Dr Dayan Driss

Liu

Prof Michel Zasadzinski

Prof Mohammed M'saad

Prof Catherine Bonnet

Prof. Taous-Meriem Laleg-Kirati

Marylene Vallee

Omar Tahri

Khaled Chetehouna

Antoine Ferreira

Toufik Arnaud Paris

Bassem Chieb

Yunhui Hou

Julien Thuillier

Xing Wei

Yanqiao Wei

Ruipeng Chen

Lifei Wang

Tingting Zhang

Bainan Liu

Driss Boutat

This thesis aims to design a boundary observer-based output feedback controller for a class of systems modelled by linear coupled parabolic PDEs by using the backstepping method. Roughly speaking, the backstepping method for PDEs mainly consists of transforming some kinds of PDEs into some particular PDEs, that are easy to analyze and stabilize by using controllers or observers. This kind of particular PDEs will be called target systems. Firstly, it considers an easy case of coupled reaction-diffusion equations with the same constant diffusion parameter. For this case, it proposes a more relaxed stability condition for the target system of the backstepping transformation. Moreover, for the same case, it designs a backstepping boundary observer-based output feedback controller. Then, it takes an example to verify the proposed method.

It also deals with a class of systems modelled by reaction-advection-diffusion equa-

tions with the same constant diffusion parameter, which are realized by proposing particular conditions on the target systems. Secondly, it deals with a kind of systems modelled by coupled reaction-diffusion equations with different diffusions. In a similar way, it designs a boundary observer for this kind of systems. However, due to the fact that the constant diffusions are not the same, it is more difficult to solve the kernel functions of the backstepping transformation than the same diffusion case. For this, an assumption on the kernel functions is made to enable us to solve the problem. Moreover, it also designs a backstepping boundary controller based on the proposed stability conditions. Those stability conditions are more relaxed than the conditions we can find in the literatures on this topic. Then, based on the Separation Principle, it designs an observer-based output feedback controller. It takes a simplified model of Chemical Tubular Reactor to highlight the proposed method. Thirdly, this thesis designs a boundary observer as a more general extension by studying a class of systems modelled by coupled reaction-advection-diffusion equations with spatially-varying coefficients, which is more challenged to solve kernel functions of the backstepping transformation. To achieve this, it transforms the parabolic kernel equations into a set of hyperbolic equations. Then, it proves the well-posedness by setting suitable boundary conditions for the kernel functions. Moreover, it also provides the stability conditions for the target systems. The performance of the proposed observer is illustrated by taking a numerical model. Fourthly, it extends the above backstepping observer-based output feedback controller to fractional-order PDE systems. Finally, conclusions are outlined with some perspectives. Index Terms: Parabolic PDEs, Coupled system, Dirichlet-type boundary, Neumanntype boundary, Backstepping method, Boundary controller, Boundary observer, Output feedback control, Lyapunov stability, Fractional-order system.

Résumé étendu en français

Cette thèse vise à concevoir un contrôleur basé observateur au bord pour une classe de systèmes modélisés par des équations aux dérivées partielles (EDP) paraboliques couplées en utilisant la méthode dite backstepping. Grosso modo, la méthode du backstepping pour les EDP consiste principalement à les transformer sous certaines formes faciles à analyser et à stabiliser à l'aide de contrôleurs ou d'observateurs. Ces formes seront appelées les systèmes cibles. Tout d'abord, ce travail considère un cas simple d'équations couplées avec des paramètres de diffusion constants. Pour ce cas, on met en évidence des conditions de stabilité moins contraignantes que les conditions proposées dans la littérature sur ce sujet. De plus, pour le même cas, on conçoit une commande par retour d'état basé observateur. Ensuite, on donne une simulation sur un exemple pour prouver la consistance de la méthode proposée. Ce travail traite également d'une classe de systèmes modélisés par équations de réaction-advectiondiffusion avec le même paramètre de diffusion constant en proposant des conditions particulières sur les systèmes cibles. Dans un second temps, on traite le cas des équations couplées réaction-diffusion avec différentes diffusions. Cependant, comme les termes de diffusions sont différents, il est plus difficile de calculer le noyau de la transformation backstepping. Pour surmonter cette difficulté, on fait une hypothèse sur le noyau qui définit la transformation backstepping. De plus, on conçoit également un contrôleur basé observateur avec les mêmes conditions de stabilité proposées pour les deux premières situations. Ensuite, on utilise le principe de séparation pour concevoir un contrôleur basé observateur. Enfin, on utilise un modèle simplifié de réacteur tubulaire pour mettre en évidence la cohérence de la méthode proposée. Dans une troisième partie, cette thèse étend ces résultats à une classe de systèmes modélisés par des équations couplées de réaction-advection-diffusion à coefficients dépendant de la variable d'espace, ce qui rend la détermination du noyau de la transformation backstepping plus difficile. Pour ce faire, on transforme les équations aux dérivées partielles paraboliques qui définissent le noyau de la transformation en un ensemble d'équations hyperboliques. Par conséquent, on peut prouver que le problème est bien posé en fixant des conditions aux limites appropriées pour la fonction noyau. De plus, on fournit également les conditions de stabilité pour les systèmes cibles. La performance de l'observateur proposé est illustrée sur un modèle numérique. Puis, on étend le contrôleur basé observateur aux systèmes EDP d'ordre fractionnaire. Enfin, des conclusions sont présentées avec quelques perspectives.

Mots clés

Background and Motivations

There are many industrial systems modelled by Partial Differential Equations (PDEs) such as flexible machines [7], solar collector systems [START_REF] Elmetennani | Bilinear reduced order approximate model of parabolic distributed solar collectors[END_REF], drilling systems [9, 10], chemical reaction systems [START_REF] Laabissi | Trajectory analysis of nonisothermal tubular reactor nonlinear models[END_REF][START_REF] Winkin | Dynamical analysis of distributed parameter tubular reactors[END_REF] and so on (Fig. 1.1). If these models are assumed to be accurate enough to reflect the real systems, they can be used to design controllers that enable the systems to achieve desired performances. To design such controllers, we always need to know the states of the system, which can usually be measured by physical sensors. However, they are sometimes difficult or even impossible to measure. In order to overcome this issue, observers also known as software sensors can be designed to estimate the unmeasurable states. Then, an observer-based output feedback controller can be used to obtain the desired performances. Recently, many researchers devoted themselves to this topic [START_REF] Boskovic | Boundary control of an unstable heat equation via measurement of domain-averaged temperature[END_REF]- [20]. As a kind of PDEs, coupled parabolic PDEs can be used to model many processes in applications [START_REF] Kunish | Control of coupled partial differential equations[END_REF]- [START_REF] Mitra | Wave propagation in imperfectly bonded single walled carbon nanotube-polymer composites[END_REF]. In the present work, our main

Parabolic PDEs

Partial Differential Equations are differential equations that contain unknown multivariable functions and their successive partial derivatives. They are used to formulate physical problems involving multivariable functions and their successive derivatives.

There are many different kinds of PDEs including parabolic PDEs, hyperbolic PDEs and other "odd" equations such as Navier-Stokes, Kuramoto-sivashinsky and so on. Parabolic PDEs is a main kind of PDEs. In order to define the simplest kind of parabolic PDEs, we consider the following linear second-order constant-coefficient PDE:

au xx + 2bu x y + cu y y + d u x + eu y + f = 0, (1.1)
where u(x, y) is a real-valued function with two independent real variables x and y, and a, b, c, d, e and f are constant coefficients. If the coefficients satisfy:

b 2 -ac = 0, (1.2)
then the above PDE is called parabolic PDE.

The basic example of a parabolic PDE is the one-dimensional heat equation:

u t (x, t ) = αu xx (x, t ), (1.3
)

u(0) = 0, (1.4)
u(1) = 0, (1.5) where u(x, t ) represents the temperature at time t and position x along a thin rod, α is a positive constant that represents the thermal diffusivity. (1.4) and (1.5) are boundary conditions of (1.3). For PDEs, there are three basic types of boundary conditions in one dimension:

• Dirichlet: u(0) = 0 (fixed temperature at x = 0).

• Neumann: u x (0) = 0 (fixed heat flux at x = 0).

• Robin: pu x (0) + qu(0) = 0 (mixed).

Within this work , the solution u(x, t ) will be called the state of the system.

Coupled Parabolic PDEs

The heat system (1.3)-(1.5) is a scalar system. In reality, there are many systems that are modelled by coupled parabolic PDEs, for which the states U(x, t ) or their partial derivatives appear in their equations. In this present work, we will consider the following linear coupled parabolic PDEs system:

U t (x, t ) = Θ(x)U xx (x, t ) + Λ(x)U x (x, t ) + Ψ(x)U(x, t ), (1.6)
U(0, t ) = 0 or U x (0, t ) = 0, (1.7)

U(1, t ) = U c (t ) or U x (1, t ) = U c (t ), (1.8)
where

• U(x, t ) = (u 1 (x, t ), • • • , u n (x, t )) T ∈ [L 2 (0, 1)
] n is the state vector, where L 2 is the space of square integrable functions,

• U c (1, t ) = u c 1 (1, t ), • • • , u c n (1, t ) T ∈ [L 2 (0, 1)
] n is the input vector,

• Θ(x) is a diagonal n × n matrix, whose diagonal components θ i (x) for i = 1, . . . , n, represent the diffusion term coefficients,

• Λ(x) is a n × n matrix, whose components λ i j (x) for i , j = 1, . . . , n, represent the advection term coefficients and are assumed to be two times differentiable,

• Ψ(x) is a n × n matrix, whose components ψ i j (x) for i , j = 1, . . . , n, represent the reaction term coefficients and are assumed to be differentiable, with n ∈ N * being the number of the coupled equations. We also call the above system reaction-advection-diffusion system.

This thesis deals with boundary controller and observer design for system (1.6)- (1.8). Due to the fact that system (1.6)-(1.8) processes different kinds of coefficients, we will adopt different methods to design observers and controllers based on the structures of their coefficients. Indeed, the present work will consider the following cases:

• Reaction-diffusion equations with constant coefficients having the same diffusion: for this kind of systems, the coefficient Θ(x) = θ × I n×n where θ is a positive constant and I n×n is identity matrix, the coefficient Λ(x) is a zero matrix and Ψ(x) is a n × n constant matrix.

• Reaction-advection-diffusion equations with constant coefficients having the same diffusion: for this kind of systems, the coefficient Θ(x) = θ × I n×n where θ is a positive constant and I n×n is identity matrix, both Λ(x) and Ψ(x) are constant matrices.

Background and Motivations

• Reaction-diffusion equations with constant coefficients having different diffusions: for this kind of systems, the coefficient Θ(x) is a diagonal constant matrix with different components, Λ(x) is a zero matrix and Ψ(x) is a constant matrix.

• Reaction-advection-diffusion equations with spatially varying coefficients: for this kind of systems, the coefficient Θ(x) is a diagonal multivalued matrix, both Λ(x) and Ψ(x) are multivalued matrices.

This thesis will deal with the boundary observer and controller design for the above coupled parabolic PDEs. In order to realize this, we will extend the existing backstepping method for scalar parabolic PDEs to the above coupled cases [START_REF] Boskovic | Boundary control of an unstable heat equation via measurement of domain-averaged temperature[END_REF]15,[START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF].

Firstly, we start by providing some definitions on the controllability and observability of infinite-dimensional systems.

Controllability and observability of parabolic PDEs

This subsection introduces some definitions on controllability and observability of infinite-dimensional systems, which come from [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]. For this, we consider the following infinite-dimensional system: ż(t ) = Az(t ) + Bu(t ), (1.9) y(t ) = Cz(t ) + Du(t ), (1.10) z(0) = z 0 , (1.11) where z(t ) is the state, u(t ) is the input, y(t ) is the output, A is the infinitesimal generator of the strongly continuous semigroup T(t ) on a Hilbert space Z, B is a bounded linear operator from a Hilbert space U to Z, C is a bounded linear operator from Z to a Hilbert space Y, and D is a bounded operator from U to Y. In the following, Σ(A, B, C, D) denotes the linear system (1.9)- (1.11). It is well-known that the mild solution of (1.9) is:

z(t ) = T(t )z 0 + t 0 T(t -s)Bu(s)d s, (1.12)
where 0 ≤ t ≤ τ and τ is a positive constant. Now we will introduce a definition on the controllability of Σ(A,B,C,D).

Definition 1.1 [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF] For the linear system Σ(A,B,C,D) with the following linear controllability map:

B := τ 0 T(τ -s)Bu(s)d s, (1.13)
we say, (1.17)

C T z := CT(•)z, (1.16
Besides the above concepts, more details related with this topic can be found in literatures [START_REF] Curtain | An introduction to infinite-dimensional linear systems theory[END_REF]- [START_REF] Kang | Partial Observability and Its Consistency for Linear PDEs[END_REF]. Moreover, there are also discussions on the boundary controllability and observability of parabolic PDEs in [6].

Literature review

This subsection reviews some existing studies on control and observer design for PDEs.

It firstly introduces some applications of PDEs. Then, it reviews some main methods especially the backstepping method for controller and observer design of PDEs.

PDE is an important mathematics tool that can be used to model many industrial or forming processes [START_REF] Kiefer | An analytical approach for modelling asymmetrical hot rolling of heavy plates[END_REF]. In recent years, quantum systems have been identified as a future key technology, for which can be modelled by PDEs [START_REF] Boscain | Multi-input Schrodinger equation: controllability, tracking, and application to the quantum angular momentum[END_REF]- [START_REF] Azouit | Well-posedness and convergence of the Lindblad master equa-tion for a quantum harmonic oscillator with multiphoton drive and damping[END_REF].

With the created models for the above systems, it often needs to operate these systems to obtain a desired performance that needs to design controllers and observers for such kind systems. For this, we have two approaches. The first approach is to reduce the PDE system to an ODE system by approximation techniques, which can be called early-lumping approach. This approach permits us to directly extend the existed methods of controller and observer design for ODE systems to PDE systems. However, it can not cover the full system dynamics due to the initial approximation. The second one is to develop dedicated PDE-based controller and observer design, which can be called late-lamping approach. Compared with the early-lumping approach, the late-lamping approach is usually more accurate to reflect the dynamics of the system without discretization. Now, we will introduce some typical methods of the controller and observer design for PDEs. Model predictive control (MPC) is a key control concept for nonlinear finite-dimensional systems and relies on the successive solution of an optimal control problem [43,[START_REF] Grune | Nonlinear Model Predictive Control[END_REF]. This method has been extended to systems governed by PDEs by adopting the early-lamping approach [START_REF] Edouard | Observer based multivariable control of a catalytic reverse flow reactor: comparison between LQR and MPC approaches[END_REF]- [START_REF] Shang | Computationally efficient model predictive control for convection dominated parabolic systems[END_REF]. The port-Hamiltonian system concept provides a theoretical and methodical framework, which has proven to be well-suited for the modelling, analysis and control of complex systems, that enables us to shape the control loop by assigning desired physical properties [START_REF] Brogliato | Dissipative Systems Analysis and Control[END_REF]- [START_REF] Seslija | Discrete exterior geometry approach to structure-preserving discretization of distributed-parameter port-Hamiltonian systems[END_REF]. Slidingmode control has long been recognized as a powerful control method to counteract non-vanishing external disturbances and unmodeled dynamics [53], and it has been extended to PDE systems to deal with the problems of boundary disturbances and system uncertainties [START_REF] Meng-Bicheng | Sliding mode boundary control of a parabolic PDE system with parameter variations and boundary uncertainties[END_REF]- [56]. Fuzzy control is also widely studied for PDEs, especially for nonlinear PDEs [48]- [START_REF] Wu | A multiobjective optimization based fuzzy control for nonlinear spatially distributed processes with application to a catalytic rod[END_REF] and coupled PDEs [START_REF] Wang | Fuzzy control design for nonlinear ODEhyperbolic PDE-cascaded systems: A fuzzy and entropy-like Lyapunov function approach[END_REF][START_REF] Wang | Hinfini Fuzzy Control for a Class of Nonlinear Coupled ODE-PDE Systems With Input Constraint[END_REF].

Backstepping method for PDEs is an efficient method to design controllers and observers for PDEs. This method was firstly adopted to the stabilization of unstable parabolic PDEs with constant parameters in [START_REF] Boskovic | Boundary control of an unstable heat equation via measurement of domain-averaged temperature[END_REF]. The proposed backstepping algorithm transforms the studied system into a target system with desired stability properties.

Inspired by this concept, in [START_REF] Liu | Boundary feedback stabilization of an unstable heat equation[END_REF] the authors built a backstepping feedback controller for a parabolic PDE with spatially-dependent parameters endowed with Neumann 1.1. Background and Motivations boundary conditions. Based on this work, a similar backstepping method was used to stabilize a more general partial integro-differential equation in [15], where the control law was obtained in an explicit form. Then, in [START_REF] De Queiroz | Lyapunov-Based Control of Mechanical Systems[END_REF] the authors designed an observer for a class of parabolic partial integro-differential equations endowed with boundary sensors. In the same work, by combining the designed observer with the backstepping controller provided in [15], a backstepping observer-based output feedback controller was designed. Later, the above work was extended in [START_REF] Smyshlyaev | On control design for PDEs with space-dependent diffusivity or time-dependent reactivity[END_REF] to the PDEs with space-dependent diffusions and time-varying parameters. Then, by using the flatness concept together with backstepping state-feedback control, a tracking controller was designed in [19]. The authors of [START_REF] Izadi | PDE backstepping control of onedimensional heat equation with time-varying domain[END_REF] developed a backstepping controller for one-dimensional unstable heat equations in time-varying domain. In [START_REF] Tsubakino | Backstepping observer design for parabolic PDEs with measurement of weighted spatial averages[END_REF], the authors designed a backstepping-based observer for one-dimensional linear parabolic PDEs, where the output was a weighted spatial average of the state. Then, the backstepping method was extended to multi-dimensional PDEs in [START_REF] Jadachowki | Backstepping observers for linear PDEs on higher-dimensional spatial domains[END_REF]62]. In [63]- [START_REF] Nikolaosbekiaris-Liberis | Stabilization of linear strict-feedback systems with delayed integrators[END_REF], the authors dealt with the backstepping stabilization for some kinds of PDEs with input or output delays. In [START_REF] Vazquez | Control of 1-D parabolic PDEs with Volterra nonlinearities, Part I: Design[END_REF], a nonlinear hyperbolic PDE system was stabilized by using the backstepping method, which is directly extended from the finite-dimension feedback backstepping approaches. The boundary controller and observer were designed for the time fractional-order PDEs by using the backstepping method in [85]- [START_REF] Zhou | Boundary feedback stabilization for an unstable time fractional reaction diffusion equation[END_REF]. Based on the works on the stabilization and observation for systems modelled by single PDE, many researchers were devoted to the ones modelled by coupled PDEs.

In [START_REF] Tang | Stabilization of a coupled PDE-ODE system by boundary control[END_REF], a controller for a scalar coupled PDE-ODE system was designed by using the backstepping method. In [START_REF] Tang | Stabilization for a coupled PDE-ODE control system[END_REF], the output feedback control problem was addressed for a coupled PDE-ODE. Moreover, a boundary output feedback controller was proposed to stabilize a coupled PDE-ODE with the interaction at the interface in [81]. Then, in [START_REF] Zhou | Boundary stabilization of a coupled wave-ODE system with internal anti-damping[END_REF], by decoupling a coupled wave-ODE system into a stable cascaded wave-ODE system, the plant system was stabilized with the backstepping method. In [66], a class of systems modelled by a set of linear first-order hyperbolic equations were stabilized by a singular control input and the authors also designed an observer for the studied system. A backstepping controller was designed for a more general case in [START_REF] Hu | Control of homodirectional linear coupled hyperbolic PDEs[END_REF],

where the number of hyperbolic PDEs in either direction is arbitrary. In [START_REF] Hu | Boundary exponential stabilization of 1-D inhomognous quasilinear hyperbolic system[END_REF], the authors solved the problem of boundary stabilization for a class of systems modelled by n × n inhomogeneous quasilinear hyperbolic equations. In [68], the authors designed a minimum time controller for a system modelled by coupled heterodirectional linear first order hyperbolic equations. The problem of stabilization for high-dimensional coupled PDEs were considered in [69,70]. In [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF], the backstepping method was used to stabilize a class of linear coupled parabolic PDEs with the same diffusion coefficient.

Based on [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF], a backstepping boundary controller was designed for coupled parabolic

Methodology

PDEs with different diffusions in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF], where the kernel functions of the backstepping transformation were firstly assumed to be identity multiplied with a constant, then the solution of the kernel functions can be obtained by using successive approximations as done for the scalar case. As an extension of [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF], the authors of [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF] designed a state observer for a system modelled by a set of coupled reaction-diffusion PDEs processing the same diffusion by using backstepping method. Based on the above works, a boundary observer for a class of coupled parabolic PDEs processing different diffusions was designed in [START_REF] Liu | Backstepping observer-based output feedback control for a class of coupled parabolic PDEs with different diffusions[END_REF] by using the backstepping method, then combined with the controller designed in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF], an observer-based output feedback controller was designed. Moreover, some more relaxed conditions for the stability of the plant systems of the backstepping transformation were deduced. In the recent work [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF], the authors solved the problem of boundary stabilization for unstable linear coupled RAD PDEs with spatially-varying coefficients. Moreover, this work also successfully unveiled the connection between the control kernels for the parabolic and hyperbolic systems.

This work is very challenging since the solutions of the kernel functions obtained for the constant-coefficient case [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF] are not applicable anymore. Very interesting, the authors of [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF] solved the problem by transforming the kernel functions into a set of well-posed first-order hyperbolic equations. Then, the authors of [START_REF] Camacho-Solorio | Boundary observer desgin for coupled reaction-diffusion systems with spatially-varying reaction[END_REF] designed a backstepping boundary observer for a class of systems modelled by two coupled reactiondiffusion equations with spatially-varying reaction coefficients, which can be used to model the diffusion phenomena in lithium-ion batteries with electrodes that comprise multiple active materials. In [START_REF] Orlov | Output feedback stabilization of coupled reaction-diffusion processes with constant parameters[END_REF], the authors designed boundary observers and output feedback controllers for parabolic PDEs with constant diffusion coefficients and mixed Dirichlet-type and Neumann-type boundary conditions as well as collocated and anti-collocated setups. Furthermore, the authors of [78] designed controllers for linear parabolic coupled PIDEs with spatially-varying coefficients and mixed boundary conditions, where the convection term was characterized by a diagonal matrix such that it can be omitted by using an invertible transformation.

Methodology

This subsection gives a brief introduction of some methods that will be adopted by this thesis mainly including the boundary control of PDEs, backstepping method for PDEs and Lyapunov stability for PDEs. Moreover, it also reviews a work on the backstepping controller and observer design for scalar parabolic PDEs, by which some methods such as Volterra transformation, variable change and successive approximations are introduced.

Boundary control of PDEs

We need to decide the positions of actuators and sensors when we design controllers and observers for PDEs systems. Roughly speaking, there are two settings for the locations of the actuators and sensors. The first setting is to put the actuators or sensors in the domain of a PDEs system, which realizes the in-domain control. The other one is to set the actuators or sensors only on the boundary of the system, which realizes the boundary control. In reality, the boundary control is generally considered to be more realistic because the actuation and sensing are nonintrusive.

Backstepping method for PDEs

The backstepping method was firstly proposed for the feedback linearization for nonlinear systems around 1990, by which the control input can compensate for the the nonlinearity. Around 2000, the backstepping method was extended to the boundary stabilization of PDEs by [START_REF] Boskovic | Boundary control of an unstable heat equation via measurement of domain-averaged temperature[END_REF]. Compared with other methods for PDEs such as optimal control, pole placement, MPC and so on, the backstepping controller gains or observer gains can be evaluated by using symbolic computation or even can be given explicitly in some cases. Actually, the backstepping method can achieve Lyapunov stabilization by collectively shifting all the eigenvalues in a favorable direction in the complex plane.

The main idea for this method is to transform a given PDE to a target PDE with desired stability properties by using an invertible integral transformation. More specifically, there are three concerns for adopting this method. The first concern is the selection of an invertible integral transformation corresponding to a given PDEs. The second one is the choice of the target PDEs with desired stability properties. The last one is to determine the kernel function of the transformation in order to realize the selected transforamtion. This thesis aims to extend the backstepping method of boundary controller and observer design for scalar parabolic PDEs to coupled parabolic PDEs based on their different structures of coefficients.

Lyapunov stability for PDEs

The Lyapunov stability for PDEs is quite different from the one for ODEs. Indeed, the state space of PDEs is a functional space, therefore the scalar product is defined by means of an integral. Thus, general Lyapunov stability offers almost no practical values for PDEs. Instead, some new "energy estimates" in different forms should be derived. For this, this subsection will introduce the design of Lyapunov functions for scalar parabolic PDEs and coupled parabolic PDEs, respectively.

Methodology

Lyapunov stability for scalar reaction-diffusion equation

In the following, we will provide some inequalities that are needed to prove the stability of PDEs. Let us firstly give the Young' s Inequality [83]:

ab ≤ γ 2 a 2 + 1 2γ b 2 , (1.18)
where a and b are two real-valued functions and γ is a positive real-valued number.

Secondly, we introduce the well-known Cauchy-Schwarz Inequality:

1 0 u(x)w(x)d x ≤ 1 0 u 2 (x)d x 1/2 1 0 w 2 (x)d x 1/2 , ( 1.19)
where u(x) and w(x) are real-valued functions. Now, we will give an example to show how the Lyapunov method works on the following reaction-diffusion equation:

u t (x, t ) = θu xx (x, t ) + ψu(x, t ), (1.20) u(0, t ) = 0, (1.21) u(1, t ) = 0, (1.22)
where u(x, t ) is the state of the system with x ∈ [0, 1] and t ≥ 0, θ is a positive constant and ψ is an arbitrary constant. In the following part, we will show the stability condition for the above system by using the Lyapunov method.

Consider the following Lyapunov candidate function for system (1.20)-(1.22):

V(t ) = 1 2 1 0 u 2 (x, t )d x. (1.23)
Then, calculate the time derivative of V:

V(t ) = 1 0 u(x, t )u t (x, t )d x. (1.24)
Substitute (1.20) into (1.24), we obtain: We conclude that the system described by (1.20)-(1.22) is unstable with any positive parameter ψ, which also means that the term ψu is the source of instability. This is a simple case of the Lyapunov method for a system modelled by scalar reaction-diffusion equation. However, when the considered system is more general with coupled states or spatially-varying coefficients, the Lyapunov functions should be designed in different ways.

V(t ) = θ 1 0 u(x, t )u xx (x, t )d x + ψ 1 0 u 2 (x, t )d x. ( 1

Lyapunov stability for coupled reaction-diffusion equations

In order to introduce the Lyapunov method for a system modelled by coupled reactiondiffusion equations, let us firstly extend the Young's Inequality and Cauchy-Schwarz Inequality to vector cases.

Young's Inequality:

AB T ≤ γ 2 AA T + 1 2γ BB T , (1.28)
where A and B are two n-dimensional multivalued vectors:

A = [a 1 (x), a 2 (x), • • • , a n (x)], B = [b 1 (x), b 2 (x), • • • , b n (x)
] and γ is a positive real-valued number.

Cauchy-Schwarz Inequality: Now, let us consider the following coupled reaction-diffusion equations:

1 0 U(x)W T (x)d x ≤ 1 0 U(x)U T (x)d x 1/2 1 0 W(x)W T (x)d x 1/2 , ( 1
U t (x, t ) = θu xx (x, t ) + Ψu(x, t ), (1.30) U(0, t ) = 0, (1.31) U(1, t ) = 0, (1.32) where U(x, t ) = (u 1 (x, t ), u 2 (x, t ), • • • , u n (x, t )) T ∈ [L 2 (0, 1)
] n is the n dimensional vector state, θ is a positive constant parameter, Ψ is a n ×n matrix, whose components ψ i j for i , j = 1, 2, . . . , n represent the reaction terms.

In the following part, we will show the stability condition for system (1.30)-(1.32).

For this, let us consider the following candidate Lyapunov function:

V(t ) = 1 2 1 0 U T (x, t )U(x, t )d x. (1.33)
Then, calculate the time derivative of V(t ), we get:

V(t ) = 1 0 U T (x, t )U t (x, t )d x.
(1.34)

θ 1 0 U T (x, t )U xx (x, t )d x = θ[U T (x, t )U x (x, t )] 1 0 -θ 1 0 U T x (x, t )U x (x, t )d x = -θ 1 0 U T x (x, t )U x (x, t )d x.
(1.36)

With (1.36), (1.35) becomes:

V(t ) ≤ σ mi n (S[Ψ]) 1 0 U T (x, t )U(x, t )d x ≤ 2σ mi n (S[Ψ])V(t ), (1.37)
where

∥ U(•, t ) ∥ 2,n ≤∥ U(•, 0) ∥ 2,n e σ mi n (S[Ψ])t . (1.38)
We conclude that the system described by (1.30)

-(1.32) is unstable if σ mi n (S[Ψ]) > 0,
which also means that the term ΨU is the source of instability. This is also a simple example because the considered coupled system only processes the same constant diffusion parameter without advection terms. However, it will be challenged to pursue the stability conditions if the system processes spatially-varying coefficients and advection terms, that will be studied in this thesis.

Backstepping output feedback control for scalar reactiondiffusion equation with spatially varying coefficients

This subsection introduces the backstepping controller and observer design for a generalized scalar reaction-diffusion equation with spatially varying coefficients proposed by [15] and [START_REF] De Queiroz | Lyapunov-Based Control of Mechanical Systems[END_REF], respectively. The main purpose is to review some basic methods for the backstepping controller and observer design for parabolic PDEs including Volterra transformation, variable change, successive approximations and so on. These methods will be extended to coupled cases in this thesis.

Consider the following linear parabolic PIDE:

u t (x, t ) = au xx (x, t ) + b(x)u x (x, t ) + λ(x)u(x, t ) + g (x)u(0, t ) + x 0 f (x,
u x (0, t ) = qu(0, t ), (1.40) u(1, t ) = u c (t ) or u x (1, t ) = u c (t ), (1.41)
where a is a positive constant, b(x), λ(x) and g (x

) are C 2 ([0, 1]) functions, f (x, y) is a C 1 ([0, 1] × [0, 1]) function, u c (t ) is the control input.
For the above system, the advection terms b(x)u x (x, t ) can be omitted without loss of generality by using the following transformation:

v(x, t ) = u(x, t )e -1 2a x 0 b(τ)d τ .
(1.42)

Backstepping boundary controller design

This subsection introduces the boundary controller design for system (1.39)-(1.41) by using the backstepping method. For this, the following Volterra transformation is adopted,

w(x, t ) = u(x, t ) - x 0 k(x, y)u(y, t )d y, (1.43)
where the kernel function k(x, y) should be determined such that this transformation can put system (1.39)-(1.41) into the following target system:

w t (x, t ) = aw xx (x, t ) -c w(x, t ), (1.44)
w x (0, t ) = q w(0, t ), (1.45) w(1, t ) = 0 or w x (1, t ) = 0.

(1.46)

As introduced before, the above target system (1.44)-(1.46) is exponentially stable with a positive parameter c.

By setting x = 1 for (1.43), the boundary controller is obtained:

u c (t ) = u(1, t ) = 1 0 k 1 (y)u(y, t )d y, (1.47)
for the Dirichlet actuation and For this, we adopt the following change of variables [15]: where M = 1 a ( λ+c+ f + ḡ )(1+e -q ) with parameters:

u c (t ) = u x (1, t ) = k 1 (1)u(1, t ) + 1 0 k 2 (
ξ = x + y, (1.54) η = x -y, (1.55) G(x, y) = k(ξ, η), (1.56
λ = sup x∈[0,1] | λ(x) |, ḡ = sup x∈[0,1] | g (x) | and f = sup (x,y)∈[0,1]×[0,1] | f (x, y) |.
The inverse transformation of (1.43) is [15]: where M is defined as the same as (1.58).

u(x, t ) = w(x, t ) + x 0 l (x,

Backstepping boundary observer design

For the boundary observer design of PDEs, there are mainly two setups: anti-collocated case for which the sensor and actuator are placed at the opposite ends and collocated case for which the sensor and actuator are placed at the same end. For the considered scalar case, there is not much technical difference between the anti-collocated and collocated setups [START_REF] De Queiroz | Lyapunov-Based Control of Mechanical Systems[END_REF]. Therefore, we only introduce the backstepping observer design for the anti-collocated setup with Dirichlet-type boundary condition as an example.

In order to design the boundary observer for system (1.39)-(1.41), let us consider the following system:

ût (x, t ) = a ûxx (x, t ) + λ(x) û(x, t ) + g (x)u(0, t ) + x 0 f (x, y) û(y, t )d y + p 1 (x)[u(0) -û(0)], (1.64) ûx (0, t ) = qu(0, t ) + p 10 [u(0) -û(0)], (1.65)
û(1, t ) = U(t ), (1.66) where p 1 (x) and p 10 are parameters that needs to be designed to make system (1.64)-(1.66) be an observer for system (1.39)-(1.41).

Then, define the observation error states:

ũ(x, t ) = u(x, t ) -û(x, t ). ( 1
d d x p(x, x) = 1 2a (λ(x) + c), (1.76)
p(1, y) = 0, (1.77)

p 1 (x) = ap y (x, 0), (1.78
) where r (x, y) is the kernel function that is also well-posed.

p 10 = p(0, 0), (1.79) where (1
a p x x ( x, ȳ) -a p ȳ ȳ ( x, ȳ) = (λ(x) + c) p( x, ȳ) -f ( x, ȳ) + x ȳ p( x, τ) f (τ, ȳ)d τ, (1.83) p( x, x) = - 1 2a x 0 (λ(τ) + c)d τ, (1.84
1.4. Objectives

Backstepping output feedback controller design

We find that the designed boundary controllers (1.47) and (1.48) depend on the full states on space of the system. However, for the systems modelled by PDEs, it is difficult even impossible to measure the full states on space point by point. For this, we can adopt the boundary observer designed in Subsection 1.3.2. This observer is independent of the controller developed in Subsection 1.3.1. Therefore, we can combine the designed observer and controller to realize a backstepping observer-based output feedback controller.

Theorem 1. 4 [15] If the parameters c and c in (1.44) and (1.72) are designed to be positive, then the controller defined by:

u c (t ) = 1 0 k 1 (y) û(y, t )d y (1.87)
stabilizes the system modelled by (1.39)-(1.41), where k 1 (y) is given by (1.49) and û(x, t )

is the estimated state obtained from the following observer:

ût (x, t ) = a ûxx (x, t ) + λ(x) û(x, t ) + g (x)u(0, t ) + x 0 f (x, y) û(y, t )d y + p 1 (x)[u(0) -û(0)], (1.88) ûx (0, t ) = qu(0, t ) + p 10 [u(0) -û(0)], (1.89) û(1, t ) = 1 0 k 1 (y) û(y, t )d y, (1.90)
with the observer gains p 1 (x) and p 10 given by (1.78) and (1.79), respectively.

We should note that the above backstepping observer-based output feedback controller is designed for the Dirichlet-type actuation case. Actually, the backstepping observer-based output feedback controller for the Neumann-type actuation case can be realized in a similar way, which is skipped here.

Objectives

This thesis aims to design observer-based output feedback controllers for coupled parabolic PDEs by using the backstepping method. However, due to the fact that coupled parabolic PDEs process different structures of coefficients, which makes the adopted backstepping method should be realized in different ways. More specifically, this thesis mainly considers the following cases:

• Coupled reaction-diffusion equations with constant coefficients having the same diffusion, 1.4. Objectives

• Coupled reaction-advection-diffusion equations with constant coefficients having the same diffusion,

• Coupled reaction-diffusion equations with constant coefficients having different diffusions,

• Coupled reaction-advection-diffusion equations with spatially varying coefficients,

• Coupled time fractional-order reaction-diffusion equations with constant coeffiients.

1.5. Outline

Outline

This thesis summarizes my PhD work from 2015-2018, which is organized as follows.

Chapter 2 designs an observer-based output feedback controller for a class of systems modelled by coupled reaction-diffusion equations with constant coefficients having the same diffusion. Moreover, it proposes a more relaxed condition for the stability of the target systems of the backstepping transformation by using the well-known Poincaré Inequality. Then, it considers a more general case with advection term, for which a backstepping controller and observer are desgined by proposing certain conditions on the parameters of the target systems.

Chapter 3 considers a class of systems modelled by coupled reaction-diffusion equations with different diffusions. It designs a boundary observer for this considered system by using the backstepping method. Due to the different diffusions, an assumption for the kernel function is made to make it easy to solve the kernel function. Moreover, it also designs a boundary controller for this considered system based on the proposed stability condition on the parameters of the target system, which is more relaxed than the existing one. Then, it proposes a backstepping observer-based output feedback controller for the considered system. And the proposed method is verified by taking a simplified model of Chemical Tubular Reactor.

Chapter 4 designs a backstepping boundary observer for a class of systems modelled by coupled reaction-advection-diffusion equations with spatially-varying coefficients. Due to the advection term and the spatially-varying coefficients, the backstepping transformation adopted in the above chapters is not valid anymore. For this, the kernel equations are transferred into a set of well-posed hyperbolic PDEs. And it also proposes the stability conditions on the target system of the backstepping transformation. The proposed observer is verified by taking a numerical example.

Chapter 5 considers a class of systems modelled by coupled reaction-diffusion equations with time fractional-order derivative. And it designs a backstepping observerbased output feedback controller for this considered system.

Finally, conclusions are outlined in Chapter 6 with some perspectives. Abstract: This section solves the problem of observer-based output feedback stabilization of coupled reaction-advection-diffusion equations with constant coefficients and the same diffusion parameter. More specifically, it firstly considers a class of systems modelled by coupled reaction-diffusion equations with constant coefficients and the same diffusion. For this case, it proposes a more relaxed condition on the parameters of the target systems of the backstepping transformation compared to the ones in [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF] and [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF], and a backtepping observer-based output feedback controller is designed for this system. Secondly, it designs a boundary controller and observer for a kind of systems modelled by coupled reaction-advection-diffusion equations with 2.1. Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient constant coefficients and the same diffusion parameter, for which an additional advection term is considered compared to the first case. And it also designs an observerbased output feedback controller. Finally, the above proposed methods are highlighted by numerical simulations.

Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient

This subsection considers the following coupled reaction-diffusion equations with constant coefficients and the same diffusion parameter:

U t (x, t ) = θU xx (x, t ) + ΨU(x, t ), (2.1)
with Neumann-type boundary conditions and Dirchlet-type actuation:

U x (0, t ) = 0, (2.2)
U(1, t ) = U c (t ), (2.3)
where

• U(x, t ) = (u 1 (x, t ), • • • , u n (x, t )) T ∈ [L 2 (0, 1)] n is the state vector, • U c (1, t ) = u c 1 (1, t ), • • • , u c n (1, t ) T ∈ [L 2 (0, 1)
] n is the input vector,

• θ ∈ R + is the diffusion of the system,

• Ψ is a n × n matrix, whose components ψ i j for i , j = 1, 2, . . . , n, represent the reaction terms, with n ∈ N * being the number of the coupled PDEs.

backstepping controller design

This subsection aims to design a backstepping boundary controller for a system modelled by (2.1)-(2.3), where a more relaxed condition for the stability of the target system of the backstepping transformation is proposed than the one given in [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF].

It has been shown in [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF] that the following backstepping transformation:

W(x, t ) = U(x, t ) - x 0 K(x, y)U(y, t )d y (2.4)
2.1. Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient can transform the plant system (2.1)-( 2.3) into the following target system:

W t (x, t ) = θW xx (x, t ) -CW(x, t ), (2.5) W x (0, t ) = 0, (2.6) W(1, t ) = 0, (2.7)
where

• W(x, t ) = (w 1 (x, t ), • • • , w n (x, t )) ∈ [L 2 (0, 1)] n ,
• C is a n × n matrix with components c i j for i , j = 1, 2, . . . , n.

Moreover, the stability features of the target system (2.5)-(2.7) is established.

Theorem 2.1 [71] If the symmetric matrix S[C] is positive, the target system (2.5)-(2.7)
is exponentially stable with the following convergence rate:

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e -σ mi n (S[C])t . (2.8)
A more relaxed condition on S[C] is provided in the following theorem, which improves the convergence rate obtained in Theorem 2.1.

Theorem 2.2 If the matrix C is chosen such that the smallest eigenvalue of the symmetric part S[C] satisfies the following condition:

σ mi n (S[C]) ≥ - θ 4 , (2.9)
the target system (2.5)-(2.7) is exponentially stable with the following convergence rate:

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e -[ θ 4 +σ mi n (S[C])]t . (2.10)
Before providing the proof of Theorem 2.2, we will introduce the following wellknown Poincaré Inequalities. Firstly, it introduces the scalar Poincaré Inequality:

Lemma 2.1 [83] For any function w(x) that is continuously differentiable on [0, 1], it fulfills: 1 0 w 2 (x)d x ≤ 2w 2 (1) + 4 1 0 w 2 x (x)d x, (2.11) 1 0 w 2 (x)d x ≤ 2w 2 (0) + 4 1 0 w 2 x (x)d x.
(2.12)

Then, we extend the scalar Poincaré Inequality to vector case:

2.1. Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient

Lemma 2.2 For any vector function W(x) which is continuously differentiable on [0, 1],

it fulfills:

1 0 W(x)W T (x)d x ≤ 2W(1)W T (1) + 4 1 0 W x (x)W T x (x)d x, (2.13) 1 0 W(x)W T (x)d x ≤ 2W(0)W T (0) + 4 1 0 W x (x)W T x (x)d x, (2.14)
where

W(x) = [w 1 (x), w 2 (x), • • • , w n (x)].
Proof of Theorem 2.2. The objective is to find a condition on the parameter C such that the following function:

V(t ) = 1 2 1 0 W T (x, t )W(x, t )d x, (2.15)
is a Lyapunov function for the system (2.5)-(2.7). For this purpose, the time derivative of V(t ) is calculated as follows:

V(t ) = 1 0 W T (x, t )W t (x, t )d x.
(2.16) Substitute (2.5) into (2.16), we obtain:

V(t ) = θ 1 0 W T (x, t )W xx (x, t )d x - 1 0 W T (x, t )CW(x, t )d x.
(2.17)

Then, by applying integration by parts and using the boundary conditions (2.6) and

(2.7), we get:

1 0 W T (x, t )W xx (x, t )d x = W T (x, t )W x (x, t ) 1 0 - 1 0 W T x (x, t )W x (x, t )d x = - 1 0 W T x (x, t )W x (x, t )d x. (2.18)
Thanks to the Poincaré Inequality and the zero boundary conditions, we deduce:

θ 1 0 W T x (x, t )W x (x, t )d x ≥ θ 4 1 0 W T (x, t )W(x, t )d x. (2.19)
Then, using (2.18) and (2.19), we get:

V(t ) ≤ - θ 4 1 0 W T (x, t )W(x, t )d x -σ mi n (S[C]) 1 0 W T (x, t )W(x, t )d x ≤ -( θ 2 + 2σ mi n (S[C]))V(t ). (2.20)
Finally, according to the condition on C given in (2.10), it is obvious that the target system (2.5)-(2.7) is exponentially stable with the following convergence rate:

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e -[ θ 4 +σ mi n (S[C])]t . (2.21)
Thus, this proof is completed.

Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient

Remark 2.1 For the stability of the considered system with the designed controller, the designed parameter C can be chosen with more freedom than the one in [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF]. This is due to the fact that the considered system in this paper processes the same diffusion.

To realize the backstepping controller, let us recall that the kernel matrix K(x, y) in transformation (2.4) has been solved in [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF] as follows:

K(x, y) = - ∞ n=0 (x 2 -y 2 ) n (2x) n!(n + 1)! 1 4θ n+1 × n i =0 n i C i (Ψ + C)Ψ n-i . (2.22)
Now, by setting x = 1 in (2.4), we get the following boundary control input:

U c (t ) = 1 0 K(1, y)U(y, t )d y, (2.23)
where using (2.22) we have:

K(1, y) = -2 ∞ n=0 (1 -y 2 ) n n!(n + 1)! 1 4θ n+1 × n i =0 n i C i (Ψ + C)Ψ n-i . (2.24)
Moreover, it is proven in [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF] that the transformation 2.4 has an inverse transformation, which is skipped here.

Backstepping observer design

This subsection aims to design a boundary observer for the plant system modelled by (2.1)-(2.3) by using a more relaxed condition than the one given in [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF]. For this purpose, we consider the following system:

Ût (x, t ) = θ Ûxx (x, t ) + Ψ Û(x, t ) + P(x)[U(0, t ) -Û(0, t )],
(2.25)

Ûx (0, t ) = Q[U(0, t ) -Û(0, t )], (2.26)
Û(1, t ) = U c (t ), (2.27)
where the n × n matrices P(x) and Q are parameters, which are designed such that system (2.25)-(2.27) is an observer for the plant system.

Let us denote the observation error between the real states U(x, t ) and the estimated states Û(x, t ) by: In order to obtain the observer gains which make the observation error converge to 0 with the time tending to infinity, the following backstepping transformation is considered:

Ũ(x, t ) = U(x, t ) -Û(x, t ). ( 2
Ũ(x, t ) = W(x, t ) - x 0 Z(x, y) W(y, t )d y, (2.32)
which can transform the system (2.29)-(2.31) into the following target system:

Wt (x, t ) = θ Wxx (x, t ) -C W(x, t ), (2.33) Wx (0, t ) = 0, (2.34) W(1, t ) = 0. (2.35)
The stability features of the target error system (2.33)-(2.35) is established in [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF],

which is recalled in the following theorem.

Theorem 2.3 [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF] If the symmetric matrix S[ C] is positive definite, the target system is exponentially stable with the following convergence rate:

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e -σ mi n (S[ C])t . ( 2
∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e -[ θ 4 +σ mi n (S[ C])]t . (2.38)
The proof of Theorem 2.4 is similar to the one of Theorem 2.2 thereby it is skipped here.

Remark 2.2

For the existence of the observer system (2.25)-(2.27), the designed parameter C can be chosen with more freedom than the one given in [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF]. This is because the considered system in this paper processes the same diffusion.

The kernel matrix Z(x, y) in the transformation (2.32) is the same as the one given in [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF]:

Z(x, y) = - ∞ n=0 2(1 -x)((1 -y) 2 -(1 -x) 2 ) n n!(n + 1)! 1 4θ n+1 × n i =0 n i Ψ i (Ψ + C) Cn-i .
(2.39)

2.1. Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient

Then, using Z(x, y), the observer gains are obtained as follows:

Q = - Ψ + C 2θ , (2.40)
P(x) = θ ∞ n=0 4n(1 -x)(2x -x 2 ) n-1 n!(n + 1)! 1 4θ n+1 × n i =0 n i Ψ i (Ψ + C) Cn-i .
(2.41)

Observer-based output feedback controller

A backstepping controller and a backstepping observer have been designed in subsection 2.1.1 and subsection 2.1.2, respectively, which are independent and satisfy the separation principle. This subsection combines the above controller and observer to realize a backstepping observer-based output feedback controller, whose convergence stability is also established.

Theorem 2.5

If the parameter matrices C and C are chosen such that:

σ mi n (S[ C]) ≥ σ mi n (S[C]) > - θ 8 , (2.42)
the following controller:

U(1, t ) = 1 0 K(1,
Ût (x, t ) = θ Ûxx (x, t ) + Ψ Û(x, t ) + P(x)[U(0, t ) -Û(0, t )],
(2.44)

Ûx (0, t ) = Q[U(0, t ) -Û(0, t )], (2.45)
Û(1, t ) = 1 0 K(1, y) Û(y, t )d y, (2.46)
where the observer gains P(x) and Q given by (2.40) and (2.41), respectively.

Proof. Let us consider the following invertible backstepping transformation:

Ŵ(x, t ) = Û(x, t ) - x 0 K(x, y) Û(y, t )d y, (2.47)
where K(x, y) is given in (2.22). Then, as done in subsection 2.1.1, the system (2.44)-

(2.46) can be transformed into the following target system: In order to prove the stability of the cascade system ( Ŵ(x, t ), W(x, t )), the following function is considered:

Ŵt (x, t ) = θ Ŵxx (x, t ) -C Ŵ(x, t ) + [P(x) - x 0 K(x, y)P(y)d y] W(0, t ), (2.48) Ŵx (0, t ) = Q W(0, t ), (2.49)
V(t ) = β 2 1 0 WT (x, t ) W(x, t )d x + 1 2 1 0 ŴT (x, t ) Ŵ(x, t )d x, ( 2.51)
where β ∈ R + will be designed later.

The objective is to prove V(t ) is a Lyapunov function. For this purpose, the time derivative of V(t ) is calculated as follows:

V(t ) = -βθ 1 0 WT x (x, t ) Wx (x, t )d x -β 1 0 WT (x, t ) C W(x, t )d x -θ ŴT (0, t )Q W(0, t ) -θ 1 0 ŴT x (x, t ) Ŵx (x, t )d x - 1 0 Ŵ(x, t )C Ŵ(x, t )d x + 1 0 ŴT (x, t )[P(x) - x 0 K(x, y)P(y)d y] W(0, t )d x. (2.52)
Then, by using the trivial inequality, we obtain:

1 0 WT (x, t ) C W(x, t )d x ≥ σ mi n (S[ C]) 1 0 WT (x, t ) W(x, t )d x, (2.53) 1 0 ŴT (x, t )C Ŵ(x, t )d x ≥ σ mi n (S[C]) 1 0 ŴT (x, t ) Ŵ(x, t )d x.
(2.54)

By using the Poincaré Inequality, the Young Inequality and the trivial inequality, we obtain:

-θ ŴT (0, t )Q Ŵ(0, t ) ≤ θ 4 1 0 ŴT x (x, t ) Ŵx (x, t )d x + θq max 1 0 WT x (x, t ) Wx (x, t )d x, (2.55) and 1 0 ŴT (x, t ) P(x) - x 0 K(x, y)P(y)d y W(0, t )d x ≤ θ 4 1 0 ŴT x (x, t ) Ŵx (x, t )d x + b max θ 1 0 WT x (x, t ) Wx (x, t )d x, ( 2.56)
where

q max = σ max (Q T Q), b max = max x∈[0,1] σ max (B T (x)B(x)).
Hence, by combining (2.52)-(2.56) and using the Poincaré Inequality, we obtain:

V(t ) ≤ - θβ 4 - θq max 4 - b max 4θ + σ mi n (S[ C])β × 1 0 WT (x, t ) W(x, t )d x - θ 8 + σ mi n (S[C]) 1 0 ŴT (x, t ) Ŵ(x, t )d x.
(2.57)

2.1. Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient

Then, we set:

θ 8 + σ mi n (S[C]) β = θβ 4 - θq max 4 - b max 4θ + σ mi n (S[ C])β. (2.58)
Thus, according to the condition (2.42), we have:

β = 2(θ 2 q max + b max ) θ(θ + 8σ mi n (S[ C]) -8σ mi n (S[C])) > 0. (2.59)
Consequently, we get:

V(t ) ≤ - θ 4 + 2σ mi n (S[C]) V(t ), (2.60)
which means that the cascade system ( Ŵ(x, t ), W(x, t )) is exponentially stable. Therefore, the cascade system ( Û(x, t ), Ũ(x, t )) is also exponentially stable. Indeed, it is related to ( Ŵ(x, t ), W(x, t )) by the invertible coordinate transformation of (2.4) and (2.32), which directly means that the closed-loop system (U(x, t ), Û(x, t )) is stable. Thus, this proof is completed.

Remark 2.3

The condition (2.42) guarantees that the observer converges faster than the controller for the stability of the observer-based output feedback control system (2.43)-(2.46).

Simulation

In order to verify the obtained results, a coupled temperature-concentration system of Chemical Tubular Reactor is considered, which is given by [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF]:

u 1t (x, t ) = 0.167u 1xx (x, t ) + 1.018u 1 (x, t ) + 0.154u 2 (x, t ), (2.61
)

u 2t (x, t ) = 0.167u 2xx (x, t ) + 2.037u 1 (x, t ) + 0.308u 2 (x, t ), (2.62) u 1x (0, t ) = 0, (2.63) u 2x (0, t ) = 0, (2.64) u 1 (1, t ) = u 1c (t ), (2.65) u 2 (1, t ) = u 2c (t ), (2.66)
where the state variables u 1 (x, t ) and u 2 (x, t ) denote the normalized temperature and concentration, respectively.

In this example, the observer designed in Subsection 2.1.2 will be applied to estimate the states of the system (2.61)-(2.66). Then, a backstepping observer-based output feedback controller will be designed based on the results obtained in Subsection 2.1.3. In order to realize this, the parameters C and C should be designed firstly. diffusion coefficient This subsection considers a class of coupled parabolic PDEs with constant coefficients and the same diffusion. Based on this case, the following subsection considers a more general case with an additional advection term.

Backstepping control for coupled reaction-advectiondiffusion equations having the same diffusion coefficient

This section considers a class of systems modelled by the following coupled reactionadvection-diffusion with constant coefficients and the same diffusion parameter:

U t (x, t ) = θU xx (x, t ) + ΛU x (x, t ) + ΨU(x, t ), (2.67)
with the Neumann-type boundary condition and the Dirichlet-type actuation:

U x (0, t ) = 0, , (2.68)
U(1, t ) = U c (t ), , (2.69)
where

• U(x, t ) = (u 1 (x, t ), • • • , u n (x, t )) T ∈ [L 2 (0, 1)] n is the state vector, • U c (1, t ) = u c 1 (1, t ), • • • , u c n (1, t ) T ∈ [L 2 (0, 1)
] n is the input vector,

• θ ∈ R + is the diffusion of the system,

• Λ is a n × n matrix, whose components λ i j ∈ R + for i , j = 1, . . . , n, represent the advection terms,

• Ψ is a n×n matrix, whose components ψ i j for i , j = 1, . . . , n, represent the reaction terms,

with n ∈ N * being the number of the coupled RAD equations.

The objective is to design a backstepping observer-based output feedback controller for the system modelled by (2.67)-(2.69). It is assumed that the output measurement is available at x = 0 and the input actuator is placed at x = 1, which are denoted by U(0, t ) and U(1, t ), respectively. The latter corresponds to the case of Dirichlet-type actuation with anti-collocated sensors and actuators setting. The case of Neumann-type actuation with collocated sensors and actuators setting can be done in a similar way.

If the matrix Λ in (2.67) is identity, which means the advection processes of the coupled system are same, the controller and observer design can be easily realized by a simple invertible transformation as done in the following remark [88].

Remark 2.4

If the matrix Λ is identity with:

λ 1 = λ 2 = • • • = λ n = λ, (2.70)
2.2. Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient the following invertible transformation:

Z(x, t ) = U(x, t )e λ 2θ x
(2.71) can be implemented to transform the system (2.67)-( 2.69) into the following advectionfree system:

Z t (x, t ) = θZ xx (x, t ) + Ψ - λ 2 4θ I n×n Z(x, t ), (2.72)
Z(0, t ) = 0, (2.73)

Z(1, t ) = U(t )e λ 2θ , (2.74)
whose controller and observer design have been addressed in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF] and [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF], respectively.

However, the method in Remark 2.4 is not feasible for the general case, where the plant system (2.67)-(2.69) has coupled advection processes with the advection parameter Λ as an arbitrary constant matrix. This general case will be focused in the sequel.

Backstepping observer design

This section deals with the observer design for the studied coupled system described by (2.67)-(2.69). For this purpose, the following system is considered:

Ût (x, t ) = θ Ûxx (x, t ) + Λ Ûx (x, t ) + Ψ Û(x, t ) + P(x)[U(0, t ) -Û(0, t )],
(2.75)

Ûx (0, t ) = Q[U(0, t ) -Û(0, t )], (2.76)
Û(1, t ) = U c (t ), (2.77)
where the n × n matrices P(x) and Q are two design gains to be determined such that (2.75)-(2.77) is an observer for the studied system.

The observation error between the real state U(x, t ) and the estimated one Û(x, t ) is denoted by:

Ũ(x, t ) = U(x, t ) -Û(x, t ). ( 2
Ũt (x, t ) = θ Ũxx (x, t ) + Λ Ũx (x, t ) + Ψ Ũ(x, t ) -P(x) Ũ(0, t ), (2.79) Ũx (0, t ) = -Q Ũ(0, t ), (2.80)
Ũ(1, t ) = 0.

(2.81) Thus, the observer gains P(x) and Q need to be determined such that the observation error Ũ(x, t ) converges to 0 as t → ∞. In order to realize this, the following backstepping transformation is applied: where Problem 2: C is a matrix parameter to be designed to guarantee the stability of the target system i .e. W(x, t ) tends to 0 asymptotically as t → ∞.

Ũ(x, t ) = W(x, t ) - x 0 H(x,

Remark 2.5 It is worth to know that the transformation (2.82) is an one-to-one map,

and its inverse is given by:

W(x, t ) = Ũ(x, t ) + x 0 R(x, y) Ũ(y, t )d y, (2.86)
where the kernel matrix R(x, y) is the unique inverse of H(x, y).

In the followings, Problem 1 and Problem 2 will be addressed.

Determinations of the kernel function and observer gains

2θ d d x H(x, x) + ΛH(x, x) -H(x, x)Λ = Ψ + C, (2.87)
H xx (x, y) -H y y (x, y) + 1 θ ΛH x (x, y) + 1 θ H y (x, y)Λ = - 1 θ ΨH(x, y) - 1 θ H(x, y)C, (2.88) H(1, y) = 0, (2.89
)

θH y (x, 0) -H(x, 0)Λ = P(x), (2.90)
H(0, 0) = Q, (2.91)

where d d x H(x, x) := H x (x, x) + H y (x, x) and h i j (x, y) for i , j = 1, • • • , n, are the components of H(x, y).
Proof. In this proof, some conditions on H(x, y), P(x) and Q are imposed to transform the observation error system (2.79)-(2.81) into the target system (2.83)-(2.85) in the following three steps.

Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient

• Step 1. Transformation to (2.83): By differentiating both sides of (2.82) with respect to x and t , and applying the Leibniz's differential rule, we obtain:

Ũx (x, t ) = Wx (x, t ) -H(x, x) W(x, t ) - x 0 H x (x, y) W(y, t )d y, (2.92) Ũxx (x, t ) = Wxx (x, t ) - d d x H(x, x) W(x, t ) -H(x, x) Wx (x, t ) -H x (x, x) W(x, t ) - x 0 H xx (x, y) W(y, t )d y, (2.93) Ũt (x, t ) = Wt (x, t ) -θH(x, x) Wx (x, t ) + θH(x, 0) Wx (0, t ) + θH y (x, x) W(x, t ) -θH y (x, 0) W(0, t ) -θ x 0 H y y (x, y) W(y, t )d y -H(x, x)Λ W(x, t ) + H(x, 0)Λ W(0, t ) + x 0 H y (x, y)Λ W(y, t )d y + x 0 H(x, y)C W(y, t )d y. (2.94)
Then, by substituting (2.92)-(2.94) into (2.79), we get:

Wt (x, t ) -θ Wxx (x, t ) -ΛW x (x, t ) + C W(x, t ) = -θ d d x H(x, x) + θH x (x, x) + θH y (x, x) + ΛH(x, x) -H(x, x)Λ -Ψ -C W(x, t ) + x 0 H y y (x, y) -θH xx (x, y) -ΛH x (x, y) -H y (x, y)Λ -H(x, y)C -ΨH(x, y) W(y, t )d y + θH y (x, 0) -P(x) -H(x, 0)Λ W(0, t ). ( 2
H(x, x) = e -Λ 2θ x H(0, 0)e Λ 2θ x + 1 2θ x 0 e Λ 2θ (y-x) (Ψ + C) e -Λ 2θ (y-x) d y. (2.103)
Consequently, this proof can be completed by substituting (2.91) in (2.103).

According to (2.89), we have H(1, 1) = 0. Hence, the following corollary can be obtained using (2.99).

Corollary 2.1 If the conditions given in (2.87), (2.89) and (2.91) hold, the observer gain

Q is given by:

Q = - 1 2θ 1 0 e Λ 2θ y (Ψ + C)e -Λ
2θ y d y.

(2.104)

Remark 2.6

The result obtained in Corollary 2.1 is consistent with the one established in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF] in the case without advection i .e. Λ = 0. More generally, if we choose C such that Ψ+C commutes with Λ, where the commutativity means

[Ψ + C, Λ] = 0, we get Q = -1 2θ (Ψ + C).
Thus, in this case, (2.99) becomes:

H(x, x) = 1 2θ (Ψ + C)(1 -x). ( 2
≤ y ≤ x ≤ 1,
which is twice continuously differentiable.

Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient

Proof. Inspired by [START_REF] De Queiroz | Lyapunov-Based Control of Mechanical Systems[END_REF], this proof can be completed by realizing the following two steps.

• Step 1. Transformation to an integral equation: Firstly, by applying the following change of variables: x → 1y and ȳ → 1x, we define:

H( x, ȳ) := H(1 -ȳ, 1 -x) = H(x, y), ( 2.106)
where 0 ≤ x ≤ ȳ ≤ 1. Then, (2.88), (2.89) and (2.99) are transformed into:

H x x ( x, ȳ) -H ȳ ȳ ( x, ȳ) + 1 θ H x ( x, ȳ)Λ + 1 θ Λ H ȳ ( x, ȳ) = 1 θ Ψ H( x, ȳ) + 1 θ H( x, ȳ)C, (2.107)
H( x, 0) = 0, (2.108)

H( x, x) = e -Λ 2θ (1-x) Qe Λ 2θ (1-x) + 1 2θ 1- x 0 e Λ 2θ (y+ x-1) (Ψ + C)e -Λ
2θ (y+ x-1) d y.

(2.109)

Secondly, by applying the following change of variables: ξ → x + ȳ and η → xȳ, we define:

G(ξ, η) := H ξ + η 2 , ξ -η 2 = H( x, ȳ), (2.110)
where 0 ≤ η ≤ ξ ≤ 2. Then, the following equations can easily be obtained:

H x ( x, ȳ) = G ξ (ξ, η) + G η (ξ, η), (2.111)
H x x ( x, ȳ) = G ξξ (ξ, η) + 2G ξη (ξ, η) + G ηη (ξ, η), (2.112)
H ȳ ( x, ȳ) = G ξ (ξ, η) -G η (ξ, η), (2.113)
H ȳ ȳ ( x, ȳ) = G ξξ (ξ, η) -2G ξη (ξ, η) + G ηη (ξ, η). ( 2
G ξη (ξ, η) + 1 4θ G ξ (ξ, η)Λ + ΛG ξ (ξ, η) + G η (ξ, η)Λ -ΛG η (ξ, η) = 1 4θ ΨG(ξ,η) + 1 4θ G(ξ, η)C, (2.115) G(ξ, ξ) = G(η, η) = 0, (2.116) G(ξ, 0) = e -Λ 4θ (2-ξ) Qe Λ 4θ (2-ξ) + 1 2θ 1- ξ 2 0 e Λ 4θ (2y+ξ-2) (Ψ + C)e -Λ 4θ (2y+ξ-2) d y.
(2.117)

By integrating (2.115), we obtain:

G ξ (ξ, η) = G ξ (ξ, 0) + 1 4θ η 0 ΨG(ξ,τ) + G(ξ, τ)C d τ - 1 4θ η 0 G ξ (ξ, τ)Λ + ΛG ξ (ξ, τ) d τ - 1 4θ G(ξ, η)Λ -ΛG(ξ,η) + 1 4θ G(ξ, 0)Λ -ΛG(ξ,0) . ( 2
G(ξ, η) = G(ξ, 0) -G(η, 0) + 1 4θ ξ η η 0 ΨG(s,τ) + G(s, τ)C d τd s - 1 4θ ξ η G(s, η)Λ -ΛG(s,η) d s + 1 4θ ξ η G(s, 0)Λ -ΛG(s,0) d s - 1 4θ η 0 G(ξ, τ)Λ + ΛG(ξ,τ) d τ + 1 4θ η 0 G(η, τ)Λ + ΛG(η,τ) d τ.
(2.119)

•
Step 2. Existence of solution: In this step, the method of successive approximations is applied to show (2.119) has an unique solution which is twice continuously differentiable. For this purpose, an initial approximation is set as follows:

G 0 (ξ, η) = 0.

(2.120)

Then, the recursive formula for (2.119) is set by:

G n+1 (ξ, η) := G(ξ, 0) -G(η, 0) + 1 4θ ξ η G(s, 0)Λ -ΛG(s,0) d s + 1 4θ ξ η η 0 ΨG n (s, τ) + G n (s, τ)C d τd s - 1 4θ ξ η G n (s, η)Λ -ΛG n (s, η) d s - 1 4θ η 0 G n (ξ, τ)Λ + ΛG n (ξ, τ) d τ + 1 4θ η 0 G n (η, τ)Λ + ΛG n (η, τ) d τ.
(2.121)

If the above recursion converges, the solution of (2.119) is given by:

G(ξ, η) = lim n→∞ G n (ξ, η). (2.122)
By considering the following difference:

G n (ξ, η) := G n+1 (ξ, η) -G n (ξ, η), (2.123)
we get:

G(ξ, η) = lim n→∞ G n (ξ, η) = ∞ n=0 G n (ξ, η). (2.124)
Then, the following comparison inequality:

∀ n ∈ N, G n (ξ, η) ≤ M n+1 (ξ + η) n n! (2.125)
is used to prove the convergence of the series

∞ n=0
G n (ξ, η), where we have:

M = 2 + Λ θ G(•, 0) + 1 2θ ( Ψ + C + 3 Λ ).
(2.126)

Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient

Remark that according to (2.117), the norm G(•, 0) exists. Now, the inequality given in (2.125) is proven by mathematical induction for n ∈ N.

Initial step:

For n = 0, using (2.120) and (2.121), we obtain the following recursive equations:

G 0 (ξ, η) = G 1 (ξ, η) -0 = G(ξ, 0) -G(η, 0) + 1 4θ ξ η G(s, 0)Λ -ΛG(s,0) d s.
(2.127)

Then, it is obvious that:

G 0 (ξ, η) ≤ M. ( 2
G n+1 (ξ, η) = 1 4θ ξ η η 0 Ψ G n (s, τ) + G n (s, τ)C d τd s - 1 4θ ξ η G n (s, η)Λ -Λ G n (s, η) d s - 1 4θ η 0 G n (ξ, τ)Λ + Λ G n (ξ, τ) d τ + 1 4θ η 0 G n (η, τ)Λ + Λ G n (η, τ) d τ.
(2.129)

Then, it yields:

G n+1 (ξ, η) ≤ 1 4θ ( Ψ + C ) M n+1 n! ξ η η 0 (s + τ) n d τd s + 1 2θ Λ M n+1 n! ξ η (s + η) n d s + 1 2θ Λ M n+1 n! η 0 (ξ + τ) n d τ + 1 2θ Λ M n+1 n! η 0 (η + τ) n d τ . ( 2

.130)

According to Eq. (2.14) in [START_REF] Guo | Sliding mode control and active disturbance rejection control to the stabilization of one-dimensional Schrodinger equation[END_REF], we know that:

ξ η η 0 (s + τ) n d τd s ≤ 2 (ξ + η) n+1 n + 1 . (2.131)
Then, by direct calculation, it is obvious that:

ξ η (s + η) n d s ≤ (ξ + η) n+1 n + 1 , (2.132) η 0 (ξ + τ) n d τ ≤ (ξ + η) n+1 n + 1 , (2.133) η 0 (η + τ) n d τ ≤ (η + η) n+1 n + 1 ≤ (ξ + η) n+1 n + 1 . ( 2
G n+1 (ξ, η) ≤ 1 2θ ( Ψ + C + 3 Λ ) M n+1 n! (ξ + η) n+1 n + 1 ≤ M n+2 (ξ + η) n+1 (n + 1)! . ( 2

Convergence of the target system

In this part, the condition on C to guarantee the stability of the target system (

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e -σ min (S[C])-∥Λ∥ 2 2θ t . ( 2

.137)

Proof. The objective of this proof is to find a condition on C such that the following function:

V(t ) = 1 2 1 0 WT (x, t ) W(x, t )d x (2.
V(t ) = θ 1 0 WT (x, t ) Wxx (x, t )d x + 1 0 WT (x, t )Λ Wx (x, t )d x.
-

1 0 WT (x, t )C W(x, t )d x.
(2.140)

On the one hand, by applying the integration by parts formula with the boundary conditions (2.84)-(2.85), we get:

1 0 WT (x, t ) Wxx (x, t )d x = - 1 0 WT x (x, t ) Wx (x, t )d x.
(2.141)

On the other hand, by applying the trivial inequality, we obtain:

1 0 WT (x, t )Λ Wx (x, t )d x ≤ 1 θ 1 0 WT (x, t )ΛΛ T W(x, t )d x + θ 1 0 WT x (x, t ) Wx (x, t )d x ≤ ∥ Λ ∥ 2 θ 1 0 WT (x, t ) W(x, t )d x + θ 1 0 WT x (x, t ) Wx (x, t )d x.
(2.142) Thus, using (2.141)-(2.142), and by applying the trivial inequality, we get:

V(t ) ≤ ∥ Λ ∥ 2 θ 1 0 W T (x, t )W(x, t )d x -σ min (S[C]) 1 0 WT (x, t ) W(x, t )d x ≤ -2σ min S[C] - 2 ∥ Λ ∥ 2 θ V(t ). ( 2

Backstepping controller design

This section deals with the controller design for the studied system described by (2.67)-(2.69). For this purpose, the following backstepping transformation is apllied: where Problem 4: C is a matrix parameter to be designed to guarantee the stability of the above target system.

W(x, t ) = U(x, t ) - x 0 Z(x,

Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient

Remark 2.9 Similar to Remark 2.5, It is worth to know that the transformation (2.144) is an one-to-one map, and its inverse is given by:

U(x, t ) = W(x, t ) + x 0 L(x, y)W(y, t )d y, (2.148)
where the kernel matrix L(x, y) is the unique inverse of Z(x, y).

In the next two parts, Problem 3 and Problem 4 will be addressed.

Determinations of the kernel function

In

2θ d d x Z(x, x) + ΛZ(x, x) -Z(x, x)Λ = -Ψ -C, (2.149) Z xx (x, y) -Z y y (x, y) + 1 θ ΛZ x (x, y) + 1 θ Z y (x, y)Λ = 1 θ ΨZ(x, y) + 1 θ Z(x, y) C, (2.150)
Z(x, 0) = 0, (2.151)

where d d x Z(x, x) = Z x (x, x) + Z y (x, x
) and z i j (x, y) for i , j = 1, . . . , n, are the components of Z(x, y).

Lemma 2.4 If the conditions given in (2.149) and (2.151) hold, the solution of (2.149) is

given as follows:

Z(x, x) = - 1 2θ x 0 e Λ 2θ (y-x) (Ψ + C)e -Λ 2θ (y-x) d y.
(2.152)

Remark 2.10 Proposition 2.2 is actually a revision of Theorem 1 in [88]

, where a questionable transformation is given as follows:

K y (x, y)Λ + ΛK x (x, y) = Λ(K x (x, y) + K y (x, y)), (2.153)
which appears in Eq. ( 23) of [88].

Then, similar to Theorem 2.6, the following theorem can be obtained.

Theorem 2.8

The conditions given in (2.149)-( 2.151) admit a solution Z(x, y) for 0 ≤ y ≤ x ≤ 1, which is twice continuously differentiable.

Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient

Proof. Firstly, by applying the following change of variables: α → x + y and β → xy, we define:

K(α, β) := Z α + β 2 , α -β 2 = Z(x, y), ( 2
K αβ (α, β) + 1 4θ ΛK α (α, β) -K α (α, β)Λ + K β (α, β)Λ + ΛK β (α, β) = 1 4θ ΨK(α,β) + 1 4θ K(α, β) C, (2.156) K(α, α) = K(β, β) = 0. (2.157)
By integrating (2.156), we get:

K α (α, β) = K α (α, 0) + 1 4θ β 0 ΨK(α,τ) + K(α, τ) C d τ - 1 4θ β 0 ΛK α (α, τ) -K α (α, τ)Λ d τ - 1 4θ K(α, β)Λ + ΛK(α,β) + 1 4θ K(α, 0)Λ + ΛK(α,0) . (2.158)
Then, by integrating (2.158) and using (2.157), we obtain:

K(α, β) = K(α, 0) -K(β, 0) + 1 4θ α β β 0 ΨK(s,τ) + K(s, τ) C d τd s - 1 4θ α β K(s, β)Λ + ΛK(s,β) d s + 1 4θ α β K(s, 0)Λ + ΛK(s,0) d s - 1 4θ β 0 ΛK(α,τ) -K(α, τ)Λ d τ + 1 4θ β 0 ΛK(β,τ) -K(β, τ)Λ d τ. (2.159)
It is obvious that (2.159) is in a similar form to (2.119). Thus, by using successive approximations method as done in the proof of Theorem 2.6, we can easily find that K(α, β) in (2.149)-(2.151) has a continuous solution, which is twice continuously differentiable. Hence, the proof of this part is skipped here for brevity.

Convergence of the target system and controller design

In this subsection, the condition on C to guarantee the stability of the target system (2.145)-(2.147) is given in the following theorem.

Theorem 2.9 If the matrix C is chosen such that the smallest eigenvalue of the symmetric part S[ C] satisfies the following condition: It means that the cascade system ( Ŵ, W) is exponentially stable. Therefore, the system ( Û, Ũ) is also exponentially stable since it is related to ( Ŵ, W) by the invertible coordinate transformations (2.86) and (2.148). Thus, the system (U, Û) is exponentially stable.

σ min (S[ C]) ≥ ∥ Λ ∥ 2 2θ , ( 2
Hence, this proof is completed.

Numerical calculations of design gains and kernel functions

To realize the backstepping observer, controller and output feedback controller, it is essential to calculate the kernel functions and design gains. In this subsection, the numerical methods to calculate the design gains and kernel functions are provided.

Calculation of the design gain Q:

Due to e Λ 2θ y , it is difficult to numerically compute Q by using (2.104). Therefore, some assumptions are needed to simply the calculations. For this purpose, C can be chosen by the following two ways:

1. as shown in Remark 2.6, if we choose C to make Ψ + C commute with Λ, it is easy

to obtain Q = -1 2θ (Ψ + C).
2. if we want more degrees of freedom on C, it can be chosen such that Λ,[Λ,Ψ+C] = 0.

In this case, thanks to the well-known Baker Campbell Hausdorff formula, we can

obtain: Q = 1 0 Ψ + C + y Λ 2θ , Ψ + C d y.
Moreover, C should satisfy the stability condition (2.136).

Calculations of the kernel functions H and Z:

According to the proof of Theorem 2.6, the kernel function H(x, y) is obtained using G(ξ, η) which is numerically approximated by applying (2.120) and (2.121), where

• G(•, 0) is calculated by applying (2.117),

• C is previously chosen in the calculation of Q, which can also simply the calculation of G(•, 0) in (2.117).

Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient

By using a similar way, the kernel function Z(x, y) is obtained based on the proof of Theorem 2.8, where C is chosen using a similar way as C and satisfies (2.162).

Calculation of the design gain P:

After calculating the kernel function H(x, y), P(x) is obtained using (2.90).

In the following section, the previous numerical methods are applied.

Simulation results

In order to verify the obtained theory results, a numerical example is considered as follows:

u 1t (x, t ) = u 1xx (x, t ) + 2u 1x (x, t ) + u 2x (x, t ) -u 1 (x, t ) + 12u 2 (x, t ), (2.171) u 2t (x, t ) = u 2xx (x, t ) + u 1x (x, t ) + 2u 2x (x, t ) + 12u 1 (x, t ) -u 2 (x, t ), (2.172) u 1x (0, t ) = 0, (2.173) u 2x (0, t ) = 0, (2.174) u 1 (1, t ) = u 1c (t ), (2.175
)

u 2 (1, t ) = u 2c (t ). (2.176)
In order to simulate the studied system, the finite-difference scheme is firstly used to discrete the system. Then, the ODE45 scheme is adopted with the initial conditions u 1 (x, 0) = u 2 (x, 0) = 0.5 cos(πx).

In this example, the states of the system (2.171)-(2.176) will be estimated by applying the observer designed in Subsection 2. with different diffusions. The considered system is endowed with a sensor at the left boundary and an actuation at the right boundary. Firstly, a backstepping observer is designed for the considered system. Secondly, compared to the existing literature [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF] a weaker condition is deduced to guarantee the stability of the plant system by means of a controller. Thirdly, by using the separation principle, an observer-based output feedback controller is set up based on the designed observer and controller. Finally, the efficiency of the proposed method is shown by taking a simplified model of Chemical Tubular Reactor.

Introduction

This chapter considers the following class of systems modelled by the following n coupled parabolic PDEs with different diffusions:

U t (x, t ) = ΘU xx (x, t ) + ΨU(x, t ), (3.1)
with the Neumann-type boundary conditions and the Dirichlet-type actuation:

U x (0, t ) = 0, (3.2) U(1, t ) = U c (t ), (3.3)
where

• the state vector U(x, t ) = (u 1 (x, t ), u 2 (x, t ), • • • , u n (x, t )) T ∈ [L 2 (0, 1)] n , • U c (1, t ) = u c 1 (1, t ), u c 2 (1, t ), • • • , u c n (1, t ) T ∈ [L 2 (0, 1)] n is the input vector,
• Θ is a positive diagonal n × n matrix, whose components θ i for i = 1, 2, . . . , n, represent different diffusions of the system,

• Ψ is a n × n matrix, whose components ψ i j for i , j = 1, 2, . . . , n, represent the reaction terms, with n ∈ N * being the number of the coupled PDEs.

Then, the following two definitions are needed.

Definition 3.1 Let u i (x, t ) and u j (x, t ) be two different states of a system modelled by (3.1)- (3.3). If ψ i j = 0 or ψ j i = 0, u i (x, t ) and u j (x, t ) are said directly coupled.

Definition 3.2 A system modelled by (3.1)-(3.3) is said fully coupled, if for any two dif-

ferent states u i (x, t ) and u j (x, t ), there exists a series of other states u r

1 , u r 2 , • • • , u u r m with m ≤ n -2, such that the states u i , u r 1 , u r 2 , • • • , u u r n
, u j are directly coupled in turn, which means that the system cannot be decomposed into any independent subsystems.

The main goal is to design a backstepping observer-based output feedback controller for the systems modelled by (3.1)-(3.3). To achieve this goal, it assumes that these systems are fully coupled. Moreover, it also assumes that the output measurement is available at x = 0 and the input actuator is placed at x = 1, which are denoted by U(0, t ) and U(1, t ), respectively. This corresponds to the case of Dirichlet-type actuation with anti-collocated sensors and actuators setting. The case of Neumann-type actuation with collocated sensors and actuators setting can be done in a similar way.

Backstepping boundary observer

Backstepping boundary observer

This section deals with the observer design for the fully coupled systems described by (3.1)- (3.3). For this purpose, the following system is considered:

U t (x, t ) = Θ U xx (x, t ) + Ψ U(x, t ) + P(x)[U(0, t ) -U(0, t )],
(3.4)

U x (0, t ) = Q[U(0, t ) -U(0, t )], (3.5) U(1, t ) = U c (t ), (3.6)
where the n × n matrices P(x) and Q are the design gains which are determined such that (3.4)-(3.6) is an observer for the studied systems.

Let us denote the observation error between the real states U(x, t ) and the estimated states U(x, t ) by: 3), a straightforward calculation leads to:

U(x, t ) = U(x, t ) -U(x, t ). ( 3
U t (x, t ) = Θ U xx (x, t ) + Ψ U(x, t ) -P(x) U(0, t ), (3.8)
U x (0, t ) = -Q U(0, t ), (3.9)
U(1, t ) = 0.

(3.10) Thus, the observer gains P(x) and Q need to be determined such that the observation error converges to 0 when the time tends to infinity. To do so, the following backstepping transformation is applied, .11) such that the observation error system (3.8)-(3.10) can be transformed into the following target system:

U(x, t ) = W(x, t ) - x 0 Z(x, y) W(y, t )d y, ( 3
W t (x, t ) = Θ W xx (x, t ) -C W(x, t ), (3.12) W x (0, t ) = 0, (3.13) W(1, t ) = 0, (3.14)
where C is a matrix parameter to be designed such that the system (3.12)-(3.14) tends to 0 asymptotically when the time tends to infinity.

In order to obtain the transformation (3.11), the kernel matrix Z(x, y) needs to be determined. As the dimensions of U(x, t ) and W(x, t ) are n, Z(x, y) is a n×n square matrix. In order to make the transformation (3.11) easy to apply, it is assumed throughout this paper that Z(x, y) is in the following form: Z(x, y) = z(x, y)I n×n , (3.15)

Backstepping boundary observer

which is similar to the one given in [START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF]. And the kernel function z(x, y) will be determined in the following subsection.

Remark 3.1 It is worth to know that the transformation (3.11) is a one-to-one map, and its inverse is given by:

W(x, t ) = U(x, t ) + x 0 R(x, y) U(y, t )d y, (3.16)
where the kernel matrix R(x, y) is the unique inverse of Z(x, y), which can be straightforwardly deduced as the scalar case done in [15] and [START_REF] De Queiroz | Lyapunov-Based Control of Mechanical Systems[END_REF].

Determinations of the kernel function and observer gains

In this subsection, the kernel function z(x, y) in the backstepping transformation (3.11) and the observer gains P(x) and Q in (3.4)-(3.5) will be determined. Firstly, z(x, y) is characterized by a set of PDEs involving P(x) and Q. Then, explicit expressions are deduced for z(x, y), P(x) and Q, respectively.

Proposition 3.1 If the kernel function z(x, y) fulfills the following conditions:

z xx (x, y)z y y (x, y) = κz(x, y), (3.17) where d d x z(x, x) = z x (x, x) + z y (x, x), κ is a constant parameter defined by the following algebraic equation:

d d x z(x, x) = κ 2 , ( 3
κI n×n = (C + Ψ)Θ -1 , (3.22)
the transformation (3.11) can transform the observation error system (3.8)-(3.10) into the target system (3.12)- (3.14).

Remark 3.2

As mentioned previously, the condition on the parameter matrix C is to make the target system (3.12)- (3.14) converging to 0. Thanks to (3.22), we have C = κΘ-Ψ, which can be described as follows:

c i j =      κ θ i -ψ i i , if i = j , -ψ i j ,
else.

(3.23)

Consequently, since matrices Ψ and Θ are known, C can be determined by choosing the constant parameter κ. In the following subsection, a specific condition on κ will be given to ensure that the system (3.12)- (3.14) converges to 0.

Proof of Proposition 3.1.

In this proof, it will be shown how to transform the observation error system (3.8)-(3.10) into the target system (3.12)- (3.14), where the conditions on z(x, y) are imposed. Hence, the following three steps are needed.

• Step 1. Transformation to (3.12): By differentiating both sides of (3.11) with respect to x and t , and applying the Leibniz's differential rule, we obtain:

U x (x, t ) = W x (x, t ) -Z(x, x) W(x, t ) - x 0 Z x (x, y) W(y, t )d y, (3.24) U xx (x, t ) = W xx (x, t ) - d d x Z(x, x) W(x, t ) -Z(x, x) W x (x, t ) -Z x (x, x) W(x, t ) - x 0 Z xx (x, y) W(y, t )d y, (3.25) U t (x, t ) = W t (x, t ) -Z(x, x)Θ W x (x, t ) + Z(x, 0)Θ W x (0, t ) + Z y (x, x)Θ W(x, t ) -Z y (x, 0)Θ W(0, t ) - x 0 Z y y (x, y)Θ W(y, t )d y + x 0 Z(x, y)C(x) W(y, t )d y. (3.26)
Then, by substituting (3.25)-(3.26) into (3.8), we get:

W t (x, t ) -Θ W xx (x, t ) + C W(x, t ) = -Θ d d x Z(x, x) + Z y (x, x)Θ + ΘZ x (x, x) -Ψ -C W(x, t ) + [Z(x, x)Θ -ΘZ(x, x)] W x (x, t )
+ [Z y (x, 0)Θ -P(x)] W(0, t )

+ x 0 Z y y (x, y)Θ -ΘZ xx (x, y) -Z(x, y)C -ΨZ(x, y) W(y, t )d y.
(3.27) Thus, (3.12) can be obtained using (3.27) if the following conditions hold: Based on Proposition 3.1, the explicit expressions of the kernel function z(x, y), the observer gains P(x) and Q can be given in the following corollary.

ΘZ xx (x, y) -Z y y (x, y)Θ + Z(x, y)C + ΨZ(x, y) = 0, (3.28) Θ d d x Z(x, x) + Z y (x, x)Θ + ΘZ x (x, x) -Ψ -C = 0, (3.29) Z(x, x)Θ = ΘZ(x, x), ( 3

Corollary 3.1

The kernel function z(x, y) that fulfills the conditions (3.17)- (3.19) is given by:

z(x, y) = -κ(1 -x) I 1 κ(2 -x -y)(x -y) κ(2 -x -y)(x -y) . (3.35)
Moreover, the observer gains P(x) and Q are given as follows:

P(x) = κ(1 -x) x(2 -x) I 2 κx(2 -x) Θ, (3.36) Q = - κ 2 I n×n , (3.37)
where I 1 (•) and I 2 (•) are the well-known modified Bessel functions of the first kind [START_REF] De Queiroz | Lyapunov-Based Control of Mechanical Systems[END_REF].

Proof. First, using a similar way as done in [START_REF] De Queiroz | Lyapunov-Based Control of Mechanical Systems[END_REF], (3.35) can be obtained. Then, by calculating z y (x, y) using (3.35), (3.36) can be obtained from (3.21).

Recall that the expression of the Bessel function is given by I

1 (x) = ∞ n=0 ( x 2 ) 2n+1 n!(n + 1)! .
Then, it is easy to obtain that lim

x→0 I 1 (x) x = 1 2
. Thus, we get:

Q = Z(0, 0) = lim x→0 y→0 -κ(1 -x) I 1 ( κ(2 -x -y)(x -y)) κ(2 -x -y)(x -y) I n×n = - κ 2 I n×n . (3.38)
Thus, this proof is completed.

Consequently, according to Corollary 3.1, it only needs to set the parameter κ to obtain the kernel function z(x, y), the observer gains P(x) and Q. The way to set κ will be presented in the following subsection.

Convergence of the target system

It is shown in Remark 3.2 that since C = κΘ-Ψ, a condition on κ should be given to guarantee the convergence of the target system (3.12)- (3.14). This condition is provided in the following theorem.

Backstepping boundary observer

Theorem 3.1 If the parameter κ is chosen such that the following matrix inequality holds:

-S[Ψ] + θ mi n 4 I n×n + κΘ 0, (3.39)

the target system (3.12)- (3.14) exponentially converges to 0 with the following rate:

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e - θ mi n 4 +σ mi n (κΘ-S[Ψ]) t , (3.40)
where θ mi n is the smallest diagonal element of Θ.

Proof. The objective is to find a condition on the parameter κ such that the following function:

V(t ) = 1 2 1 0 W T (x, t ) W(x, t )d x, (3.41)
is a Lyapunov function for the system (3.12)- (3.14). For this purpose, the time derivative of V(t ) is calculated as follows:

V(t ) = 1 0 W T (x, t ) W t (x, t )d x. (3.42)
Then, using (3.12) in (3.42), we obtain:

V(t ) = 1 0 W T (x, t )Θ W xx (x, t )d x - 1 0 W T (x, t )C W(x, t )d x. (3.43)
By applying integration by parts with the boundary conditions given in (3.13)-(3.14), the first term on the right side of (3.43) becomes:

1 0 W T (x, t )Θ W xx (x, t )d x = W T (x, t )Θ W x (x, t ) 1 0 - 1 0 W T x (x, t )Θ W x (x, t )d x = - 1 0 W T x (x, t )Θ W x (x, t )d x. ( 3.44)
Thanks to the Poincaré inequality and the zero boundary conditions, we get:

1 0 W T x (x, t )Θ W x (x, t )d x ≥ 1 4 1 0 W T (x, t )Θ W(x, t )d x. ( 3.45)
Hence, by using (3.44) and (3.45), and applying the trivial inequality, (3.43) becomes:

V(t ) ≤ - θ mi n 4 1 0 W T (x, t ) W(x, t )d x -σ mi n (S[C]) 1 0 W T (x, t ) W(x, t )d x ≤ - θ mi n 2 + 2σ mi n (S[C]) V(t ). (3.46)
Thus, based on (3.46), the following convergence rate is deduced: Remark that the convergence condition (3.39) is deduced based on the inequality (3.46). Hence, it is sufficient rather than necessary.

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e - θ mi n 4 +σ mi n (S[C]) t . ( 3
At the end of this subsection, two ways are provided to show how to choose the parameter κ fulfilling (3.39).

Algorithm 1

The parameter κ can be chosen to be positive and to satisfy the following condition:

κ > σ max (S[Ψ]) θ mi n - 1 4 . (3.48)
Indeed, since -S[Ψ]+ θ mi n 4 I n×n +κΘ is a symmetric matrix, we have:

∀ κ > 0, v ∈ R n \{0}, v T -S[Ψ] + θ mi n 4 I n×n + κΘ v ≥ σ mi n (-S[Ψ]) + θ mi n 4 + κθ mi n v T v. (3.49) Hence, if the condition σ mi n (-S[Ψ])+ θ mi n 4 +κθ mi n > 0 holds, -S[Ψ]+ θ mi n 4 I n×n +κΘ 0. Finally, as -σ mi n (-S[Ψ]) = σ max (S[Ψ]
), the former condition is equivalent to (3.48).

Algorithm 2 The parameter κ can also be chosen by solving the following inequality by means of the well-known Linear Matrix Inequality (LMI) algorithm:

M + I T n×n XΘ + Θ T X T I n×n 0, (3.50) where M = -S[Ψ] + θ mi n 4 I n×n and X = κ 2 I n×n .
These two algorithms will be applied in the numerical simulation part.

Backstepping boundary controller

The objective of this section is to design a backstepping controller for a system modelled by (3.1)-(3.3) using a weaker condition than the one given in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF]. For this purpose, similar techniques are applied as done for the observer design in the previous section.

Backstepping boundary controller

It is shown in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF] that the following backstepping transformation:

W(x, t ) = U(x, t ) - x 0 K(x, y)U(y, t )d y (3.51)
can transform the system modelled by (3.1)-(3.3) into the following target system:

W t (x, t ) = ΘW xx (x, t ) -CW(x, t ), (3.52)
W x (0, t ) = 0, (3.53) W(1, t ) = 0, (3.54)
where

• W(x, t ) = (w 1 (x, t ), w 2 (x, t ), • • • , w n (x, t )) ∈ [L 2 (0, 1)] n ,
• C is a n × n matrix with components ci j for i , j = 1, 2, . . . , n,

• the kernel matrix is in the form:

K(x, y) = k(x, y)I n×n , (3.55)
• the kernel function is given by: k(x, y) = -κx

I 1 κ(x 2 -y 2 ) κ(x 2 -y 2 ) , (3.56)
I 1 (•) is the well-known modified Bessel function of the first kind,

• the parameter κ is defined by the following algebraic equation:

κI n×n = ( C + Ψ)Θ -1 . (3.57)
Moreover, the stability features of the target system (3.52)-(3.54) is established.

Theorem 3.2 [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF] If the parameter κ defined in (3.57) is chosen such that S[ C] is positive definite, the target system (3.52)-(3.54) is exponentially stable with the following convergence rate:

∥ W(•, t ) ∥ 2,n ≤∥ W(•, 0) ∥ 2,n e -σ mi n (S[ C])t . (3.58)
Inspired by Theorem 3.1, a relaxed condition on κ is provided in the following theorem, which improves the convergence rate obtained in Theorem 3.2.

3.4. Backstepping observer-based output feedback controller

Backstepping observer-based output feedback controller

An observer and a controller are designed in Section 3.2 and Section 3.3, respectively, which are exponentially convergent. Moreover, they are independent and satisfy the separation principle. In this following theorem, we combine them to realize a backstepping observer-based output feedback controller, whose convergence stability is also established.

Theorem 3.4

If the parameters κ and κ defined in (3.22) and (3.57) are chosen such that the following condition holds:

σ mi n (S[C]) ≥ σ mi n (S[ C]) > - θ mi n 8 , (3.64)
the controller defined by:

U c (t ) = 1 0 K(1, y) U(y, t )d y, (3.65)
stabilizes the system modelled by (3.1)- (3.3), where K(1, y) is given by (3.55)-(3.56) with x = 1, and U(•, t ) is the estimated state obtained by the following observer:

U t (x, t ) = Θ U xx (x, t ) + Ψ U(x, t ) + P(x)[U(0, t ) -U(0, t )], (3.66)
U x (0, t ) = Q[U(0, t ) -U(0, t )], (3.67)
U(1, t ) = 1 0 K(1, y) U(y, t )d y, (3.68)
with the observer gains P(x) and Q given by (3.36) and (3.37), respectively. In other words, an observe-based output feedback control system is provided for the system modelled by (3.1)- (3.3), which is stable under the condition (3.64).

Remark 3.3 The condition (3.64) is given to guarantee that the observer converges faster than the controller for the stability of the observer-based output feedback control system (3.65)-(3.68).

Proof of Theorem 3.4. This proof contains the following steps.

• Step 1. Transformation to a cascade target system: Let us consider the following backstepping transformation:

W(x, t ) = U(x, t ) - x 0 K(x, y) U(y, t )d y, (3.69)
where K(x, y) is given in (3.55)-(3.56). Then, using a similar way as done in Section 3.2, the system (3.66)-(3.68) can be transformed into the following target system:

W t (x, t ) = Θ W xx (x, t ) -C W(x, t ) + [P(x) - x 0 K(x, y)P(y)d y] W(0, t ), (3.70)
W x (0, t ) = Q W(0, t ), (3.71)
W(1, t ) = 0. (

By combining the target system obtained in (3.70)-(3.72) and the one obtained in (3.12)-(3.14), a cascade system of ( W(x, t ), W(x, t )) is constructed.

• Step 2. Stability of the cascade target system:

The following function is introduced to prove the stability of the cascade system ( W(x, t ), W(x, t )):

V(t ) = α 2 1 0 W T (x, t ) W(x, t )d x + 1 2 1 0 W T (x, t ) W(x, t )d x, (3.73)
where α ∈ R + will be designed later. Then, the objective is to prove V(t ) is a Lyapunov function for the cascade system ( W(x, t ), W(x, t )), whose time derivative is given as follows:

V(t ) = -α 1 0 W T x (x, t )Θ W x (x, t )d x -α 1 0 W T (x, t )C W(x, t )d x - 1 0 W T x (x, t )Θ W x (x, t )d x - 1 0 W T (x, t ) C W(x, t )d x + 1 0 W T (x, t ) P(x) - x 0 K(x, y)P(y)d y W(0, t )d x -W T (0, t )ΘQ W(0, t ). (3.74)
Then, the following useful inequalities are obtained.

• Using the Poincaré inequality and the Young inequality, we get:

1 0 W T (x, t )B(x) W(0, t )d x ≤ 1 4 1 0 W T x (x, t )Θ W x (x, t )d x + [σ sup (B(x))] 2 θ mi n 1 0 W T x (x, t ) W x (x, t )d x, (3.75)
where B(x) = P(x)-

x 0 K(x, y)P(y)d y, σ sup (B(x)) = max 1≤i ≤n sup 0≤x≤1 |σ i (B(x))|, and σ i (B(x))
is the eigenvalue of B(x).

• Using the Poincaré inequality and the Young inequality, we obtain:

-W T (0, t )ΘQ W(0, t ) ≤ 1 4 1 0 W T x (x, t )Θ W x (x, t )d x + (ΘQ) 2 max θ mi n 1 0 W T x (x, t ) W x (x, t )d x, (3.76)
where (ΘQ) max denotes the largest component of the diagonal matrix ΘQ.

Simulations

Hence, by combining (3.74)-(3.76) and using the Trivial inequality, we obtain:

V(t ) ≤ -αθ mi n - (ΘQ) 2 max θ mi n - [σ sup (B(x))] 2 θ mi n 1 0 W T x (x, t ) W x (x, t )d x -ασ mi n (S[C]) 1 0 W T (x, t ) W(x, t )d x -σ mi n (S[ C]) 1 0 W T (x, t ) W(x, t )d x - θ mi n 2 1 0 W T x (x, t ) W x (x, t )d x. ( 3.77)
Assume that α fulfills the following condition:

αθ mi n - (ΘQ) 2 max θ mi n - [σ sup (B(x))] 2 θ mi n > 0. (3.78)
Thus, by assuming σ mi n (S[C]) ≥ σ mi n (S[ C]) and using the Poincaré inequality, we obtain:

V(t ) ≤ - αθ mi n 4 - (ΘQ) 2 max 4θ mi n - [σ sup (B(x))] 2 4θ mi n + σ mi n (S[ C])α 1 0 W T (x, t ) W(x, t )d x - θ mi n 8 + σ mi n (S[ C]) 1 0 W T (x, t ) W(x, t )d x. ( 3.79)
Then, α is set to be the solution of the following algebraic equation:

αθ mi n 4 - (ΘQ) 2 max 4θ mi n - σ sup (B(x)) 2 4θ mi n + σ mi n (S[ C])α = θ mi n 8 + σ mi n (S[ C]) α, (3.80)
which leads to α = 2

σ max (ΘQ) θ mi n 2 + 2 σ sup (B(x)) θ mi n 2 satisfying (3.78).
Consequently, using (3.73), (3.79) becomes:

V(t ) ≤ - θ mi n 4 + 2σ mi n (S[ C]) V(t ), (3.81)
which means that the cascade target system ( W(x, t ), W(x, t )) is exponentially stable if (3.64) holds. Therefore, the cascade system ( U(x, t ), U(x, t )) is also exponentially stable, since it is related to ( W(x, t ), W(x, t )) by the invertible coordinate transformation of (3.11) and (3.51), which directly means that the closed-loop system (U(x, t ), U(x, t )) is exponentially stable.

Simulations

In order to verify the efficiency of the obtained theory results, a linearlized Chemical Tubular Reactor model is studied in this section.

Simulations

We begin with the following coupled temperature-concentration system of Chemical Tubular Reactor given by [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF] and simplified by [START_REF] Liu | Backstepping observer-based output feedback control for a class of coupled parabolic PDEs with different diffusions[END_REF]:

u 1t (x, t ) = D 1 u 1xx (x, t ) + k 0 δ(1 -u 2 (x, t ))e - µ 1+u 1 (x,t ) , (3.82
)

u 2t (x, t ) = D 2 u 2xx (x, t ) + k 0 (1 -u 2 (x, t ))e - µ 1+u 1 (x,t ) , (3.83)
u 1x (0, t ) = 0, (3.84
)

u 2x (0, t ) = 0, (3.85
)

u 1 (1, t ) = u 1c (t ), (3.86
)

u 2 (1, t ) = u 2c (t ), (3.87)
where the state variables u 1 (x, t ) and u 2 (x, t ) denote the normalized temperature and concentration, respectively, and the physical parameters are taken such as: D 1 = 0.14, D 2 = 0.16, k 0 = 2.426 × 10 7 , δ = 0.5, µ = 20.

Since the system modelled by (3.82)-(3.87) is nonlinear, it should be linearized around a steady-state profile in order to apply our results. For this purpose, let u s 1 and u s 2 denote the steady-state variables which are the solutions of the following differential equations:

D 1 d 2 u s 1 d x 2 + k 0 δ(1 -u s 2 )e - µ 1+u s 1 = 0, (3.88)
D 2 d 2 u s 2 d x 2 + k 0 (1 -u s 2 )e - µ 1+u s 1 = 0. ( 3.89)
Then, by setting v i (x, t ) = u i (x, t )u s i (x), the corresponding linearized system around the steady-state (u s 1 , u s 2 ) can be obtained as follows:

v 1t (x, t ) = D 1 v 1xx (x, t ) + a 11 (x)v 1 (x, t ) + a 12 (x)v 2 (x, t ), (3.90)
v 2t (x, t ) = D 2 v 2xx (x, t ) + a 21 (x)v 1 (x, t ) + a 22 (x)v 2 (x, t ), (3.91)
where

a 11 (x) = ∂F 1 ∂u 1 (u s 1 ,u s 2 ) = k 0 δ 1 -u s 2 (x) (1 + u s 1 (x)) 2 e - µ 1+u s 1 (x) , (3.92)
a 12 (x) = ∂F 1 ∂u 2 (u s 1 ,u s 2 ) = -k 0 δu s 2 (x)e - µ 1+u s 1 (x) , (3.93)
a 21 (x) = a 11 (x)
δ and a 22 (x) = a 12 (x) δ with Thus, by taking the average values of the coefficients a i j (x), the following linear PDEs are obtained:

   v 1t (x, t ) v 2t (x, t )    = Θ    v 1xx (x, t ) v 2xx (x, t )    + Ψ    v 1 (x, t ) v 2 (x, t )    ( 3
v 1x (0, t ) = 0, (3.95) v 2x (0, t ) = 0, (3.96) v c 1 (t ) = 5 sin(t ), (3.97)
v c 2 (t ) = 10 sin(2t ). (3.98)
In this simulation, the state of the system (3.94)-(3.98) will be estimated by applying the observer designed in Section 3.2. Moreover, a backstepping observer-based output feedback controller will be designed by applying Theorem 3.4. To do so, the parameters κ and κ need to be set to obtain the controller U c using (3.55)-(3.56) and the observer gains P(x) and Q using (3.36)-(3.37), respectively. Hence, Algorithm 1 and Algorithm 2 are applied. On the one hand, we have σ max (S[Ψ])

θ mi n
-1 4 = -0.2487. Hence, according to Algorithm 1, we should set κ ≥ κ > 0. On the other hand, according to Algorithm 2, we should set κ ≥ κ > 1.6829 × 10 -31 . Consequently, it is worth to note that the criteria on setting κ and κ theoretically given by Algorithm 1 is close to the one numerically given by Algorithm 2. Moreover, remark that according to Condition 1 given in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF], the parameter κ should be set greater than 7.6538.

In order to simulate the studied system, the finite-difference scheme is first used to discrete the system. Then, the ODE45 scheme is adopted with the initial conditions v 1 (x, 0) = v 2 (x, 0) = sin(πx) + sin(3πx).

Finally, the spatiotemporal evolution of the state in the open-loop is shown in Fig.

Conclusions

In this chapter, a backstepping observer-based output feedback controller for a class of coupled parabolic PDEs with different diffusions was designed. Firstly, a backstepping observer for the considered system was designed, where explicit expressions of the kernel function and observer gains were obtained, and the convergence condition based on the parameter κ was provided. Secondly, a backstepping controller designed by [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF] was introduced, where the condition for the controller was relaxed. Thirdly, a backstepping observer-based output feedback controller was realized by combining the previously designed observer and controller. Finally, the obtained results were applied to a linearlized Chemical Tubular Reactor model. Compared to the systems considered in the above chapters, the considered system of this chapter processes spatially-varying coefficients for all the terms of reaction, advection and diffusion. This makes it more challenged to solve kernel functions of the backstepping transformation. Thanks to the recent work [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF], a very important relationship between the control kernels for the parabolic system and the hyperbolic system has been found. This significantly unveils the connection between the backstepping controllers for the parabolic and the hyperbolic systems. This result permits to obtain the control kernel for the parabolic system by transferring the kernel equation into a set of coupled hyperbolic equations which have been proven well-posed. Using a similar way, we can find that this relationship is also useful for the observation kernel for the parabolic system. Then, by proposing some artificial boundary conditions, we can show the well-posedness of the observation kernel for the parabolic system. Moreover, it also proves that target system of the backstepping transformation is H 1 stable by proposing certain conditions on the parameters of the target system. Finally, the proposed method is highlighted by a numerical example.

Introduction

This chapter considers a class of linear systems modelled by the following n coupled reaction-advection-diffusion equations with spatially-varying coefficients:

U t (x, t ) = Θ(x)U xx (x, t ) + Λ(x)U x (x, t ) + Ψ(x)U(x, t ),
U x (0, t ) = 0, (4.2) U(1, t ) = U c (t ), (4.3)
where

• U(x, t ) = (u 1 (x, t ), • • • , u n (x, t )) T ∈ [L 2 (0, 1)] n is the state vector, • U c (1, t ) = u c 1 (1, t ), • • • , u c n (1, t ) T ∈ [L 2 (0, 1)
] n is the input vector,

• Θ(x) is a diagonal n × n matrix, whose diagonal components θ i (x) for i = 1, . . . , n, represent the diffusion term coefficients, are assumed to be three times differentiable and be bounded in order such that

θ max > θ 1 (x) > θ 2 (x) > • • • > θ n (x) > θ mi n > 0,
• Λ(x) is a n × n matrix, whose components λ i j (x) for i , j = 1, . . . , n, represent the advection terms coefficients and are assumed to be two times differentiable, 4.2. Backstepping observer design

• Ψ(x) is a n × n matrix, whose components ψ i j (x) for i , j = 1, . . . , n, represent the reaction terms coefficients and are assumed to be differentiable, with n ∈ N * being the number of the coupled RAD equations.

The objectiver is to design a backstepping observer for the system modeled by (4.1)-(4.3). It is assumed that the output measurement is available at x = 0, which is denoted by U(0, t ). The latter corresponds to the case of Neumann-type observation with anticollocated sensors and actuators setting.

Backstepping observer design

This section deals with the observer design for the studied coupled system described by (4.1)-(4.3). For this purpose, the following system is considered:

Ût (x, t ) = Θ(x) Ûxx (x, t ) + Λ(x) Ûx (x, t ) + Ψ(x) Û(x, t ) + P(x)[U(0, t ) -Û(0, t )], (4.4)
Ûx (0, t ) = Q[U(0, t ) -Û(0, t )], (4.5)
Û(1, t ) = U c (t ), (4.6)
where P(x) and Q are two n × n matrices to be determined such that (4.4)-(4.6) will be an observer for the studied system.

The observation error between the real state U(x, t ) and the estimated one Û(x, t ) is denoted by: Ũ(x, t ) = U(x, t ) -Û(x, t ). (4.7)

By subtracting (4.4)-(4.6) from (4.1)-( 4.3), a straightforward calculation leads to:

Ũt (x, t ) = Θ(x) Ũxx (x, t ) + Λ(x) Ũx (x, t ) + Ψ(x) Ũ(x, t ) -P(x) Ũ(0, t ), (4.8)
Ũx (0, t ) = -Q Ũ(0, t ), (4.9)

Ũ(1, t ) = 0. (4.10) Thus, the observer gains P(x) and Q need to be determined such that the observation error Ũ(x, t ) converges to 0 as t → ∞. In order to achieve this, two stages will be proceeded in the following two subsections, respectively.

In the first stage, the kernel function H(x, y) is computed such that the following backstepping transformation: where C is a constant n × n matrix with the components c i j , and L(y) is a n × n matrix with the varying components l i j (y) for i , j = 1, • • • , n. In particular, certain conditions will be given on L(y) for the well-posedness of H(x, y).

Ũ(x, t ) = W(x, t ) - x 0 H(x,
In the second stage, both of C and L(y) should be designed to guarantee the stability of the target system, i .e. W(x, t ) tends to 0 asymptotically as t → ∞.

Remark 4.1 It is worth knowing that the transformation (4.11) is an one-to-one map, and its inverse is given by:

W(x, t ) = Ũ(x, t ) + x 0 R(x, y) Ũ(y, t )d y, ( 4.15)
where the kernel matrix R(x, y) is the unique inverse of H(x, y).

In order to realize the transformation (4.11), the kernel function H(x, y) is determined in the next subsection.

Determinations of the kernel function and observer gains

In this subsection, the kernel function H(x, y) in the backstepping transformation (4.11) and the observer gains P(x) and Q in (4.8)-(4.9) are determined. First, the conditions on H(x, y), P(x) and Q are given in the following proposition. where d d x H(x, x) := H x (x, x)+H y (x, x), and h i j (x, y) for i , j = 1, • • • , n, are the components of H(x, y), then the transformation (4.11) can transform the system (4.8)-(4.10) into the target system (4.12)- (4.14).

∂ y y [H(x, y)Θ(y)] -Θ(x)H xx (x, y) -Λ(x)H x (x, y) -∂ y [H(x, y)Λ(y)] = H(x, y)C + Ψ(x)H(x, y), (4.16) Θ(x) d d x H(x, x) + Θ(x)H x (x, x) + H y (x, x)Θ(x) + H(x, x)Θ (x) + Λ(x)H(x, x) -H(x, x)Λ(x) = Ψ(x) + C, (4.17
Proof. In this proof, some conditions on H(x, y), P(x) and Q are imposed to transform the observation error system (4.8)-(4.10) into the target system (4.12)-(4.14) in the following three steps.

• Step 1. Transformation to (4.12): By differentiating both sides of (4.11) with respect to x and t , and applying the Leibniz's differential rule, we obtain: A very important relationship between the control kernels for the parabolic system and the hyperbolic system has been found by [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF], which significantly unveils the connection between the backstepping controllers for the parabolic and the hyperbolic systems. This result permits to obtain the control kernel for the parabolic system by transferring the kernel equation into a set of coupled hyperbolic equations which have been proven well-posed. Using a similar way, we can find that this relationship is also useful for the observation kernel for the parabolic system. Then, with some added artificial boundary conditions, we can show the well-posedness of the observation kernel for the parabolic system, which is given by the following theorem. In addition, the transformation (4.11) is invertible, and both of the transformation and its inverse map H 1 functions into H 1 functions.

Ũx (x, t ) = Wx (x, t ) -H(x, x) W(x, t ) - x 0 H x (x, y) W(y, t )d y, (4.22) Ũxx (x, t ) = Wxx (x, t ) - d d x H(x, x) W(x, t ) -H(x, x) Wx (x, t ) -H x (x, x) W(x, t ) - x 0 H xx (x, y) W(y, t )d y, (4.23) Ũt (x, t ) = Wt (x, t ) -H(x, x)Θ(x) Wx (x, t ) + H(x, 0)Θ(0) Wx (0, t ) + H y (x, x)Θ(x) + H(x, x)Θ (x) W(x, t ) -H y (x, 0)Θ(0) + H(x, 0)Θ (0) W(0, t ) - x 0 ∂ y y [H(x, y)Θ(y)] W(y, t )d y -H(x, x)Λ(x) W(x, t ) + H(x, 0)Λ(0) W(0, t ) + x 0 ∂ y [H(x,
Wt (x, t ) -Θ(x) Wxx (x, t ) -Λ(x)W x (x, t ) + C W(x, t ) = -Θ(x) d d x H(x, x) + Θ(x)H x (x, x) + H y (x, x)Θ(x) + H(x, x)Θ (x) + Λ(x)H(x, x) -H(x, x)Λ(x) -Ψ(x) -C W(x, t ) + x 0 ∂ y y [H(x, y)Θ(y)] -Θ(x)H xx (x, y) -Λ(x)H x (x, y) -∂ y [H(x, y)Λ(y)] -H(x, y)C -Ψ(x)H(x, y) W(y, t )d y + H y (x, 0)Θ(0) + H(x, 0)Θ (0) -P(x) -H(x, 0)Λ(0) W(0, t ) + H(x, x)Θ(x) -Θ(x)H(x, x) Wx (x, t ). ( 4
Inspired by [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF], we will transform (4.16) into 2n 2 ×2n 2 coupled hyperbolic equations as given in the following lemma due to [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF]. Then, by calculating the boundary conditions of the above coupled hyperbolic equations and using the methods of characteristics and successive approximations, we can prove the above theorem.

Θ(x)H x + H y Θ(y) = Y -G 1 (x)H -HG 2 (y), (4.29) Θ(x)Y x -Y y Θ(y) = HG 3 (y) + G 4 (x)H -G 1 (x)Y + YG 2 (y), ( 4
G 1i j (x) = λ i j (x) -1 2 δ i j θ i (x) θ i (x) + θ j (x) , (4.31) G 2i j (y) = 3 2 δ i j θ i (y) -λ i j (y) θ i (y) + θ j (y) , (4.32) G 3 (y) = -C -Λ (y) + Θ (y) -G 2 (y) Θ(y) -G 2 2 (y), (4.33) G 4 (x) = -Ψ(x) + Θ(x)G 1 (x) + G 2 1 (x). ( 4

.34)

Proof. Inspired by [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF], this proof is to transform (4.16) into a n 2 × n 2 coupled wellposed hyperbolic system. Firstly, (4.16) can be transformed into the following form:

Θ(x)H xx (x, y) -H y y (x, y)Θ(y) = -H(x, y)[Λ (y) + C -Θ (y)] -Ψ(x)H(x, y) -Λ(x)H x (x, y) + H y (x, y)[2Θ (y) -Λ(y)]. (4.35)
Then, we define a new state Y(x, y):

Y(x, y) = Θ(x)H x (x, y) + H y (x, y) Θ(y) + G 1 (x)H(x, y) + H(x, y)G 2 (y), (4.36)
where G 1 (x) and G 2 (y) are given by (4.31) and (4.32).

Then, the following equation is obtained:

Θ(x)H xx (x, y) -H y y (x, y)Θ(y) = Θ(x)Y x -Y y Θ(y) + G 1 (x)Y(x, y) -Y(x, y)G 2 (y) + H(x, y) G 2 (y) Θ(y) + G 2 2 (y) + Θ(x)G 1 (x) -G 2 1 H(x, y) + Θ (x) 2 + Θ(x)G 1 (x) + G 1 (x) Θ(x) H x (x, y) + H y 1 2 Θ (y) -G 2 (y) Θ(y) -Θ(y)G 2 (y) , (4.37)
where the terms H x y (x, y) and H y x (x, y) cancel out. Firstly, we will seek boundary conditions of H(x, y) and Y(x, y).

For i = j , (4.17) leads to:

2θ i (x) d d x H i i (x,x) + θ i (x)H i i (x, x) = ψ i i (x) + c i i . ( 4
Y i i (x, x) = ψ i i (x) + c i i 2 θ i (x) . ( 4

.39)

For i = j , (4.18) leads to:

H i j (x, x) = 0, (4.40)
hence, (4.17) leads to:

θ i (x) -θ j (x) H i j x (x, x) + λ i j (x) H j j (x, x) -H i i (x, x) = ψ i j (x) + c i j , (4.41)
Then, using (4.29), we obtain:

Y i j (x, x) = ψ i j (x) + c i j θ i (x) + θ j (x) . ( 4.42)
Consequently, we get:

• if i = j , we have:

H i i (1, y) = L i i (y), (4.43)
Y i i (x, x) = ψ i i (x) + c i i 2 θ i (x) , (4.44)
• if i = j , we have:

H i j (1, y) = L i j (y), (4.45)
H i j (x, x) = 0, (4.46)
Y i j (x, x) = ψ i j (x) + c i j θ i (x) + θ j (x) , (4.47)
Then, we can design characteristics for equations (4.29) and (4.30) respectively as done as [START_REF] Hu | Control of homodirectional linear coupled hyperbolic PDEs[END_REF].

For equation (4.29), we multiply both sides of (4.29) by -1, then we can define the following characteristic lines x i j (x, y; •), y i j (x, y;

•) with i , j = 1, • • • , n. d x i j (x, y; s) d s = θ i (x i j (x, y; s)), s ∈ [0, s F i j (x, y)],
x i j (x, y; 0) = x, x i j (x, y; s F i j (x, y)) = x F i j (x, y), (4.48)

d y i j (x, y; s) d s = θ j (y i j (x, y; s)), s ∈ [0, s F i j (x, y)],
y i j (x, y; 0) = y, y i j (x, y; s F i j (x, y)) = x F i j (x, y), (4.49)

Backstepping observer design

These characteristic curves originate at the point (x, y) and terminate on different boundary lines according to the following conditions:

• If i = j , the case is similar to the one solved in [66] but on the inverse direction,the characteristic curves will hit on the line x = 1.

• If i < j , the characteristic curves will hit on the line x = 1 because the following condition always fulfills:

θ i (x) -θ j (y) ≥ y θ i (x) -x θ j (y). ( 4.50)
• If i > j and (x, y) fulfills the above condition (4.50), the characteristic curves will hit on the line x = 1, else the characteristic curves will hit on the line y = x.

For equation (4.30), we define the following characteristic lines χ i j (x, y; •), ξ i j (x, y; •)

for i , j = 1, • • • , n. d χ i j (x, y; v) d v = -θ i (χ i j (x, y; v)), v ∈ [0, v F i j (x, y)], χ i j (x, y; 0) = x, χ i j (x, y; v F i j (x, y)) = χ F i j (x, y), (4.51)
d ξ i j (x, y; v) d v = θ j (ξ i j (x, y; v)), v ∈ [0, v F i j (x, y)],
ξ i j (x, y; 0) = y, ξ i j (x, y; v F i j (x, y)) = ξ F i j (x, y), (4.52)

Different from the characteristic curves of (4.29), the curves of (4.30) will originate at the point (x, y) and terminate on the line y = x.

Then, by using the method of successive approximations as similar as done in [START_REF] Hu | Control of homodirectional linear coupled hyperbolic PDEs[END_REF],

we conclude that the kernel equations (4. instead of H i j (x, 0) = 0 for i ≤ j , which is different from [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF] and [START_REF] Hu | Control of homodirectional linear coupled hyperbolic PDEs[END_REF].

Now, we will give the conditions for the stability of the target system (4.12)-(4.14) after an assumption.

∥ W(•, t ) ∥ H 1 ≤∥ W(•, 0) ∥ H 1 e -(σ min (S[C])-max(d
with c max = C 2 , l max = sup x∈[0,1] L(x) 2 , λ max = sup x∈[0,1] Λ(x) 2 , α max = sup x∈[0,1] Λ(x)-Θ (x) 2 .
Proof. The objective of this proof is to find conditions on C and L(y) such that the following function: Firstly, differentiate V 1 (t ) with respect to time, consider (4.12) and apply integration by parts. Then, substitute (4.14) and use the trivial inequality, we get:

V(t ) = V 1 (t ) + V 2 (t ) ( 4
V1 (t ) ≤ θ max | W T (1, t )W x (1, t ) | + (α max -θ mi n ) 1 0 WT x (x, t ) Wx (x, t )d x + (α max -σ min (S[C])) 1 0 WT (x, t ) W(x, t )d x, ( 4.61)
On the one hand, we consider the following useful equation [START_REF] Vazquez | Boundary control of coupled reaction-advectiondiffusion systems with spatially-varying coefficients[END_REF]:

Wx (1, t ) = 1 0 [x Wx (x, t )] x d x. ( 4
| W(1, t ) | 2 < 2(V 1 + V 2 ). (4.65)
Consequently, using the inequalities (4.61), (4.63) and (4.65), and by applying the wellknown Cauch-Schwarz's inequality, we obtain:

∀ β 1 > 0, V1 (t ) ≤ -2θ mi n V 2 (t ) + 2θ max l 2 max β 1 V 1 (t ) + 2θ max β 1 (V 2 (t ) + V 3 (t )) + 2α max (V 1 (t ) + V 2 (t )) -2σ min (S[C])V 1 (t ), (4.66)
Secondly, differentiate V 2 (t ) with respect to time, consider (4.12) and apply integrations by parts, we get:

V2 (t ) = - 1 0 WT xx (x, t )Θ(x) Wxx (x, t )d x - 1 0 WT xx (x, t )Λ(x) Wx (x, t )d x - 1 0 WT x (x, t )C Wx (x, t )d x + Wx (1, t )C W(1, t ) + Wx (1, t ) Wt (1, t ). (4.67)
On the one hand, using the boundary condition (4.14) and (4.12), we get:

WT x (1, t ) Wt (1, t ) = WT x (1, t ) 1 0 L(x)Θ(x) Wxx (x, t )d x + WT x (1, t ) 1 0 L(x)Λ(x) Wx (x, t )d x -WT x (1, t ) 1 0 L(x)C W(x, t )d x. (4.68)
Hence, by applying the well-known Cauch-Schwarz's inequality and trivial inequality,

we get: ∀ β 2 , β 3 , β 4 > 0, WT x (1, t ) Wt (1, t ) ≤ 1 2β 2 + 1 2β 3 + 1 2β 4 (V 2 (t ) + V 3 (t )) + 2β 2 l 2 max θ 2 max V 3 (t ) + 2β 3 l 2 max λ 2 max V 2 (t ) + 2β 4 l 2 max c 2 max V 1 (t ). ( 4

.69)

On the other hand, using the boundary condition (4.14) and (4.63), we obtain:

∀ β 5 > 0, Wx (1, t )C W(1, t ) ≤ 1 β 5 (V 2 (t ) + V 3 (t )) + β 5 l 2 max c 2 max V 1 (t ). ( 4
β i > 0 for i = 1, . . . , 6, V2 (t ) ≤ β 5 l 2 max c 2 max + 2β 4 l 2 max c 2 max V 1 (t ) + -2σ min (S[C]) + β 6 λ max + 1 β 5 + 1 2β 2 + 1 2β 3 + 1 2β 4 + 2β 3 l 2 max c 2 max V 2 (t ) + -2θ mi n + λ max β 6 + 1 β 5 + 1 2β 2 + 1 2β 3 + 1 2β 4 + 2β 2 l 2 max θ 2 max V 3 (t ). (4.71)
Thirdly, combine (4.71) with (4.66), we get:

V(t ) ≤ -2σ min (S[C]) - 12l 2 max c 2 max θ mi n - 24θ 2 max l 2 max θ mi n -2α max V 1 (t ) -2σ min (S[C]) + 7θ mi n 6 - 6λ 2 max θ mi n - 6l 2 max c 2 max θ mi n -2α max V 2 (t ) -θ mi n - 6l 2 max θ 2 max θ mi n V 3 (t ). ( 4

.72)

where

β 1 = θ mi n 12θ max , β 2 = β 3 = β 4 = 3 θ mi n , β 5 = 6 θ mi n and β 6 = 6λ max θ mi n
. Then, with (4.54) and (4.55), we get:

V(t ) ≤ σ min (S[C]) -max(d 1 , d 2 ) V(t ), (4.73)
Consequently, it is obvious that (4.73) can lead to (4.53).

We will end this subsection with the following remark:

Conclusions

This chaper designed a backstepping boundary observer for a class of systems modelled by coupled RAD equations processing spatially-varying coefficients with Dirichlettype left boundary. For this purpose, the observer gains were firstly obtained with the kernel functions of the backstepping transformation. Then, the well-posedness of the kernel equations was proven by transforming the kernel equations into a set of wellposed hyperbolic equations. Thirdly, the conditions on the designed parameters were provided to guarantee the stability of the error observation system. Finally, the obtained results were verified by considering a numerical example. for the stabilization of scalar time fractional-order parabolic PDEs to coupled cases.

Firstly, a backstepping observer is designed with a sensor at the left boundary. Then, a backstepping controller is designed with an actuation at the right boundary. Based on the designed observer and controller, an observer-based output feedback controller is set up. The proposed method is verified by taking a numerical example.

Introduction

We consider the following system modelled by n coupled reaction-diffusion equations with different diffusion coefficients and fractional-order time derivative: U x (0, t ) = 0, (

U(1, t ) = U c (t ), (5.3) where n ∈ N * is the number of the coupled PDEs, and

• C o D α t U(x, t ) =        ∂ ∂t U(x, t ), α = 1 0 I 1-α t ∂ ∂t U(x, t ), 0 < α < 1
denotes the Caputo time fractionalorder derivative.

• 0 I t = 1 Γ(α) t 0 (t -s) α-1 U(x, t )d t , α > 0 denotes the Riemann-Liouville time fracitonalorder.

• U(x, t ) = [u 1 (x, t ), u 2 (x, t ), • • • , u n (x, t )] T ∈ [L 2 (0, 1)] n assumed to be square integrable denotes the state vector,

• U(0, t ) = [u 1 (0, t ), u 2 (0, t ), • • • , u n (0, t )] T ∈ [L 2 (0, 1)] n denotes the outputs or the measurements of the system,

• U c (1, t ) = [u c1 (1, t ), u c2 (1, t ), • • • , u cn (1, t )] T ∈ [L 2 (0, 1)] n is the input vector,
• Θ is a positive diagonal n ×n matrix that represents diffusions of the system with the components θ i for i = 1, 2, . . . , n,

• Ψ is a n ×n matrix that represents the reaction term with the components ψ i j for i , j = 1, 2, . . . , n.

Keep in mind that the available measurements are placed at x = 0 and the actuators are placed at x = 1, which are denoted by U(0, t ) and U(1, t ) respectively. The plant system in this chapter is fully coupled.

Backstepping observer design

Then, we can obtain the following theorem. Proof.

• L 2 Mittag-Leffler stability:The objective is to find a condition on the parameter κ such that the following function: (5.45)

V(t ) = 1

Backstepping observer design

• H 1 Mittag-Leffler stability: As done in the above, the objective is to find a condition on the parameter κ such that the following function:

J(t ) = 1
Then, by applying integration by parts and using the boundary conditions (5.13)- Here, a remark is given for the convergence rate of the realized observer: The methods to design κ is the same with Algorithm 1 and Algorithm 2 given by Section 3, which are skipped here.

Backstepping controller design

This section aims to design a backstepping controller for a system modelled by (5.1)-(5.3) by using the following transformation: W(x, t ) = U(x, t ) - • C is a n × n matrix with components ci j for i , j = 1, 2, . . . , n,

• the kernel matrix is in the form: K(x, y) = k(x, y)I n×n , (5.68)

• the kernel function is given by: k(x, y) = -κx

I 1 κ(x 2 -y 2 )
κ(x 2y 2 ) , (5.69) I 1 (•) is the well-known modified Bessel function of the first kind,

• the parameter κ is defined by the following algebraic equation:

κI n×n = ( C + Ψ)Θ -1 .
(5.70) Remark 5.3 It is worth to know that the transformation (5.64) is an one-to-one map, and its inverse is given by: U(x, t ) = W(x, t ) + The proof of Theorem 5.2 is similar with the one of Theorem 5.1, which is skipped here.

Conclusions

In this chapter, a backsteppinng observer-based output feedback controller was designed for a class of systems modelled coupled time fractional-order reaction-diffusion equations with different diffusion coefficients. A backstepping observer was firstly designed for the considered system. Then, a backstepping controller was designed. With the designed observer and controller, the output feedback controller was designed. Finally, it highlighted the proposed method by taking a numerical example.

Chapter 6

General conclusions and perspectives

Conclusions

This PhD thesis was devoted to designing boundary observer-based output feedback controllers for a kind of coupled parabolic PDEs by using the backstepping method. It has adopted different methods to design controllers and observers for different structures of the coefficients of the considered system. The main contributions of this work can be summarized as follows:

• It firstly considered a kind of systems modelled by coupled reaction-diffusion equations with constant coefficients having the same diffusion. The boundary backstepping controller and observer for this system has been designed in [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF]- [START_REF] Baccoli | Anti-collocated backstepping observer design for a class of coupled reaction-diffusion PDEs[END_REF]. Compared with the existing works, a more relaxed condition on the parameters of the target system for its stability was proposed in this thesis. Then, a backstepping boundary observer-based output feedback controller was designed.

Furthermore, an additional advection term was considered compared to the first case, for which some conditions were proposed on the parameters of the target system in order to design backstepping controllers and observers.

• Secondly, it dealt with the observer and control design for a class of systems modelled by reaction-diffusion equations with constant coefficients and different diffusion parameters. By using the backstepping method, it designed a boundary backstepping observer for the considered system. An assumption was proposed on the kernel function in order to solve it. Then, compared to the existing literature [START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF], a weaker condition was deduced to guarantee the stability of the plant system by means of a controller. Based on the designed observer and controller, it designed an observer-based output feedback controller. The efficiency of the proposed method was shown by taking a simplified model of Chemical Tubular Reactor.

6.2. Future works

• Thirdly, it designed a backstepping boundary observer for a class of linear coupled reaction-advection-diffusion equations with spatially-varying coefficients. Due to the advection terms and spatially-varying coefficients, we transformed the parabolic kernel equations into a set of hyperbolic equations. By proposing some artificial boundary conditions for the kernel functions, we showed that the hyperbolic equations are well-posed. Moreover, we proved that the target system of the backstepping transformation is H 1 stable by proposing certain conditions on the parameters of the target system. the proposed method is highlighted by a numerical example.

• Finally, it dealt with the boundary stabilization for the time fractional-order coupled reaction-diffusion equations with different diffusion coefficients by using the backstepping method. The backstepping stabilization for the scalar time fractionalorder reaction-diffusion equation was extended to the coupled cases.

Future works

Backstepping method has been proven to be efficient to design controllers and observers for PDEs. In the future, several more general PDEs should be considered:

• Coupled reaction-advection-diffusion equations with spatially-varying coefficients and mixed boundary conditions.

• Coupled parabolic PDEs and hyperbolic PDEs.

• Coupled time-fractional and spatial-fractional parabolic PDEs.

:

  EDP paraboliques, Système couplé, Conditions aux bords de type Dirichlet, Méthode de Backstepping, Contrôle aux bords, Observation aux bords, Command par retour de sortie, Stabilité de Lyapunov, Système d'ordre fractionnaire.

Contents 0 . 1 1 Introduction 1 . 1

 01111 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Background and Motivations . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Backstepping output feedback control for scalar reaction-diffusion equation with spatially varying coefficients . . . . . . . . . . . . . . . . . . . . . . 1.4 Objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Backstepping output feedback control for coupled parabolic PDEs with constant coefficients having the same diffusion coefficient 2.1 Backstepping control for coupled reaction-diffusion equations having the same diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient . . . . . . . . . . . . . . . . . . . . . . 2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Boundary control of coupled reaction-diffusion equations with different diffusion parameters 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Figure 1 . 1 :

 11 Figure 1.1: Applications of Partial Differential Equation

. 29 )

 29 where U(x) = [u 1 (x), u 2 (x), • • • , u n (x)] and W(x) = [w 1 (x), w 2 (x), • • • , w n (x)] are two ndimensional multivalued vectors.

  ) that transforms equations (1.51)-(1.53) into the following integral equation of G: G(ξ, η) = G 0 (ξ, η) + F[G](ξ, η), (1.57) where G 0 and F[G] are provided in [15]. By using the method of successive approximations, the following results can be obtained [15]: Theorem 1.1 [15] The equations (1.51)-(1.53) on kernel function k(x, y) have a unique solution for 0 ≤ y ≤ x ≤ 1 wih the following bound: | k(x, y) |≤ Me 2Mx , (1.58)

  .75)-(1.77) are conditions on kernel function p(x, y), and (1.78)-(1.79) are the observer gains. If equations (1.75)-(1.77) are well-posed, the observer gains can be obtained from (1.78)-(1.79). In order to solve (1.75)-(1.77), we define the following change of variables, x = 1y, (1.80) ȳ = 1x, (1.81) p( x, ȳ) = p(x, y), (1.82) by which (1.75)-(1.77) can be transferred into,

  83)-(1.85) with (1.51)-(1.53), we get the following theorem. Theorem 1.3 [15] There exists a unique solution of p(x, y) of (1.75)-(1.77) on 0 ≤ y ≤ x ≤ 1, and the backstepping transformation (1.71) is invertible with the following inverse transformation, w(x, t ) = ũ(x, t ) + x 0 r (x, y) ũ(y, t )d y, (1.86)

  Ce chapitre conçoit un observateur des EDP couplées de type réaction-advection-diffusion avec des coefficients de diffusions constants de même paramètre. Plus spécifiquement, il considère tout d'abord un système modélisé par des équations de réaction-diffusion couplées à coefficients constants et à diffusion identique. Pour ce cas, il propose une condition moins restrictive sur les paramètres des systèmes cibles pour concevoir un contrôleur basé observateur par rapport à celle proposée dans les travaux[START_REF] Baccoli | On the boundary control of coupled reaction-diffusion equations having the same diffusivity parameters[END_REF] et[START_REF] Baccoli | Boundary control of coupled reactiondiffusion processes with constant parameters[END_REF]. Deuxièmement, il conçoit un contrôleur basé observateur pour un système modélisé par des équations couplées de type réactionadvection-diffusion avec des coefficients constants et de même paramètre de diffusion, le terme d'advection étant pris en compte par rapport au premier cas. Il termine par un exemple numérique pour mettre évidence la consistance de la méthode proposée.

Figure 2 . 1 :

 21 Figure 2.1: Spatiotemporal evolution of u 1 (x, t ) and u 2 (x, t ) in the open loop

2. 1 .Figure 2 . 2 :Figure 2 . 4 :

 12224 Figure 2.2: Spatiotemporal evolution of estimated states û1 (x, t ) and û2 (x, t ) in the open loop

  .154) where 0 ≤ β ≤ α ≤ 2. Then, (2.149)-(2.151) are transformed into: -α) (Ψ + C)e -Λ 4θ (2y-α) d y, (2.155)

2 . 1 .Figure 2 . 8 :

 2128 Figure 2.6: Spatiotemporal evolution of u 1 (x, t ) and u 2 (x, t ) in the open-loop

  .7) Then, by subtracting (3.4)-(3.6) from (3.1)-(3.

3 . 2 .• Step 3 . 1 0Z( 1 ,

 32311 (3.15) and(3.22) in(3.28) and(3.29),(3.17) and(3.18) can be obtained as two equivalent conditions, respectively.• Step 2. Transformation to (3.13): By taking x = 0 in(3.11) and in(3.24), we obtain:U(0, t ) = W(0, t ), (3.32) U x (0, t ) = W x (0, t ) -Z(0, 0) W(0, t ). (3.33) Backstepping boundary observer Thus, (3.13) can be obtained using (3.9) and (3.32) by setting Q = Z(0, 0) = z(0, 0)I n×n in (3.33). Transformation to (3.14): By taking x = 1 in (3.11), we obtain: U(1, t ) = W(1, t )y) W(y, t )d y. (3.34) Thus, (3.14) can be obtained using (3.10) by setting z(1, y) = 0 in (3.34).

.47) 3 . 3 .

 33 Backstepping boundary controller On the one hand, using C = κΘ -Ψ, -S[Ψ] + θ mi n 4 I n×n + κΘ 0 can be written as S[C]+ θ mi n 4 I n×n 0. On the other hand, since S[C] is symmetric, it is diagonalizable i .e., there exists an inverse matrix R such that D = R -1 S[C]R, where D is a diagonal matrix and has the same eigenvalues to S[C]. Thus, D + θ mi n 4 I n×n = R -1 S[C] + θ mi n 4 I n×n R 0 is equivalent to S[C] + θ mi n 4 I n×n 0, and leads to θ mi n 4 +σ mi n (S[C]) > 0. Consequently, if the condition (3.39) holds, θ mi n 4 +σ mi n (S[C]) > 0 fulfills, W(•, t ) exponentially converges to 0. Thus, this theorem is proven. According to Theorem 5.1, if κ fulfills the condition (3.39), the target system converges to 0. Then, thanks to (3.11) the observer error in (3.8)-(3.10) also converges to 0. Thus, the system (3.4)-(3.6) is an observer for the system modelled by (3.1)-(3.3).

  Moreover, the following boundary conditions and open-loop actuations are considered:

3. 1 Figure 3 . 1 :

 131 Figure 3.1: Spatiotemporal evolution of v 1 (x, t ) and v 2 (x, t ) in the open-loop.

Figure 3 . 2 :Figure 3 . 3 :Figure 3 . 4 :

 323334 Figure 3.2: Spatiotemporal evolution of the estimated state variables v 1 (x, t ) and v 2 (x, t )

  Ce chapitre vise à concevoir un observateur pour une classe de processus de réaction-advection-diffusion couplés avec coefficients variant dans l'espace. Le système considéré est doté d'un capteur au bord gauche. Comparé aux systèmes considérés dans les chapitres précédents, ce système a des coefficients variants dans l'espace pour tous les termes de réaction, d'advection et de diffusion. Cela rend plus difficile la détermination de la fonction noyau de la transformation backstepping. Grâce au récent travail [75], une relation très importante a été établie entre le système parabolique qui définit le noyau et un système hyperbolique. Cela révèle de manière significative la connexion entre les contrôleurs pour les systèmes paraboliques et hyperboliques. Ce résultat permet d'obtenir le noyau de contrôle du système parabolique en transformant l'équation du noyau dans un ensemble d'équations hyperboliques couplées qui se sont avérées bien posées. En utilisant une méthode similaire, nous pouvons constater que cette relation est également utile pour le noyau de la conception d'observateur du système parabolique. Ensuite, en proposant des conditions aux limites artificielles, nous pouvons montrer que les équations du noyau d'observation pour le système parabolique est un problème bien posé. De plus, cela prouve également que le système cible de la transformation backstepping est H 1 stable en mettant certaines conditions sur les paramètres du système cible. Enfin, la méthode proposée est a été mise en évidence par un exemple numérique.Abstract: This chapter aims to design a backstepping boundary observer for a class of linear coupled reaction-advection-diffusion processes with spatially-varying 4.1. Introduction coefficients. The considered system is endowed with a sensor at the left boundary.

(4. 1 )

 1 with the Neumann-type left boundary condition and the Dirichlet-type right actuation:

Proposition 4 . 1

 41 If the kernel function H(x, y) and the observer gains P(x) and Q fulfill the following conditions:

  ) H(x, x)Θ(x) -Θ(x)H(x, x) = 0, (4.18) H(1, y) = L(y),(4.19)H y (x, 0)Θ(0) + H(x, 0)Θ (0) -H(x, 0)Λ(0) = P(x), (4.20) H(0, 0) = Q, (4.21)

Lemma 4 . 1

 41 The n ×n equation(4.16) is equivalent to the following 2n 2 ×2n 2 hyperbolic equations with the two states H(x, y) and Y(x, y):

Finally, by substituting ( 4 .

 4 [START_REF] Azouit | Well-posedness and convergence of the Lindblad master equa-tion for a quantum harmonic oscillator with multiphoton drive and damping[END_REF]) into (4.37), and using (4.31)-(4.34), (4.30) can be ob-

Remark 4 . 4

 44 Thanks to the exponential stability of the target system (4.12)-(4.14), the boundedness and invertibility of the backstepping transformation shown by Theorem 4.1 implies also the exponential stability of the observer error dynamics (4.8)-(4.10).

Figure 4 . 1 :

 41 Figure 4.1: Spatiotemporal evolution of u 1 (x, t ) and u 2 (x, t ) in the open-loop.

Figure 4 . 2 :Figure 4 . 3 :

 4243 Figure 4.2: Spatiotemporal evolution of û1 (x, t ) and û2 (x, t ) in the open-loop.

  Ce chapitre vise à explorer la stabilisation pour l'équation de réaction-diffusion couplées d'ordre fractionnaire avec différents coefficients de diffusion en utilisant la méthode du backstepping. Le système considéré est un système de réaction-diffusion couplés ayant différents coefficients, où la dérivée temporelle est d'ordre fractionnaire dans l' intervalle (0,1] de type Caputo. En fait, ce chapitre prolonge le travail [85] sur la stabilisation des EDP paraboliques scalaire d'ordre fractionnaire pour le cas couplé. Tout d'abord, un observateur est conçu avec un capteur au bord gauche. Ensuite, un contrôleur est conçu avec un actionneur au bord droit. Sur la base de l'observateur et du contrôleur conçus, une commande par retour de sortie basé sur l'observateur est mise en place. La méthode proposée est vérifiée en prenant un exemple numérique.Abstract: This chapter aims to explore the boundary stabilization for the time fractional-order coupled reaction-diffusion equations with different diffusion coefficients by using the backstepping method. The considered system is a class of coupled reaction-diffusion systems with different coefficients, where the time derivative is a Caputo time fractional of order α ∈ (0, 1]. Actually, this chapter extends the work[85]

C

  o D α t U(x, t ) = ΘU xx (x, t ) + ΨU(x, t ),(5.1) and the Neumann-type left boundary conditions and the Dirichlet-type right actuations:

1 0W

 1 x, t )B W(x, t )d x,(5.42) is a Mittag-Leffler function for the system (5.12)-(5.14) in L 2 , where B ∈ R n×n is a positive diagonal matrix. For this purpose, the fractional time derivative of V(t ) is calculated t W T (x, t )B W(x, t )d x.(5.43)Then, because of Lemma 5.1 and substitute (5.12) into (5.43), we can obtain:x, t )BΘ W xx (x, t )d x -T (x, t )BC W(x, t )d x (5.44)Then, by applying integration by parts and using the boundary conditions (5.13)-(5.14), we get:1 0 W T (x, t )BΘ W xx (x, t )d x = -1 0 W T x (x, t )BΘ W x (x, t )d x.

( 5 .mi n 2 +

 52 14) meanwhile considering the trial inequality, we get: xx (x, t )BC W(x, t ) ≤ -σ mi n (S[C]) t )B W x (x, t )d x, (5.60) Thanks to the Poincaré Inequality and the boundary condition (5.14) meanwhile consider the trial inequality, we deduce: (x, t )BΘ W xx (x, t ) ≥ t )B W x (x, t ), (5.62) Thus, with (5.57) and (5.62), we obtain: C 0 D α t J(t ) ≤ -( θ 2σ mi n (S[C]))J(t ). (5.63) In similar ways done in (5.48)-(5.54), (5.41) can be obtained. The proof is finished.

Figure 5 . 1 :Remark 5 . 2

 5152 Figure 5.1: The features of the Mittag-Leffler function with different fractional order α.

x 0 K 5 . 3 .

 053 (x, y)U(y, t )d y,(5.64) which can transform the studied system into the following target system:C 0 D α t W(x, t ) = ΘW xx (x, t ) -CW(x, t ), (5.65) W x (0, t ) = 0, (5.66) W(1, t ) = 0,(5.67)where• W(x, t ) = (w 1 (x, t ), • • • , w n (x, t )) ∈ [L 2 (0,1)] n , Backstepping controller design

x 0 LTheorem 5 . 2 mi n 2 +

 0522 (x, y)W(y, t )d y,(5.71) where the kernel matrix L(x, y) is the unique inverse of K(x, y).The stability features of the target system (5.65)-(5.67) is established in the following theorem. If the parameter κ defined in(5.70) is chosen such that the following inthe target system (5.65)-(5.67) satisfies the following L 2 Mittag-Leffler stability with the following convergence rate:∥ W(•, t ) ∥ 2 2,n ≤ ∥ W(•, 0) ∥ 2 2,n ×E α -( θ 2σ mi n (S[C]))t α . (5.73)Moreover, the target system (5.65)-(5.67) also satisfies the following H 1 Mittag-Leffler stability with the following convergence rate:∥ W(•, t ) ∥ 2 H 1 ≤∥ W(•, 0) ∥ 2 H 1 ×E α -( θ mi n 2 + 2σ mi n (κΘ -S[Ψ]))t α .(5.74)

5. 4 .Figure 5 . 2 :

 452 Figure 5.2: Spatiotemporal evolution of u 1 (x, t ) and u 2 (x, t ) in the open-loop

Figure 5 . 3 :

 53 Figure 5.3: Spatiotemporal evolution of the state variables û1 (x, t ) and û2 (x, t ) obtained by the observer.

Figure 5 . 4 :Figure 5 . 5 :

 5455 Figure 5.4: Spatiotemporal evolution of the state variables u 1 (x, t ) and u 2 (x, t ) obtained in the closed-loop.

well-posedness of kernel equations (1.51)-(1.53).

	1.3. Backstepping output feedback control for scalar reaction-diffusion equation with
	spatially varying coefficients
	In order to realize the controllers (1.47) and (1.48), we need to determine the corres-
	ponding kernel functions (1.49) and (1.50). For this, we differentiate (1.43) with respect
	to time variable t and space variable x, then substitute the obtained equations into
	(1.44)-(1.45), and compare them with (1.39)-(1.41). This leads to a set of kernel equa-
	tions:
						x
	ak xx (x, y) -ak y y (x, y) = (λ(y) + c)k(x, y) -f (x, y) +	k(x, τ) f (τ, y)d τ,	(1.51)
						y
					x
	ak y (x, 0) = aqk(x, 0) + g (x) -	k(x, y)g (y)d y,	(1.52)
					0
	k(x, x) = -	1 2a	0	x	(λ(y) + c)d y,		(1.53)
	where 0 ≤ y ≤ x ≤ 1.
	Now a natural question is the
							y)u(y, t )d y,	(1.48)
	for the Neumann actuation where,
					k 1 (y) = k(1, y),	(1.49)
					k 2 (y) = k x (1, y).	(1.50)

  Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficientwhere Problem 1: the kernel function H(x, y) should be determined on 0 ≤ y ≤ x ≤ 1 such that the observation error system (2.79)-(2.81) can be transformed into the fol-

	lowing target system:
	Wt (x, t ) = θ Wxx (x, t ) + Λ Wx (x, t ) -C W(x, t ),	(2.83)
	Wx (0, t ) = 0,	(2.84)
	W(1, t ) = 0,	(2.85)

y) W(y, t )d y, (2.82) 2.2.

  .105)

	Remark 2.7 If the parameter Λ is diagonal as done in [88], the solution of H(x, x) is
	similar to the one of K(x, x) given by Eqs. (28)-(29) in [88].
	Then, it is shown in the following theorem that the conditions given in (2.87)-(2.89)

can also be realizable, i .e. these conditions admit a solution H(x, y).

Theorem 2.6

The conditions given in (2.87)-(2.89) admit a solution H(x, y) for 0

  .118) 2.2. Backstepping control for coupled reaction-advection-diffusion equations having the same diffusion coefficient Then, by integrating (2.118) from η to ξ and using (2.116), we get the following integral equaiton:

  this subsection, the kernel function Z(x, y) in the backstepping transformation(2.144)

is determined. By applying similar techniques as done for the observer design in last Subsection, the following proposition and lemma can be obtained, where the proofs are skipped here for brevity.

Proposition 2.2 The transformation (2.144) can transform the system (2.67)-(2.69) into the target system (2.145)-(2.147), if the kernel matrix function Z(x, y) fulfills the following conditions:

  Theorem 5.1 If the parameter κ is chosen such that the following matrix inequality

	holds:
	-S[Ψ] +	θ mi n 4	I n×n + κΘ 0,	(5.38)
	the target system (5.12)-(5.14) satisfies the following L 2 Mittag-Leffler stability with the
	following convergence rate::
	∥ W(•, t ) ∥ 2 2,n ≤∥ W(•, 0) ∥ 2 2,n ×E α -(	θ mi n 2	+ 2σ mi n (κΘ -S[Ψ]))t α .	(5.39)
	where θ mi n is the smallest diagonal element of Θ and
	E α (t ) :=	∞ t =0	t i Γ(αi + 1)	, α > 0	(5.40)

is the well-known Mittag-Leffler function in one parameter.

Moreover, the target system (5.12)-(5.14) also satisfies the following H 1 Mittag-Leffler stability with the following convergence rate:

∥ W(•, t ) ∥ 2 H 1 ≤∥ W(•, 0) ∥ 2 H 1 ×E α -( θ mi n 2 + 2σ mi n (κΘ -S[Ψ]))t α .

(5.41)

  Leffler function for the system (5.12)-(5.14) in H 1 . For this purpose, the fractional time derivative of V(t ) is calculated as follows:

					2	0	1	W T x (x, t )B W x (x, t )d x,	(5.55)
	is a Mittag-C 0 D α t J(t ) =	1 2	0	1	C 0 D α t W T x (x, t )B W x (x, t )d x,	(5.56)
	Then, because of Lemma 5.1, we obtain:
				C 0 D α t J(t ) ≤	0	1	W T x (x, t )B C 0 D α t W x (x, t )d x,	(5.57)
	By multiplying W T xx (x, t )B and integrating for both sides of (5.12), we obtain:
	0	1	W T xx (x, t )B C 0 D α t W(x, t )
	=		1	W T
			0

xx (x, t )BΘ W xx (x, t ) -1 0 W T xx (x, t )BC W(x, t )d x, (

5.4. Backstepping observer-based output feedback controller design

Acknowledgements

The PhD work presented in this thesis was performed by the author under the supervision of ProfThe author wishes to acknowledge the financial support for this thesis by the China Scholarship Council (CSC).

Backstepping boundary controller

Theorem 3. [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] If the parameter κ defined in (3.57) is chosen such that the following condition holds: Theorem 3.3 can be proven similarly to Theorem 1, thereby it is skipped here for brevity. However, it can be remarked that different from Theorem 3.2, the condition (3.59) depends on θ mi n . Hence, C can be negative definite. This is in fact due to the application of the Poincaré inequality in the proof.

Then, based on Theorem 3.3, the following result is obtained.

Proposition 3.2 If the parameter κ fulfills the following matrix inequality:

-S[Ψ] + θ mi n 4 I n×n + κΘ 0, (3.61) the controller defined by:

κ(1y 2 ) U(y, t )d y (3.62)

stabilizes the system modelled by (3.1)- (3.3).

Proof. On the one hand, as done in the proof of Theorem 3.1 it can be shown that the condition (3.59) is equivalent to the condition S[ C] + θ mi n 4 I n×n 0. Since C = κΘ -Ψ, the latter condition is equivalent to the condition (3.61). On the other hand, by setting x = 1 in (3.51), the following backstepping controller is obtained:

where K(1, y) can be given by κ using (3.56). Consequently, according to Theorem 3. Consequently, in order to realize the controller defined by (3.62), the parameter κ needs to be chosen such that (3.61) holds. This can be similarly achieved by two ways as done in Algorithm 1 and Algorithm 2.

Simulation

In the simulation part, we consider the Dirichlet-type right boundary condition case to verify the proposed methods. For this, we consider the following system:

)

)

)

)

In order to simulate the considered system, the finite-difference scheme is firstly used to discrete the system. Then, we adopt ODE45 scheme with the initial conditions

In this example, the states of the considered system (4.74)-(4.79) will be estimated by applying the observer designed in Section 2.2.1. For this purpose, the parameters L and C need to be set firstly, by which we can obtain a set of coupled hyperbolic PDEs for the kernel functions. Here we set L(y) = [0, 0; 0, 0] and C = [15, 0; 0, 20]. As done in [START_REF] Hu | Control of homodirectional linear coupled hyperbolic PDEs[END_REF], we transform the obtained coupled hyperbolic PDEs into a set of integral equations. Then, we adopt successive approximations to calculate the kernel functions.

Although it is not available to obtain the kernel functions in explicit forms, numerical computation is always possible. With the obtained kernel functions, the observer can be realized.

Backstepping observer design

This section aims to design a backstepping observer for a system modelled by (5.1)-( 5.3), the following system is considered:

(5.4)

where the n × n matrices P(x) and Q are parameters, which are designed such that the system (5.4)-(5.6) is an observer for the plant system.

Let us denote the observation error between the real states U(x, t ) and the estimated states Û(x, t ) by:

Then, by subtracting (5.4)-(5.6) from (5.1)-( 5.3), a straightforward calculation leads to:

Ũ(1, t ) = 0.

(5.10)

In order to obtain the observer gains which make the observation error converge to 0 when the time tends to infinity, the following backstepping transformation is considered:

which can transform the system (5.8)-(5.10) into the following target system:

(5.12)

It is assumed that Z(x, y) is in the following form: Z(x, y) = z(x, y)I n×n .. (5.15) Remark 5.1 It is worth to know that the transformation (5.11) is an one-to-one map, and its inverse is given by:

where the kernel matrix R(x, y) is the unique inverse of Z(x, y).

Backstepping observer design

In order to realize the above transformation, the conditions on the kernel function z(x, y) and observer gains P(x) and Q are given in the following proposition.

Proposition 5.1 If the kernel function Z(x, y) fulfills the following conditions: z xx (x, y)z y y (x, y) = κz(x, y), (5.17)

)

, κ is a constant parameter defined by the following algebraic equation:

the transformation (5.11) can transform the observation error system (5.8)-( 5.10) into the target system (5.12)- (5.14).

Proof. In this proof, it will be shown how to transform the observation error system (5.8)-(5.10) into the target system (5.12)- (5.14), where the conditions on z(x, y) are imposed. Hence, the following three steps are needed.

• Step 1. Transformation to (5.12): By differentiating both sides of (5.13) with respect to x and t , and applying the Leibniz's differential rule, we obtain:

(5.23)

(5.25)

Then, by substituting (5.23)-(5.25) into (5.8), (5.12) can be obtained, if the following conditions hold:

Z y (x, 0)Θ = P(x).

(5.29)

Backstepping observer design

Consequently, using (5.15) and (5.22) in (5.26) and (5.27), (5.17) and (5.18) can be obtained as two equivalent conditions, respectively.

• Step 2. Transformation to (5.82): By taking x = 0 in (5.11) and in (5.23), we obtain: U(0, t ) = W(0, t ), (5.30) U x (0, t ) = W x (0, t ) -Z(0, 0) W(0, t ).

(5.31) Thus, (5.82) can be obtained using (5.9) and (5.30) by setting Q = Z(0, 0) = z(0, 0)I n×n in (5.31).

• Step 3. Transformation to (5.14): By taking x = 1 in (5.11), we obtain:

(5.32) Thus, (5.14) can be obtained using (5.10) by setting z(1, y) = 0 in (5.32). This proof is finished.

Based on (5.22), we have C = κΘ -Ψ, which can be described as follows:

-ψ i j , else.

(

Consequently, since matrices Ψ and Θ are known, C can be determined by choosing the constant parameter κ.

The explicit expressions of the kernel function z(x, y), the observer gains P(x) and Q in Proposition 5.1 have been given by [START_REF] Liu | Backstepping observer-based output feedback control for a class of coupled parabolic PDEs with different diffusions[END_REF] such as:

(5.34)

where I 1 (•) and I 2 (•) are the well-known modified Bessel functions of the first kind.

Consequently, it only needs to set the parameter κ to obtain the kernel function z(x, y), the observer gains P(x) and Q. The way to set κ will be presented in the following Theorem. Before giving this theorem, an useful lemma is introduced.

Lemma 5.1 [START_REF] Aguila | Lyapunov functions for fractional order systems[END_REF] Let x(t ) ∈ R n be a vector of differentiable functions. Then, for any time instant t > t 0 , the following relationship holds:

where ∀α ∈ (0, 1] and M ∈ R n×n is a constant, square, symmetric and positive definite matrix.

Backstepping observer design

Thanks to the Poincaré Inequality and the zero boundary conditions, we deduce:

(5.46) Then, by using (5.45) and (5.46), and applying the trivial inequality, we get:

(5.47)

Then, we set:

It has been shown in [START_REF] Liu | Backstepping observer-based output feedback control for a class of coupled parabolic PDEs with different diffusions[END_REF] that if (5.38) holds,

) > 0 fulfills. Thus, M(t ) is obviously positive. Assume that the Laplace transform W T (x, t )B W(x, t ) exists, then the Laplace transform of C 0 D α t V(t ) and M(t ) exist. Then, by taking the Laplace transform on both sides of (5.48), we obtain: are the Laplace transform of M(t ) and V(t ), respectively. Then, we can obtain:

Based on the uniqueness and existence theorem given by [85] and the inverse Laplace transform, the unique solution of (5.48) is:

where t > 0 and * denotes the convolution operator. Because E α,α (-

) and t α-1 are non-negative, we can obtain:

which implies (5.39).

Proposition 5.2 If the parameter κ fulfills the condition given by (5.72), the controller defined by:

κ(1y 2 ) U(y, t )d y (5.75)

stabilizes the system modelled by (5.1)- (5.3).

Proof. By setting x = 1 in (5.64), the following backstepping controller is obtained:

where K(1, y) can be given by κ using (5.69). Consequently, according to Theorem 5.2, if the parameter κ fulfills the condition (5.72), the Mittag-Leffler stability of target system (5.65)-(5.67) is guaranteed. This leads to that the controller U c (t ) realized by such κ stabilizes the system modelled by (5.1)-( 5.3).

The parameter κ needs to be chosen such that (5.72) holds. This can be similarly achieved by two ways as done in Algorithm 1 and Algorithm 2 in Section 3.

Backstepping observer-based output feedback controller design

A backstepping observer and a backstepping controller for the studied system have been designed in the above two Subsections, which are independent and satisfy the separation principle. This Subsection combines the above observer and controller to realize a backstepping observer-based output feedback controller.

Theorem 5.3 if the parameter matrices C and C are chosen such that:

the following controller:

(5.79)

Û(1, t ) = where the observer gains P(x) and Q given by (5.35) and (5.36), respectively.

Proof. Let us consider the following invertible backstepping transformation:

where K(x, y) is given in (5.68). Then, the system (5.79)-(5.81) can be transformed into the following target system:

Ŵ(1, t ) = 0, (5.85)

The W(x, t ) system (5.12)-(5.14) is proven to be stable with the condition (5.34) on C.

Then, it is obvious that the system (5.83)-(5.85) including the homogeneous part as (5.12)-(5.14) and W(0, t ) is stable. It means that the cascade system ( Ŵ, W) is stable. Therefore, the system ( Û, Ũ) is also exponentially stable because it is related to ( Ŵ, W) by the invertible coordinate transformation (5.16) and (5.71). Thus, the system (U, Û) is stable.

Remark 5. [START_REF] Zuazua | Controllability and observability of partial differential equations: some results and open problems[END_REF] The condition (5.77) guarantees that the observer converges faster than the controller for the stability of the observer-based output feedback control system (5.79)-(5.81).

Simulation results

In order to verify the obtained theory results, a numerical example is considered as follows:

C o D 0.9 t u 1 (x, t ) = 0.2u 1xx (x, t ) -0.1u 1 (x, t ) -0.12u 2 (x, t ), (

C o D 0.9 t u 2 (x, t ) = 0.5u 2xx (x, t ) -0.15u 1 (x, t ) -0.3u 2 (x, t ), (5.87) u 1x (0, t ) = 0, (5.88) u 2x (0, t ) = 0, (5.89) In order to simulate the studied system, the finite-difference scheme is firstly used to discrete the system. Then, the ODE45 scheme is adopted with the initial conditions u 1 (x, 0) = u 2 (x, 0) = sin(πx)0 + sin(3πx).

In this example, the states of the system (5.86)-(5.91) will be stabilized by the observerbased output feedback controller designed in chapter. The spatiotemporal evolution