Hugo Lhachemi
email: hugo.lhachemi@centralesupelec.fr

Christophe Prieur
email: christophe.prieur@gipsa-lab.fr

Saturated boundary stabilization of partial differential equations using control-Lyapunov functions

Saturated boundary stabilization of partial differential equations

3

This chapter reviews some recent results on the boundary stabilization of different classes of partial differential equations. In order to provide a self-content chapter with consistent control objectives and notation, we first review the finite-dimensional case. Controllability and observability conditions for linear ordinary differential equations are recalled together with some basic Lyapunov theory for the stability analysis and the design of saturated controllers. Then we address the boundary control problem for the stabilization of a reaction-diffusion equation by means of numerically tractable design methods while considering different norms and possible constraints on the amplitude of the inputs. Finally similar control design problems will be studied for the stabilization of the Korteweg-de-Vries equation and the wave equation.

Introduction

The goal of this chapter is to review some recent results on boundary stabilization of distributed parameter systems as those modeled by parabolic partial differential equations or hyperbolic partial differential equations. No prerequisite on control theory will be necessary, only basic knowledge on control objectives. However, background in nonlinear dynamical systems and essentials on partial differential equations (PDEs) would be helpful, even if some references will be given throughout the text.

The topics covered in this chapter embrace different potential applications such as control and stability theory of reaction-diffusion phenomenon as those modeled by parabolic PDEs. Some control techniques presented in this chapter will be useful for stability theory of physical dynamics described by balance laws and modeled by hyperbolic partial differential equation. Different control objectives will be studied and solved such as the design of stabilizing control laws ensuring that all the trajectories of the closedloop systems converge to a given equilibrium. Different control schemes are considered, covering in-domain control (the control input appears directly in the main part of the PDE) and boundary control (the control input applies at the boundary of the domain as it appears through the boundary conditions). Moreover, when possible, the described control laws will be designed based on the only knowledge of a prescribed and limited part of the state, the so-called output.

For each of the different numerical illustrations reported in this chapter, the Python code of the numerical simulations is provided, allowing the readers to easily modify the control objectives and further experience the control theory of the considered dynamical systems.

The outline of this chapter is as follows. First finite-dimensional control systems will be considered and some basic definitions will be given on stability, attractivity, etc., providing a sharp introduction to basics of control systems theory. Then in Section 3, parabolic PDEs are considered for the design of finite-dimensional output-feedback controllers towards saturated control schemes. Section 4 is devoted to the wave and Korteweg-de-Vries equation, and the use of finite-dimensional controllers to solve the stabilization problems. In these both sections, linear feedback laws and also cone-bounded controllers are designed. Section 5 contains a concluding discussion on current research activities and presents some possible research directions emanating from this chapter.

This chapter has been written following an online course given in LIASFMA school by the second author in April 2021. We would like to thank the Organizing Committee of this school that was composed of Jean-Michel Coron (Sorbonne Université), Tatsien Li (Fudan University), and Zhiqiang Wang (Fudan University). The help of Xinyue Feng has been very much appreciated.

Notation used in this chapter

Spaces R n are endowed with the Euclidean norm denoted by • . The associated induced norms of matrices are also denoted by • . Given two vectors X and Y , col(X, Y ) denotes the vector [X , Y ] . L 2 (0, 1) stands for the space of square integrable functions on (0, 1) and is endowed with the inner product f, g = 1 0 f (x)g(x) dx with associated norm denoted by • L 2 . For an integer m ≥ 1, the m-order Sobolev space is denoted by H m (0, 1) and is endowed with its usual norm denoted by • H m . For a symmetric matrix P ∈ R n×n , P 0 (resp. P 0) means that P is positive semi-definite (resp. positive definite) while λ M (P ) (resp. λ m (P )) denotes its maximal (resp. minimal) eigenvalue. For a symmetric matrix, stands for the symmetric term. For instance, A B C stands for A B B C .

For any Hilbert basis {φ n , n ≥ 1} of L 2 (0, 1) and any integers 1 ≤ N < M , we define the operators of projection π N : L 2 (0, 1) → R N and π N,M : L 2 (0, 1) → R M -N by setting π N f = f, φ 1 . . . f, φ N and π N,M f = f, φ N +1 . . . f, φ M . We also define R N : L 2 (0, 1) → L 2 (0, 1) by R N f = f -2. Finite-dimensional systems 2.1. Stability notions of nonlinear finite-dimensional systems

This section is devoted to the introduction of control theory for finitedimensional systems, as those described by nonlinear dynamics. To be more specific, let us consider the following dynamical system:

ż(t) = f (z(t)) (1)
where the state z(t) is a vector from a finite-dimensional state-space R n and f is a nonlinear function from R n to R n . Under suitable regularity assumptions, such as locally Lipchitz continuity of f with respect to z, for any given initial condition z 0 ∈ R n there exists a unique solution x : [0, T ) → R n to the Cauchy problem:

ż(t) = f (z(t)), t > 0 z(0) = z 0 (2)
defined on a maximal interval of existence [0, T ) for some T > 0 (which depends on z 0 ). See e.g. [30, Theorem 3.1] for such a existence and uniqueness result. The value z 0 is called the initial condition and, at any time t ∈ [0, T ), the value z(t) is called the state at time t. Assume further that f (0) = 0. This implies that the constant trajectory z(t) = 0, for all t ≥ 0, is a particular solution to (1) associated with the initial condition z 0 = 0. The point 0 ∈ R n , sometimes referred to as the origin, is called an equilibrium for (1). In control theory, the nature of an equilibrium is characterized by certain "stability" properties. Some basic definitions related to the concept of "stability" are introduced in the following definition. Definition 1. Assume that f (0) = 0. Then the equilibrium 0 of (1) is said to be • stable if for any ε > 0, there exists δ > 0 such that

|z(0)| ≤ δ ⇒ |z(t)| ≤ ε , ∀t ≥ 0 .
• attractive if there exists δ > 0 such that |z(0)| ≤ δ ⇒ z(t) → t→+∞ 0 .

• asymptotically stable if it is both stable and attractive.

In the previous definition, it is implicitly required that the solutions exist, are unique, and are well defined for all t ≥ 0. Even implicit, these requirements are of primary importance. Some of them can be difficult to check in practice depending upon the nature of the studied system.

Assuming that 0 is an attractive equilibrium of (1), an important concept is the notion of basin of attraction. This is defined as the set of all initial conditions z 0 ∈ R n such that the solution to (2) tends to 0 as t → ∞.

In addition, we say that the equilibrium is globally attractive if it is attractive and the basin of attraction coincides with the whole state-space R n . When 0 is not globally attractive, we often write that 0 is locally asymptotically stable (LAS) to emphasize the "local" nature of the property. Finally, we say that 0 is globally asymptotically stable (GAS) if it is asymptotically stable and globally attractive. It is worth being noted that the notions of attractivity and stability are disconnected. More specifically, there exist systems for which 0 is stable but not attractive (the most simple example being ż = 0) while there are also systems such that 0 is attractive but not stable (see for instance the example of [24, Paragraph 40]).

Instead of (1), let us now consider the case where the dynamics depends on an external signal, called the control or the input. More specifically, consider the dynamics described by ż(t) = f (z(t), u(t))

where u(t) is a vector of R m . The input u is seen as a way to influence the dynamics of the system, which can significantly vary depending on the choice of the control. As an example, consider the following control system described by

ż(t) = u(t)z(t) (4)
with u(t) ∈ R. If u(t) = u ∈ R is constant control, the trajectories of the system stating at time t = 0 from the initial condition z 0 ∈ R n can be expressed as z(t) = e ut z 0 for all t ≥ 0. For u = -1 (more generally for any constant control u < 0), the equilibrium 0 is globally asymptotically stable. For u = 0, any point of R n is an equilibrium (they are stable but not attractive). For u = 1 (more generally for any constant control u > 0), all solutions to (4) with non zero initial condition z 0 = 0 diverges to infinity (the equilibrium 0 is neither stable nor attractive).

In the more general setting of a time-varying control, i.e., u = u(t) for a suitable function u of the time, (3) is a time-varying system. This implies that the solution starting from an initial condition z 0 at time t 0 differs, in general, from the trajectory starting from the same initial condition z 0 but at a different time t 1 = t 0 . The behavior of these different solutions can be very different.

Assuming that f in (1) is linear, the system dynamics reduces to

ż(t) = Az(t) (5)
where A is a matrix in R n×n . In this case, the stability of the origin is intimately related to the position of the eigenvalues of the matrix A in the complex plane (see [26, theorem 6.1]). More specifically, it can be proven that the origin of ( 5) is stable if and only if (i) all eigenvalues of A have a non-positive real part and (ii) for all eigenvalues with a zero real part, their algebraic multiplicity (exponent associated with the eigenvalue when computing the characteristic polynomial) coincides with their geometric multiplicity a (dimension of the eigenspace associated with the eigenvalue). Moreover, the origin of ( 5) is asymptotically stable if and only if all eigenvalues of A have a negative real part. In that case we say that the matrix A is Hurwitz. Finally, for such linear systems, the attractivity of the origin of (5) implies that the origin is stable and also asymptotically stable.

In this lecture notes, we will first study finite-dimensional control systems, and then dynamical control systems described by linear partial differential equations (PDEs) for which some nonlinear control problems will be solved.

Control systems: a basic tour

We focus in the the first part of this section on systems described by ż = Az + Bu (6) where z ∈ R n is the state, u ∈ R m is the control, A, B are two matrices of appropriate dimensions. One natural question is the design of a so-called stabilizing state feedback law. That is, can we compute state-feedback law z → u(z) so that the resulting closed-loop system

ż = Az + Bu(z) (7)
is asymptotically stable? In this context, due to the linearity of the system, it is natural to try to determine a state-feedback law z → u(z) that is also linear, i.e., which tales the form u = Kz where K is a matrix that is referred to as the feedback gain. In this setting, the closed-loop system dynamics reads

ż = (A + BK)z (8)
Consequently the stability properties of the closed-loop system are fully characterized by the spectrum of the closed-loop matrix A + BK. The question is: can we compute a matrix K in order to impose the spectrum of A + BK to ensure stability properties for the closed-loop system? For linear finite-dimensional systems, the control theory is complete and the design of stabilizing state feeback laws is fully solved. [START_REF] Panos | Linear systems[END_REF][START_REF] Hespanha | Linear systems theory[END_REF] More specifically, assuming the following Kalman rank condition (for controllability) rank B, AB, . . . , A n-1 B = n , there exists a matrix K so that u(z) = Kz makes the system (7) asymptotically stable. Furthermore, the matrix gain K can be selected to impose any arbitrary spectrum assignment for the closed-loop matrix A + BK. This result is not only an existence result, but it is also a practical design method. Indeed, it is the base of efficient numerical algorithms to compute the control matrix K. This is the so-called pole-shifting theorem (see 73 for an existence result and 71 for a constructive algorithm), which is stated in the next result.

Theorem 1. Under the Kalman rank condition assumption, for any polynomial Π of degree n and with unit dominant coefficient, there exists a matrix K such that the characteristic polynomial of A + BK is Π.

With the previous result, computing a matrix K so that the linear state feedback law z → Kz renders the origin of the closed-loop system (8) asymptotically stable is numerically tractable.

Example 1. Let us see how to solve this control problem in practice using the programming language Python. In the next lines, with dimension n = 3, first a randomly chosen control system is selected (lines 7-8), the controllability condition is checked and a pole-placement controller is computed using the Python Control Systems Library (lines 10-18). Then the differential equation is integrated numerically and the phase-portrait of the solution is plotted (lines 27-33). This givea Figure 1 where it can be checked that a solution converges to the equilibrium 0 in R 3 . We repeat the same procedure for 10 randomly chosen initial conditions. See the lines 35-41 of the code and the corresponding Figure 1.

Lyapunov direct method

The first part of this section was devoted to linear systems for which the situation is relatively simple as the stability of the origin is fully characterized by the spectrum of the matrix A. When considering general nonlinear systems such as (1), the situations becomes much more complex. Here we need tools that allow studying the stability properties of an equilibrium condition without being able to write down the system trajectories in closed form (in general, very few nonlinear systems can be analytically integrated to obtain the closed form of the trajectories). In this context, an important tool to prove the attractivity of the equilibrium is the so-called Direct Lyapunov method which relies on the concept of Lyapunov functions. To explain this method, let us come back to the nonlinear system described by (1). The so-called Lyapunov stability theorem can be stated as follows (see [30, Theorem 4.1] for a proof.)

Theorem 2. Assume that f (0) = 0 and let D be an open and connected subset of R n containing 0. Assume that V : Then z = 0 is stable. Moreover, if we have

D → R is a C 1 function such that V (0) = 0 and V (z) > 0 , ∀z ∈ D \ {0} ∂V ∂z (z) • f (z) ≤ 0 , ∀z ∈ D .
∂V ∂z (z) • f (z) < 0 , ∀z ∈ D \ {0} ,
then z = 0 is locally asymptotically stable.

We often denote

V = ∂V ∂z (z) • f (z) since ∂V ∂z (z(t)) • f (z(t)
) is the time-derivative of V (z(t)) along the solutions to (1).

If the Lyapunov theorem applies with the domain D specified as

D = {z, V (z) < r}
for some given r > 0, then the level set {z, V (z) < r} is contained in the basin of attraction. Hence V can be used in order to estimate the basin of attraction while trying to maximize the value of r > 0 such that Theorem 2 applies with D = {z, V (z) < r}.

It is worth noting that, for finite-dimensional systems as the ones that are considered in this section, all norms are equivalent and, somehow, V is "equivalent" to any norm (say, e.g., the Euclidian norm). Thus establishing a stability by considering a particular norm is actually the same as establishing a stability by considering any other norm. Such an equivalence fails in infinite-dimension, which will be the topic of the next sections.

As we saw, Lyapunov functions are very convenient to prove asymptotic stability since all we need is to consider a suitable Lyapunov function candidate

V : D → R, that is a C 1 function such that V (0) = 0 and V (z) > 0 , ∀z ∈ D \ {0}
and then compute the following vectors in R n :

∂V ∂z (z) • f (z) , ∀z ∈ D \ {0} .
and evaluate its sign. Obviously, finding such functions V highly depends on the nature of the studied nonlinear system and can be very complex in practice. Some basic techniques for finding such functions will be reviewed in this notes, as well as associated numerically tractable methods.

In the context of linear systems as described by (5), the Lyapunov theorem is rewritten as follows. Using the Lyapunov function candidate V (z) = z P z for some symmetric positive definite matrix P , and computing its time derivative along the system trajectories, the origin of ( 5) is asymptotically stable if and only if there exists such a symmetric positive definite matrix P such that

A P + P A -I
Let us emphasize the "if and only if" condition from the previous statement, as well as the class of quadratic function V (z) = z P z as sufficient Lyapunov function candidates. In other words, for linear systems, there is not need to consider other class of Lyapunov function candidates. This result is one of so-called converse Lyapunov theorems. Such converse results of the direct method also exist for nonlinear systems under certain regularity conditions on the function f (see e.g. [2, Theorem 2.4]). Note however that converse Lyapunov theorems can hardly be applied to actually find Lyapunov function candidates since these converse results are generally not constructive (even if some design methods exist as reviewed in particular in the references [START_REF] Coron | Control and Nonlinearity[END_REF][START_REF] Sepulchre | Constructive nonlinear control. Communications and Control Engineering Series[END_REF] ). print ( ' Largest real part for the closed -loop system : ' ," {:.2 f } " . format ( m ) ) P = control . lyap ( AA .T , np . eye ( n ) )

Separation principle for linear systems

Up to now we only considered the control problem of dynamical systems such as the ones described by (3). In this context, we made the implicit assumption that the full state z(t) is known in real time at any time t ≥ 0 so that we can use this information to implement the control law z → u(z). We say that this control strategy takes the form of a state-feedback. However in many applications the full state is not available in real-time. Only partial information are available under the form of sensor measurements y(t) ∈ R p which are somehow related to the state x(t) ∈ R n of the system. For control linear system described by (6), the relation between the output y and the state x generally takes the form:

y = Cz (9)
where C is a matrix of appropriate dimensions. We say that y is the output of the system. This output represents the measurements that are assumed to be available at each time instant. In this context, a natural question is whether the knowledge of the system (i.e., the matrices A, B, and C), of the control u(t), and of the measurements y(t), is sufficient to asymptotically estimate the state z(t). This problem is a so-called observation problem.

For linear systems, this problem is also fully solved and is strongly connected to the so-called Kalman rank condition for observability, which is written as

rank      C . . . CA CA n-1      = n.
Note that this assumption is equivalent to the controllability of the pair (A , C ). This is why observability and controllability properties are said to be dual properties.

Consider now the dynamics described by ż

= Aẑ + Bu + L(C ẑ -y) ( 10
)
where L is a matrix with suitable dimensions. We say that ( 10) is an observer for (6). The observer mimics the dynamics of the system (6) while adding an extra term used to correct the dynamics of the observation in function of the error between the actual measurement y(t) and its estimation ŷ(t) = C x(t) obtained from the observer. Introducing the error of observation e = z -ẑ, this error satisfies the dynamics described by

ė = (A + LC)e. (11)
Under the abovementionned observability assumption, there exists a matrix L so that A + LC is Hurwitz. Selecting this way the observer gain L, the origin of ( 11) is asymptotically stable, meaning that the observation error e(t) = z(t)-ẑ(t) asymptotically converges to zero. In other words, the state of the observer ẑ(t) "asymptotically observes" the actual (unmeasured) stated of the system z(t). We say that ( 10) is an observer for (6). So far, we detailed (i) how an state feedback u = Kz can be designed to stabilize the linear system ( 6) and (ii) how an observer of the form (10) can be designed in order to compute ẑ an estimate of the state z of the system (6) from its outputs y given by (9). A natural question is whether we can reunite these two approaches to obtain a stabilizing output feedback. In other words, under the controllability assumption of (A, B) and the observability assumption of (A, C), can we separately design a feedback gain K and an observer gain L so that the origin of the system (6) in closed-loop with u = K ẑ where the dynamics of ẑ is given by ( 10) is asymptotically stable? The answer to this question is positive and is referred to as the separatation principle for linear finite-dimensional systems.

Theorem 3. Let us consider the dynamics:

ż = Az + Bu y = Cz (12)
where z ∈ R n , y ∈ R p and A, B, C are matrices with suitable dimensions. Assume that the pair (A, B) is controllable and the pair (A, C) is observable. Then for any matrices K and L such that A+BK and A+LC are Hurwitz, the equilibrium (0, 0) of

ż = Az + BK ẑ ż = (A + BK)ẑ + L(C ẑ -y) (13)
is asymptotically stable.

This theorem provides a design method for a stabilizing dynamic output feedback controller whose architecture is described by ż

= Aẑ + Bu + L(C ẑ -y) u = K ẑ
Proof of Theorem 3. For proving Theorem 3, it is convenient not to study the asymptotic stability of the origin of (13) in the coordinates (z, ẑ), but rather in the coordinates (ẑ, e) which give ż

= (A + BK)ẑ + L(y -C ẑ) ė = (A + LC)e (14)
Since A + LC is Hurwitz, there exists a symmetric positive definite matrix along the trajectories of ż = Aẑ + BK ẑ + L(y -C ẑ). Invoking now Young inequality and the fact that e(t) → 0 gives that ẑ(t) goes to 0 as well when time goes to +∞. Therefore, the origin of the linear system ( 14) is attractive and thus asymptotically stable. Note that another proof of the asymptotic stability of the origin of ( 14) is based on proving that V + 4 P LC 2 W is actually a Lyapunov function. To do that denote V(ẑ, e) = V (ẑ) + 4 P LC 2 W (e) and compute the time derivative of V along the solutions to (14):

Q such that (A + LC) Q + Q(A + LC) -I (
V = ẑ ((A + BK) P + P (A + BK))ẑ + 2ẑ P LCe

+4 P LC 2 e ((A + LC) Q + (A + LC)Q)e ≤ - 1 2 ẑ 2 + 2 P LCe 2 -4 P LC 2 e 2 ,
where Young inequality, (15) and ( 16) have been used for the previous inequality. Therefore V -1 2 I, and V is a Lyapunov function for (14).

The computation done in the proof of Theorem 3 will be generalized for PDEs in the next sections.

Example 3. (Example 1 continued) In this part of the example, we first select a ramdomnly chosen matrix, and we check the Kalman rank condition for observability (lines 54-57). Then we compute a matrix L by placing the eigenvalues of the matrix A + LC (line 60), and finally we plot solutions of (13) for 10 ramdomnly chosen initial conditions (lines 72-79). print ( ' observable system ') else : print ( ' unobservable system ') q = np . linspace ( -n -1 , -2 , n ) # choice of the eigenvalues of the closed -loop system L = -control . place ( A .T , C .T , q ) . T ax1 . plot ( zhat [0] , zhat [1] , zhat [2]) ; ax0 . set_title ( 'z ') ; ax1 . set_title ( '$ \ hat z$ ') plt . savefig ( ' solutions2 . png ' , bbox_inches = ' tight ') Figure 3 presents several solutions to (14) for ranmdonly chosen initial conditions (z(0), ẑ(0)), and confirms the attractivity of the origin for this system.

The Lyapunov function that is considered at the end of the proof of Theorem 3 is computed on lines 49, 85 and 86. It is checked on Figure 3 that this function decreases and converge to 0 along the solutions to (14) for the initial conditons used for Figure 3. ax . plot (t , lyapu ) plt . savefig ( ' lyapu2 . png ' , bbox_inches = ' tight ')

Saturated control

For many applications of control problems, the input values are limited in amplitude. Instead of applying u = Kz, only u = sat(Kz) can actually be applied, where sat: R m → R m is the saturation map defined componentwise by, for all i = 1, . . . , m, for a fixed vector s in R m with positive components s i > 0. Such function is a decentralized nonlinear map that makes the closed-loop system as follows:

sat i (σ i ) = σ i if |σ i | < s i sign(σ i )s i , else , (17)
ż = Az + Bsat(Kz) (18)
In the presence of a saturation, system (18) can exhibit various behaviors. Even if the matrix A + BK is Hurwitz, there may exist several equilibrium points, some limit cycle may appear, and there may exist diverging trajectories. See [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF]74 for introductory references on stability of such dynamical systems.

Example 4. As an example, consider

ż = Az + Bsat(Kz) (19)
with A = 0 1 1 0 , B = 0 -1 , K = 13 7 , and s = 5 as saturation level.

The matrix A is unstable (eigenvalues located at -1 and +1), and the matrix A + BK is Hurwitz (eigenvalues located at -1 and -13). As noted in [70, Example 1.1], the nonlinear system (19) exhibits several equilibriums and presents different behaviors depending on the initial condition. These behaviors are illustrated on Figure 21 based on different initial conditions. The first trajectory converges to 0 in R 2 , the second trajectory converges to the non zero equilibrium point -5 0 , and the last trajectory diverges as the time increases. The simulation code is given below. To analyze the stability of the equilibrium 0 of (18), let us consider the following Lyapunov function candidate V : z → z P z, where P ∈ R n×n is a symmetric definite positive matrix. The computation of its time derivative along the solutions of (18) gives V = z (A P + P A)z + 2z P Bsat(Kz) .

To ease the comparison of the dynamics of (19) and of (18), we introduce the deadzone function φ defined by

φ(σ) = sat(σ) -σ , ∀σ ∈ R m . (20)
Using this notation we get V = z ((A + BK) P + P (A + BK))z + 2z P Bφ(Kz)

= z φ(Kz) (A + BK) P + P (A + BK) P B 0 z φ(Kz)
Note that the matrix (A + BK) P + P (A + BK) P B 0 can not be in general negative semidefinite (except, e.g., for the trivial case B = 0). Consequently, in order to use the Lyapunov function candidate V to analyze the stability of the origin of (18), we need to find a relation between z and sat(Kz). This can be done by using the geometric conditions of the saturation map, as described by the so-called local and global sector conditions. As introduced in, [START_REF] Gomes Da | Anti-windup design with guaranteed regions of stability: an LMI-based approach[END_REF] for any given G ∈ R m×n and any given diagonal positive definite matrix T ∈ R m , the following local sector condition holds:

(sat(Kz)-Kz)T (sat(Kz)-(K-G)z) ≤ 0 , ∀z such that |((K-G)z) i | ≤ s i , (21) where
(K -G) (i) denotes the ith row of K -G.
Letting in particular G = K in ( 21), the following global sector condition holds for any diagonal positive definite matrix T

(sat(Kz) -Kz)T sat(Kz) ≤ 0 , ∀z ∈ R m (22)
From the local sector condition, we obtain that for any G ∈ R n×m and any diagonal positive definite matrix T , as long as

|((K -G)z) i | ≤ s i , V ≤ z ((A + BK) P + P (A + BK))z + 2z P Bφ(Kz) -2φ(Kz) T (φ(Kz) + Gz) ≤ z φ(Kz) (A + BK) P + P (A + BK) P B-G T -2T z φ(Kz)
Considering the special case where G = K, we obtain the following theorem.

Theorem 4. If there exist a symmetric definite matrix P in R n×n and a diagonal positive definite matrix T in R m such that

(A + BK) P + P (A + BK) P B -K T -2T ≺ 0
then the origin of ( 19) is globally asymptotically stable.

Remark 1. Some observations are in order.

Checking the existence of such matrices P and T is numerically tractable. This is a convex problem that could be solved using different solvers and method as interior-point method, [START_REF] Gahinet | LMI control toolbox[END_REF] or a primal/dual method. [START_REF] De | Aspects of semidefinite programming: interior point algorithms and selected applications[END_REF] See also. [START_REF] Boyd | Linear matrix inequalities in system and control theory[END_REF] As discussed in, [START_REF] Sontag | A general result on the stabilization of linear systems using bounded controls[END_REF] the existence of a globally stabilizing saturating control is subject to a number of strong conditions such as: (i) A has no eigenvalues with positive real part, and (ii) the pair (A, B) is stabilizable in the ordinary sense, that there exists a matrix K such that A + BK is asymptotically stable.

Setting G = K is generally restrictive since global asymptotic stability is generally a too strong property for saturated systems. To derive a sufficient condition for the weaker property of local asymptotic stability, we use the local sector condition (21). To ensure the condition

|((K -G)z) i | ≤ s i , we note that {z, z P z ≤ 1} ⊂ {z, |((K -G)z) i | ≤ s i } provided the LMI condition P (K -G) (i) s 2 i 0
holds. This result is a direct consequence of the Schur complement (see [6, Page 7]). Returning now to the LMI

(A + BK) P + P (A + BK) P B-G T -2T ≺ 0
we note that there is a product G T of unknown variables, making the problem nonlinear. Nevertheless, the problem can be made linear by introducing a simple change of variable. Indeed, using the change of variables

S = T -1 , W = P -1
, and H = GP -1 , we obtain the equivalent condition

W (A + BK) + (A + BK)W BS -H -2S ≺ 0
We have thus proven the following sufficient condition for local asymptotic stability of (19).

Theorem 5. If there exist W = W > 0, S diagonal definite positive and G such that

W W K (i) -H (i) s 2 i 0 (23)
W (A + BK) + (A + BK)W BS -H -2S ≺ 0 ( 24
)
then the origin of ( 19) locally asymptotically stable with a basin of attraction containing {z, z W -1 z ≤ 1}.

Remark 2. Checking the condition of Theorem 5 reduces to solving a convex problem. Different optimization criterion can be considered in order to maximize the estimation of the basin of attraction, as e.g., maximizing the trace of the matrix W . This idea is illustrated in Example 5 below. Note that this sufficient condition for local asymptotic stability of the closed-loop system can also be used in order to compute the matrix of feedback gain K. See [70, Chapter 3] and in the next sections for infinitedimensional dynamics. 23) and ( 24) hold is solved in line 56, using the default solver. For the first initial condition of Figure 4, that is with z 0 = [-1 -3] , we have

n = len ( A ) ; m = 1 W = cp . Variable (( n , n ) , PSD = True ) S = cp . Variable (( m , m ) , diag = True ) H = cp . Variable (( m , n ) ) B = B . reshape (2 ,1) K = K . reshape (1 ,2) M11 = W @ ( np . transpose ( A + np . dot (B , K ) ) ) M11 = M11 + M11 . T M12 = B @ S -H . T matrixConstr1 = cp . bmat ([[ M11 , M12 ] , [ M12 .T , -2 * S ]]) M22 = W @ K . T -H . T matrixConstr2 = cp . bmat ([[ W , M22 ] , [ M22 . T , s0 ** 2* np . array ([[1]]) ]]) constr = [ S >> 0] constr += [ matrixConstr1 < <0] + [ matrixConstr2 > >0] prob = cp
z 0 P z 0 = 0.99 < 1.
Thus z 0 is indeed in the basin of attraction, as confirmed by the timeevolution of the first solution in Figure 4.

Section conclusion

This section was devoted to finite-dimensional control systems by recalling some basic definitions and techniques for the stability analysis of equilibrium of such dynamical systems. In particular we reviewed the direct Lyapunov method for the asymptotic stability analysis. The control systems with saturated inputs have been also considered, and some sufficient conditions for the local (and global) asymptotic stability of the origin have been recalled. The next section will develop these techniques for the boundary stabilization of parabolic and hyperbolic systems.

Parabolic equations

Introduction

This section considers parabolic partial differential equations modeling reaction-diffusion phenomenon. This class of dynamical systems may be unstable in open loop. We focus on 1D parabolic equations for which spectral decomposition can be easily handled since the eigenvalues are simple and the eigenfunctions form a Hilbert basis of the state-space. For further studies on abstract parabolic PDEs in several dimensional spaces, see [START_REF] Boyer | Controllability of linear parabolic equations and systems[END_REF] in particular for controllability properties of such systems.

Based on the basic tools presented in the previous section, we present design methods for the design of output-feedback laws rendering the equilibrium asymptotically stable. The approach is based on modal approximation methods that have been shown to be efficient for other control problems related to parabolic PDEs; see [START_REF] Russell | Controllability and stabilizability theory for linear partial differential equations: recent progress and open questions[END_REF] as well as more recent references including. [START_REF] Coron | Global steady-state controllability of onedimensional semilinear heat equations[END_REF][START_REF] Coron | Global steady-state stabilization and controllability of 1D semilinear wave equations[END_REF][START_REF] Lhachemi | PI regulation of a reaction-diffusion equation with delayed boundary control[END_REF][START_REF] Lhachemi | An LMI condition for the robustness of constant-delay linear predictor feedback with respect to uncertain time-varying input delays[END_REF][START_REF] Orlov | Discontinuous unit feedback control of uncertain infinitedimensional systems[END_REF][START_REF] Prieur | Feedback stabilization of a 1D linear reaction-diffusion equation with delay boundary control[END_REF] The rationale behind the design method presented in this section is split into several steps. First a finite-dimensional statefeedback is computed only with a finite number of selected modes of the model. Then a finite-dimensional observer is designed in a separate fashion in order to estimate a finite number of modes that include in particular the modes used for the state-feedback design. Such a control design approach roots back to the pioneer papers [START_REF] Mark | Finite-dimensional controllers for linear distributed parameter systems: exponential stability using residual mode filters[END_REF][START_REF] Curtain | Finite-dimensional compensator design for parabolic distributed systems with point sensors and boundary input[END_REF][START_REF] Harkort | Finite-dimensional observer-based control of linear distributed parameter systems using cascaded output observers[END_REF][START_REF] Sakawa | Feedback stabilization of linear diffusion systems[END_REF] which essentially rely on small gain arguments. Taking advantage of the controller architecture reported in, [START_REF] Sakawa | Feedback stabilization of linear diffusion systems[END_REF] the possibility to recast this control design problem into a LMI framework was shown in [START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF] for a particular set of input/output maps and specific norms for the asymptotic stability estimates. This procedure was enhanced and generalized in a systematic manner in [START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF][START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF] for general reaction-diffusion PDEs with Dirichlet/Neumann/Robin boundary control and Dirichlet/Neumann measurement while performing the control design directly with the control input instead of it time derivative (see [START_REF] Curtain | Introduction to infinite-dimensional systems theory: a state-space approach[END_REF] for an introduction to boundary control systems). This generalized and systematic approach has been shown to be key and very efficient for the predictorbased compensation of arbitrarily long input and output delays, [START_REF] Lhachemi | Boundary output feedback stabilization of reaction-diffusion PDEs with delayed boundary measurement[END_REF][START_REF] Lhachemi | Predictor-based output feedback stabilization of an input delayed parabolic PDE with boundary measurement[END_REF] the domination of state-delays, [START_REF] Lhachemi | Boundary output feedback stabilization of state delayed reaction-diffusion pdes[END_REF] the local output feedback stabilization of linear reaction-diffusion PDEs in the presence of a saturation, [START_REF] Lhachemi | Local output feedback stabilization of a reactiondiffusion equation with saturated actuation[END_REF] the global stabilization of linear-reaction-diffusion PDEs in the presence of a Lipchitz continuous sector nonlinearity in the application of the boundary control, [START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF] as well as the global stabilization of semilinear reaction-diffusion PDE with globally Lipchitz nonlinearity. [START_REF] Lhachemi | Global output feedback stabilization of semilinear reaction-diffusion PDEs[END_REF] In this framework, the proof of stability of the closed-loop system (composed of the PDE, the finite-dimensional observer, and the state-feedback) is assessed using the Lyapunov direct method presented in the previous section, but adapted to the distributed nature of the state. This approach can be seen as an alternative output feedback design method for reaction parabolic PDEs to other very efficient tools, such as backstepping transformations for PDEs (see the introductory textbook [START_REF] Krstic | Boundary control of PDEs: A course on backstepping designs[END_REF] ) for which a form of separation principle between controller and observer designs generally exists. Nevertheless, the infinite-dimensional nature of the observer obtained using backstepping methods implies the necessity to resort to late lumping approximations in order to obtain a finite-dimensional control strategy that is suitable for practical implementation, inducing in general the loss of the stability performance guarantees originally obtained during the synthesis phase. The benefit of the approach reported in this section is that the observer obtained during the synthesis phase is directly finite-dimensional.

The material presented in this Section of the lecture notes is widely inspired from [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF] in the linear case and from [START_REF] Lhachemi | Local output feedback stabilization of a reactiondiffusion equation with saturated actuation[END_REF] for the saturated input scenario.

The rest of this section is organized as follows. After introducing a number of notations and properties, the case of Dirichlet boundary control with a bounded observation operator is considered in Section 3.2. The control design procedure is then extended to the cases of a boundary Dirichlet observation in Section 3.3. The case of in-domain control in the presence of an input saturation in discussed in Section 3.4.

Reminders on Sturm Liouville theory

Let us conclude this introduction with some reminders on Sturm Liouville theory for parabolic operators in one space dimension. See 51 for a reference on the mathematical properties that will be extensively used in this section.

Let

θ 1 , θ 2 ∈ [0, π/2], p ∈ C 1 ([0, 1]
), and q ∈ C 0 ([0, 1]) with p > 0 and q ≥ 0. Consider the Sturm-Liouville operator A :

D(A) ⊂ L 2 (0, 1) → L 2 (0, 1) defined by Af = -(pf ) + qf on the domain D(A) = {f ∈ H 2 (0, 1) : cos(θ 1 )f (0) -sin(θ 1 )f (0) = 0, cos(θ 2 )f (1) + sin(θ 2 )f (1) = 0}.
The eigenvalues (λ n ) n≥1 of A are simple, non negative, and form an increasing sequence with λ n → +∞ as n → +∞. The associated unit eigenvectors Φ n ∈ L 2 (0, 1) form a Hilbert basis. The operator A and its domain can be characterized by there eigenstructures in the sense that

Af = n≥1 λ n f, Φ n , ∀f ∈ D(A)
and

D(A) = {f ∈ L 2 (0, 1) : n≥1 |λ n | 2 | f, Φ n | 2 < +∞} where f, g = 1 0 f (x)g(x)
, dx, for any f, g ∈ L 2 (0, 1), stands for the inner product of L 2 (0, 1). Hence, using an integration by parts, it can be seen that, for any f ∈ D(A),

n≥1 λ n f, Φ n 2 = Af, f = p(0)f (0)f (0) -p(1)f (1)f (1) + 1 0 p(x)f (x) 2 + q(x)f (x) 2 dx.
Using the boundary conditions involved in the definition of D(A), we infer the existence of a constant C 2 > 0 such that

n≥1 λ n f, Φ n 2 = Af, f ≤ C 2 f H 1 .
Moreover, if either (i) θ 1 , θ 2 ∈ {0, π/2} with θ i = 0 for at least one i ∈ {0, 1}; or (ii) q > 0, this implies the existence of a constant C 1 > 0 such that

C 1 f H 1 ≤ n≥1 λ n f, Φ n 2 = Af, f ≤ C 2 f H 1 . (25)
Hence, for any f ∈ D(A), the series expansion f = n≥1 f, Φ n Φ n holds in H 1 (0, 1) norm. Then, using the definition of A and the fact that it is a Riesz-spectral operator, we obtain that the latter series expansion holds in H 2 (0, 1) norm. Due to the continuous embedding H 1 (0, 1) ⊂ L ∞ (0, 1), we obtain that

f (0) = n≥1 f, Φ n Φ n (0), f (0) = n≥1 f, Φ n Φ n (0).
Let p * , p * , q * ∈ R be such that 0 < p * ≤ p(x) ≤ p * and 0 ≤ q(x) ≤ q * for all x ∈ [0, 1]. Then we have:

51 0 ≤ π 2 (n -1) 2 p * ≤ λ n ≤ π 2 n 2 p * + q * ( 26
)
for all n ≥ 1. If we further assume that p ∈ C 2 ([0, 1]), we have (see again [START_REF] Orlov | On general properties of eigenvalues and eigenfunctions of a Sturm-Liouville operator: comments on "ISS with respect to boundary disturbances for 1-D parabolic PDEs[END_REF] ) that

Φ n (0) = O n→+∞ (1), Φ n (0) = O n→+∞ ( λ n ). ( 27
)

Bounded observation operator

We first consider the reaction-diffusion system described by

z t (t, x) = (p(x)z x (t, x)) x + (q c -q(x))z(t, x) (28a) z x (t, 0) = 0, z(t, 1) = u(t) (28b) z(0, x) = z 0 (x) (28c) y(t) = 1 0 c(x)z(t, x) dx (28d)
for t > 0 and x ∈ (0, 1). Here

q c ∈ R is a constant, u(t) ∈ R is the command input, y(t) ∈ R with c ∈ L 2 (0, 1) is the measurement, z 0 ∈ L 2 (0, 1)
is the initial condition, and z(t, •) ∈ L 2 (0, 1) is the state.

Spectral reduction

In (28), the control input u appears in the right boundary condition. Let us transfer the control input from the boundary into the PDE by invoking the change of variable:

w(t, x) = z(t, x) -x 2 u(t). (29)
It has been specifically selected in order to ensure that we still have the left boundary condition w x (t, 0) = 0 while enforcing w(t, 0) = 0. Hence, we have

w t (t, x) = (p(x)w x (t, x)) x + (q c -q(x))w(t, x) + a(x)u(t) + b(x) u(t) (30a) w x (t, 0) = 0, w(t, 1) = 0 (30b) w(0, x) = w 0 (x) (30c) ỹ(t) = 1 0 c(x)w(t, x) dx (30d)
Here a, b ∈ L 2 (0, 1) are defined by a(x) = 2p(x) + 2xp (x) + (q c -q(x))x 2 and b(x) = -x 2 , respectively, while ỹ(t) = y(t) -

1 0 x 2 c(x) dx u(t) and w 0 (x) = z 0 (x) -x 2 u(0).
The parabolic equation (30) presents homogeneous boundary conditions (30b) that are much easier to deal with. However, the price of this transfer is the occurrence of the time derivative u of the control input u in the PDE (30a). This is why we introduce the auxiliary command input v(t) = u(t), that will be used as the control input for control design. In other words, v will be used as the control input for the design of the control strategy. However, for final implementation of the control strategy, u remains the actual control input of the plant. In this context, the dynamics of the system reads

u(t) = v(t) (31a) dw dt (t, •) = -Aw(t, •) + q c w(t, •) + au(t) + bv(t) (31b) with D(A) = {f ∈ H 2 (0, 1) : f (0) = f (1) = 0}. Introducing the coeffi- cients of projection w n (t) = w(t, •), Φ n , a n = a, Φ n , b n = b, Φ n , and c n = c, Φ n , the projection of the PDE solutions into the Hilbert basis of eigenfunctions (Φ n ) n≥1 gives u(t) = v(t) (32a) ẇn (t) = (-λ n + q c )w n (t) + a n u(t) + b n v(t), n ≥ 1 (32b) ỹ(t) = i≥1 c i w i (t) (32c)
Note that (32) has been obtained from (31) by 1) multiplying (31) by Φ n ; 2) integrating on the space domain; and 3) performing two integration by parts why using the boundary conditions coming from the definition of D(A).

Control design

We start by fixing an integer N 0 ≥ 1 and positive real number δ > 0 such that -λ n + q c < -δ < 0 for all n ≥ N 0 + 1. Let N ≥ N 0 + 1 be arbitrary.

The general idea, borrowed to, [START_REF] Sakawa | Feedback stabilization of linear diffusion systems[END_REF] is to compute a stabilizing output-feedback controller in three steps. First an observer to estimate the N first modes of the plant is designed. Secondly the state-feedback is only performed on the N 0 first estimated modes of the plant. Finally a dedicated stability analysis is performed to prove that the origin of the closed-loop is asymptotically stable. In this context, inspired by the controller architecture first reported Saturated boundary stabilization of partial differential equations 29 in, 62 the adopted control strategy takes the form:

ẇn (t) = (-λ n + q c ) ŵn (t) + a n u(t) + b n v(t) + l n (ŷ(t) -ỹ(t)) , 1 ≤ n ≤ N 0 (33a) ẇn (t) = (-λ n + q c ) ŵn (t) + a n u(t) + b n v(t), N 0 + 1 ≤ n ≤ N (33b) ŷ(t) = 1 0 c(x) N i=1 ŵi (t)Φ i (x) dx = N i=1 c i ŵi (t) (33c) v(t) = u(t) = N0 i=1 k i ŵi (t) + k u u(t) (33d)
where k i , k u ∈ R are the feedback gains while l n ∈ R are the observer gains. Signals ŵn stand for the estimations of the modes w n for 1 ≤ n ≤ N . These estimations are used for the computation of ŷ that represents the estimation of the actual system measurement ỹ. Note that the feedback law (33d) is computed only based on the observations ŵn for 1 ≤ n ≤ N 0 . The remaining observations, namely ŵn for N 0 + 1 ≤ n ≤ N , are only used in (33c) to improve the estimation ŷ of the actual system output ỹ. This estimation ŷ is used to introduce a correction term in the observation dynamics (33a) related to the mismatch between the estimation ŷ and the measurement ỹ. Note that no such correction is applied in (33b) for the observed modes associated with

N 0 + 1 ≤ n ≤ N .
In order to study the validity of the control strategy (33), we need to introduce a number of definitions. Introducing

W N0 (t) =    w 1 (t) . . . w N0 (t)    , B 0,a =    a 1 . . . a N0    , B 0,b =    b 1 . . . b N0    , A 0 = diag(-λ 1 + q c , . . . , -λ N0 + q c ),
we have from (32b) that

Ẇ N0 (t) = A 0 W N0 (t) + B 0,a u(t) + B 0,b v(t). (34)
Hence, defining

W N0 a (t) = u(t) W N0 (t) , A 1 = 0 0 B 0,a A 0 , B 1 = 1 B 0,b , we obtain that Ẇ N0 a (t) = A 1 W N0 a (t) + B 1 v(t). H.

Lhachemi and C. Prieur

We now define for 1 ≤ n ≤ N the observation error as

e n (t) = w n (t) -ŵn (t). ( 35
)
With

ζ(t) = i≥N +1 c i w i (t), we infer from (33a) that ẇn (t) = (-λ n + q c ) ŵn (t) + a n u(t) + b n v(t) -l n N i=1 c i e i (t) -l n ζ(t) (36)
for 1 ≤ n ≤ N 0 . Inspired by Section 2, we write the dynamics in coordinates of the observer state and of the error variable. To do so we introduce

Ŵ N0 (t) =    ŵ1 (t) . . . ŵN0 (t)    , E N0 (t) =    e 1 (t) . . . e N0 (t)    , E N -N0 (t) =    e N0+1 (t) . . . e N (t)    , C 0 = c 1 c 2 . . . c N0 , C 1 = c N0+1 . . . c N , L =    l 1 . . . l N0    . Hence we have Ẇ N0 (t) = A 0 Ŵ N0 (t) + B 0,a u(t) + B 0,b v(t) (37)
-LC 0 E N0 (t) -LC 1 E N -N0 (t) -Lζ(t). With Ŵ N0 a (t) = u(t) Ŵ N0 (t) , L = 0 L (38) we deduce that Ẇ N0 a (t) = A 1 Ŵ N0 a (t) + B 1 v(t) -LC 0 E N0 (t) -LC 1 E N -N0 (t) -Lζ(t) (39) In view of (33d) we deduce that v(t) = K Ŵ N0 a (t), (40)
where

K ∈ R 1×(N0+1) . Hence we obtain that Ẇ N0 a (t) = (A 1 + B 1 K) Ŵ N0 a (t) -LC 0 E N0 (t) -LC 1 E N -N0 (t) -Lζ(t) (41)
and, using ( 34) and (37),

ĖN0 (t) = (A 0 + LC 0 )E N0 (t) + LC 1 E N -N0 (t) + Lζ(t). ( 42
)
Claim 1. The pair (A 1 , B 1 ) is controllable.

Proof of Claim 1. Let us compute the Kalman matrix C for controllability of (A 1 , B 1 ) as introduced in Section 1. Denoting by µ n = -λ n + q c , we get

C =         1 0 . . . 0 b 1 a 1 + µ 1 b 1 . . . (a 1 + µ 1 b 1 )µ N0-1 1 b 2 a 2 + µ 2 b 2 . . . (a 2 + µ 2 b 2 )µ N0-1 2 . . . . . . b N0 a N0 + µ N0 b N0 . . . (a 1 + µ N0 b N0 )µ N0-1 N0         whose determinant is det(C) = Π N0 n=1 (a n + µ n b n ) 1 µ 1 . . . µ N0-1 1 1 µ 2 . . . µ N0-1 2 . . . . . . 1 µ N0 . . . µ N0-1 N0
.

The second determinant appearing in the latter equations is known as the Vandermonde determinant. Since the µ n are distinct, the Vandermonde determinant is non zero hence the pair

(A 1 , B 1 ) is controllable if and only if Π N0 n=1 (a n + µ n b n ) = 0.
To check this latter condition, let us compute, for each n = 1, . . . , N 0 , the quantity a n + µ n b n . Recalling µ n = -λ n + q c and from the definitions of the function a and b, we obtain that

a n + µ n b n = 1 0 [2p(x) + 2xp (x) + (q c -q(x))x 2 ]Φ n (x)dx +(-λ n + q c ) 1 0 -x 2 Φ n (x)dx = 1 0 [(2p(x) + 2xp (x))Φ n (x) -x 2 q(x)Φ n (x)]dx + 1 0 [-x 2 (p(x)Φ n (x)) + x 2 q(x)Φ n (x)]dx = 1 0 (2p(x) + 2xp (x))Φ n (x)dx - 1 0 x 2 (p(x)Φ n (x)) dx = -p(1)Φ n (1).
Recalling that Φ n is a non-trivial solution to a second order ODE with Φ n (1) = 0, we must have Φ n (1) = 0. Therefore

a n + µ n b n = 0 hence the pair (A 1 , B 1 ) is controllable. Claim 2. Assuming c n = 0 for all 1 ≤ n ≤ N 0 , the pair (A 0 , C 0 ) is observable.
Proof of Claim 2. Let us compute the Kalman matrix for observation of the pair (A 0 , C 0 ):

    c 1 . . . c N0 µ 1 c 1 . . . µ N0 c N0 . . . µ N0-1 1 c 1 . . . µ N0-1 N0 c N0     .
Since n = m implies µ n = µ m , this matrix is full rank if an only if c n = 0 for all n = 1, . . . , N 0 .

We now define the vectors and matrices:

Ŵ N -N0 (t) =    ŵN0+1 (t) . . . ŵN (t)    , B 2,a =    a N0+1 . . . a N    , B 2,b =    b N0+1 . . . b N    , A 2 = diag(-λ N0+1 + q c , . . . , -λ N + q c ).
From (33b) and ( 40) we obtain that

Ẇ N -N0 (t) = A 2 Ŵ N -N0 (t) + B 2,a u(t) + B 2,b v(t) = A 2 Ŵ N -N0 (t) + B 2,b K + B 2,a 0 Ŵ N0 a (t) (43)
and, using in addition (32b) and (35),

ĖN-N0 (t) = A 2 E N -N0 (t). ( 44
)
Putting together (41-44), we obtain with

X(t) = col( Ŵ N0 a (t), E N0 (t), Ŵ N -N0 (t), E N -N0 (t)) (45) that Ẋ(t) = F X(t) + Lζ(t) (46)
where

F =     A 1 + B 1 K -LC 0 0 -LC 1 0 A 0 + LC 0 0 LC 1 B 2,b K + B 2,a 0 0 A 2 0 0 0 0 A 2     , L =     - L L 0 0     . (47)
Defining E = 1 0 . . . 0 and K = K 0 0 0 , we obtain from ( 38), (40), and (45) that

u(t) = EX(t), v(t) = KX(t). (48)
Finally, defining g = a 2

L 2 + b 2 L 2 K 2 , we can introduce G = a 2 L 2 E E + b 2 L 2 K K gI. (49)
3.2.3. Stability analysis Theorem 6. Let p ∈ C 1 ([0, 1]) with p > 0, q ∈ C 0 ([0, 1]) with q ≥ 0, q c ∈ R, and c ∈ L 2 (0, 1). Consider the reaction-diffusion system described by (28). Let N 0 ≥ 1 and δ > 0 be given such that -λ n + q c < -δ < 0 for all

n ≥ N 0 + 1. Assume that c n = 0 for all 1 ≤ n ≤ N 0 . Let K ∈ R 1×(N0+1)
and L ∈ R N0 be such that A 1 + B 1 K and A 0 + LC 0 are Hurwitz with eigenvalues that have a real part strictly less than -δ < 0. Assume that there exist N ≥ N 0 + 1, P 0, α > 1, and β, γ > 0 such that

Θ = F P + P F + 2δP + αγG P L -β 0, (50a) Γ 1,N +1 = -λ N +1 + q c + δ + 1 α + β c 2 L 2 2γ ≤ 0, ( 50b
) Γ 2,N +1 = -1 - 1 α λ N +1 + q c + δ + β c 2 L 2 2γλ N +1 ≤ 0, ( 50c
)
for all n ≥ N + 1. Then, for the closed-loop system composed of the plant (28) and the controller (33) (1) the origin is asymptotically stable in L 2 -norm, that is there exists M > 0 such that, for any ŵn (0) ∈ R, any z 0 ∈ L 2 (0, 1) and any u(0) ∈ R, the mild solution of the closed-loop system satisfies

u(t) 2 + N n=1 ŵn (t) 2 + z(t, •) 2 L 2 ≤ M e -2δt u(0) 2 + N n=1 ŵn (0) 2 + z 0 2 L 2
for all t ≥ 0. (2) the origin is asymptotically stable in H 1 -norm, that is there exists M > 0 such that, for any ŵn (0) ∈ R, any z 0 ∈ H 2 (0, 1) and any u(0) ∈ R such that z 0 (0) = 0 and z 0 (1) = u(0), the classical solution of the closed-loop system satisfies

u(t) 2 + N n=1 ŵn (t) 2 + z(t, •) 2 H 1 ≤ M e -2δt u(0) 2 + N n=1 ŵn (0) 2 + z 0 2 H 1 for all t ≥ 0.
Moreover, the above constraints are always feasible for N large enough.

Remark 3. The feasibility problem of Theorem 6 is not linear due to the presence of some terms such as αγ and 1 α involving the decision variables. However the use of Schur complement allows to rewrite (50b) as follows:

-λ N +1 + q c + δ + β c 2 L 2 2γ 1 -α ≤ 0,
and similarly for (50c). Therefore, as soon as γ is fixed, checking the conditions of Theorem 6 reduces to check linear matrix inequalities (LMIs). Thus, given a desired exponential decay rate δ > 0 and a number of modes N ≥ N 0 + 1 for the observer, the sufficient conditions of the previous theorem can be recasted as an efficient optimization problem to solve LMIs.

Proof of Theorem 6. Consider the Lyapunov function candidate

V (X, w) = X P X + γ n≥N +1 w, Φ n 2 for X ∈ R 2N +1
and w ∈ L 2 (0, 1). The first term accounts for the dynamics of the N first modes of the PDE and the dynamics of the observer, while the series accounts for the dynamics of the modes corresponding to n ≥ N + 1.

The computation of the time derivative of V along the system solutions (32b) and ( 46) gives V + 2δV =X F P + P F + 2δP X + 2X P Lζ

+ 2γ n≥N +1 (-λ n + q c + δ)w 2 n + 2γ n≥N +1 (a n u + b n v)w n .
The use of Young's inequality gives

2 n≥N +1 a n w n u ≤ 1 α n≥N +1 w 2 n + α a 2 L 2 u 2 , 2 n≥N +1 b n w n (t)v(t) ≤ 1 α n≥N +1 w 2 n + α b 2 L 2 v 2 .
for any α > 0. From (48-49), we infer that

V + 2δV ≤ X ζ F P + P F + 2δP + αγG P L 0 X ζ + 2γ n≥N +1 -λ n + q c + δ + 1 α w 2 n .
Recalling the definition ζ(t) = n≥N +1 c n w n (t), we obtain from Cauchy-Schwarz inequality that

ζ(t) 2 ≤ c 2 L 2 n≥N +1 w n (t) 2 .
Hence, for any β > 0,

β c 2 L 2 n≥N +1 w 2 n -βζ 2 ≥ 0. ( 51
)
Combining the two latter inequalities, we obtain that

V + 2δV ≤ X ζ Θ X ζ + 2γ n≥N +1 Γ 1,n w 2 n ≤ 0 where Γ 1,n = -λ n + q c + δ + 1 α + β c 2 L 2 2γ
≤ Γ 1,N +1 for all n ≥ N + 1. The assumptions (50) imply that V (t) ≤ e -2δt V (0) for all t ≥ 0, giving the claimed stability estimate for PDE trajectories evaluated in L 2 -norm.

We now address the stability assessment of the system trajectories when evaluated in H 1 -norm. To do so, in view of (25), we introduce the Lyapunov functional candidate:

V (X, w) = X P X + γ n≥N +1 λ n w, Φ n 2 (52)
with X ∈ R 2N +1 and w ∈ D(A). The computation of the time derivative of V along the system solutions (32b) and (46) gives

V + 2δV = X F P + P F + 2δP X + 2X P Lζ (53) + 2γ n≥N +1 λ n (-λ n + q c + δ)w 2 n + 2γ n≥N +1 λ n (a n u + b n v)w n (t).
Using again Young's inequality, we obtain

2 n≥N +1 λ n a n w n u ≤ 1 α n≥N +1 λ 2 n w 2 n + α a 2 L 2 u 2 (54a) 2 n≥N +1 λ n b n w n v ≤ 1 α n≥N +1 λ 2 n w 2 n + α b 2 L 2 v 2 (54b)
for any α > 0. Hence, owing to (48-49) and ( 51), we deduce that

V + 2δV ≤ X ζ Θ X ζ + 2γ n≥N +1 λ n Γ 2,n w 2 n ≤ 0 with Γ 2,n = -λ n + q c + δ + λn α + β c 2 L 2
2γλn ≤ Γ 2,N +1 for all n ≥ N + 1 where it has been used that α > 1. Thus (50) implies that V (t) ≤ e -2δt V (0) for all t ≥ 0. The claimed stability estimate in H 1 -norm is now obtained from (25), (29), and (52).

We conclude the proof by showing that one can always select the order of the observer N ≥ N 0 + 1 large enough and find P 0, α > 1, and β, γ > 0 such that Θ 0, Γ 1,N +1 ≤ 0, and Γ 2,N +1 ≤ 0. Owing to the Schur complement, we have Θ 0 if and only if F P + P F + 2δP + αγG + 1 β P LL P 0. We now note that A 1 + B 1 K + δI and A 0 -LC 0 + δI are Hurwitz and e (A2+δI)t ≤ e -κ0t with κ

0 = λ N0+1 -q c - δ > 0. Moreover, LC 1 ≤ L c L 2 , LC 1 ≤ L c L 2 , and B 2,b K + B 2,a 0 ≤ b L 2 K + a L 2 .
The right-hand sides of all the previous inequalities are independent of the order of the observer N ≥ N 0 + 1. Hence, Lemma 1, which is reported immediately after this proof, applied to F + δI shows for any N ≥ N 0 + 1 the existence of P 0 such that F P + P F + 2δP = -I with P = O(1) as N → +∞. Finally, we have (49) and L = √ 2 L with g and L that are independent of N . Hence, with α = N 1/4 , β = N , and γ = N -1/2 , we infer from (26) the existence of a sufficiently large integer N ≥ N 0 +1, independent of the initial conditions, such that Θ 0, Γ 1,N +1 ≤ 0, and Γ 2,N +1 ≤ 0.

A technical lemma

The following lemma generalizes the statement of a result presented in [START_REF] Katz | Constructive method for finite-dimensional observer-based control of 1-D parabolic PDEs[END_REF] while the proof, reported below, remains essentially identical.

Lemma 1. Let n, m, N ≥ 1, M 11 ∈ R n×n and M 22 ∈ R m×m Hurwitz, M 12 ∈ R n×m , M N 14 ∈ R n×N , M N 24 ∈ R m×N , M N 31 ∈ R N ×n , M N 33 , M N 44 ∈ R N ×N , and
F N =     M 11 M 12 0 M N 14 0 M 22 0 M N 24 M N 31 0 M N 33 0 0 0 0 M N 44     .
We assume that there exist constants C 0 , κ 0 > 0 such that e M N 33 t ≤ C 0 e -κ0t and e M N 44 t ≤ C 0 e -κ0t for all t ≥ 0 and all N ≥ 1. Moreover, we assume that there exists a constant

C 1 > 0 such that M N 14 ≤ C 1 , M N 24 ≤ C 1 , and M N 31 ≤ C 1 for all N ≥ 1.
Then there exists a constant C 2 > 0 such that, for any N ≥ 1, there exists a symmetric matrix P N ∈ R n+m+2N with P N 0 such that (F N ) P N + P N F N = -I and

P N ≤ C 2 .
Proof of Lemma 1. It is sufficient to show the existence of constants C0 , η > 0 such that e F N t ≤ C0 e -ηt for all t ≥ 0 and all N ≥ 1. Indeed, in that case, P N = ∞ 0 e (F N ) t e F N t dt is well defined and satisfies the claimed properties. We introduce

F N = F N 1 + F N 2 with F N 1 =     M 11 M 12 0 0 0 M 22 0 0 0 0 M N 33 0 0 0 0 M N 44     , F N 2 =     0 0 0 M N 14 0 0 0 M N 24 M N 31 0 0 0 0 0 0 0     .
Then there exist constants κ, C1 , C2 > 0 such that e F N 1 t ≤ C1 e -κt and

F N 2
≤ C2 for all t ≥ 0 and all N ≥ 1. One can check that (F N 2 ) 3 = 0 and

(F N 1 ) ni =     • • 0 0 0 • 0 0 0 0 • 0 0 0 0 •    
for any n i ≥ 0 and where "•" denotes a possibly non zero element, that is not needed in this proof. Hence

(F N 1 ) ni F N 2 =     0 0 0 • 0 0 0 • • 0 0 0 0 0 0 0    
for any n i ≥ 0. We deduce that

3 i=1 (F N 1 ) ni F N 2 =     0 0 0 • 0 0 0 • • 0 0 0 0 0 0 0     3 = 0
for any n i ≥ 0. Therefore,

3 i=1 e F N 1 ti F N 2 = k1≥0 k2≥0 k3≥0 t k1 1 t k2 2 t k3 3 k 1 !k 2 !k 3 ! 3 i=1 (F N 1 ) ki F N 2 = (55)
for all t 1 , t 2 , t 3 ≥ 0. Now we note that b , for any square matrices A, B, e (A+B)t = e At + t 0 e A(t-τ ) Be (A+B)τ dτ . Hence we have, using the last identity three times consecutively,

e F N t = e F N 1 t + t 0 e F N 1 (t-t1) F N 2 e F N t1 dt 1 = e F N 1 t + t 0 e F N 1 (t-t1) F N 2 e F N 1 t1 dt 1 + t 0 t1 0 e F N 1 (t-t1) F N 2 e F N 1 (t1-t2) F N 2 e F N t2 dt 2 dt = e F N 1 t + t 0 e F N 1 (t-t1) F N 2 e F N 1 t1 dt 1 + t 0 t1 0 e F N 1 (t-t1) F N 2 e F N 1 (t1-t2) F N 2 e F N 1 t2 dt 2 dt
where the last identity has been obtained by using (55). Recalling that e F N 1 t ≤ C1 e -κt and F N

2

≤ C2 for all t ≥ 0 and all N ≥ 1, the claimed conclusion holds. b x(t) = e (A+B)t x 0 is such that ẋ(t) = Ax(t) + u(t) with u(t) = Bx(t). The claimed formula follows from x(t) = e At x 0 + t 0 e A(t-τ ) u(τ ) dτ .

Dirichlet boundary measurement

We extend the result of the previous section to the case of a reactiondiffusion PDE with Dirichlet boundary observation described by

z t (t, x) = (p(x)z x (t, x)) x + (q c -q(x))z(t, x) (56a)
z x (t, 0) = 0, z(t, 1) = u(t) (56b) z(0, x) = z 0 (x) (56c)
y(t) = z(t, 0) (56d)
for t > 0 and x ∈ (0, 1). We make throughout this subsection the assumption that p ∈ C 2 ([0, 1]) in order to use the asymptotic behavior (27).

Spectral reduction

The only change compared to the previous subsection is the modification of the nature of the observation operator. Hence, the spectral reduction is conducted identically but the observation (30d) is replaced by ỹ(t) = w(t, 0) = y(t). Considering classical solutions for the PDE, we have w(t, •) ∈ D(A) for all t ≥ 0. Hence, (32c) is simply replaced by ỹ(t) = i≥1 Φ i (0)w i (t).

Control design

Let N 0 ≥ 1 and δ > 0 be fixes so that -λ n + q c < -δ < 0 for all n ≥ N 0 + 1.

Let N ≥ N 0 + 1 be arbitrary and to be determined later. We proceed as in the previous subsection: we design an observer to estimate the N first modes of the plant while the state-feedback is performed on the N 0 first modes of the plant. Hence, the controller dynamics is described by ẇn

(t) = (-λ n + q c ) ŵn (t) + a n u(t) + b n v(t) + l n (ŷ(t) -ỹ(t)) , 1 ≤ n ≤ N 0 (57a) ẇn (t) = (-λ n + q c ) ŵn (t) + a n u(t) + b n v(t), N 0 + 1 ≤ n ≤ N (57b) ŷ(t) = N i=1 Φ i (0) ŵi (t) (57c) v(t) = u(t) = N0 i=1 k i ŵi (t) + k u u(t) (57d)
which is the same as the one described by (33) but with measurement, originally given by (33a), replaced by (57a). In this context, (36) is replaced by the following, defined for 1

≤ n ≤ N 0 , ẇn (t) = (-λ n + q c ) ŵn (t) + a n u(t) + b n v(t) -l n N0 i=1 Φ i (0)e i (t) + l n N i=N0+1 Φ i (0) √ λ i ẽi (t) + l n ζ(t).
Here ζ(t) is defined by ζ(t) = i≥N +1 Φ i (0)w i (t) while, following, [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF] we introduced the scaled error of observation ẽn (t) = √ λ n e n (t) with e n given by (35). The definitions of C 0 and C 1 are replaced by

C 0 = Φ 1 (0) . . . Φ N0 (0) , C 1 = Φ N0+1 (0) λ N0+1 . . . Φ N (0) √ λ N (58)
and defining ẼN-N0 (t) = ẽN0+1 (t) . . . ẽN (t) ,

we obtain in replacement of ( 37) and ( 39) that

Ẇ N0 (t) = A 0 Ŵ N0 (t) + B 0,a u(t) + B 0,b v(t) (59)
-LC 0 E N0 (t) -LC 1 ẼN-N0 (t) -Lζ(t) and Ẇ N0 a (t) = A 1 Ŵ N0 a (t) + B 1 v(t) -LC 0 E N0 (t) -LC 1 ẼN-N0 (t) -Lζ(t), (60)
respectively. In this framework, the command input is still given by (40).

Using now (34) and ( 59), the error dynamics originally given by ( 42) is now replaced by

ĖN0 (t) = (A 0 + LC 0 )E N0 (t) + LC 1 ẼN-N0 (t) + Lζ(t). (61)
Moreover, since ėn (t) = (-λ n + q c )e n (t), we have ėn (t) = (-λ n + q c )ẽ n (t) for all N 0 + 1 ≤ n ≤ N . Then ( 44) is replaced by

ĖN-N0 (t) = A 2 ẼN-N0 (t). ( 62
)
Putting together ( 40), (43), and (60-62), the introduction of

X(t) = col( Ŵ N0 a (t), E N0 (t), Ŵ N -N0 (t), ẼN-N0 (t)),
shows that (46) holds with the different matrices defined by (47).

Remark 4. Based on the arguments of Claim 1 and Claim 2, we have that (A 1 , B 1 ) is controllable and (A 0 , C 0 ) is observable.

Stability analysis

We introduce the constant M 1,Φ = n≥2 Φn(0) 2 λn . Note that this constant is well defined (i.e., finite) when p ∈ C 2 ([0, 1]) due to (26-27).

Theorem 7. Let p ∈ C 2 ([0, 1]) with p > 0, q ∈ C 0 ([0, 1]) with q ≥ 0, and q c ∈ R. Consider the reaction-diffusion system described by (56). Let N 0 ≥ 1 and δ > 0 be given such that -λ n + q c < -δ < 0 for all n ≥ N 0 + 1. Let K ∈ R 1×(N0+1) and L ∈ R N0 be such that A 1 + B 1 K and A 0 -LC 0 are Hurwitz with eigenvalues that have a real part strictly less than -δ < 0. Assume that there exist N ≥ N 0 + 1, P 0, α > 1, and β, γ > 0 such that Θ 0, where Θ is defined by (50a), and

Γ 3,N +1 = -1 - 1 α λ N +1 + q c + δ + βM 1,Φ 2γ ≤ 0. ( 63
)
Then the origin of the closed-loop system composed of the plant (56) and the controller ( 57) is exponentially stable in H 1 -norm in the sense that there exists M > 0 such that, for any ŵn (0) ∈ R, for any z 0 ∈ H 2 (0, 1) and any u(0) ∈ R such that z 0 (0) = 0 and z 0 (1) = u(0), the classical solution of the closed-loop system satisfies

u(t) 2 + N n=1 ŵn (t) 2 + z(t, •) 2 H 1 ≤ M e -2δt u(0) 2 + N n=1 ŵn (0) 2 + z 0 2 H 1 .
for all t ≥ 0. Moreover, the above constraints are always feasible for N large enough.

Remark 5. The previous result deals with the exponential stability of the closed-loop system in H 1 -norm. This type of approach can be extended to a number of control design problems such as:

• L 2 stability using the same control strategy; [START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF] • Robin boundary conditions; [START_REF] Lhachemi | Local output feedback stabilization of a reactiondiffusion equation with saturated actuation[END_REF][START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF] • Neumann boundary observations; [START_REF] Lhachemi | Finite-dimensional observer-based boundary stabilization of reaction-diffusion equations with either a Dirichlet or Neumann boundary measurement[END_REF] • input/output delayed boundary control; [START_REF] Lhachemi | Boundary output feedback stabilization of reaction-diffusion PDEs with delayed boundary measurement[END_REF][START_REF] Lhachemi | Predictor-based output feedback stabilization of an input delayed parabolic PDE with boundary measurement[END_REF] • nonlinearities [START_REF] Lhachemi | Local output feedback stabilization of a reactiondiffusion equation with saturated actuation[END_REF][START_REF] Lhachemi | Global output feedback stabilization of semilinear reaction-diffusion PDEs[END_REF][START_REF] Lhachemi | Nonlinear boundary output feedback stabilization of reaction diffusion PDEs[END_REF] • regulation problems. [START_REF] Lhachemi | PI regulation of a reaction-diffusion equation with delayed boundary control[END_REF] Proof. Consider again the Lyapunov function candidate defined by ( 52). The computation of its time derivative along the system solutions (32b) and ( 46) gives (53). Since [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF] -βζ(t) 2 ≥ 0 for any β > 0. Using this latter estimate into (53) and invoking Young's inequality as in (54) along with (48-49), we deduce that

ζ(t) = n≥N +1 Φ n (0)w n (t), we have by Cauchy-Schwarz inequality that ζ(t) 2 ≤ M 1,Φ n≥N +1 λ n w n (t) 2 hence βM 1,Φ n≥N +1 λ n w n (t)
V + 2δV ≤ X ζ Θ X ζ + 2γ n≥N +1 λ n Γ 3,n w n (t) 2 ≤ 0 where Γ 3,n = -1 -1 α λ n + q c + δ + βM1,Φ 2γ
≤ Γ 3,N +1 for all n ≥ N + 1. Hence the assumptions give V (t) ≤ e -2δt V (0) for all t ≥ 0. Proceeding as in the previous proof, we obtain the claimed estimate.

To complete the proof, it remains to show that one can always select N ≥ N 0 + 1 large enough, P 0, α > 1, and β, γ > 0, such that Θ 0 and Γ 3,N +1 ≤ 0. Owing to the Schur complement, Θ 0 is equivalent to F P + P F + 2δP + αγG + 1 β P LL P 0. Applying Lemma 1 to c F + δI, we have for any N ≥ N 0 + 1 the existence of P 0 such that F P + P F + 2δP = -I with P = O(1) as N → +∞. Moreover, we have (49) and L = √ 2 L with g and L that are independent of N . Hence, setting α = β = √ N and γ = N -1 , we obtain from ( 26) the existence of a sufficiently large integer N ≥ N 0 + 1 such that Θ 0 and Γ 3,N +1 ≤ 0. Example 6. Consider the Dirichlet boundary measurement setting described by (56). Let p = 1, q = 0, and q c = 3, giving an unstable open-loop system. To obtain the closed-loop exponential decay rate δ = 0.5, we set N 0 = 1. Then we run the following Python code. On lines 14-18, we compute the eigenvalues and eigenvectors of the problem. On lines 26-29, we check whether N 0 is selected adequately.

import numpy as np import control import scipy . integrate as integrate import cvxpy as cp import matplotlib as mpl import matplotlib . pyplot as plt from mpl_toolkits . mplot3d import Axes3D # Parameters of the PDE p = 1 q = 3 # this is q_c ( q is zero ) delta = 0.5 On line 36, we set define the number N of modes for the observer. Then we start building the matrices necessary to check the conditions of Theorem 7 after line 49. The control matrix K and the observation matrix L are chosen separately on lines 84 and 88. The feedback gain is K = -5.0058 -2.7748 , and the observer gain is L = 1.4373. The matrix inequalities in Theorem 7 are built after line 104. The Schur complement is used to rewrite (63) into a linear matrix inequality in the unknown variables as described in Remark 3. The time-evolutions of the control and output variables are given in Figure 7.

# E ig e ns tr uc t ur es

Saturated control with internal measurement

In this section we consider the stability analysis of parabolic PDEs when controlled in the presence of input saturations. In this setting, the control inputs apply in the domain by means of a bounded operator while the observation can take the form of either a bounded or an unbounded measurement operator. As in the previous section, the adopted approach relies on spectral-reduction methods. The presence of the input saturation is handled in the stability analysis by invoking the generalized sector condition reported in Section 2. This type of control design problem was reported in [START_REF] Mironchenko | Local stabilization of an unstable parabolic equation via saturated controls[END_REF] in the case of a state-feedback. We consider here the case of an output feedback by combining the Lyapunov-based analysis procedure discussed in the previous sections and the previously generalized sector condition. This allows the derivation of a set of sufficient conditions ensuring the local exponential stability of the origin of the closed-loop system. A subset of the domain of attraction is characterized by the decision variables of the abovementionned sufficient constraints.

Problem description

Let the reaction-diffusion equation with Robin boundary conditions described by

z t (t, x) = (p(x)z x (t, x)) x -(q(x) -q c )z(t, x) + m k=1 b k (x)u sat,k (t) (64a) cos(θ 1 )z(t, 0) -sin(θ 1 )z x (t, 0) = 0 (64b) cos(θ 2 )z(t, 1) + sin(θ 2 )z x (t, 1) = 0 (64c) z(0, x) = z 0 (x) (64d)
with measurement equation

y(t) = 1 0 c(x)z(t, x) dx. ( 65
)
Here we have

θ 1 , θ 2 ∈ [0, π/2], p ∈ C 1 ([0, 1]) with p > 0, q ∈ C 0 ([0, 1]
) with q ≥ 0, q c ∈ R, and b k ∈ L 2 (0, 1). The scalar control input u sat,k (t) ∈ R act on the system. Hence, ( 64) can be written as

z t (t, •) = -Az(t, •) + q c z(t, •) + m k=1 b k u sat,k (t) (66a) z(0, •) = z 0 ( 66b
)
where A is the Sturm-Liouville operator defined at the beginning of this section.

The control input is assumed to be subject to saturations; for a given vector s = s 1 s 2 . . . s m ∈ (R * + ) m , we define sat : R m → R m by ( 17). Hence, the input u sat,k (t) that is applied to the plant is expressed in function of the actual control inputs u k (t) as u sat (t) = sat(u(t)).

with

u sat (t) = u sat,1 (t) u sat,2 (t) . . . u sat,m (t) and
u(t) = u 1 (t) u 2 (t) . . . u m (t) .
In this context and similarly to 48 in the case of a state-feedback, the objective is to study the local stabilization of (64) with measurement (65) for the controller architecture studied in the first part of this section but in the presence of the saturating control inputs while estimating the associated domain of attraction.

Spectral analysis

Consider again the coefficients of projection z

n (t) = z(t, •), Φ n , b n,k = b k , Φ n ,
and c n = c, Φ n . As done for (31) without saturation, the projection of the system solutions (66) and the output equation ( 65) into the Hilbert basis {Φ n : n ≥ 1} gives the following representation:

żn (t) = (-λ n + q c )z n (t) + m k=1 b n,k u sat,k (t) (67a) y(t) = n≥1 c n z n (t) (67b)
Proceeding as in the previous subsection, we consider the feedback law taking the form of a finite-dimensional state-feedback coupled with a finitedimensional observer. More precisely, let δ > 0 and N 0 ≥ 1 be such that -λ n + q c < -δ for all n ≥ N 0 + 1. For a given integer N ≥ N 0 + 1 to be selected later, the controller architecture takes the form:

żn (t) = (-λ n + q c )ẑ n (t) + m k=1 b n,k u sat,k (t) (68a) + L n N k=1 c k ẑk (t) -y(t) , 1 ≤ n ≤ N u k (t) = N0 l=1 K k,l ẑl (t), 1 ≤ k ≤ m (68b) with L n , K k,l ∈ R where L n = 0 for N 0 + 1 ≤ n ≤ N .
We define the errors of estimation e n (t) = z n (t) -ẑn (t). As in the previous subsection, we introduce the vectors and matrices defined by ẐN0 = ẑ1 . . . ẑN0 , ẐN-N0 = ẑN0+1 . . . ẑN ,

E N0 = e 1 . . . e N0 , E N -N0 = e N0+1 . . . e N , A 0 = diag(-λ 1 +q c , . . . , -λ N0 +q c ), A 1 = diag(-λ N0+1 + q c , . . . , -λ N +q c ), B 0 = (b n,k ) 1≤n≤N0,1≤k≤m , B 1 = (b n,k ) N0+1≤n≤N,1≤k≤m , C 0 = c 1 . . . c N0 , C 1 = c N0+1 . . . c N , L = L 1 . . . L N0 , and K = (K k,l ) 1≤k≤m,1≤l≤N0
. This leads to

ŻN0 = A 0 ẐN0 + B 0 u sat -LC 0 E N0 -LC 1 E N -N0 -Lζ ĖN0 = (A 0 + LC 0 )E N0 + LC 1 E N -N0 + Lζ ŻN-N0 = A 1 ẐN-N0 + B 1 u sat ĖN-N0 = A 1 E N -N0 u = K ẐN0 where ζ(t) = n≥N +1 c n z n (t)
is the residue of measurement. Owing to the definition of the deadzone nonlinearity (20), we infer that

ŻN0 = (A 0 + B 0 K) ẐN0 -LC 0 E N0 -LC 1 E N -N0 -Lζ + B 0 φ(K ẐN0 ) ĖN0 = (A 0 + LC 0 )E N0 + LC 1 E N -N0 + Lζ ŻN-N0 = A 1 ẐN-N0 + B 1 K ẐN0 + B 1 φ(K ẐN0 ) ĖN-N0 = A 1 E N -N0 .
Introducing the state-vector

X = col( ẐN0 , E N0 , ẐN-N0 , E N -N0 )
and the matrices

F =     A 0 + B 0 K -LC 0 0 -LC 1 0 A 0 + LC 0 0 LC 1 B 1 K 0 A 1 0 0 0 0 A 1     , L =     -L L 0 0     , L φ =     B 0 0 B 1 0,     we deduce that Ẋ = F X + Lζ + L φ φ(K ẐN0 ). ( 69
)
We finally define E = I 0 0 0 and K = K 0 0 0 , which are such that ẐN0 = EX, u = KX.

Stability results

For z ∈ L 2 (0, 1) and ẑ ∈ R N , we define

Π(z, ẑ) =     Π 1 (z, ẑ) Π 2 (z, ẑ) Π 3 (z, ẑ) Π 4 (z, ẑ)     with Π 1 (z, ẑ) =    ẑ1 . . . ẑN0    , Π 2 (z, ẑ) =    z, Φ 1 -ẑ1 . . . z, Φ N0 -ẑN0    , and
Π 3 (z, ẑ) =    ẑN0+1 . . . ẑN    , Π 4 (z, ẑ) =    z, Φ N0+1 -ẑN0+1 . . . z, Φ N -ẑN    .
Stabilization in L 2 norm Let us now state and prove a result providing a stabilization for (64) in L 2 -norm. This result is extracted from. [START_REF] Lhachemi | Local output feedback stabilization of a reactiondiffusion equation with saturated actuation[END_REF] Theorem 8.

Let θ 1 , θ 2 ∈ [0, π/2], p ∈ C 1 ([0, 1]) with p > 0, q ∈ C 0 ([0, 1]
) with q ≥ 0, q c ∈ R, and s ∈ (R * + ) m . Let c ∈ L 2 (0, 1) and b k ∈ L 2 (0, 1) for 1 ≤ k ≤ m. Consider the reaction-diffusion system described by (64) with measured output (65). Let N 0 ≥ 1 and δ > 0 be given such that -λ n + q c < -δ < 0 for all n ≥ N 0 + 1. Assume that 1) for any 1 ≤ n ≤ N 0 , there exists 1 ≤ k = k(n) ≤ m such that b n,k = 0; 2) c n = 0 for all 1 ≤ n ≤ N 0 . Let K ∈ R m×N0 and L ∈ R N0 be such that A 1 + B 1 K and A 0 +LC 0 are Hurwitz with eigenvalues that have a real part strictly less than -δ < 0. Assume that there exist N ≥ N 0 + 1, a symmetric positive definite P ∈ R 2N ×2N , α, β, γ, µ, κ > 0, a diagonal positive definite T ∈ R m×m , and

G ∈ R m×N0 such that Θ 1 (κ) 0, Θ 2 0, Θ 3 (κ) ≤ 0 (70) where Θ 1 (κ) =   Θ 1,1,1 (κ) P L -E G T + P L φ -β 0 αγ m k=1 R N b k 2 L 2 I -2T   Θ 2 = P E (K -G) µ diag(s) 2 , Θ 3 (κ) = 2γ -λ N +1 + q c + κ + 1 α + β R N c 2 L 2 with Θ 1,1,1 (κ) = F P + P F + 2κP + αγ m k=1 R N b k 2 L 2 K K. Define the ellipsoid E 1 = (z, ẑ) ∈ L 2 (0, 1) × R N : Π N (z, ẑ) P Π N (z, ẑ) + γ R N z 2 L 2 ≤ 1 µ .
Then, the origin of the closed-loop system composed of the plant (64) with measured output (65) and the control law ( 68) is locally exponentially stable in L 2 -norm with exponential decay rate κ and with a basin of attraction including E 1 . More precisely, there exists M > 0 such that for any initial condition (z 0 , ẑ(0)) ∈ E 1 , the solution satisfies

z(t, •) 2 L 2 + N n=1 ẑn (t) 2 ≤ M e -2κt z 0 2 L 2 + N n=1 ẑn (0) 2 (71)
for all t ≥ 0. Moreover, for any fixed κ ∈ (0, δ], the constraints (70) are always feasible for N large enough.

Proof of Theorem 8. Let the Lyapunov function candidate be defined by V (X, z)

= X P X + γ n≥N +1 z, Φ n 2 for X ∈ R 2N and z ∈ L 2 (0, 1).
The computation of the time derivative of V along the system solutions to ( 67) and ( 69) gives V + 2κV = X F P + P F + 2κP X + 2X P Lζ

+ 2X P L φ (K ẐN0 ) + 2γ n≥N +1 (-λ n + q c + κ)z 2 n + 2γ n≥N +1 z n L b n KX + 2γ n≥N +1 z n L b n φ(K ẐN0 )
where L b n = b n,1 . . . b n,m . From Young's inequality, we obtain for any α > 0 and any

w ∈ R m that 2 n≥N +1 z n L b n w ≤ 1 α n≥N +1 z 2 n + α m k=1 R N b k 2 L 2 w 2 . Hence, introducing X = col(X, ζ, φ(K ẐN0 )), we deduce that V + 2κV ≤ X   Θ 1,1,1 P L P L φ 0 0 αγ m k=1 R N b k 2 L 2 I   X + 2γ n≥N +1 -λ n + q c + κ + 1 α z 2 n . Since, by definition, ζ = n≥N +1 c n z n , we obtain that ζ 2 ≤ R N c 2 L 2 n≥N +1 z 2 n . Moreover, if ẐN0 ∈ R N0 satisfies |(K -G) ẐN0 | ≤ s,
we deduce from (21) that φ(K ẐN0 ) T (φ(K ẐN0 ) + G ẐN0 ) ≤ 0. Combining the latter estimates, we obtain for all

X ∈ R 2N satisfying |(K -G)EX| ≤ s that V + 2κV ≤ X Θ 1 (κ) X + n≥N +1 Γ n z 2 n where Γ n = 2γ -λ n + q c + κ + 1 α + β R N c 2 L 2 ≤ Θ 3 (κ) for all n ≥ N + 1. Hence the assumptions imply that V + 2κV ≤ 0 for all X ∈ R 2N is such that |(K -G)EX| ≤ s.
We now need to give a sufficient condition such that |(K -G)EX| ≤ s holds. To do so, consider X ∈ R 2N and z ∈ L 2 (0, 1) such that V (X, z) ≤ 1/µ. Applying the Schur complement to Θ 2 0, we obtain that P

1 µ E (K -G) diag(s) -2 (K -G)E. This implies that diag(s) -1 (K -G)EX ≤ 1, giving in particular that |(K -G)EX| ≤ s hence V + 2κV ≤ 0.
From now it is easy to show that, for any initial condition selected such that (z 0 , ẑ0 ) ∈ E 1 with z 0 ∈ D(A), we have V (X(t), z(t, •)) ≤ 1/µ and V (X(t), z(t, •)) + 2κV (X(t), z(t, •)) ≤ 0 for all t ≥ 0. The claimed stability estimate (71) follows from the definition of V . The extension of this result to mild solutions associated with any (z 0 , ẑ0 ) ∈ E 1 follows from a classical density argument [53, Thm. 6.1.2].

The rest of the proof, which concerns the feasibility of the constraints, is reported in. [START_REF] Lhachemi | Local output feedback stabilization of a reactiondiffusion equation with saturated actuation[END_REF] Stabilization in H 1 norm The following result deals with the exponential stability of the system trajectories evaluated in H 1 -norm. Theorem 9. In the context of Theorem 8, we further assume that q > 0. Assume that there exist N ≥ N 0 + 1, a symmetric positive definite P ∈

R 2N ×2N , α > 1, β, γ, µ, κ > 0, a diagonal positive definite T ∈ R m×m , and G ∈ R m×N0 such that Θ 1 (κ) 0, Θ 2 0, Θ 3 (κ) ≤ 0 ( 72
)
where Θ 1 (κ) and Θ 2 are defined as in Theorem 8 while

Θ 3 (κ) = 2γ -1 - 1 α λ N +1 + q c + κ + β R N c 2 L 2 λ N +1 .
Define the ellipsoid

E 2 = (z, ẑ) ∈ D(A) × R N : Π(z, ẑ) P Π(z, ẑ) + γ R N A 1/2 z 2 L 2 ≤ 1 µ .

Saturated boundary stabilization of partial differential equations 57

Then, the origin of the closed-loop system composed of the plant (64) with measured output (65) and the control law ( 68) is locally exponentially stable in H 1 -norm with exponential decay rate κ and with a basin of attraction including E 1 . In other words, there exists M > 0 such that for any initial condition (z 0 , ẑ(0)) ∈ E 1 , the solution satisfies

z(t, •) 2 H 1 + N n=1 ẑn (t) 2 ≤ M e -2κt z 0 2 H 1 + N n=1 ẑn (0) 2
for all t ≥ 0. Moreover, for any fixed κ ∈ (0, δ], the constraints (72) are always feasible for N large enough.

Proof of Theorem 9. We introduce the Lyapunov functional candidate

V (X, z) = X P X + γ n≥N +1 λ n z, Φ n 2 when X ∈ R 2N and z ∈ D(A).
The computation of the time derivative of V along the system solutions to ( 67) and ( 69) gives V + 2κV = X F P + P F + 2κP X + 2X P Lζ

+ 2X P L φ φ(K ẐN0 ) + 2γ n≥N +1 λ n (-λ n + q c + κ)z 2 n + 2γ n≥N +1 λ n z n L b n KX + 2γ n≥N +1 λ n z n L b n φ(K ẐN0 )
where L b n = b n,1 . . . b n,m . Invoking Young's inequality, we obtain for any α > 0 and any

w ∈ R m that 2 n≥N +1 λ n z n L b n w ≤ 1 α n≥N +1 λ 2 n z 2 n + α m k=1 R N b k 2 L 2 w 2 . Let X = col(X, ζ, φ(K ẐN0 )).
Proceeding as in the proof of Theorem 8, we deduce that

V + 2κV ≤ X Θ 1 (κ) X + n≥N +1 λ n Γ n z 2 n for all X ∈ R 2N satisfying |(K -G)EX| ≤ s and where Γ n = 2γ -1 -1 α λ n + q c + κ + β R N c 2 L 2 λn ≤ Θ 3 (κ) for all n ≥ N + 1.
The proof now follows similar arguments that the ones employed in the proof of Theorem 8. Remark 6. The conditions in Theorem 8 and in Theorem 9 are nonlinear in the unknown variables, due to, in particular the product G T . Some nonlinearity could be transformed into linear conditions as for the variable α, as discussed in Remark 3. To deal with the particular case m = 1, or to deduce convex constraints from these theorems, see. [START_REF] Lhachemi | Local output feedback stabilization of a reactiondiffusion equation with saturated actuation[END_REF]

Section conclusion

This section has discussed the topic of output feedback stabilization of a reaction-diffusion equation by means of in-domain or boundary control inputs. The controllers that are considered in this section are output feedback laws where the output is defined from a boundary measure or an internal measurement of the state. The control strategy takes the form of a finite-dimensional controller composed of an observer coupled with a finite-dimensional partial state-feedback. The control can be either linear or subject to a saturation map. In the latter scenario, only a local asymptotic stability can be obtained in general along with an estimation of the basin of attraction. The reported stability analysis takes advantage of Lyapunov functionals coupled with the generalized sector condition that has been recalled in Section 2 in the context of finite-dimensional systems. The obtained sets of constraints ensuring the stability of the closed-loop system take an explicit form and have been shown to be feasible when the order of the controller is selected large enough. An explicit subset of the domain of attraction of the closed-loop system has also been derived.

Stabilization of wave and KdV equations

Two classes of particular equations are considered in this section: first the wave equation and then the Korteweg-de-Vries (KdV) equation. We particularly focus on the boundary stabilization problem. The interest of the first equation is that it gives a transition towards the boundary control of general hyperbolic systems, whereas the second one allows to show perspectives in terms of stabilization of nonlinear partial differential equations, and give a highlighting example of what could be done for boundary control of other classes of hyperbolic PDEs (as considered e.g. in [START_REF] Prieur | Stabilization of a 1-D tank containing a fluid modeled by the shallow water equations[END_REF]75 ).

For both equations, we solve the common objectives of well-posedness assessment and asymptotic stabilization by means of of distributed or boundary control that can be either linear or subject to a nonlinear map (e.g., a saturation).

This section is organized as follows. First, in Section 4.1, the stabilization of the linear wave equation with linear and with nonlinear in-domain control is presented. The topic of boundary control is then considered for the same equation. Finally the nonlinear KdV equation is considered in Section 4.2 with in-domain control. This result is illustrated with some numerical simulations.

Wave equation with a bounded control operator

Motivated by the illustration depicted in Figure 8, where z stands for the deflection of a membrane with respect to the rest and horizontal axis and that is subject to a distributed force u, we start this section by considering the following wave equation:

u(x, t) z(x, t) x = 0 x = 1
z tt (t, x) = z xx (t, x) + u(t, x), ∀t ≥ 0 , x ∈ (0, 1), (73)
We assume that the membrane is clamped at both extremities. This implies the following boundary conditions, for all t ≥ 0, z(t, 0) = 0 , z(t, 1) = 0 .

The initial condition are given, for all x ∈ (0, 1), by

z(0, x) = z 0 (x) , z t (0, x) = z 1 (x) , (75)
where z 0 and z 1 stand for the initial deflection and the initial deflection speed, respectively. Let us note that the function defined by z(t, x) = 0, for all (t, x) in (0, 1) × [0, ∞) is a particular solution to (73) and (74) in the uncontrolled scenario (u = 0). Hence the orgin is an equilibrium for the studied wave equation. The objective is to render this equilibrium asymptotically stable by designing an adequate feedback control u.

Internal linear control

Let us define the linear control by

u(t, x) = -az t (t, x), t ≥ 0 , x ∈ (0, 1), (76) and consider
V 1 = 1 2 (z 2 x + z 2 t )dx. (77)
A formal computation gives, along the solutions to (73), ( 74) and ( 76), V1 =

1 0 (z x z xt -az 2 t + z t z xx )dx = - 1 0 az 2 t dx + [z t z x ] x=1 x=0 = - 1 0 az 2 t dx Thus, if a > 0, V 1 is a (non strict) Lyapunov function.
Using standard technics, such as Lumer-Phillips thereom for the wellposedness (see e.g., [14, Theorem A.4.]) and Huang-Prüss theorem for the exponential stability (see [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and 58 ), we may prove the following result: Theorem 10. For a > 0 and (z 0 , z 1 ) ∈ H 1 0 (0, 1) × L 2 (0, 1), there exists a unique (weak) solution z: [0, ∞) → H 1 0 (0, 1) × L 2 (0, 1) to ( 73)-(76). Moreover, the origin of H 1 0 (0, 1) × L 2 (0, 1) is an exponentially stable equilibrium, that is there exist two positive values C and µ > 0 such that, for any initial condition (z 0 , z 1 ) ∈ H 1 0 (0, 1) × L 2 (0, 1), it holds, for all t ≥ 0, z H 1 0 (0,1) + z t L 2 (0,1) ≤ Ce -µt ( z 0

H 1 0 (0,1) + z 1 L 2 (0,1) ).
Proof of Theorem 10. Let us first prove the well-posedness. Let A l be the linear unbounded operator

A l f g = g f xx -ag
with the domain D(A l ) = (H 2 (0, 1) ∩ H 1 0 (0, 1)) × H 1 0 (0, 1). This domain is dense is H 1 0 (0, 1) × L 2 (0, 1), and the operator A l is closed. Let us rewrite (73)-(76) as

∂ ∂t z z t = A l z z t , z z t (t = 0, •) = z 0 z 1 ( 78
)
Our objective is to prove that A l generates a contraction semigroup T l (t), that is the solution of ( 78) is T l (t) z 0 z 1 and satisfies

T l (t) z 0 z 1 2 ≤ z 0 z 1 2 , ∀t ≥ 0. ( 79
)
Informally, one can try to prove (79) by differentiating the right-handside with respect to the time. Using

z 0 z 1 2 = z 0 z 1 , z 0 z 1 , we get d dt T l (t) z 0 z 1 2 = A l T l (t) z 0 z 1 , z 0 z 1 + z 0 z 1 , A l T l (t) z 0 z 1 = 2Re A l T l (t) z 0 z 1 , z 0 z 1
where Re denotes the real part. This gives, at time t = 0,

d dt T l (t) z 0 z 1 2 (t = 0, •) = 2Re A l z 0 z 1 , z 0 z 1
This formal computation tends to show that in order to obtain (79), a necessary condition is to have Re A l z 0 z 1 , z 0 z 1 ≤ 0. This condition is one of the two key elements of the Lumer-Phillips theorem which provides a characterization of the unbounded operators generating a contraction semigroup. Specifically, in order to apply the Lumer-Phillips theorem, we need to show that the two following points hold true:

(1) Re A l z 0 z 1 , z 0 z 1 ≤ 0, for all z 0 z 1 in D(A l ) (2) there exists λ > 0 such that Ran(I -λA l ) = H 1 0 (0, 1) × L 2 (0, 1), where Ran is the range set. Under these two conditions, the unbounded operator A l generates a semigroup of contraction and the Cauchy problem (78) is well-posed for strong and weak solutions as considered in Theorem 10.

Even if we do not give here a complete proof of these both properties, note that the interest of the second item is that it replaces the timedependent Cauchy problem (78) by

∀ f g ∈ H 1 0 (0, 1) × L 2 (0, 1), find f g ∈ D(A l ) such that (I -λA l ) f g = f g
which is a stationary Cauchy problem of a linear ODE with prescribed boundary conditions.

Let us now sketch the proof of the exponential stability. According to Huang-Prüss theorem (see [START_REF] Huang | Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces[END_REF] and 58 ), it is sufficient to check the two conditions

iR ⊂ ρ(A), (80) sup β∈R
(iβ -A l ) -1 < ∞. (81)
Inspired by, [START_REF] Gall | Exact controllability and output feedback stabilization of a bimorph mirror[END_REF] let us prove these both properties succesively. To prove (80), we argue by contradiction, assuming the existence of an eigenvalue of A l of the form iβ. Pick

f g in D(A l ) \ {0} such that (iβ -A l ) f g = 0. Then 0 = (iβ -A l ) f g , f g H 1 0 (0,1)×L 2 (0,1) , (82)
= iβ(

1 0 |f | 2 dx + 1 0 |g| 2 dx) + a 1 0 |g| 2 dx. ( 83
)
Thus, inspecting the real part of the previous equation, with a = 0, we get g = 0. Moreover, inspecting the imaginary part, we get f = 0 which gives f = 0 using the definition of D(A l ) and the boundary conditions of f . This is a contradiction with f g = 0. Therefore (80) holds.

Let us now prove (81), by proceeding again with a contradiction. If 81 is false, then there exists a sequence (β n ) n∈N and a sequence

f n g n n∈N in D(A l ) such that f n g n n∈N H 1 0 (0,1)×L 2 (0,1) = 1 , (84)
β n → n→∞ +∞ (85)
and fn gn H 1 0 (0,1)×L 2 (0,1)

→ n→∞ 0 ( 86
)
where fn gn

= (iβ n -A l ) f n g n n∈N . We compute fn gn , f n g n H 1 0 (0,1)×L 2 (0,1) = (iβ n -A l ) f n g n , f n g n H 1 0 (0,1)×L 2 (0,1) = iβ n ( 1 0 |f n | 2 dx + 1 0 |g n | 2 dx) + a 1 0 |g n | 2 dx.
Therefore, with (84) and ( 86), inspecting the imaginary part in the last equation,we get

β n ( 1 0 f 2 n dx + 1 0 g 2 n dx) → n→∞ 0 ,
thus with (84), we get β n → 0 which is a contradiction with (85). Therefore (81) holds. This concludes the proof of the exponential stability and of the proof of Theorem 10.

Internal saturating control

We now study the nonlinear control u(t, x) = -sat(az t (t, x)), x ∈ (0, 1), ∀t ≥ 0, (87

)
where sat is the nonlinear function defined in (17) with m = 1 and level s 0 . Following the terminology of, [START_REF] Marx | Global stabilization of a Korteweg-De Vries equation with saturating distributed control[END_REF] we call this nonlinearity the localized saturated map. The wave equation (73) in closed loop with the control (87) gives the dynamics

z tt = z xx -sat(az t ) (88)
A formal computation of the time derivative of V 1 defined by (77) along the solutions to the wave PDE (88) with boundary conditions (74) gives

V1 = - 1 0 z t sat(az t )dx.
Hence, in order to conclude on the possible stability of the closed-loop system, one needs to handle the nonlinearity z t sat(az t ). Note that other choices of saturation mechanisms can also be considered instead of the localized saturation studied in (87). For instance, papers [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF] and 32 deal with the L 2 saturation denoted by sat L2 and defined for any σ ∈ L 2 (0, 1) by

sat L2 (σ)(x) = σ(x) if σ L 2 (0,1) < 1 σ(x) σ L 2 (0,1) , else (89)
Even if all the different saturation mechanisms are of interest, we focus here on the localized saturation used in (87), which is generally more relevant from a physical point of view and in practical applications.

The well-posedness of the nonlinear PDE (88), which is borrowed from, [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] is assessed by the following theorem.

Theorem 11. For all a ≥ 0 and (z 0 , z 1 ) in (H 2 (0, 1)∩H 1 0 (0, 1))×H 1 0 (0, 1), there exists a unique solution z: [0, ∞) → H 2 (0, 1) ∩ H 1 0 (0, 1) to (88) with the boundary conditions (74) and the initial condition (75).

Proof of Theorem 11. We only provide a sketch of the proof reported in. [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] Consider the nonlinear operator

A 1 f g = g f xx -sat(ag)
with the domain D(A 1 ) = (H 2 (0, 1) ∩ H 1 0 (0, 1)) × H 1 0 (0, 1). We are want to invoke here a generalization of the Lumer-Phillips theorem, which is the so-called Crandall-Liggett theorem. A precise statement of this theorem can be found; 4 see also [START_REF] Brezis | Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert[END_REF] and. [START_REF] Miyadera | Nonlinear Semigroups. Translations of mathematical monographs[END_REF] To apply this theorem, two conditions need to be checked:

(1) A 1 is dissipative, that is for any two elements of D(A 1 ),

Re A 1 f g -A 1 f g , f g - f g ≤ 0 (2) For all λ > 0, D(A 1 ) ⊂ Ran(I -λA 1 )
Let us prove the first item. To do that, given f g and f g in

H 1 0 (0, 1) × L 2 (0, 1),in H 1 0 (0, 1) × L 2 (0, 1), we denote ∆ = Re A 1 f g -A 1 f g , f g - f g ≤ 0 .
Let us check that ∆ ≤ 0. Using the definition of A 1 and of the hermitian product in H 1 0 (0, 1) × L 2 (0, 1), we compute

∆ = Re 1 0 (g x (x) -gx (x))(f x (x) -fx (x))dx + 1 0 (f xx (x) -fxx (x))(g(x) -g(x))dx -Re 1 0 (sat(a g(x)) -sat(a g(x)))(g(x) -g(x))dx , = -Re 1 0
(sat(a g(x))sat(a g(x)))(g(x) -g(x))dx .

Note that, for all a ≥ 0 and for all (s, s) in C × C, Re (sat(a s)sat(a s))(s -s) ≥ 0.

Thus A 1 is dissipative.

The second item requires to deal with a nonlinear ODE. To be more specific, let λ > 0 and f g ∈ H 1 0 (0, 1) × L 2 (0, 1) be arbitrarily given. Our objective is to find

f g ∈ D(A 1 ) such that (I -λA 1 ) f g = f g , that is f -λg = f , g -λ( fxx -sat(a g)) = g ,
Using the first identity to express g in function of f and f , we only have to find f such that fxx -

1 λ 2 f -sat( a λ ( f -f )) = - 1 λ g - 1 λ 2 f f (0) = f (1) = 0
holds. The existence of a solution to this nonhomogeneous nonlinear ODE with two boundary conditions is provided by the following lemma. Lemma 2. For any a ≥ 0 and λ > 0, there exists f solution to

fxx -1 λ 2 f -sat( a λ ( f -f )) = -1 λ g -1 λ 2 f f (0) = f (1) = 0 (90)
To prove this lemma, let us introduce the mapping:

T : L 2 (0, 1) → L 2 (0, 1) , y → z ,
where z = T (y) is the unique solution to

z xx -1 λ 2 z = -1 λ v -1 λ 2 u + sat( a λ (y -u)) , z(0) = z(1) = 0 .
It can be proven that T is a well defined mapping. Then, it is possible to invoke the Schauder fixed-point theorem (see e.g., [START_REF] Coron | Control and Nonlinearity[END_REF] ) to deduce the existence of y such that T (y) = y. After doing so, we obtain that f = y solves (90)

After having assessed the well-poseness of the closed-loop system dynamics, we can focus on the study of its stability. The global asymptotic stability of this nonlinear PDE is stated in the following result.

Theorem 12. For all a > 0, the origin of the PDE (88) with the boundary conditions (74) is globally asymptotically stable. More specifically, for all (z 0 , z 1 ) in (H 2 (0, 1) ∩ H 1 0 (0, 1)) × H 1 0 (0, 1), the solution to (88) with the boundary conditions (74) and the initial condition (75) satisfies, ∀t ≥ 0, z(t, .) H 1 0 (0,1) + z t (t, .) L 2 (0,1) ≤ z 0

H 1 0 (0,1) + z 1 L 2 (0,1)
together with the attractivity property z(t, .) H 1 0 (0,1) + z t (t, .) L 2 (0,1) → 0, as t → ∞ . Proof of Theorem 12. Due to Theorem 11, the formal computation of the time derivative of V 1 previously computed is rigorously justified. Hence we have V1 = -1 0 z t sat(az t )dx. This is a weak Lyapunov function because V1 ≤ 0 which guarantees the stability of the origin. In order to prove the attractivity of the origin, we are going to invoke LaSalle's Invariance Principle [17, Chapter 11] for infinite-dimensional systems. To apply LaSalle's Invariance Principle, we have to check that the set of solutions is precompact. This result can be obtained here by following the approach reported in [START_REF] Dafermos | Asymptotic behavior of nonlinear contraction semigroups[END_REF][START_REF] Novel | Feedback stabilization of a hybrid PDE-ODE system: Application to an overhead crane[END_REF] and relies on the following lemma (see below for a sketch of proof). Lemma 3. The canonical embedding from D(A 1 ), equipped with the graph norm, into H 1 0 (0, 1) × L 2 (0, 1) is compact. for all t ≥ 0 and with the initial condition

u(t) z(x, t) x = 0 x = 1
z(0, x) = z 0 (x) , z t (0, x) = z 1 (x) . ( 93
)
for all x in (0, 1). We define the linear control

u(t) = -bz t (t, 1), x ∈ (0, 1), ∀t ≥ 0 (94)
and we consider

V 2 = 1 2 (e µx (z t + z x ) 2 dx + (e -µx (z t -z x ) 2 dx.
A formal computation of the time derivative along the solutions to (91), (92) and (94) gives

V2 = -µV 2 + 1 2 e µ (1 -b) 2 -e -µ (1 + b) 2 z 2 t (t, 1)
Assuming that b > 0 and letting µ > 0 such that d e µ (1 -b) 2 ≤ e -µ (1 + b) [START_REF] Bacciotti | Liapunov functions and stability in control theory[END_REF] , it holds that V2 ≤ -µV 2 . Hence V 2 is a strict Lyapunov function and thus the origin of (91) with boundary conditions (92) and command (94) is exponentially stable.

A boundary saturating control

Let us consider now the nonlinear control u(t) = -sat(bz t (t, 1)), for all t ≥ 0. The boundary conditions become:

z(t, 0) = 0 , z x (t, 1) = -sat(bz t (t, 1)) . (95)
Inspired by [14, Sec. 2.4], we introduce H 1 (0) (0, 1) = {u ∈ H 1 (0, 1), u(0) = 0} and u H 1 (0) (0,1) =

= -µV 2 + e µ (σ -sat(bσ)) 2 -e -µ (σ + sat(bσ)) 2 ≤ -µV 2 + e µ ((1 -b)σ -φ) 2 -e -µ ((1 + b)σ + φ) 2 -2φ(φ + cσ) ≤ -µV 2 + σ φ M σ φ where M(µ, c) = e µ (1 -b) 2 -e -µ (1 + b) 2 -e µ (1 -b) -e -µ (1 + b) + c -2 + e µ -e -µ .
In particular we have at µ = 0 that

M(0, c) = -4b -2 + c -2 .
We have to select c close to b such that M(0, c) is symmetric semi-definite negative. Of course, c = b is not convenient since M(0, c) is not semidefinite negative (moreover the choice c = b would yield the global section condition which does not hold, confirming that the choice c = b is not suitable for c). But c < b and close to b exists such that det(M) > 0. Thus M < 0.

Given r > 0, we consider initial condition such that A 2 z(., 0) z t (., 0) ≤ r. This implies, for a suitable c ensuring that (b -c)|z t (t, 1)| ≤ 1 for all t ≥ 0, that V2 ≤ -µV 2 . The semi-global exponential stability follows.

Note here that the exponential stability is only achieved on bounded sets of initial conditions. An open question is whether we have (or not) the global exponential stability of the origin of the PDE (91) with the boundary conditions (95).

KdV equation with a bounded control operator

In the previous section we reviewed two classes of controllers for the linear wave equation with linear and nonlinear feedback. Different methods for proving asymptotic stability have been reported, one using LaSalle's Invariance Principle and another one establishing semi-global exponential stability based on a local sector condition. In this section, we move to a control problems for a nonlinear PDE. Specifically, let us consider the following nonlinear Korteweg-de-Vries (KdV) PDE:

   z t + z x + z xxx + zz x + u = 0, x ∈ [0, L], t ≥ 0, z(t, 0) = z(t, L) = z x (t, L) = 0, t ≥ 0, z(0, x) = z 0 (x), x ∈ [0, L], (99)
where z stands for the state and u for the control.

As shown in, [START_REF] Rosier | Exact boundary controllability for the korteweg-de vries equation on a bounded domain[END_REF] in the uncontrolled scenario (u = 0) and for a length L of the spatial domain such that

L ∈ 2π k 2 + kl + l 2 3 k, l ∈ N * , (100)
there exist solutions of the linearized version of (99) for which the L 2 (0, L) norm of the state does not decay to zero. This can be observed, for instance, in the particular case for the first critical length L 1 = 2π (obtained by letting k = l = 1 in (100)) by considering the initial condition z 0 (x) = 1 -cos(x) for all x ∈ [0, L]. Let us denote the second critial by L 2 = 2π 7 3 (obtained by letting k = l = 1 in (100))We refer the reader to the papers [START_REF] Cerpa | Exact controllability of a nonlinear korteweg-de vries equation on a critical spatial domain[END_REF][START_REF] Cerpa | Boundary controllability for the nonlinear korteweg-de vries equation on any critical domain[END_REF][START_REF] Rosier | Exact boundary controllability for the korteweg-de vries equation on a bounded domain[END_REF] for controllability results of (99) and the role of the so-called critical lengths (100).

In these notes we are interested in the stabilization problem of the origin of the KdV. We refer the reader to [START_REF] Tang | Stabilization of linearized korteweg-de vries systems with anti-diffusion[END_REF] for the stabilization of the origin of the linearized KdV equation with anti-diffusion. In 9 and in, 54 localized damping are considered for the linearized KdV equation. Specifically, when setting linear control u = a(x)z, for a non-negative continuous function a

: [0, 1] → R, we obtain    z t + z x + z xxx + a(x)z = 0, x ∈ [0, L], t ≥ 0, z(t, 0) = z(t, L) = z x (t, L) = 0, t ≥ 0, z(0, x) = z 0 (x), x ∈ [0, L] (101)
The following theorem is proven in. [START_REF] Menzala | Stabilization of the korteweg-de vries equation with localized damping[END_REF] Theorem 14. The following results hold true for (101).

• When L is not a critical length (i.e., (100) does not hold) and a ≡ 0, the origin of ( 101) is asymptotically stable. To be more specific, there exist M and µ such that

z(t) L 2 ≤ M e -µt z(0) L 2 .
• When a > 0 on an non-empty subset of [0, L], then the same conclusion holds.

Let us now shortly review the stabilization results of the origin of the nonlinear KdV PDE (99) when using a control given by u = a(x)z. The papers [START_REF] Chu | Asymptotic stability of a nonlinear korteweg-de vries equation with critical lengths[END_REF][START_REF] Menzala | Stabilization of the korteweg-de vries equation with localized damping[END_REF][START_REF] Tang | Asymptotic stability of a korteweg-de vries equation with a two-dimensional center manifold[END_REF] consider the following closed-loop dynamics:

   z t + z x + z xxx + zz x + a(x)z = 0, x ∈ [0, L], t ≥ 0, z(t, 0) = z(t, L) = z x (t, L) = 0, t ≥ 0, z(0, x) = z 0 (x), x ∈ [0, L], (102)
The following theorem summarizes some of the contributions contained in these papers (see [START_REF] Chu | Asymptotic stability of a nonlinear korteweg-de vries equation with critical lengths[END_REF] and 68 for the proof of the first item, respectively for L = L 1 and L = L 2 , and see [START_REF] Menzala | Stabilization of the korteweg-de vries equation with localized damping[END_REF] for the proof of the second item).

Theorem 15. The following results hold true for (102).

• When L = L 1 or L = L 2 and a ≡ 0, the origin of (99) is locally asymptotically stable. More precisely e , there exist r > 0, M > 0, and µ > 0 such that the solutions to (102) issuing from z(0) with z(0) L 2 ≤ r satisfy

z(t) L 2 ≤ M e -µt z(0) L 2
• For all L > 0, when a > 0 on an non-empty subset of [0, L], then the origin of (102) is globally asymptotically stable. More precisely f , for all r > 0, there exist M > 0, and µ > 0 such that

z(t) L 2 ≤ M e -µt z(0) L 2

Saturating control for KdV

Let us now consider the case of a saturating control. To symplify the presentation, we will consider the case where the function a(x) in ( 102) is a constant denoted by a. The localized control is subject to a saturation map. To be more specific, let the KdV equation controlled by a saturated distributed control be described by

       z t + z x + z xxx + zz x + sat(az) = 0, z(t, 0) = z(t, L) = 0, z x (t, L) = 0, z(0, x) = z 0 (x). ( 103
)
where sat is the saturation map defined in (17) with m = 1, and with level s. The corresponding nonlinear equation ( 103) is studied in. [START_REF] Marx | Global stabilization of a Korteweg-De Vries equation with saturating distributed control[END_REF] The case of L 2 saturation, defined in (89), is also considered. In these notes we focus on the nonlinear equation (103), but some numerical simulations will also be performed with the L 2 saturation in the next numerical example.

The well-posedness result is proven in [START_REF] Marx | Global stabilization of a Korteweg-De Vries equation with saturating distributed control[END_REF] by proving first existence of solution for small time following the approach of, [START_REF] Chapouly | Global controllability of a nonlinear korteweg-de vries equation[END_REF][START_REF] Rosier | Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain[END_REF] and then removing the smallness property of the time existence using a priori estimates. It yields the following theorem.

Theorem 16. For any initial condition z 0 ∈ L 2 (0, L), there exists a unique solution

z ∈ C([0, T ]; L 2 (0, L)) ∩ L 2 (0, T ; H 1 (0, L)) to (103).
The global asymptotic stability of the origin, which is also proven in the same paper, can be stated as follows.

Theorem 17. The origin of (103) is globally asymptotically stable. More precisely there exist µ > 0 and a class K function g α : [0, ∞) → [0, ∞) such that for any z 0 ∈ L 2 (0, 1), any solution z to (103) satisfies, for all t ≥ 0,

z(t) L 2 (0,1) ≤ α( z 0 L 2 (0,1) )e -µt .
This result is proved by following the approaches of [START_REF] Cerpa | Control of a Korteweg-de Vries equation: a tutorial[END_REF][START_REF] Rosier | Global stabilization of the generalized Korteweg-de Vries equation posed on a finite domain[END_REF] by showing that the origin of (103) is semi-globally exponentially stable. Proposition 1. For any given r > 0, there exist positive values C and µ such that for all initial condition z 0 satisfying z 0 L 2 (0,L) ≤ r, the solution to (103) satisfies, for all t ≥ 0,

z(t) L 2 (0,L) ≤ C z 0 L 2 (0,L) e -µt .
Proof of Proposition 1. To prove this proposition, a key result is the following claim.

Claim 3. For all T > 0 and r > 0, there exists C > 0 such that for any solution z to (103) starting from z 0 ∈ L 2 (0, L) with z 0 L 2 (0,L) ≤ r, it g A class K function is a continuous and increasing function that is zero at zero. holds

z 0 2 L 2 (0,L) ≤ C T 0 |z x (t, 0)| 2 dt + 2 T 0 L 0 sat(az)zdtdx . (104)
Assume Claim 3 holds for the time being. Then with (104) it holds

z(t, .) 2 L 2 (0,L) = z 0 2 L 2 (0,L) - T 0 |z x (t, 0)| 2 dt -2 T 0 L 0 sat(az)zdxdt we get z(., kT ) 2 L 2 (0,L) ≤ γ k z 0 2 L 2 (0,L) ∀k ≥ 0
where γ ∈ (0, 1). From the dissipativity property, we have z(t, .) L 2 (0,L) ≤ z(., kT ) L 2 (0,L) for kT ≤ t ≤ (k + 1)T . Thus we obtain, for all t ≥ 0,

z(t, .) 2 L 2 (0,L) ≤ 1 γ z 0 L 2 (0,L) e log γ T t
We conclude the proof of the semi-global exponential stability, as stated in Proposition 1.

Let us now prove Claim 3 that has been used in the proof of Proposition 1.

Proof of the Claim 3. We prove (104) by contradiction Assume that there exists a sequence of solution z n to (103) with

z n (., 0) L 2 (0,L) ≤ r (105)
and such that

lim n→+∞ z n 2 L 2 (0,T ;L 2 (0,L)) T 0 |z n x (t, 0)| 2 dt + 2 T 0 L 0 sat(az n (t, x))z n (t, x)dtdx = +∞.
(106) By dissipativity property, there exists β > 0 such that sup

t∈[0,T ] z n (t, .) L 2 (0,L) ≤ r , sup x∈[0,L] T 0 |z n (t, x)| 2 dt ≤ β. ( 107
) Now let us define Ω i := t ∈ [0, T ], sup x∈[0,L] |z(t, x)| > i ⊂ [0, T ]. We have β ≥ T 0 sup x∈[0,L] |z n (t, x)| 2 dt ≥ Ωi sup x∈[0,L] |z n (t, x)| 2 dt ≥ i 2 ν(Ω i ),
Therefore, denoting the Lebesgue mesure by ν, and the complementary set of Ω c i ) by ν(Ω c i ), we obtain, with (107), ν(Ω i ) ≤ β i 2 , and thus ν(Ω c i ) ≥ max T -β i 2 , 0 . Let us note that denoting, k(i) = min( s ai , 1), for each i in N, it holds for all z in Ω c i , |z| ≤ i, and thus h (sat(az) -k(i)az)z ≥ 0 (108)

Moreover, using again the local sector condition, we have

T 0 L 0 sat(az n )z 2 dtdx = Ωi L 0 sat(az n )z n dtdx + Ω c i L 0 sat(az n )z n ≥ 0 + Ω c i L 0 ak(i)(z n ) 2 dtdx. ( 109
)
where (108) has been used in the last inequality. Thus, with (105), for all i in N \ {0},

z n (t, .) 2 L 2 (0,L) ≤ z n (., 0) 2 L 2 (0,L) - T 0 |z n x (t, 0)| 2 dt -2 Ω c i L 0 ak(i)(z n ) 2 dtdx.
Let λ n := z n L 2 (0,T ;L 2 (0,L)) and v n (t, x) = z n (t,x) λ n . Due to (105), up to extracting a subsequence, we may assume that λ n → λ ≥ 0. Due to (106) and (109), we have, for all i ∈ N T 0 |v n x (t, 0)| 2 dt + 2

Ω c i L 0 ak(i)(v n ) 2 dtdx → 0 ( 110
)
Using Aubin-Lions lemma in, [START_REF] Simon | Compact sets in the space l p (o, t; b)[END_REF] we get {v n } n∈N converges strongly in L 2 (0, T ; L 2 (0, L)). Thus, with (110), we have, for all i ∈ N v x (t, 0) = 0, ∀t ∈ (t, 0) and v(t, x) = 0, ∀x ∈ [0, L], ∀t ∈ Ω c i . We know that ν i∈N Ω c i = T . We get a contradiction with v L 2 (0,T ;L 2 (0,L)) = 1. This concludes the proof of Claim 3. h To prove (108), assume first that ai ≤ s, then k(i) = 1, and sat(az) = az, which gives (108). Second, if ai > s and sat(az) = az, then 1 -k(i) > 0 and (sat(az) -k(i)az) = (1 -k(i))az, which gives (108). Third, if ai > s and az > s, then (sat(az) -k(i)az) = s -s ai az = s(1 -i z ) ≥ 0, which gives (108). The fourth case ai > s and az < -s is studied in a simular way as the third one. On line 14, we set the first critical length L = 2π and the initial condition z 0 (x) = 1 -cos(x) is chosen on line 32 so that its energy is constant along the linearized KdV equation, without any control. The function a is chosen as the constant value 1 on line 15, and the level of the saturation map is set at 0.5 on line 27. The space and time discretization steps are selected respectively at lines 20 and 23. The localized saturation sat and the L 2 saturation sat L2 are defined after line 37 and line 46 respectively. Between lines 55 and 63 the initialization of the state and of its norm for both the linear control (thus with (102)) and the saturated control (thus with (103) with either the saturation map sat or with sat L2 . To discretize (103) and (102), we follow the approach of, [START_REF] Fernando Pazoto | Uniform stabilization of numerical schemes for the critical generalized kortewegde vries equation with damping[END_REF] and solve, at each time-step, a fixed point problem. No proof of convergence of the numerical scheme is garanteed in the context of (103), since another nonlinearity is considered in. [START_REF] Fernando Pazoto | Uniform stabilization of numerical schemes for the critical generalized kortewegde vries equation with damping[END_REF] In particular the term z xxx is discretized as follows It yields two discretizations, for respectively the equations ( 102) and ( 103), between lines 66-93 and lines 95-123 respectively. It asks to solved a fixed point problem that is solved using a iteration scheme wit 100 steps (see after lines 85 and 115). The choice of the saturation map (either sat or sat L2 is made on line 121). In the python code given here, sat is considered. The discretization in time is done after line 125, where an Euler scheme is used. The figures are drawn after line 135. It yields Figures 7 and7 where the time-evolutions of the solutions to (103) and to (102) are respectively given. It is observed that the solutions converge to the origin, as predicted in Theorem 17 and the second item of Theorem 15.

D + D + D -z i

Conclusion so far

In this section we have reviewed the well-posedness and the asymptotic stability of the origin of the wave equation and of the Korteweg-de-Vries equation in presence of (possibly saturating) control. Different proofs have

Conclusion

This chapter has reviewed some recent results on stability analysis of distributed parameter systems as those modeled by parabolic partial differential equations, or the wave equation or the Korteweg-de-Vries equation. The suggested approach succeeds to design boundary stabilizing controllers, possibly subject to amplitude constraint, ensuring an asymptotic stability of the closed-loop equation. The constructive approach is based on Lyapunov function, and numerically tractable conditions. Some simulations have illustrated our results and design methods. More recent works follow the present chapter as the control of reaction-diffusion equation coupled with ordinary differential equations (see [START_REF] Lhachemi | Stability analysis of reaction-diffusion PDEs coupled at the boundaries with an ODE[END_REF] ), or control of such partial differential equation by means of delayed control (see [START_REF] Lhachemi | Predictor-based output feedback stabilization of an input delayed parabolic PDE with boundary measurement[END_REF] ) to cite just a few. As far as hyperbolic system are considered, nonlinear controllers could be also designed as done in. [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF]72 Finally, let us cite the papers [START_REF] Lhachemi | PI regulation control of a 1-d semilinear wave equation[END_REF][START_REF] Lhachemi | PI regulation of a reaction-diffusion equation with delayed boundary control[END_REF] dealing with regulation problems, that could be seen as generalizations of stabilization problems for both the parabolic equations and the wave equation.

  fig = plt . figure () ; ax = fig . gca ( projection = '3 d ') x , y , z = sol . T ax . plot (x , y , z , label = ' solution ') ; ax . legend () plt . savefig ( ' solution . png ' , bbox_inches = ' tight ')

Saturated boundary stabilization of partial differential equations 9 Fig. 1 .

 91 fig = plt . figure () ; ax = fig . gca ( projection = '3 d ') for i in range (10) : z0 = np . random . random ([ n ,1]) ; z0 = z0 . reshape (n ,) sol = odeint ( ode , z0 , t ) x , y , z = sol . T ax . plot (x , y , z , label = ' solution '+ str ( i ) ) ; ax . legend () plt . savefig ( ' solutions . png ' , bbox_inches = ' tight ')

Fig. 2 .

 2 Fig. 2. Time-evolution of solutions to (7) with u = Kz for 10 randomly chosen initial conditions

Example 2 .

 2 (Example 1 continued) In this extension of Example 1, we compute the eigenvalues of the previous closed-loop system (see line45) and we compute a Lyapunov matrix P .

  15) and so W (e) = e Qe satisfies Ẇ ≤ -e e along the trajectories of ė = (A + LC)e. Hence the e-component of (14) converges to 0 as time goes to +∞. Now pick a symmetric positive definite matrix P such that (A + BK) P + P (A + BK) -I. (16) Letting V (ẑ) = ẑ P ẑ we have Ẇ ≤ -ẑ ẑ + 2ẑLCe

C

  = np . random . random ([1 , n ]) ObsvMatrix = control . obsv (A , C ) # compute the observability matrix if np . linalg . matrix_rank ( ObsvMatrix ) == n :

  def ode2 ( ztot , t ) : z = ztot [: n ]; zhat = ztot [ n :] u = np . dot ( np . dot (B , K ) , zhat ) return np . concatenate (( np . dot (A , z ) + u , np . dot (A , zhat ) +u -np . dot (L , np . dot (C , z ) -np . dot (C , zhat ) ) ) ) # set up a figure twice as wide as it is tall fig = plt . figure ( figsize = plt . figaspect (0.5) ) ax0 = fig . add_subplot (1 , 2 , 1 , projection = '3 d ') ax1 = fig . add_subplot (1 , 2 , 2 , projection = '3 d ') for i in range (10) : z0 = np . random . random ([ n ,1]) ; z0 = z0 . reshape (n ,) zhat0 = np . random . random ([ n ,1]) ; zhat0 = zhat0 . reshape (n ,) ztot0 = np . concatenate (( z0 , zhat0 ) ) sol = odeint ( ode2 , ztot0 , t ) ztot = sol . T ; z = ztot [: n ]; zhat = ztot [ n :]; ax0 . plot ( z [0] , z [1] , z [2]) ;

  fig , ax = plt . subplots ()

Fig. 3 .

 3 Fig. 3. Time-evolution of solutions to (13) for 10 randomnly chosen initial conditions

Fig. 4 .

 4 Fig.4. Time-evolution of the designed Lyapunov function along several solutions to(14)

Fig. 5 .

 5 Fig. 5. Time-evolutions of three solutions to (19) for three different initial conditions

  fig = plt . figure () ; ax = fig . gca ( projection = '3 d ') x , y , z = sol . T ax . plot (x , y , z , label = ' solution ') ; ax . legend () plt . savefig ( ' solution . png ' , bbox_inches = ' tight ') # for 10 randomnly chose initial conditions fig = plt . figure () ; ax = fig . gca ( projection = '3 d ') for i in range (10) : z0 = np . random . random ([ n ,1]) ; z0 = z0 . reshape (n ,) sol = odeint ( ode , z0 , t ) x , y , z = sol . T ax . plot (x , y , z , label = ' solution '+ str ( i ) ) ; ax . legend () plt . savefig ( ' solutions . png ' , bbox_inches = ' tight ')

Example 5 .

 5 (Example 4 continued) Solving the matrix inequalities of Theorem 5 is done with the code below, where the Python cvxpy Library has been used to write the matrix conditions in lines 52-64 with the unknown variables introduced in lines 46-48. The optimization problem max W,S,H trace(W ) such that (

  . Problem ( cp . Maximize ( cp . trace ( W ) ) , constr ) prob . solve () P = np . linalg . inv ( W . value ) z0 = z0tot [0] print ( " z0 ^T P z0 is " + str ( np . dot ( np . dot ( z0 .T , P ) , z0 ) ) ) due to Theorem 5, {z, z P z ≤ 1} is included in the basin of attraction.

c 2 #

 2 This is possible because, owing to the definition (58) of the matrix C 1 , it ensures that C 1 = O(1) as N → +∞. This remark is key to allow the application of Lemma 1. H. Lhachemi and C. Prieur def lam ( n ) : return (n -1/2) **2* np . pi **2* p def phi (n , x ) : return np . sqrt (2) * np . cos (( n -1/2) * np . pi * x ) # Equivalent bounded input operators def input_a ( x ) : return 2* p + q * x **2 # case of p constant and q =0 def input_b ( x ) : return -x **Number of modes to be stabilized N0 = 0 while ( -lam ( N0 +1) + q ) >= -delta : N0 = N0 + 1;

  if N0 == 0: print ( ' All the modes of the open -loop system are < -delta ') else : print ( ' The number of modes to be stabilized is N_0 = '+ str ( N0 ) ) # Select the number of modes for the observer N = N0 +2 # Matrices of the truncated model tmp =[] for i in range (1 , N +1) : # print ( i ) tmp . append ( -lam ( i ) + q ) A0 = np . diag ( tmp [0: N0 ]) A2 = np . diag ( tmp [ N0 : N +1]) B0a = []; B0b = []; B2a = []; B2b = []; C0 = []; C1 = [] for k in range (1 , N0 +1) : def fun ( x ) : return input_a ( x ) * phi (k , x ) y , err = integrate . quad ( fun ,0 ,1) B0a . append ( y ) def fun ( x ) : return input_b ( x ) * phi (k , x ) y , err = integrate . quad ( fun ,0 ,1) B0b . append ( y ) C0 . append ( phi (k ,0) ) for k in range (1 ,N -N0 +1) : def fun ( x ) : return input_a ( x ) * phi ( N0 +k , x ) y , err = integrate . quad ( fun ,0 ,1) B2a . append ( y ) def fun ( x ) : return input_b ( x ) * phi ( N0 +k , x ) y , err = integrate . quad ( fun ,0 ,1) B2b . append ( y ) for k in range ( N0 +1 , N +1) : C1 . append ( phi (k ,0) / np . sqrt ( lam ( k ) ) ) B0a = np . array ( B0a ) . reshape (( N0 ,1) ) B0b = np . array ( B0b ) . reshape (( N0 ,1) ) B2a = np . array ( B2a ) . reshape (( N -N0 ,1) ) B2b = np . array ( B2b ) . reshape (( N -N0 ,1) ) C0 = np . array ( C0 ) . reshape ((1 , N0 ) ) C1 = np . array ( C1 ) . reshape ((1 , N -N0 ) ) A1 = np . vstack (( np . zeros ((1 , N0 +1) ) , np . hstack (( B0a , A0 ) ) ) ) B1 = np . vstack (( np . ones ((1 ,1) ) , B0b ) )

#

  fig , ax = plt . subplots () ax . set_title ( ' To check ') ax . plot ( space , w0 ( space ) , 'g -' , label = ' w0 ') ax . plot ( space , check , 'r . ' , label = ' approx ') ax . legend () # plt . savefig ( ' to_check . png ', bbox_inches = ' tight ') # time dis cretizat ion Tsim = 6; def ode (z , t ) : u = float ( z [0]) zsim = z [1: Nsim +1] zhat = z [ Nsim +1:] whata = np . vstack (( u , zhat [0: N0 ]) ) v = float ( np . dot (K , whata ) ) udot = v zsimdot = np . dot ( Ass , zsim ) + np . dot ( Bssa , u ) + np . dot ( Bssb , v ) zhatdot = np . dot ( Aobs , zhat ) + np . dot ( Bobsa , u ) + np . dot ( Bobsb , v ) zhatdot += -Lobs * float ( np . dot ( Css , zsim ) -np . dot ( Cobs , zhat ) ) zdot = np . hstack (( np . hstack (( udot * np . ones (1) , zsimdot ) ) , zhatdot ) ) return zdot t = np . linspace (0 , Tsim ,60) # Initial condition of the full state zhat0 = np . zeros (( N ,1) ) z0tot = np . vstack (( np . vstack (( u0 , zsim0 ) ) , zhat0 ) ) z0tot = z0tot . reshape ( len ( z0tot ) ,) sol = integrate . odeint ( ode , z0tot , t ) Mstate_z = np . zeros (( len ( space ) , len ( t ) ) ) # PDE in original coordinates z Mstate_w = np . zeros (( len ( space ) , len ( t ) ) ) # PDE in homogeneous coordinates w MstateObs = np . zeros (( len ( space ) , len ( t ) ) ) # State of the observer for k_time in range ( len ( t ) ) :for k_space in range ( len ( space ) ) :for k in range ( Nsim ) :

Fig. 6 .Fig. 7 .

 67 Fig. 6. State z and observation error e in closed-loop with Dirichlet boundary measurement feedback control for the reaction-diffusion system (56)

Fig. 8 .

 8 Fig. 8. Wave equation with bounded control operator

Fig. 9 .

 9 Fig. 9. Wave equation with unbounded control operator

Example 7 .

 7 Let us discretize (103) and illustrate Theorem 17. Moreover we will discretize this equation using the saturation map sat L2 instead of sat and without any saturation map (the equation becomes (102)).

5 #

 5 with saturating control . Use of central difference in space and forward Euler in time schemes . Code originally written by S . Marx for [ S . Marx et al , SIAM J . Control Opt . , 2017] """ import numpy as np import matplotlib as mpl import matplotlib . pyplot as plt from mpl_toolkits . mplot3d import Axes3D # Parameters of the PDE L =2* np . pi ; a =1.0 # Space d iscretiz ation Nx = 30 x = np . linspace (0 ,L , Nx +1) dx = L / Nx ; # Time dis cretizat ion dt = 0.06; tfinal =6 Nt = np . floor ( tfinal / dt ) . astype ( int ) # Saturation level s0 =0.Set initial condition z0 =[] for ii in range ( Nx +1) : z0 . append (1 -np . cos ( x [ ii ]) )

  range ( m ) : if np . absolute ( u [ i ]) > s0 : sigma [ i ]= s0 * np . sign ( u [ i ]) return sigma # L2 saturation function def sat2 ( u ) : L2 = np . linalg . norm ( u ) * np . sqrt ( dx ) sigma = u if not L2 < s0 : for ii in range ( Nx ) : sigma [ ii ]= s0 * u [ ii ]/ L2 return sigma L2norm =[] # L2norm of the sol with sat L2normNoSat =[] # L2norm of the sol without sat ztot = np . zeros (( Nx +1 , Nt +1) ) # to save the solution ztotNoSat = np . zeros (( Nx +1 , Nt +1) ) # to save the solution without sat ztot [: ,0]= z0 ztotNoSat [: ,0]= z0 L2norm . append ( np . linalg . norm ( ztot [: ,0]) * np . sqrt ( dx ) ) L2normNoSat . append ( np . linalg . norm ( ztotNoSat [: ,0]) * np . sqrt ( dx ) )

  def discretNoSat (z , dx , dt , a ) : """ disc retizati on of the nonlinear KdV using [ Pazoto , et al , Numer . Math . , 2010] method without saturation """ n = len ( z ) n1 =n -1 Dm =1/ dx * np . identity ( n1 ) Dp = -Dm for i in range ( n1 -1) :Dp [i , i +1]= -Dp [i , i ] Dm [ i +1 , i ]= -Dm [i , i ] D =1/2*( Dm + Dp ) I = np . identity ( n1 ) A = np . dot ( np . dot ( Dp , Dp ) , Dm ) + D C = I + dt * A NS = np . zeros (( n1 , n1 ) ) NS [ n1 -1 , n1 -1]= C [ n1 -1 , n1 -1] # Fixed -point method NIter = 100 # number of iterations J =[] J . append ( z [: -1]) tmp = J [ -1]for k in range ( NIter ) : tmp = np . linalg . solve (C -NS , z [: -1] -dt /2* np . dot (D , np . multiply ( tmp , tmp ) ) -np . dot ( dt *a , tmp ) ) J . append ( tmp ) return tmp def discret (z , dx , dt , a ) : """ disc retizati on of the nonlinear KdV using [ Pazoto , et al , Numer . Math . , 2010] method with saturation ( select sat or sat2 function ) """ n = len ( z ) n1 =n -1 Dm =1/ dx * np . identity ( n1 ) Dp = -Dm for i in range ( n1 -1) :Dp [i , i +1]= -Dp [i , i ] Dm [ i +1 , i ]= -Dm [i , i ]D =1/2*( Dm + Dp ) I = np . identity ( n1 ) A = np . dot ( np . dot ( Dp , Dp ) , Dm ) + D C = I + dt * A NS = np . zeros (( n1 , n1 ) ) NS [ n1 -1 , n1 -1]= C [ n1 -1 , n1 -1] # Fixed -point method NIter = 100 # number of iterations J =[] J . append ( z [: -1]) tmp = J [ -1] for k in range ( NIter ) : tmp = np . linalg . solve (C -NS , z [: -1] -dt /2* np . dot (D , np . multiply ( tmp , tmp ) ) -dt * sat ( np . dot (a , tmp ) ) ) J . append ( tmp ) return tmp

  where z i is the discretized version of z, and where Dp and Dm are the matrices

  mpl . rcParams [ ' legend . fontsize '] = 10 fig = plt . figure () ; ax = fig . add_subplot (111 , projection = '3 d ') ax . plot_surface ( SX , ST , ztot .T , cmap = ' jet ') ax . set_xlabel ( 'x ') ax . set_ylabel ( 't ') ax . set_zlabel ( 'z (t , x ) ') ax . set_title ( ' Solution with saturation ') ax . view_init ( elev =15 , azim =20) # adjust view so it is easy to see plt . savefig ( 'pde -3 d . png ')

Fig. 10 .

 10 Fig. 10. Solution to (103) with the saturation map sat

Fig. 11 .Fig. 12 .

 1112 Fig. 11. Solution to (102)

Fig. 13 .Fig. 14 .

 1314 Fig. 13. Solution to (103) with the saturation map sat L2

  ≤ 1 and thus, recalling the definition of the deadzone function φ in(20), the local sector condition holds φ(φ + cz t (t, 1)) ≤ 0 , see(21). Let us now go back to the Lyapunov function candidate V 2 . Given b > 0, using the previous inequality, we infer that V2

	is a non-increasing function. Moreover by continuity of the trace function
	on H 1 (0) (0, 1), it holds
	|z t (t, 1)| ≤ z tx (t, .) L 2 (0,1) ≤ A 2	z(t, .) z t (t, .)	≤ A 2	z(0, .) z t (0, .)
	where the decreasing property of the function in (97) has been used for the
	last inequality. Thus, for all t ≥ 0,
	|z t (t, 1)| ≤ A 2	z(0, .) z t (0, .)	.	(98)
	Now, given r > 0, for an initial conditions satisfying (96), we have
	|z
	1 0 |u | 2 (x)dx for all u ∈ H 1 (0) . We are now in

t (t, 1)| ≤ r and thus there exists c = b such that, for all t ≥ 0, (b -c)|z t (t, 1)|

AA = A + np . dot (B , K ) ; e , v = np . linalg . eig ( AA ) # eigen -values ,vectors

d This constraint is always satisfied for µ > 0 small enough by a continuity argument at µ = 0.

e This property is the definition of the global exponential stability of the origin. f This property is the definition of the semi-global asymptotic stability of the origin.

* This work has been partially supported by MIAI@Grenoble Alpes (ANR-19-P3IA-0003).

The code to numerically compute the behavior of the closed-loop system associated with the initial condition z 0 (x) = 1 + x 2 , and with zero initial condition for the observer, obtained based on the 50 dominant modes of the plant is given after line 167. Using the dissipativity of A 1 and Lemma 3, the trajectory z(t, .) z t (t, .) is precompact in H 1 0 (0, 1) × L 2 (0, 1). Moreover the ω-limit set ω z(0, .) z t (0, .) ⊂ D(A 1 ) is not empty and is invariant with respect to the nonlinear semigroup T (t) (see [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF] ). We these elements in hand, we can indeed apply LaSalle's Invariance Principle to show that ω z(0, .)

This shows that the origin of the equation ( 88) with the boundary conditions (74) is attractive. This concludes the proof of Theorem 12.

Let us now give the main steps of the proof of Lemma 3.

Proof of Lemma 3. Consider a sequence f n g n n∈N in D(A 1 ), which is bounded in graph norm, that is there exists M > 0 such that, for all n ∈ N,

From that, we deduce that

bounded. Hence, with compact injection of H 1 0 (0, 1) in L 2 (0, 1), and of H 2 (0, 1) in H 1 0 (0, 1) we infer the existence of a subsequence of f n g n n∈N which converges in H 1 0 (0, 1) × L 2 (0, 1), giving the precompactness of the set of solutions to equation (88) with the boundary conditions (74).

A boundary linear control

We now consider the wave equation with a boundary control, as depicted in Figure 9. The system dynamics reads

with the boundary conditions

Saturated boundary stabilization of partial differential equations 69 position to state the following well-posedness and asymptotic stability result (see [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] for a complete proof).

Theorem 13. For all b > 0, the origin of the PDE (91) with the boundary conditions ( 95) is globally asymptotically stable. More specifically, for all (z 0 , z 1 ) ∈ {(f, g) ∈ H 2 (0, 1) × H 1 (0) (0, 1) : f x (1) + sat(bg(1)) = 0, f (0) = 0}, there exists a unique solution to (91) with the boundary conditions (95) and the initial condition (75). Moreover it satisfies the following stability property, for all t ≥ 0, z(t, .) H 1 (0) (0,1) + z t (t, .) L 2 (0,1) ≤ z 0

together with the attractivity property z(t, .) H 1 (0) (0,1) + z t (t, .) L 2 (0,1) → 0, as t → ∞ . Proof of Theorem 13. To prove the well-posedness of the Cauchy problem we can show that A 2 defined by

is a semigroup of contraction by applying Lumer-Phillips thereom. The global stability property is immediately inferred from contraction property (consequence of the dissipativity of A 2 ). Finally, the global attractivity property comes from the following lemma establishing that the origin of the PDE (91) with the boundary conditions (95) is semi-globally exponentially stable. This completes the proof of the theorem.

Lemma 4. For all r > 0, there exists µ > 0 such that, for all initial condition satisfying

it holds

along the solutions to (91) with the boundary conditions (95).

Proof of Lemma 4. First note that by dissipativity of A 2 , it holds that

Saturated boundary stabilization of partial differential equations 83

been provided for the attractivity, using either direct Lyapunov method or a LaSalle invariance principle or a contradiction argument. Let us emphasize that the approaches presented in this section are also useful for certain other classes of equations such as hyperbolic systems. See 5 for the stabilization of linear and quasilinear hyperbolic systems. See also [START_REF] Shreim | Design of saturated boundary control for hyperbolic systems[END_REF] for the stabilization of hyperbolic systems using saturated control.

In [START_REF] Marx | Output feedback stabilization of the Korteweg-de Vries equation[END_REF] an output feedback control has been computed for the linearized KdV equation. It would be relevant to evaluate the impact of the saturation map on the obtained result.

Finally, in addition to the stabilization control problem, the impact of disturbances could be studied. It would be relevant to obtain Input-to-State Stability results in the context of this section (see [START_REF] Karafyllis | Input-to-State Stability for PDEs[END_REF][START_REF] Mironchenko | Input-to-state stability of infinite-dimensional systems: recent results and open questions[END_REF] for introductory presentations on this subject for infinite-dimensional systems.)