Controllability of Nonlinear Diffusion System

Ahmed Maidi1 and Jean-Pierre Corriou2*

1. Laboratoire de Conception et Conduite des Systèmes de Production, Université Mouloud MAMMERI, 15 000 Tizi-Ouzou, Algérie

2. Laboratoire Réactions et Genie des Procédés, UMR 7274-CNRS, Lorraine Université, ENSIC 1, rue Grandville, BP 20451,54001 Nancy Cedex, France

In this article, the semi-group theory is exploited to study the controllability of a particular but important class of distributed parameter systemswith a uniform distributed control variable. This class is described by a partial differential equation that characterizes the nonlinear diffusionphenomenon. The main idea consists of using the Cole–Hopf tangent transformation, under some assumptions, in order to derive an equivalentlinear model of the nonlinear diffusion system. Then, the study of the controllability of the linear model, based on the semi-group approach, allowsus to conclude about the controllability of the nonlinear diffusion system. The study is illustrated by an application example.

Keywords: distributed parameter systems, partial differential equation, Cole–Hopf tangent transformation, controllability,semi-group theory

INTRODUCTION

Controllability is a central concept in control theory andplays a key role in designing a controller to a dynamicalsystem. This fundamental control property means the exis-

tence of a control that steers the dynamical system from a givenstate to another one in finite time. Thus, if such a control existswhatever the considered initial and final states, the system is saidto be controllable.[1] The analysis of controllability constitutes aprimary question for any control strategy and the aim is to definethe space of the reachable states.[2]

The study of the controllability of dynamical systems is anactive research area especially for distributed parameter systems(DPSs), termed also infinite-dimensional systems. The dynamicalbehaviour of DPSs is described by partial differential equations(PDEs). Compared to the lumped parameter systems (LPSs),which are described by ordinary differential equations (ODEs),the analysis of the controllability of DPSs is very delicate sinceseveral types of controllability exist[1,3] and it needs sophisti-cated mathematical tools from functional analysis and semi-grouptheory.[4] In addition, for a DPS, this fundamental control prop-erty does not depend only on its dynamics described by the PDEsbut also on the type (punctual, distributed, or boundary), thegeometry and the location of the actuators.[5]

The conventional approach for the analysis of the controllabil-ity of DPSs consists of approximating the original PDE model by alow dimensional ODE model using powerful reduction methods.[6]

This approach reduces the DPS to a LPS for which very powerfuland well-established theories of the analysis of the controllabilityhave been developed.[1,7] This approach presents the drawbackthat the distributed nature of the system can be masked by thereduction or the approximation process.[8–10] Indeed, the fun-damental properties (controllability, observability and stability)depend directly on the number and the locations of the discretiza-tion points, which often leads to erroneous conclusions on thesevarious properties.[8–10] Consequently, the analysis of the control-lability using directly the PDE model without any reduction orapproximation is the right approach since the distributed nature ofthe system is entirely preserved.[9] Nevertheless, direct handling

of PDE model is difficult, in particular in the case of nonlinearmodels.

The study of controllability directly using the PDE model hasbeen widely explored and several approaches have been devel-oped in the dedicated literature though certain problems are stillopen. A synthesis of the various proposed approaches can befound in Coron,[1] Zuazua,[3] and Glass.[11] From the literature,it appears that, in contrast to the linear systems, the study of thecontrollability of nonlinear DPSs, using directly the PDE model,remains an unexplored area and few results are available.[12] Thedifficulties come from the nonlinear nature of these models.

Based on the semi-group theory,[4,13,14] various powerful con-cepts and results available in the case of linear LPSs aregeneralized and extended to linear DPSs by considering directlythe PDE model.[4,13–15] The semi-group theory has made a leapforward in both analysis and design of linear DPSs by generaliz-ing the state-space representation of linear LPSs to the linear DPSsusing the abstract formulation.[14,15]

In this article, the semi-group theory[4] is exploited to study theexact controllability of a particular but important class of non-linear DPSs. This class is modelled by a PDE that describes thenonlinear diffusion phenomenon.[16] The study is focused on thecase of a uniform distributed actuation. The main idea consistsin seeking an abstract formulation of the nonlinear model, whichsimplifies the study of the controllability. Thus, under a practi-cally acceptable assumption, an equivalent linear model to thenonlinear diffusion equation is derived thanks to Cole–Hopf tan-gent transformation.[17] Then, the obtained linear model is writtenunder the abstract form that allows the study of its exact control-lability using some concepts from semi-group theory. Thereafter,based on the properties of the used tangent transformation, a

∗Author to whom correspondence may be addressed.E-mail address: jean-pierre.corriou@univ-lorraine.frCan. J. Chem. Eng. 9999:1–5, 2014© 2014 Canadian Society for Chemical EngineeringDOI 10.1002/cjce.22030Published online in Wiley Online Library(wileyonlinelibrary.com).

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global conclusion about the controllability of the nonlinear diffu-sion system can be easily drawn from the conclusion concerningthe controllability of the equivalent linear model. The proposedapproach is illustrated by an example.

The article is structured as follows: The second section presentsthe class of the nonlinear diffusion system considered in this workand its linearization using the Cole–Hopf tangent transformation.The third section is devoted to the study of the controllabilityof linear DPSs, written under the abstract formulation, based onthe semi-group theory. The proposed approach to the study thecontrollability of the nonlinear diffusion system is presented inthe fourth section, which is illustrated by an example given in thefifth section. Finally, a conclusion is provided in the sixth section.

NONLINEAR DIFFUSION SYSTEM

The nonlinear diffusion equation constitutes one of the mostimportant parabolic equations. This equation describes thedynamical behaviour of the diffusion phenomena which occurs inmany fields for instance mass and heat transfer with in particularFourier’s and Fick’s laws.[16]

Mathematical Model

In this study, the class of DPSs, considered here, is modelledby the one-dimensional nonlinear diffusion equation, with auniform distributed control u(t), according to the following state-representation:

� Cp(x(z, t))∂x(z, t)∂t

= ∂

∂z

(k(x(z, t))

∂x(z, t)∂z

)

+ b(z)u(t) in �×]0, t[ (1)

accompanied by the Dirichlet boundary conditions

x(0, t) = x0 (2)

x(l, t) = xl (3)

and the initial condition

x(z, 0) = ϕ(z) in � (4)

where x(z, t) denotes the state, z ∈ � = [0, l] ⊂ � is the spatialdomain, t ∈ [0, ∞[ is the time variable, ϕ(z) is a given initialprofile, and u(t) ∈ L2([0, ∞[; �) is the manipulated heat flux. b(z)is a known smooth function that characterizes the distribution ofthe lumped control u(t). In this study, the manipulated heat fluxu(t) is assumed to be uniformly distributed, that is, b(z) = 1.

In the following, without loss of generality and to simplify thepresentation, the terminology from heat conduction is adopted.Thus, k(x(z, t)) > 0, � and Cp(x(z, t)) denote the thermal con-ductivity, the density, and the heat capacity, respectively. Notethat the development which follows remains valid for other typesof boundary conditions since these latter can only affect the defi-nition of the subspace of the controllable state of the system.

The objective of this work is to study the controllability of thenonlinear system (1)–(4), that is, being given two states xi andxf that belong to the state space X, is it possible to find a controlu(t) that can steer the system from the initial state xi to the finalstate xf [1,2] and specify a subspace of X where the system is exactlycontrollable.

Cole–Hopf Tangent Transformation

To simplify both the study and the solution of particular non-linear PDEs, several tangent transformations are developed inthe literature to convert a nonlinear PDE to a linear one.[18] Forthe nonlinear diffusion equation(1), when thermal diffusivity isapproximatively constant, one can use the Cole–Hopf tangenttransformation[19,17] to derive an equivalent linear diffusion ofthe nonlinear PDE (1). The thermal diffusivity is defined by thefollowing ratio:

˛ = k(x(z, t))� Cp(x(z, t))

. (5)

Indeed, in the majority of the practical cases, the variation of thethermal diffusivity ˛ according to x(z, t) is less significant thanthat of thermal conductivity k(x(z, t)), consequently the assump-tion of a constant thermal diffusivity is physically acceptable.[19]

The application of Cole–Hopf tangent transformation is based onthe following assumption, which is assumed in this study.

Assumption 1. The thermal diffusivity ˛ of system (1) is approx-imatively constant.

To linearize the nonlinear diffusion equation (1) using the Cole–Hopf tangent transformation, one seeks a transformation of theform[17]

x(z, t) = h(w(z, t)) (6)

where h( . ) is a bijective operator (map) and w(z, t) has the samedimension as x(z, t). By using the transform (6), the evaluationof the derivatives of the left hand side of (1) gives

∂h(w(z, t))∂t

= dh(w(z, t))dw(z, t)

∂w(z, t)∂t

, (7)

and for the right hand side of (1), one obtains

∂

∂z

(k (h(w(z, t)))

∂h(w(z, t))∂z

)

= M

(∂w(z, t)∂z

)2

+ k(h(w(z, t)))dh(w(z, t))

dw(z, t)∂2w(z, t)∂z2

(8)

with

M = k(h(w(z, t)))d2h(w(z, t))

dw2(z, t)

+ dk(h(w(z, t)))dh(w(z, t))

(dh(w(z, t))

dw(z, t)

)2

. (9)

Now, to make the right-hand side of (8) linear, the term M isset equal to zero. Notice that the term M, which is a differentialequation, can be expressed in the following integrable form

ddw(z, t)

[k(h(w(z, t)))

dh(w(z, t))dw(z, t)

]= 0. (10)

Integrating (10) gives

k(h(w(z, t)))dh(w(z, t))

dw(z, t)= c1 (11)

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hence∫k(h(w(z, t))) dh(w(z, t)) =

∫c1 dw(z, t)

= c1w(z, t) + c2 (12)

or equivalently

w(z, t) = h−1(x(z, t)) = 1c1

∫ x(z, t)

x(z, 0)

k(s) ds+w(z, 0) (13)

where h−1( . ) is the inverse operator of h( . ).In summary, using transformation (6), with the operator h( . )

satisfying (10) and considering the expression (5) of diffusivity˛, the nonlinear diffusion equation (1) will be converted to thefollowing linear one:

∂w(z, t)∂t

= ˛∂2w(z, t)∂z2

+ ˛

c1b(z)u(t) (14)

with the following boundary and initial conditions

w(0, t) = h−1(x(0, t)) = h−1(x0) = w0, (15)

w(l, t) = h−1(x(l, t)) = h−1(xl) = wl, (16)

w(z, 0) = h−1(ϕ(z)) = (z). (17)

Remark 1. The tangent transformations[18] represent an inter-esting tool that can be exploited for dynamical system analysisand design. For instance, the Cole–Hopf tangent transformationhas been used by Maidi and Couriou[20] to design a distributedstate-feedback controller to the nonlinear diffusion equation (1).

Dimensionless Model

To give more generality to the study of the controllability of thenonlinear diffusion system, the dimensionless model of its equiv-alent linear model (14)–(17) is derived in this subsection. Hence,by defining the dimensionless length � and the dimensionless time� as

� = z

l, � = ˛

l2t (18)

and by introducing the following control, of dimension similar tow(�, �), that is also to x(z, t),

v(�) = l2

c1b(z)u(t) (19)

the linear model (14)–(17) is reduced to the following partiallydimensionless model, with corresponding boundary conditionsand initial profile,

∂w(�, �)∂�

= ∂2w(�, �)∂�2

+ v(�) (20)

w(0, �) = w0 (21)

w(1, �) = wl (22)

w(�, 0) = (�) (23)

then by introducing the following variable

y(�, �) = w(�, �) −w0 + � (w0 −wl) (24)

the dimensionless model (20)–(23) takes the following form

∂y(�, �)∂�

= ∂2y(�, �)∂�2

+ v(�) (25)

y(0, �) = 0 (26)

y(1, �) = 0 (27)

y(�, 0) = (�) −w0 + � (w0 −wl) = y0(�) (28)

CONTROLLABILITY OF LINEAR DISTRIBUTED PARAMETERSYSTEMS

For DPSs, two types of controllability are defined:[1,13–15,21] exactand approximate controllability. Before introducing the definitionof these two concepts, first the abstract formulation of a DPS ispresented. This last consists in writing the PDE model of the DPSin the following operator form[13–15]

y(�) = A y(�) + B v(�); 0 ≤ � ≤ T (29)

y(0) = y0 ∈ D(A) (30)

where D(A) is the domain of the linear operator A ∈ L(Y), whichis assumed to be a generator of an infinitesimal strongly con-tinuous semi-group C(t) in the state space Y, and B ∈ L(Y;U)is the linear control operator assumed to be bounded. L(Y) andL(U;Y) are spaces of linear operators from Y to Y and from U toY, respectively.

The two types of controllability in the case of linear DPSs aredefined as follows:[13–15]

Definition 1. The system (29) is exactly controllable in time Tif, for every y0 = y(0) ∈ Y and yT = y(T) ∈ Y, there exits a controlv(�) ∈ L2([0, T];U) such that the solution of (29) verifies y(T) =yT .

Definition 2. The system (29) is approximately controllable intime T if, for every y0 = y(0) ∈ Y, for every yT = y(T) ∈ Y, andevery ε > 0, there exists a control v(�) ∈ L2([0, T];U) such thatthe solution of (29) verifies

∥∥y(T) − yT∥∥Y ≤ ε where ‖ . ‖ is the

norm in the space Y.

To study the controllability of a linear DPS, various methodsexist, which can be split into two classes, namely direct and dual-ity methods. A complete review of these different methods is givenby Coron[1] In the following, a theorem that allows the studyof controllability of linear DPS, developed in the framework ofsemi-group theory, is presented.

Let us assume that the operator A has eigenvalues n and thecorresponding eigenfunctions n(�) form an orthogonal basison the space Y, thus the semi-group generated by A is given byCurtain and Pritchard,[13] Curtain and Zwart,[14] and El Jai[15]

C(t) y =∞∑n=1

en trn∑j=1

〈y, n(�)〉n(�) (31)

where 〈 . 〉 is the inner product in Y and rn is the algebraicmultiplicity of the eigenvalue n.

The exact controllability condition of the linear PDE system (29)is stated by the following theorem:[13–15]

Theorem 1 If u(t) ∈ Lp([0, T];U) with 1 < p < ∞, then the sys-tem (29) is said to be exactly controllable if and only if there exist

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� > 0 such that

‖y∗‖Y∗ ≤ �‖B∗C∗( . ) y∗‖Lq([0, T];U∗) (32)

with y∗ ∈ Y∗ and 1/p + 1/q = 1.

In relation (32), B∗ and C∗( . ) are the adjoint operator B and theadjoint semi-group of C( . ), respectively. Y∗ is the dual space of Y.For the demonstration of this property, the reader can refer to ElJai[15] (page 196).

CONTROLLABILITY OF NONLINEAR DIFFUSION SYSTEM

The analysis of the controllability of nonlinear DPSs occupiesan important place in control theory and constitutes an activeresearch area. This subject has been investigated by many authorsbut few results are available. For a review of the fundamentalresults concerning the study of the controllability of nonlinearDPSs, please see Coron,[1] Glass,[11] and Balachandran andDauer.[12]

If, for linear DPSs, general results are available,[1,3] for nonlin-ear DPSs, it is very difficult, not to say impossible, to establish ageneral framework for the analysis of the controllability. For thiskind of dynamical systems, the investigation is done, in general,by considering a particular case. Most results are developed byassuming a linearized model, which leads either to local or globalcontrollability results if the nonlinearity is not very strong.[1]

In this section, the result of Theorem 1 is exploited to study thecontrollability of the nonlinear diffusion system (1)–(4). Indeed,based on the results on the study of the controllability of the equiv-alent dimensionless linear model (25)–(28), derived using theCole–Hopf tangent transformation (6), one can draw easily a con-clusion about the controllability of the nonlinear system (1)–(4).This approach presents the advantage to lead to global controlla-bility results, which makes it an interesting approach comparedto the methods based on the linearized model, proposed in theliterature, that lead to local controllability results.[1] Note that theequivalent dimensionless linear diffusion model (25)–(28) can bewritten in the abstract formulation (29) by defining

A = ∂2

∂�2, B = I (Identity operator). (33)

The Cole–Hopf tangent transformation consists in defining abijective operator h : W ⊆ Y → V ⊆ X, therefore its inverse h−1 :V ⊆ X → W ⊆ Y exists. The subspace V represents the imageunder the operator h of W, that is, V = Im(h) ≡ h(W). The sub-spaces W and V are called the domains of the bijective operatorh( . ) and its inverse h−1( . ), respectively.

After having presented the different notions used in this study,the following proposition allow us to address the exact controlla-bility of the nonlinear diffusion system (1)–(4).

Proposition 1 If the equivalent linear diffusion system (25)–(28)is exactly controllable to a subspace Z ⊆ Y, then the nonlineardiffusion system (1)–(4) is exactly controllable to a subspaceh(W ∩ Z) ⊆ V ⊆ X.

Proof. The Cole–Hopf tangent transformation yields the bijectiveoperator h( . ) with the domainW ⊆ Y. Therefore, if the equivalentdimensionless linear diffusion system (25)–(28) is exactly control-lable to a subspace Z ⊆ Y, it follows that for any controllable statey that belongs to the domain W of h( . ), that is, y ∈ W ∩ Z, it cor-responds a state x ∈ X of the nonlinear diffusion system (1)–(4),

Figure 1. Controllability of nonlinear diffusion system.

which actually is the image under the operator h( . ) of the state y(x = h(y)). Consequently, the nonlinear diffusion system (1)–(4)is exactly controllable to a subspace, which is the image underthe operator h( . ) of W ∩ Z, that is, the subspace h(W ∩ Z) as itis illustrated by Figure 1.

ILLUSTRATIVE EXAMPLE

Let us consider the nonlinear diffusion system with a thermalconductivity of the form

k(x(z, t)) = kr e(

1− x(z, t)xr

)(34)

and the following boundary conditions

x(0, t) = x(l, t) = xr (35)

The operator h( . ) that linearizes this model obtained using therelation (13) is

w(z, t) = h−1(x(z, t))

= 1kr

∫ x(z, t)

xr

kr e(1− sxr ) ds = xr

(1 − e

(1− x(z, t)

xr

))(36)

which yields

x(z, t) = h(w(z, t)) = xr

(1 − ln

(1 − w(z, t)

xr

))(37)

Remark 2. In this example, the constant parameter c1 is deter-mined following the Kirchhoff transformation, a particular caseof the Cole–Hopf tangent transformation,[17] which yields c1 = kr .More details can be found in Vadasz[17] and Maidi and Corriou[20]

This Cole–Hopf tangent transformation allows to linearize thenonlinear diffusion system with the thermal conductivity (34) andboundary conditions (35), and write its corresponding equiva-lent linear dimensionless model in the abstract formulation asexplained both in second and fourth sections.

The controllability of the equivalent linear system (25)–(28)has been studied by Curtain and Pritchard[13] (Page 58, Example3.9) using Theorem 1 and it is found that this system is exactlycontrollable to a subspace

Z = H10 (0, 1) ∈ Y = L2(0, 1) (38)

and from (37), the domain W of the bijective operator h( . ) is

W = {y ∈ Y = L2(0, 1) | y < xr}. (39)

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Thus, using the proposition 1, one deduces that the nonlineardiffusion system, with the thermal conductivity (34) and bound-ary conditions (35), is exactly controllable to the subspace, whichis the image under operator h( . ) of the following subspace:

W ∩ Z = {y ∈ H10 (0, 1) | y < xr}, (40)

that is

h(W ∩ Z) = {x = h(y) ∈ X | y ∈ H10 (0, 1) and y < xr}. (41)

Remark 3. The control of the present diffusion nonlinear systemwas studied by the authors in Maidi and Courriou[20] where a geo-metric control law that ensures output tracking, that is steeringthe output (a function of the state) from a given initial condi-tion to a desired one, in closed-loop was developed following thelate lumping approach, that is, using directly the partial differen-tial equation, and the closed-loop stability was demonstrated usingthe perturbation theorem from semi-group theory. The performanceof the developed control law was evaluated by simulation and theobtained results demonstrated the ability of the control law in steer-ing the controlled output in a finite time. Note that in Maidi andCourriou,[20] the output operator is bijective, therefore the outputcontrollability implies the state controllability.

CONCLUSION

In this article, the controllability of a class of nonlinear diffusionsystems is investigated. It is shown that the conclusion on the con-trollability of a nonlinear diffusion system can be deduced fromthe controllability of the equivalent dimensionless linear modelobtained using the Cole–Hopf tangent transformation that is abijective operator. The use of this transformation allows to exploitsome powerful theoretical tools, developed in the framework ofsemi-group theory, to study fundamental control properties oflinear DPSs. Consequently, once a conclusion about the controlla-bility of the equivalent dimensionless linear model is drawn, it isproved that the nonlinear diffusion is controllable to a subspacedefined as the image, under the Cole–Hopf tangent transforma-tion, of the intersection of the domain of this transformation(operator) and the subspace of the exact controllable states of theequivalent dimensionless linear model. The established result isillustrated by an application example.

Note that the Cole–Hopf tangent transformation and othertangent transformations,[18,22] can be used to linearize some non-linear DPSs, which allows to take advantage of the full potential ofan existing control theory for linear DPSs. For instance, the non-linear Burger’s equation also can be linearized by the Cole–Hopftangent transformation. The linearization based on the tangenttransformations constitutes an interesting approach since it leadsto global analysis and design results compared to the use of a lin-earized model, around a given profile, which yields local results.

REFERENCES

[1] J.-M. Coron, Control and Nonlinearity, AmericanMathematical Society, Providence, USA 2007.

[2] J. Klamka, Matematyka Stosowana 2008, 9, 57.[3] E. Zuazua, Handbook of Differential Equations:

Evolutionary Differential Equations, Volume 3, C. M.Dafermos, E. Feireil, Eds., Elsevier, North-Holland 2007,p. 527.

[4] A. Pazy, Semigroups of Linear Operators and Applications toPartial Differential Equations, Springer-Verlag, New York1983.

[5] A. El Jai, A. J. Pritchard, Capteurs et Actionneurs dansl’Analyse des Systèmes Distribués, Masson, Paris 1986.

[6] H. X. Li, C. Qi, J. Process Control 2010, 20, 891.[7] L. Rosier, Encyclopedia of Complexity and Systems Science,

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[10] A. Singh, J. Hahn, 8th International IFAC Symposium onDynamics and Control of Process Systems 2007, 6–8 June2007, Cancun, Mexico, p. 199.

[11] O. Glass, Encyclopedia of Complexity and Systems Science,R. A. Meyers, Ed., Springer, New York 2009, p. 755.

[12] K. Balachandran, J. Dauer, J. Optim. Theory Appl. 2002,115, 7.

[13] R. F. Curtain, J. Pritchard, Infinite Dimensional LinearSystems Theory, Springer-Verlag, Berlin 1978.

[14] R. F. Curtain, H. J. Zwart, An Introduction toInfinite-Dimensional Linear Systems Theory,Springer-Verlag, New York 1995.

[15] A. El Jai, Eléments d’Analyse et de Contrôle des Systèmes,Presses Universitaires de Perpignan, Perpignan 2004.

[16] Z. Wu, J. Zhao, J. Yin, H. Li, Nonlinear DiffusionEquations, World Scientific, Singapore 2001.

[17] P. Vadasz, J. Heat Transfer 2010, 132, 121302.1.[18] S. V. Meleshko, Methods for Constructing Exact Solutions of

Partial Differential Equations. Mathematical and AnalyticalTechniques with Applications to Engineering, Springer, NewYork 2005.

[19] H. Carslaw, J. Jaeger, Conduction of Heat in Solids, OxfordUniversity Press, Oxford 1959.

[20] A. Maidi, J.-P. Corriou, Int. J. Robust Nonlinear Control2014, 24, 386.

[21] R. Glowinski, J.-L. Lions, J. He, Exact and ApproximateControllability for Distributed Parameter Systems : ANumerical Approach, Cambridge University Press,Cambridge 2008.

[22] K. Kowalski, W.-H. Steeb, Nonlinear Dynamical Systemsand Carleman Linearization, World Scientific Publishing,Farrer Road, Singapore 9128, 1991.

Manuscript received February 26, 2014; revised manuscriptreceived April 18, 2014; accepted for publication April 28, 2014.

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