v1 = 2; from this point, it goes upwards in the corresponding segment ofNv2 , (v2,1,0) in the first case and (v2,0,1) in the second one; and so on. Being at a point Q j,kafter r steps, the path will thus go upwards in the segment (vr+1, j,k), arriving either at Q j+1,kif vr+1 = 1 or at Q j,k+1 if vr+1 = 2. This construction continues until, after going through then1+n2-th interval, it arrives at P′. An example of an element of v∈V3,2 is given in blue in Figure3.5, where we also highlight all possible paths from O to P′. This path corresponding to v passesby n2 segments of N2. We recall that each such segment can be put into correspondence withthe interval [0,L2] by the procedure described in Section 3.5.1, its lower endpoint correspondingto L2 and its upper endpoint corresponding to 0. At each passage through a segment of N2,corresponding to a certain v j = 2, this path arrives at the point corresponding to a at time

t−(

∑j−1i=1 Lvi +L2−a

)and at the point corresponding to b at a time t−

(∑

j−1i=1 Lvi +L2−b

),

which means that the decay on this interval is, by Lemma 3.28,

e−

r ∑j−1i=1 Lvi+L2−a

∑j−1i=1 Lvi+L2−b

α(t−s)ds.

Hence the total decay along the path v is

∏j | v j=2

e−

r ∑j−1i=1 Lvi+L2−a

∑j−1i=1 Lvi+L2−b

α(t−s)ds

68

and thus

αn,t =1

2n1+n2+1 ∑v∈Vn1,n2

∏j | v j=2

e−

r ∑j−1i=1 Lvi+L2−a

∑j−1i=1 Lvi+L2−b

α(t−s)ds, n1 =

⌊t−nL2

L1

⌋, n2 = n. (3.67)

The similar argument also applies to βm,t , with now n1 = m and n2 =⌊

t−mL1L2

⌋. We also need

to take into account, as we did in Section 3.5.1, the decay that may happen in the segment ofN2 containing P, which is now given by

δm,t =

1 if L2−t−mL1L2 ≥ b,

e−r b−L2+t−mL1L2

0 α(s)ds if a≤ L2−t−mL1L2 ≤ b,

e−

r b−L2+t−mL1L2a−L2+t−mL1L2

α(s)dsif L2−t−mL1L2 ≤ a.

Hence

βm,t =1

2n1+n2+1 ∑v∈Vn1,n2

∏j | v j=2

e−

r ∑j−1i=1 Lvi+L2−a

∑j−1i=1 Lvi+L2−b

α(t−s)dsδm,t , n1 = m, n2 =

⌊t−mL1

L2

⌋.

(3.68)We thus obtain the following result, whose rigorous proof is not presented here, since, as

before, such a technical proof can be done by following the same steps as in the proof ofTheorem 3.13.

Theorem 3.30. Suppose (u1,0,u2,0) ∈ D(A). The solution of (3.62) with this initial conditionis (3.64), where u1(t,0) = u2(t,0) is given for t ≥ 0 by u1(t,0) = u2(t,0) = φ(t), φ is given by(3.66), and the coefficients αn,t and βm,t are given by (3.67) and (3.68).

It is obvious that exponential convergence of the solutions of (3.62) to the origin will followif αn,t and βm,t can be shown to satisfy an exponential estimate as in Theorem 3.26, since theproof of Theorem 3.27 only exploits the properties of αn,t and βm,t through Theorem 3.26.Hence, the analysis of the exponential convergence of the solutions of (3.62) is reduced to theasymptotic study of αn,t and βm,t as t→+∞. Note that we can rewrite αn,t and βm,t as

αn,t = αn,t1(n1+n2n2

) ∑v∈Vn1,n2

∏j | v j=2

e−

r ∑j−1i=1 Lvi+L2−a

∑j−1i=1 Lvi+L2−b

α(t−s)ds, n1 =

⌊t−nL2

L1

⌋, n2 = n

and

βm,t = β m,t1(n1+n2n2

) ∑v∈Vn1,n2

∏j | v j=2

e−

r ∑j−1i=1 Lvi+L2−a

∑j−1i=1 Lvi+L2−b

α(t−s)dsδm,t , n1 = m, n2 =

⌊t−mL1

L2

⌋,

with αn,t and β m,t as in (3.47),

αn,t =

(n+⌊

t−nL2L1

⌋n

)2n+

⌊t−nL2

L1

⌋+1

, β m,t =

(m+⌊

t−mL1L2

⌋m

)2m+

⌊t−mL1

L2

⌋+1

.

69

The study of αn,t and β m,t has already been done in Lemma 3.24 through the study of γn,τ givenby 3.48, and so we are left to study the quantity

ζn1,n2 =1(n1+n2n2

) ∑v∈Vn1,n2

∏j | v j=2

e−

r ∑j−1i=1 Lvi+L2−a

∑j−1i=1 Lvi+L2−b

α(t−s)ds. (3.69)

This value represents, for the point P′=Qn1,n2 , the average decay along all the paths connectingP′ to O. In the case where α ≡ 1, all these decays are equal to ηn2 = e−n2(b−a), and so is theiraverage value, but, for a general α ∈G(T,µ), these decays depend on the particular path from P′

to O, and their influence on the solution comes only through their average value. It is possible,due to the fact that α may be zero on certain time intervals, that the decay along a given pathfrom P′ to O may be of a factor e0 = 1, i.e., that no decay may occur on a given path, but, sincewe are only interested in the average along all paths, we expect the condition of persistenceof excitation of α to allow us to obtain an estimate for ζn1,n2 . As we shall see in the nextsection, the hypothesis that α ∈ G(T,µ) is in general not sufficient to establish the exponentialconvergence of the solutions of (3.62) to the origin, and we shall also demand that L1

L2/∈Q.

Notice also that, when α ≡ 1, we have ζn1,n2 = ηn2 for η = e−(b−a) ∈ (0,1), and it is clearthat the arguments of Section 3.5.2 would apply if we were able to prove a bound of the kind

ζn1,n2 ≤ ηn2 (3.70)

for a certain η ∈ (0,1). It is not hard to see, however, considering the construction on Figure3.5, that, if b−a is small enough, one can choose α ∈ G(T,µ) to be zero on all decay intervalscorresponding to paths with n1 = 1, for instance, and thus ζ1,n2 = 1 for every n2 ≥ 0 in this case.Actually, given N1 ∈N∗, if b−a is small enough, one can find α ∈ G(T,µ) which is zero on alldecay intervals corresponding to paths with n1 ∈ [0,N1]∩N, so that ζn1,n2 = 1 for every n2 ≥ 0.However, thanks to Lemma 3.24, this is not a problem, since the behavior of αn,t and βm,t iswell-controlled by γn,τ when n ∈ [0,µ0τ]∪ [ν0τ,τ/`], and so it suffices to study the behavior ofζn1,n2 for n j ∈ [µ jt,ν jt] for certain constants 0 < µ j < ν j <

1L j

, j = 1,2. For these values of n1

and n2, we still hope to obtain a bound as (3.70).

3.6.2 The case of rationally dependent lengths L1 and L2

Section 3.4 has shown that the asymptotic behavior of the unbounded system (3.11) dependson the rationality of the ratio L1

L2: if L1

L2/∈ Q, every solution of (3.11) converges to a constant,

whereas one can find periodic solutions for (3.11) if L1L2∈ Q. The damped system studied in

Section 3.5 no longer presents this dependence on L1L2

, Theorem 3.27 holding for every valueof L1

L2in R∗+. A persistently excited damping, however, reintroduces the possibility of having a

periodic solution when L1L2∈Q, as we show in the following theorem.

Theorem 3.31. Suppose that L1L2∈ Q. Then there exists `0 > 0 such that, if b− a ≤ `0, there

exists α ∈ G(4`0, `0) for which (3.62) admits a non-zero periodic solution.

Proof. We consider here the construction of a periodic solution for (3.11) done in Propo-sition 3.11. Take p,q ∈ N∗ such that L1

L2= p

q and note ` = L1p = L2

q . Take ϕ ∈ C∞c (R) not

identically zero with support included in (0, `/2). For x ∈ [0,L1], we define u1,0 by

u1,0(x) =+∞

∑k=−∞

ϕ(x− k`),

70

and, for x ∈ [0,L2], we define u2,0 by the same expression,

u2,0(x) =+∞

∑k=−∞

ϕ(x− k`);

as in Proposition 3.11, these sums are actually reduced, for a fixed x, to at most one singleterm, and, in particular, u1,0 ∈ C∞([0,L1]) and u2,0 ∈ C∞([0,L2]). Then, as in Proposition3.11,

u1(t,x) =+∞

∑k=−∞

ϕ(x− t− k`),

u2(t,x) =+∞

∑k=−∞

ϕ(x− t− k`)

(3.71)

is a periodic non-zero solution of (3.11) with initial data z0 = (u1,0,u2,0).We now want to give conditions on the length of the decay interval [a,b] ⊂ [0,L2] and

construct a persistently exciting signal α so that α(t)χ(x)u2(t,x) = 0 for every t ≥ 0 andevery x ∈ [0,L2], and hence (3.71) will be a solution of (3.62).

Take `0 =`4 and suppose that b−a≤ `0. We construct a periodic signal α : R→0,1

defined by

α(t) =

0, if t ∈⋃

n∈Z[a− (n+ 1/2)`,b−n`] ,

1, otherwise.

This defines a periodic signal α with period T = ` = 4`0. Starting at a point of the kinda− (n+ 1/2)`, α is 0 during a time b−n`− (a− (n+ 1/2)`) = b−a+ /2 ∈ (0, 3`

4 ] and is 1during a time a− (n−1+ 1/2)`− (b−n`) = a−b− /2+ `≥ `− `0− /2 = `

4 > 0, so that

w a−(n−1/2)`

a−(n+1/2)`α(s)ds≥ `

4= `0.

By the periodicity of α ,w t+`

tα(s)ds≥ `0

for every t ∈ R, and so α ∈ G(`,`0).Consider now the product α(t)χ(x)u2(t,x). Since the support of α is

⋃n∈Z[b−n`,a−

(n− 1/2)`] and the support of χ is [a,b], the product α(t)χ(x)u2(t,x) can only be nonzero ifx∈ [a,b] and t ∈ [b−n`,a−(n−1/2)`] for a certain n∈Z. In this case, we have (n−1/2)`≤x− t ≤ n` and, since the support of ϕ(· − k`) is in (k`,(k + 1/2)`) for every k ∈ Z, weconclude that ϕ(x− t − k`) = 0 for every k ∈ Z, and thus, by (3.71), u2(t,x) = 0. Henceα(t)χ(x)u2(t,x) = 0 for every (t,x) ∈ R+× [0,L2], and so (3.71) satisfies (3.62).

The example of periodic solution constructed in the proof of Theorem 3.31 is illustratedin Figure 3.6, where we consider two circles of lengths L1 = 2 and L2 = 7

2 . The support ofthe initial condition is represented by the blue intervals, and the flow of the transport equationtranslates these intervals in the sense given by the arrows. By taking a small decay interval [a,b],we can choose a periodic persistently exciting signal α which activates the damping when thesupport of the solution does not intercept the interval [a,b] and deactivates it otherwise.

71

0 0L1 L2C1 C2

FIGURE 3.6: Example constructed in the proof of Theorem 3.31 in a case where L1 = 2 and L2 =72 .

The blue intervals represent the support of the initial condition, which are transported by the flow in thesense of the arrows. The interval [a,b] is represented in red, and the strategy is to take α(t) = 0 whenthe support of the solution intersects [a,b] and α(t) = 1 otherwise. This construction is possible and

gives rise to a persistently exciting signal α when b−a is small enough.

3.6.3 Persistently exciting signals and the flow of the transport equationAs we remarked in Section 3.6.1, the study of the stability of the solutions of (3.62) can bereduced to the study of ζn1,n2 introduced in (3.69). This quantity depends deeply on the signalα ∈G(T,µ) and, in order to better exploit its properties, we will first study the relations betweenthe signal α and the construction of Definition 3.14.

Given a signal α ∈ G(T,µ) and a time t ≥ 0, the explicit solution given by Theorem 3.30depends on the decay on intervals (2,n1,n2) with n1L1 + (n2 + 1)L2 ≤ t. The decay on aninterval (2,n1,n2) is given by

ηn1,n2 = e−r n1L1+(n2+1)L2−a

n1L1+(n2+1)L2−b α(t−s)ds= e−

r t−n1L1−(n2+1)L2+bt−n1L1−(n2+1)L2+a α(s)ds

. (3.72)

In order to estimate this decay, we would thus like to estimate the quantity

wτ+b

τ+aα(s)ds

for τ ≥ 0. The first step in this estimate is the following lemma.

Lemma 3.32. Let T ≥ µ > 0. For ρ > 0 and α ∈ G(T,µ), define

Iρ,α =

τ ∈ R+ |

wτ+b

τ+aα(s)ds≥ ρ

. (3.73)

There exist ρ > 0 and ` > 0, depending only on µ

T and b− a, such that, for every t ∈ R+ andevery α ∈ G(T,µ), Iρ,α ∩ [t, t +T ] contains an interval of length `.

Proof. We take ρ = µ(b−a)2T , ` = minρ

2 ,T. Take α ∈ G(T,µ) and define the functionA : R+→ R by

A(τ) =w

τ+b

τ+aα(s)ds.

Since α ∈ L∞(R+, [0,1]), A is a Lipschitz continuous function with Lipschitz constant 2.We also have, for every t ∈ R+,

w t+T

tA(τ)dτ =

w t+T

t

w b

aα(s+ τ)dsdτ =

w b

a

w s+t+T

s+tα(τ)dτds≥ µ(b−a). (3.74)

72

Take t ∈ R+. Then there exists t? ∈ [t, t +T ] such that A(t?) ≥ µ(b−a)T = 2ρ , for otherwise

(3.74) would not be satisfied. Since A is 2-Lipschitz, we have A(τ)≥ ρ for τ ∈ [t?− ρ

2 , t?+ρ

2 ]∩R+, and thus [t?− ρ

2 , t?+ρ

2

]∩ [t, t +T ]⊂ Iρ,α ∩ [t, t +T ].

But, since t? ∈ [t, t +T ], [t?− ρ

2 , t?+ρ

2 ]∩ [t, t +T ] is an interval of length at least `, whichconcludes the proof.

Lemma 3.32 translates the property of persistence of excitation of α into a property on theset Iρ,α , which is a step in order to better understand the decay (3.72) on a segment (2,n1,n2).The following lemma gives a further step into the understanding of ηn1,n2 , exploiting now theirrationality of L1

L2.

Lemma 3.33. Suppose L1L2

/∈Q. Let T ≥ µ > 0 and let ρ > 0 be as in Lemma 3.32. There existN1,N2 ∈ N such that, for every t ∈ R+ with t ≥ N1L1 +(N2 +1)L2 and t ≥ T , every n1,n2 ∈ Nwith (n1 +N1)L1 + (n2 +N2)L2 ≤ t − T , and every α ∈ G(T,µ), there exist r1,r2 ∈ N withn j ≤ r j ≤ N j +n j, j ∈ 1,2, such that

t− r1L1− (r2 +1)L2 ∈ Iρ,α .

Proof. Let ρ > 0 and ` > 0 be obtained from µ

T and b−a as in Lemma 3.32. Let K = 3⌈T`

⌉and consider the numbers

x j =j

KT, j = 0, . . . ,K,

which satisfy x j − x j−1 = KT ≤

`3 for j = 1, . . . ,K. The following result is an important

property of these numbers x j.

Lemma 3.34. For any interval J of length ` contained in [0,T ], there exists j ∈ 1, . . . ,Ksuch that x j−1,x j ∈ J.

Proof. Let a = infJ < T and let j0 = min j ∈ 0, . . .K | x j > a; such a j0 is well-defined since, for instance, xK = T > a, and j0 ≥ 1 since a≥ 0 = x0. Also, j0 < K sincea+ ` ≤ T and so xK−1 = T − K

T ≥ T − `3 > T − ` ≥ a. We have x j0 − a ≤ K

T ≤`3 (for

otherwise we would have x j0−1 = x j0− KT > a, contradicting the minimality of j0), and

thus a < x j0 ≤ a+ `3 < a+ `, so that x j0 ∈ J. Now a < x j0 < x j0+1 = x j0 +

KT ≤ a+ 2`

3 <a+ `, so that x j0+1 ∈ J. We thus have the desired result with j = j0 +1.

We now construct intermediate points between the x j, j = 0, . . . ,K. Since L1L2

/∈ Q, theset

n1L1 +(n2 +1)L2 |n1,n2 ∈ Z (3.75)

is dense in R. Hence we can find n1, j,n2, j ∈ Z, j = 1, . . . ,K, such that the numbers y j =n1, jL1 +(n2, j +1)L2 satisfy

0 = x0 < y1 < x1 < y2 < x2 < · · ·< yK < xK = T, (3.76)

i.e, such that y j ∈ (x j−1,x j) for every j = 1, . . . ,K. As a consequence of Lemma 3.34, weobtain

Corollary 3.35. For any interval J of length ` contained in [0,T ], there exists j ∈ 1, . . . ,Ksuch that y j ∈ J.

73

Let N?1 = max1,−n1,0,−n1,1, . . . ,−n1,K and N?

2 = max1,−n2,0,−n2,1, . . . ,−n2,Kand define

N1 = maxj=0,...,K

(n1, j +N?1 ), N2 = max

j=0,...,K(n2, j +N?

2 ).

By the definition of N?1 and N?

1 , we clearly have N1,N2 ≥ 0. Take t ∈ R with t ≥ N1L1 +(N2 + 1)L2 and t ≥ T , and take n1,n2 ∈ N with (n1 +N1)L1 + (n2 +N2 + 1)L2 ≤ t. Forj = 1, . . . ,K, define

r1, j = n1 +n1, j +N?1 , r2, j = n2 +n2, j +N?

2 ;

it is clear, by this definition, that ni ≤ ri, j ≤ Ni +ni for i = 1,2 and j = 1, . . . ,K. Set

z j = t− r1, jL1− (r2, j +1)L2, j = 1, . . . ,N;

we thus havez j = t−n1, jL1− (n2, j +1)L2−Z? = t−Z?− y j

with Z? = (n1 +N?1 )L1 +(n2 +N?

2 )L2. Since, by construction, y j ∈ (0,T ) for j = 1, . . . ,N,we have z j ∈ [t−Z?−T, t−Z?]. By hypothesis, Z? ≤ t−T , and thus t−Z?−T ≥ 0.

Take α ∈ G(T,µ). By Lemma 3.32, Iρ,α ∩ [t−Z?−T, t−Z?] contains an interval J oflength `. Consider the interval J′ = −J + t−Z?, which is a subinterval of [0,T ] of length`. By Corollary 3.35, there exists j ∈ 1, . . . ,K such that y j ∈ J′, and thus z j ∈ J ⊂ Iρ,α .Since z j = t− r1, jL1− (r2, j +1)L2 and ni ≤ ri, j ≤ Ni +ni for i = 1,2, we obtain the desiredresult.

Remark 3.36. The only moment in the proof of Lemma 3.33 where we use the hypothesisthat L1

L2/∈ Q is when we establish the existence of numbers y j, j = 1, . . . ,K, of the form y j =

n1, jL1 +(n2, j + 1)L2 with n1, j,n2, j ∈ Z satisfying (3.76), which we do by using the density ofthe set (3.75). When L1

L2∈Q and we write L1

L2= p

q for coprime p,q ∈ N∗, the set given in (3.75)is, by Bézout’s Lemma,

L2n1 p+(n2 +1)q

q|n1,n2 ∈ Z

=

k

L2

q| k ∈ Z

.

Hence the construction of y j = n1, jL1 +(n2, j +1)L2 with n1, j,n2, j ∈ Z satisfying (3.76) is stillpossible if L2

q < KT , i.e., if q > L2K

T , and thus Lemma 3.33 still holds true if L1L2

= pq with coprime

p,q ∈ N and q large enough.Recalling that K = 3

⌈T`

⌉with `= minµ(b−a)

4T ,T, we can even give a more explicit neces-sary condition for q to still have Lemma 3.33: if

q≥ 3L2

(max

4T

µ(b−a),

1T

+

1T

), (3.77)

then one can easily check that q > L2KT and hence we are in the previous situation. Condition

(3.77) depends only on the constants T , µ of the PE condition, on the length b− a of thedamping interval and on the length L2.

The consequence of Lemma 3.33 is the following property of ηn1,n2 , which gives informa-tion on the distribution of the segments where the damping term is smaller than a certain value.

74

Theorem 3.37. Suppose L1L2

/∈Q and let T ≥ µ > 0. Then there exist η ∈ (0,1) and N1,N2 ∈ Nsuch that, for every t ∈ R+ with t ≥ N1L1 +(N2 +1)L2 and t ≥ T , every n1,n2 ∈ N with (n1 +N1)L1+(n2+N2)L2≤ t−T and every α ∈G(T,µ), there exists r1,r2 ∈N with n j≤ r j≤ n j+N j,j ∈ 1,2, such that

ηr1,r2 ≤ η .

Proof. Take ρ > 0 as in Lemma 3.32 and N1,N2 ∈ N as in Lemma 3.33 and define η =e−ρ ∈ (0,1). Let t ∈ R+ be such that t ≥ N1L1 +(N2 + 1)L2 and t ≥ T , let n1,n2 ∈ N besuch that (n1 +N1)L1 + (n2 +N2)L2 ≤ t − T and let α ∈ G(T,µ). Take r1,r2 ∈ N as inLemma 3.33, so that

t− r1L1− (r1 +1)L2 ∈ Iρ,α .

By the definition (3.73) of Iρ,α , this means that

w t−r1L1−(r2+1)L2+b

t−r1L1−(r2+1)L2+aα(s)ds≥ ρ.

By the definition (3.72) of ηr1,r2 , we thus obtain that

ηr1,r2 ≤ e−ρ = η ,

which is the desired result.

Remark 3.38. Notice that the hypothesis L1L2

/∈Q is only used to apply Lemma 3.33, and thus,by Remark 3.36, Theorem 3.37 still holds true if L1

L2= p

q with coprime p,q ∈ N∗ satisfying(3.77).

O

L1 L2

bt

bt−T

x

y

FIGURE 3.7: Interpretation of Theorem 3.37 in terms of the graphical construction used to obtain theexplicit solution in Section 3.6.1. The statement of Theorem 3.37 says that, for every t large enough and

every rectangle of sizes N1,N2 whose upper extremity is not greater than t−T , there exists a segment(2,r1,r2) in which the decay factor is at least a certain η ∈ (0,1).

In order to understand the meaning of Theorem 3.37, let us consider once more the con-struction of Definition 3.14, which we represent again in Figure 3.7, with a scale factor with

75

respect to its previous representations in Figures 3.3, 3.4 and 3.5 in order to represent a largertime interval. We fix T ≥ µ > 0. Theorem 3.37 states that there exists numbers N1,N2 ∈ N,such that, for every time t large enough, every rectangle that we place in Figure 3.7 of sizesN1 and N2 and below the horizontal line corresponding to t−T contains at least one segment(2,r1,r2) ∈ N2 where ηr1,r2 ≤ η , i.e., where we can guarantee that we have a certain decaygiven by a universal constant. In Figure 3.7, all the segments (2,r1,r2) where ηr1,r2 ≤ η arerepresented in red, and one can see that every rectangle of sizes N1 = 7 and N2 = 6 contain atleast one red segment, as is the case for the rectangle highlighted in the figure, starting at n1 = 3and n2 = 1. The most important remark here is that N1, N2 and η depend only on T , µ andb− a, and so they are the same for every α ∈ G(T,µ) and every t. Hence the position of thered segments may change if we choose another persistently exciting signal α ∈ G(T,µ), but wecan guarantee that on every rectangle of sizes given by N1 and N2 there exists at least one suchsegment.

3.6.4 Perspectives

The developments presented so far in the study of (3.62) by its explicit solution are an importantstep to understand the stability of its solutions. Our study of the damped transport equation withan always active damping (3.26) in Section 3.5 is an important preliminary study, which hasshown us that the exponential convergence to the origin can be studied by the coefficients αn,tand βm,t appearing in the explicit formula of the solution through the function φ given by (3.29).In the case of the always active damping, these coefficients can be written as αn,t = αn,tη

n and

βm,t = β m,tη

⌊t−mL1

L2

⌋δm,t for αn,t and β m,t given by (3.47), and our exponential estimate comes

from the fact that η < 1, δm,t ≤ 1 and from the estimate of αn,t and β m,t obtained from Lemma3.24.

By studying the explicit formula for the solution in the case of the persistently excited damp-ing in Section 3.6.1, we were able to show that the same formula (3.66) still holds for the aux-iliary function φ , but with different coefficients αn,t and βn,t , which are now given by (3.67)and (3.68). We can still write αn,t and βn,t using the same coefficients αn,t and β m,t from thenon-PE case, which can thus be estimated by Lemma 3.24, and another coefficient ζn1,n2 givenby (3.69), which can be seen as an average decay among all possible trajectories completing n1turns in C1 and n2 turns in C2. If this average is shown to satisfy an estimate of the kind (3.70)for n j ∈ [µ jt,ν jt] for certain 0 < µ j < ν j <

1L j

, j = 1,2, then the exponential convergence of thesolutions of (3.62) to the origin will follow, and we thus concentrate in the study of (3.69).

We remark, in Section 3.6.2, that not all the values of L1L2

and b− a can lead to asymptoticstability of the origin of (3.62), since in some cases one may find periodic solutions. With thatin mind, we continued the study of ζn1,n2 in Section 3.6.3: since ζn1,n2 represents an averagedecay, we propose to study each individual decay ηn1,n2 on a segment (2,n1,n2). Our main resultconcerning these decays is Theorem 3.37, which gives properties on the location of segmentswhere ηn1,n2 ≤ η for a certain η ∈ (0,1) depending on the parameters of the problem. Thelast part of our study is thus to obtain properties for ζn1,n2 from the ηr1,r2 for 0 ≤ r1 ≤ n1 and0 ≤ r2 ≤ n2− 1. Even though we have some ideas on how to obtain estimates for ζn1,n2 fromTheorem 3.37, we have not been able to conclude so far due to the lack of time to fully developthem. We expect to obtain a conclusion in the coming months, as part of the beginning of thePhD program of the student. The motivation to our model being the study of networks of stringswith persistently excited damping, the sequence of this study will be the application of these

76

ideas to this latter case.

3.A Appendix: Lyapunov functions in Banach spacesIn this section, X denotes a Banach space and A : D(A) ⊂ X→ X is a linear operator in X thatgenerates a strongly continuous semigroup etAt≥0. The results we recall here can be found,for instance, in [29], but also in [32, 54].

Definition 3.39 (ω-limit set). For z0 ∈ X, the ω-limit set ω(z0) is the set of z∈ X such that thereis a nondecreasing sequence (tn)n∈N in R+ with tn→+∞ as n→∞ such that etnAz0→ z in X asn→ ∞.

Clearly, by the definition of ω-limit set, we have that d(etAz0,ω(z0))→ 0 as t→+∞, whered denotes the distance in X.

Definition 3.40 (Invariant set). We say that a set M ⊂ X is invariant under etAt≥0 if, for anyz0 ∈M, there exists a continuous function z : R→M with z(0) = z0 such that

etAz(s) = z(t + s), ∀s ∈ R, ∀t ≥ 0.

Note that the definition of invariant set requires the function z to be defined for every t ∈ R,whereas a solution of z = Az is in general only defined for t ≥ 0. The importance of having zdefined for all t ∈ R is discussed in [29].

By the definition of invariant set, the union of a family of invariant sets is still invariant.Hence, given E ⊂ X, we can define the maximal invariant subset M of E as the union of allinvariant sets contained in E, and such a M is thus invariant.

Theorem 3.41. Suppose that etAz0 | t ≥ 0 is precompact in X. Then ω(z0) is a nonempty,compact, connected invariant set.

Definition 3.42 (Lyapunov function). A Lyapunov function for etAt≥0 is a continuous func-tion V : X→ R+ such that

V (z) = limsupt→0+

V (etAz)−V (z)t

≤ 0, ∀z ∈ X.

Theorem 3.43 (LaSalle Principle in Banach spaces). Let V be a Lyapunov function on X, defineE = z ∈ X | V (z) = 0 and let M be the maximal invariant subset of E. If etAz0 | t ≥ 0 isprecompact in X, then ω(z0)⊂M.

Proof. Thanks to Theorem 3.41, ω(z0) is nonempty, compact, connected and invariant.To prove that ω(z0) ⊂ M, it suffices to show that ω(z0) ⊂ E, i.e., that V (z) = 0 for everyz ∈ ω(z0).

Note that t 7→ V (etAz0) is nonincreasing and bounded from below, and so the limitlimt→+∞V (etAz0) exists; let us note V0 = limt→+∞V (etAz0). We claim that V (z) = V0 forevery z ∈ ω(z0); indeed, if z ∈ ω(z0), we take (tn)n∈N a nonincreasing sequence in R+ withtn→+∞ as n→ ∞ and such that etnAz0→ z as n→ ∞. By the continuity of V , we thus getthat

V0 = limt→+∞

V (etAz0) = limn→∞

V (etnAz0) =V (z)

and so V is constant and equal to V0 in ω(z0). Since ω(z0) is invariant, V (etAz) = V (z) forevery z ∈ ω(z0) and every t ≥ 0, so that V (z) = 0 for every z ∈ ω(z0), which concludes theproof.

77

3.B Appendix: Well-posedness of a class of time-dependentdifferential equations in Banach spaces

Let X be a reflexive Banach space, A be the generator of a strongly continuous semigroupetAt≥0 on X and B ∈ L∞(R+,L(X)). We want to study the differential equation

z(t) = (A+B(t))z(t),z(s) = z0

(3.78)

for a certain s ≥ 0. We wish to find a family of operators T (t,s)t≥s≥0 in L(X) such that thesolution of (3.78) will be t 7→ T (t,s)z0, at least for regular z0 ∈X, in the sense that t 7→ T (t,s)z0 isalmost everywhere weakly differentiable and satisfies the first line of (3.78) almost everywhere,and T (s,s)z0 = z0.

In order to study the existence of solutions to (3.78), we first establish the following result,which generalizes [48, Chapter 3, Proposition 1.2].

Proposition 3.44. Let A be the generator of a strongly continuous semigroup etAt≥0 on X andB ∈ L∞(R+,L(X)). Then there exists a unique family T (t,s)t≥s≥0 of bounded operators in Xsuch that (t,s) 7→ T (t,s)z is continuous for every z ∈ X and

T (t,s)z = e(t−s)Az+w t

se(t−τ)AB(τ)T (τ,s)zdτ, ∀z ∈ X. (3.79)

Proof. Let M0 > 0 and ω ∈R be constants such that∥∥etA

∥∥L(X)≤M0eωt for every t ≥ 0 and

let M1 = ‖B‖L∞(R+,L(X)). Define, for t ≥ s≥ 0,

T0(t,s) = e(t−s)A (3.80)

and, for n≥ 1 and z ∈ X,

Tn(t,s)z =w t

se(t−τ)AB(τ)Tn−1(τ,s)zdτ; (3.81)

notice that this is an integral of a function with values in X, which we consider here as aBochner integral. For the general properties of the Bochner integral, see, for instance, [59].

Clearly, for every z ∈ X and every n ∈ N, (t,s) 7→ Tn(t,s)z is continuous for t ≥ s ≥ 0.Also, we have

‖Tn(t,s)‖L(X) ≤M0eω(t−s)Mn0Mn

1(t− s)n

n!(3.82)

for every t ≥ s ≥ 0 and every n ∈ N, which we can see by induction. Indeed, (3.82) istrivially satisfied for n = 0 and, if (3.82) is satisfied for n ∈ N, then, by (3.81), we have

‖Tn+1(t,s)z‖X ≤w t

sM0eω(t−τ)M1M0eω(τ−s)Mn

0Mn1(τ− s)n

n!dτ =

= M0eω(t−s)Mn+10 Mn+1

1 (t− s)n+1

(n+1)!,

which establishes (3.82) by recurrence.We define

T (t,s) =+∞

∑n=0

Tn(t,s),

78

which is well-defined and uniformly convergent in bounded time intervals thanks to (3.82).Hence, for every z ∈ X, (t,s) 7→ T (t,s)z is continuous for t ≥ s≥ 0, and, furthermore, (3.80)and (3.81) imply trivially that T (t,s) satisfies (3.79).

To show that this T (t,s) is unique, suppose that S(t,s)t≥s≥0 is a family of boundedoperators such that (t,s) 7→ S(t,s)z is continuous for t ≥ s≥ 0 and for every z ∈ X and suchthat

S(t,s)z = e(t−s)Az+w t

se(t−τ)AB(τ)S(τ,s)zdτ, ∀z ∈ X.

Then(S(t,s)−T (t,s))z =

w t

se(t−τ)AB(τ)(S(τ,s)−T (t,s))zdτ,

and hence

‖(S(t,s)−T (t,s))z‖X ≤M0M1

w t

seω(t−τ) ‖(S(τ,s)−T (τ,s))z‖X dτ.

Applying Gronwall’s Lemma to the function t 7→ e−ωt ‖(S(t,s)−T (t,s))z‖X, we concludethat T (t,s) = S(t,s) for every t ≥ s≥ 0.

This family T (t,s)t≥s≥0 allows us to obtain solutions for (3.78) for regular z0.

Theorem 3.45. Suppose that z0 ∈D(A). Then (3.78) admits a unique solution z(t), in the sensethat z : [s,+∞)→ X is continuous, z(s) = z0, z is almost everywhere weakly differentiable and

z(t) = (A+B(t))z(t)

for almost every t ≥ s. Furthermore, z is absolutely continuous on [s,+∞) and is given byz(t) = T (t,s)z0.

Proof. To show the existence, take z(t) = T (t,s)z0. Notice that, by (3.79), z is absolutelycontinuous on [s,+∞), z(s) = z0, and, for almost every t ≥ s, the weak derivative z(t) satis-fies z(t) = (A+B(t))z(t), so that it is a solution of (3.78).

To obtain uniqueness, suppose that z(t) is a solution of (3.78) with z(s) = 0. As in theproof of Proposition (3.44), we take constants M0,M1 > 0 and ω ∈R such that

∥∥etA∥∥L(X)≤

M0eωt for every t ≥ 0 and M1 = ‖B‖L∞(R+,L(X)). We write

z(t) = Az(t)+ f (t) (3.83)

with f (t) = B(t)z(t) and, since f ∈ L∞loc([s,+∞) ,X), (3.83) admits at most one solution,

which satisfiesz(t) =

w t

se(t−τ)A f (τ)dτ

(see, for instance, [48, Chapter 4, Corollary 2.2]). Hence

‖z(t)‖X ≤M0M1

w t

seω(t−τ) ‖z(τ)‖X dτ

and we conclude that z(t)≡ 0 by applying Gronwall’s Lemma to t 7→ e−ωt ‖z(t)‖X.

79

80

Bibliography

[1] R.A. Adams: Sobolev spaces. Academic Press, New York, London, 1975. Pure andApplied Mathematics, Vol. 65.

[2] F. Ali Mehmeti, J. von Below, and S. Nicaise (eds.): Partial differential equations onmultistructures, vol. 219 of Lecture Notes in Pure and Applied Mathematics, New York,2001. Marcel Dekker Inc.

[3] B. Anderson: Exponential stability of linear equations arising in adaptive identification.IEEE Trans. Automat. Control, 22(1):83–88, 1977.

[4] B. Anderson, R. Bitmead, C. Johnson, P. Kokotovic, R. Kosut, I. Mareels, L. Praly, andB. Riedle: Stability of adaptive systems: Passivity and averaging analysis. MIT PressSeries in Signal Processing, Optimization, and Control, 8. MIT Press, Cambridge, MA,1986.

[5] S. Andersson and P. Krishnaprasad: Degenerate gradient flows: a comparison study ofconvergence rate estimates. In Decision and Control, 2002, Proceedings of the 41st IEEEConference on, vol. 4, pp. 4712–4717. IEEE, 2002.

[6] P.J. Antsaklis: A brief introduction to the theory and applications of hybrid systems. In-troductory Article for the Special Issue on Hybrid Systems: Theory and Applications,Proceedings of the IEEE, 88(7):879–887, 2000.

[7] P.J. Antsaklis and H. Lin: Hybrid dynamical systems: Stability and stabilization. In W.S.Levine (ed.): The Control Handbook: Control System Applications. CRC Press, BocaRaton, Florida, 2nd ed., 2010.

[8] E. Asarin, O. Bournez, T. Dang, O. Maler, and A. Pnueli: Effective synthesis of switchingcontrollers for linear systems. Proceedings of the IEEE, 88(7):1011–1025, 2000.

[9] M. Balde, U. Boscain, and P. Mason: A note on stability conditions for planar switchedsystems. Internat. J. Control, 82(10):1882–1888, 2009.

[10] A. Balluchi, L. Benvenuti, M. Di Benedetto, C. Pinello, and A. Sangiovanni-Vincentelli:Automotive engine control and hybrid systems: Challenges and opportunities. Proceed-ings of the IEEE, 88(7):888–912, 2000.

[11] G. Bastin, B. Haut, J.M. Coron, and B. D’Andréa-Novel: Lyapunov stability analysis ofnetworks of scalar conservation laws. Netw. Heterog. Media, 2(4):751–759, 2007.

[12] U. Boscain: Stability of planar switched systems: the linear single input case. SIAM J.Control Optim., 41(1):89–112, 2002.

81

[13] H. Brezis: Analyse fonctionnelle. Théorie et applications. Collection Mathématiques Ap-pliquées pour la Maîtrise. Masson, Paris, 1983.

[14] R. Brockett: The rate of descent for degenerate gradient flows. In Proceedings of the 2000MTNS, 2000.

[15] A. Chaillet, Y. Chitour, A. Loría, and M. Sigalotti: Towards uniform linear time-invariantstabilization of systems with persistency of excitation. In Decision and Control, 2007 46thIEEE Conference on, pp. 6394–6399. IEEE, 2007.

[16] A. Chaillet, Y. Chitour, A. Loría, and M. Sigalotti: Uniform stabilization for linear systemswith persistency of excitation: the neutrally stable and the double integrator cases. Math.Control Signals Systems, 20(2):135–156, 2008.

[17] D. Cheng, L. Guo, Y. Lin, and Y. Wang: Stabilization of switched linear systems. IEEETrans. Automat. Control, 50(5):661–666, 2005.

[18] Y. Chitour, G. Mazanti, and M. Sigalotti: Stabilization of persistently excited linear sys-tems. In J. Daafouz, S. Tarbouriech, and M. Sigalotti (eds.): Hybrid Systems with Con-straints, ch. 4. Wiley-ISTE, London, UK, 2013.

[19] Y. Chitour and M. Sigalotti: On the stabilization of persistently excited linear systems.SIAM J. Control Optim., 48(6):4032–4055, 2010.

[20] J.M. Coron: Control and nonlinearity, vol. 136 of Mathematical Surveys and Monographs.American Mathematical Society, Providence, RI, 2007.

[21] R. Courant and F. John: Introduction to calculus and analysis. Vol. I. Springer-Verlag,New York, 1989. Reprint of the 1965 edition.

[22] J. Daafouz, S. Tarbouriech, and M. Sigalotti: Hybrid Systems with Constraints. Wiley-ISTE, London, UK, 2013.

[23] R. Dáger and E. Zuazua: Wave propagation, observation and control in 1-d flexible multi-structures, vol. 50 of Mathématiques & Applications. Springer-Verlag, Berlin, 2006.

[24] R. DeCarlo, M. Branicky, S. Pettersson, and B. Lennartson: Perspectives and results onthe stability and stabilizability of hybrid systems. Proceedings of the IEEE, 88(7):1069–1082, 2000.

[25] W. Feller: An introduction to probability theory and its applications, vol. 1. John Wiley &Sons Inc., New York, 3rd ed., 1968.

[26] W. Feller: An introduction to probability theory and its applications, vol. 2. John Wiley &Sons Inc., New York, 2nd ed., 1971.

[27] J.P. Gauthier and I.A.K. Kupka: Observability and observers for nonlinear systems. SIAMJ. Control Optim., 32(4):975–994, 1994.

[28] M. Gugat and M. Sigalotti: Stars of vibrating strings: switching boundary feedback stabi-lization. Netw. Heterog. Media, 5(2):299–314, 2010.

[29] J.K. Hale: Dynamical systems and stability. J. Math. Anal. Appl., 26:39–59, 1969.

82

[30] J.K. Hale and S.M. Verduyn Lunel: Introduction to functional differential equations,vol. 99 of Applied Mathematical Sciences. Springer-Verlag, New York, 1993.

[31] F. Hante, M. Sigalotti, and M. Tucsnak: On conditions for asymptotic stability of dis-sipative infinite-dimensional systems with intermittent damping. Journal of DifferentialEquations, 252(10):5569–5593, 2012.

[32] D. Henry: Geometric theory of semilinear parabolic equations, vol. 840 of Lecture Notesin Mathematics. Springer-Verlag, Berlin, 1981.

[33] J.L. Kelley: General topology. Springer-Verlag, New York, 1975. Reprint of the 1955edition [Van Nostrand, Toronto, Ont.], Graduate Texts in Mathematics, No. 27.

[34] D. Liberzon: Switching in Systems and Control. Birkhäuser Boston, 1st ed., 2003.

[35] H. Lin and P.J. Antsaklis: Stability and stabilizability of switched linear systems: a surveyof recent results. IEEE Trans. Automat. Control, 54(2):308–322, 2009.

[36] J. L. Lions: Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués.Tome 1. Contrôlabilité exacte., t. 8 de Recherches en Mathématiques Appliquées. Masson,Paris, 1988.

[37] A. Loría, A. Chaillet, G. Besançon, and Y. Chitour: On the PE stabilization of time-varyingsystems: open questions and preliminary answers. In Decision and Control, 2005 and2005 European Control Conference. CDC-ECC’05. 44th IEEE Conference on, pp. 6847–6852. IEEE, 2005.

[38] M. Lovera and A. Astolfi: Global spacecraft attitude control using magnetic actuators.In Advances in dynamics and control, vol. 2 of Nonlinear Syst. Aviat. Aerosp. Aeronaut.Astronaut., pp. 1–13. Chapman & Hall/CRC, Boca Raton, FL, 2004.

[39] G. Lumer: Connecting of local operators and evolution equations on networks. In Po-tential theory, Copenhagen 1979 (Proc. Colloq., Copenhagen, 1979), vol. 787 of LectureNotes in Math., pp. 219–234. Springer, Berlin, 1980.

[40] M. Malisoff and F. Mazenc: Constructions of strict Lyapunov functions. Communicationsand Control Engineering Series. Springer-Verlag London Ltd., London, 2009.

[41] M. Margaliot: Stability analysis of switched systems using variational principles: an in-troduction. Automatica, 42(12):2059–2077, 2006.

[42] G. Mazanti: Stabilization of persistently excited linear systems by delayed feedback laws.(preprint), 2013.

[43] G. Mazanti, Y. Chitour, and M. Sigalotti: Stabilization of two-dimensional persistentlyexcited linear control systems with arbitrary rate of convergence. SIAM J. Control Optim.,51(2):801–823, 2013.

[44] W. Michiels and S.I. Niculescu: Stability and stabilization of time-delay systems: Aneigenvalue-based approach, vol. 12 of Advances in Design and Control. Society for In-dustrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007.

83

[45] A.P. Morgan and K.S. Narendra: On the stability of nonautonomous differential equationsx= [A+B(t)]x with skew-symmetric matrix B(t). SIAM J. Control Optim., 15(1):163–176,1977.

[46] S. Nicaise: Spectre des réseaux topologiques finis. Bull. Sci. Math. (2), 111(4) :401–413,1987.

[47] S.I. Niculescu: Delay effects on stability: A robust control approach, vol. 269 of LectureNotes in Control and Information Sciences. Springer-Verlag London Ltd., London, 2001.

[48] A. Pazy: Semigroups of linear operators and applications to partial differential equations,vol. 44 of Applied Mathematical Sciences. Springer-Verlag, New York, 1983.

[49] S. Pettersson: Synthesis of switched linear systems. In Decision and Control, 2003. Pro-ceedings. 42nd IEEE Conference on, vol. 5, pp. 5283–5288. IEEE, 2003.

[50] J. le Rond D’Alembert: Recherches sur la courbe que forme une corde tendue mise envibration. Hist. Ac. Sci. Berlin, 3 :214–219, 1747.

[51] J. le Rond D’Alembert: Suite des recherches sur la courbe que forme une corde tendue,mise en vibration. Hist. Ac. Sci. Berlin, 3 :220–249, 1747.

[52] W.J. Rugh: Linear System Theory. Prentice Hall, 2nd ed., 1996.

[53] R. Shorten, F. Wirth, O. Mason, K. Wulff, and C. King: Stability criteria for switched andhybrid systems. SIAM Rev., 49(4):545–592, 2007.

[54] M. Slemrod: The LaSalle invariance principle in infinite-dimensional Hilbert space. InDynamical systems approaches to nonlinear problems in systems and circuits (Henniker,NH, 1986), pp. 53–59. SIAM, Philadelphia, PA, 1988.

[55] E.D. Sontag: Mathematical control theory, vol. 6 of Texts in Applied Mathematics.Springer-Verlag, New York, 2nd ed., 1998. Deterministic finite-dimensional systems.

[56] M. Tucsnak and G. Weiss: Observation and control for operator semigroups. BirkhäuserAdvanced Texts: Basler Lehrbücher. [Birkhäuser Advanced Texts: Basel Textbooks].Birkhäuser Verlag, Basel, 2009.

[57] J. Valein: Stabilité de quelques problèmes d’évolution. Thèse de doctorat, Université deValenciennes et du Hainaut-Cambrésis, 2008.

[58] J. Valein and E. Zuazua: Stabilization of the wave equation on 1-D networks. SIAM J.Control Optim., 48(4):2771–2797, 2009.

[59] K. Yosida: Functional analysis, vol. 123 of Grundlehren der Mathematischen Wis-senschaften [Fundamental Principles of Mathematical Sciences]. Springer-Verlag, Berlin,6th ed., 1980.

[60] E. Zuazua: Control and stabilization of waves on 1-d networks. In Modelling and Optimi-sation of Flows on Networks, pp. 463–493. Springer, 2013.

84