Ismaïla Balogoun

Swann Marx

Thibault Liard

Franck Plestan

Super-twisting sliding mode control for the stabilization of a linear hyperbolic system

This paper deals with the stabilization of a linear hyperbolic system subject to a boundary disturbance. Our feedback design relies on a super-twisting control algorithm, which leads to a feedback that is continuous with respect to the state, in contrast with the classical sliding mode design. Our first result is the existence of solutions of the closed-loop system. Moreover, the global asymptotic stability, that is our second result, is proved together with the guarantee that the disturbance is rejected.

I. INTRODUCTION

This paper is concerned with the stabilization of a linear hyperbolic system with a boundary control and subject to a disturbance (see e.g, [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF] for a review on this class of system). To be more precise, we aim at designing a super-twisting algorithm [START_REF] Levant | Sliding order and sliding accuracy in sliding mode control[END_REF] for that purpose.

Systems of linear transport equations have been the subject of much attention for many years because of the many physical phenomena they model: e.g pressure drilling [START_REF] Landet | Modeling and control of heave-induced pressure fluctuations in managed pressure drilling[END_REF], aeronomy [START_REF] Schunk | Transport equations for aeronomy[END_REF]. A good overview of the actual research lines concerning this topic is provided in [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]. For more details, the reader can also refer to [START_REF] Bribiesca-Argomedo | Backstepping-forwarding control and observation for hyperbolic PDEs with fredholm integrals[END_REF]- [START_REF] Di Meglio | Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input[END_REF].

Sliding mode control (SMC) strategy has been proved to be efficient for robust control of nonlinear systems of ordinary differential equations (ODEs) [START_REF] Edwards | Sliding mode control: theory and applications[END_REF], [START_REF] Shtessel | Sliding mode control and observation[END_REF], [START_REF] Utkin | Sliding modes in control and optimization[END_REF], [START_REF] Young | A control engineer's guide to sliding mode control[END_REF]. Such controllers allow to force, thanks to discontinuous terms, the system trajectories to reach in a finite time a manifold, called sliding surface, and to evolve on it. This manifold being defined from control objectives [START_REF] Shtessel | Sliding mode control and observation[END_REF]. Roughly speaking, the control design is decomposed into two steps: firstly, a sliding variable is selected such that, once this variable equals zero, global asymptotic stability is ensured; secondly, a discontinuous feedback-law is designed such that the trajectory reaches the sliding surface, that is defined thanks to the sliding variable. On this sliding surface, the disturbance is rejected. The generalization of the SMC procedure to the partial differential equations (PDE) case is not new. In [START_REF] Orlov | Discontinuous unit feedback control of uncertain infinite-dimensional systems[END_REF], [START_REF] Orlov | Sliding mode control in indefinitedimensional systems[END_REF], a definition of equivalent control (which is the control applied to the system after reaching the sliding surface, to ensure that the trajectories stays on the surface thereafter) for systems governed by semilinear differential equations in Banach spaces has been proposed. One can refer also to [START_REF] Levaggi | Infinite dimensional systems' sliding motions[END_REF], [START_REF] Levaggi | Sliding modes in banach spaces[END_REF] where differential inclusions and viability theory are combined to design sliding mode controllers for semilinear differential equations in Banach spaces. In the last decade, Ismaïla Balogoun, Swann Marx and Franck Plestan are with LS2N, Ecole Centrale de Nantes and CNRS UMR 6004, Nantes, France (e-mail: {ismaila.balogoun,swann.marx}@ls2n.fr; franck.plestan@ec-nantes.fr).

Thibault Liard is with Univ. Limoges, CNRS, XLIM, UMR 7252, F-87000 Limoges, France (e-mail:thibault.liard@unilim.fr).

a backstepping strategy has been used to select a sliding variable [START_REF] Guo | Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input[END_REF], [START_REF] Pisano | Combined backstepping/second-order sliding-mode boundary stabilization of an unstable reaction-diffusion process[END_REF], [START_REF] Wang | Sliding mode control to stabilization of cascaded heat PDE-ODE systems subject to boundary control matched disturbance[END_REF]. Note also that the SMC feedbacklaw is discontinuous, which creates chattering phenomena when implementing the control numerically. Recall that chattering phenomena are characterized by oscillations having finite amplitude. Therefore, in practical control cases, it is important to reduce this phenomena by providing continuous or smooth controller.

Based on second-order sliding mode techniques (see e.g, [START_REF] Shtessel | Sliding mode control and observation[END_REF]Chapter 4]), the super twisting (ST) algorithm has been developed for systems whose the sliding variable admits a relative degree (see [START_REF] Shtessel | Sliding mode control and observation[END_REF]Definition 1.6]) equal to 1. The essential feature of the ST control is to require that only the measurement of the sliding variable to guarantee the convergence in finite time to zero of the sliding variable and its derivative. Moreover, the ST feedback-law is continuous with respect to the state and this drastically attenuates the chattering phenomenon. The contribution of this paper is to use a super-twisting strategy to design a continuous feedback-law which allows to reject the disturbance in finite time and to ensure that the resulting closed-loop system is globally asymptotically stable. The sliding variable comes from the gradient of a Lyapunov functional that is used for hyperbolic systems [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]Section 2.3]. This imposes to measure the integral of the state with a certain weight and a boundary of the state. Such a sliding variable allows to directly use well-known results on the stabilization of hyperbolic systems.

This paper is organized as follows. Section II presents a system of linear hyperbolic equations, the super-twisting based control law and the main results of the paper. Section III contains the proofs of the main results. Finally, Section IV collects some remarks and introduces some future research lines to be followed.

Notation. The set of non-negative real numbers is denoted in this paper by R + . When a function f only depends on the time variable t (resp. on the space variable x), its derivative is denoted by ḟ (resp. f ′ ). Given any subset of R denoted by Ω (R + or an interval, for instance), L 2 (Ω; R n ) denotes the set of (Lebesgue) measurable functions f 1 , . . ., f n such that, for i = {1, . . . , n},

Ω |f i (x)| 2 dx < +∞. The associated norm is ∥(f 1 , . . . , f n )∥ 2 L 2 (Ω;R n ) := Ω |f 1 (x)| 2 dx + . . . + Ω |f n (x)| 2 dx.
Given two vector spaces E and F , L(E, F ) denotes the space of linear applications from E into F . A function α : IR + → IR + is of class K, if it is continuous, strictly increasing and satisfies α(0) = 0. A function α :

IR + → IR + is of class K ∞ , if it is of class K and unbounded. A function β : IR + × IR + → IR + is of class KL, if for each fixed t ≥ 0, β(•, t) is of class K,
and, for each fixed r ≥ 0, β(r, •) is decreasing and satisfies lim t→∞ β(r, t) = 0.

II. MAIN RESULTS

A. Problem statement

Let ℓ > 0 and λ 1 , λ 2 ∈ C 1 ([0, ℓ]) such that, for all x ∈ [0, ℓ], λ 1 (x) > 0, λ 2 (x) > 0 and k 2 ∈ IR\{0}. Consider the following linear hyperbolic system                ∂ t R 1 (t, x) + λ 1 (x)∂ x R 1 (t, x) = 0, ∂ t R 2 (t, x) -λ 2 (x)∂ x R 2 (t, x) = 0, R 1 (t, 0) = u(t) + d(t), R 2 (t, ℓ) = k 2 R 1 (t, ℓ), R 1 (0, x) = R 0 1 (x), R 2 (0, x) = R 0 2 (x), (1)
where R 0 1 , R 0 2 ∈ L 2 (0, ℓ), u denotes the control and d(•) is an unknown disturbance. Assume that d(•) is bounded and globally Lipschitz over R + . Furthermore, there exists a known positive constant C such that, for a.e t ∈ R + ,

| ḋ(t)| ≤ C.
(2)

When the system (1) is undisturbed (d = 0), it is known that the feedback law

u(t) := k 1 R 2 (t, 0), allows to stabilize the system if |k 1 k 2 | < 1, see [1, The- orem 2.11.
]. The proof, relies on the following Lyapunov functional

V (t) = ℓ 0 q 1 (x)R 2 1 (t, x) + q 2 (x)R 2 2 (t, x) dx, (3)
q 1 (x) = p 1 λ 1 (x) exp - x 0 ν λ 1 (σ) dσ , q 2 (x) = p 2 λ 2 (x) exp x 0 ν λ 2 (σ) dσ , (4)
for any ν, p 1 , p 2 > 0 selected as in [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]Theorem 2.11.]. In this paper, the goal is to develop a super-twisting based controller. To do so, we will consider a continuous robust feedback-law u which allows to reject the disturbance in finite-time and to globally asymptotically stabilize the system around the equilibrium point (0, 0) in the functional space

X := L 2 (0, ℓ; R 2 ).
More precisely, the aim is to find a sliding surface Σ on which (1) becomes in finite-time the following system

         ∂ t R 1 (t, x) + λ 1 (x)∂ x R 1 (t, x) = 0, ∂ t R 2 (t, x) -λ 2 (x)∂ x R 2 (t, x) = 0, R 1 (t, 0) = k 1 R 2 (t, 0), R 2 (t, ℓ) = k 2 R 1 (t, ℓ), (5)
with k 1 chosen such that |k 1 k 2 | < 1. From [1, Theorem 2.11], it is known that (0, 0) is exponentially stable for [START_REF] Di Meglio | Stabilization of a system of n + 1 coupled first-order hyperbolic linear PDEs with a single boundary input[END_REF]. The next section will provide a definition of this sliding surface Σ (and its related sliding variable) and the associated super-twisting sliding mode controller.

B. Control Design

Introduce the sliding surface Σ defined as follows,

Σ := (f, g) ∈ X | ℓ 0 (q 1 (x)f (x) + q 2 (x)g(x)) dx = 0 (6)
with q 1 and q 2 defined in in [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF]. Then, for any solution (R 1 , R 2 ) of (1), the sliding variable S : IR + → IR is defined by [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF] for all t ≥ 0, with ν > 0 and

S(t) := ℓ 0 (q 1 (x)R 1 (t, x) + q 2 (x)R 2 (t, x)) dx
     p 1 = 1, p 2 = k 1 = exp -ν ℓ 0 1 λ1(σ) + 1 λ2(σ) dσ k 2 . ( 8
)
Consider the super-twisting based controller u defined by, for a.e t ≥ 0,

u(t) = νS(t) + k 1 R 2 (t, 0) -α|S(t)| 1 2 sign(S(t)) + v(t), v(t) ∈ -βsign(S(t)) (9)
where k 1 is defined in [START_REF] Guo | Sliding mode and active disturbance rejection control to stabilization of one-dimensional anti-stable wave equations subject to disturbance in boundary input[END_REF], S is introduced in (7), β and α are positive constants which will be chosen later and the set-valued function sign is defined by

sign(z) :=    -1 if z < 0, [-1, 1] if z = 0, 1 if z > 0.
From [START_REF] Ho | Admissible input elements for systems in hilbert space and a carleson measure criterion[END_REF], only the measurements of R 2 (t, 0) and S(t) are required. Furthermore, the full-state is not needed to measure S(t). Indeed, it is just needed to measure the integral of the state with the weight functions q 1 and q 2 . For a.e t ≥ 0, after some formal integration by parts, one gets

Ṡ(t) = -α|S(t)| 1 2 sign(S(t)) + v(t) + d(t), v(t) ∈ -βsign(S(t)). (10)
along the trajectories of (1)- [START_REF] Ho | Admissible input elements for systems in hilbert space and a carleson measure criterion[END_REF]. Then, according to the following transformation

W (t) = d(t) + v(t), (11)
the system (10) is rewritten as

Ṡ(t) = -α|S(t)| 1 2 sign(S(t)) + W (t), Ẇ (t) ∈ ḋ(t) -βsign(S(t)). (12)
The system [START_REF] Levaggi | Sliding modes in banach spaces[END_REF] is understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF] and the existence of solutions is given in Lemma 1. From [20, Theorem 1], all trajectories of (12) converge to zero in finite time.

Proposition 1: ( [20, Theorem 1]) Assuming that

β > C and α > β + C, (13)
there exists a finite time t r > 0 such that S(t) = 0 and W (t) = 0 for any t > t r .

Then, the closed-loop system (1)-( 9) can be seen as follows:

                         ∂ t R 1 (t, x) + λ 1 (x)∂ x R 1 (t, x) = 0, ∂ t R 2 (t, x) -λ 2 (x)∂ x R 2 (t, x) = 0, R 1 (t, 0) = k 1 R 2 (t, 0) + νS(t) -α|S(t)| 1 2 sign(S(t)) + W (t), Ẇ (t) ∈ ḋ(t) -βsign(S(t)) R 2 (t, ℓ) = k 2 R 1 (t, ℓ), R 1 (0, x) = R 0 1 (x), R 2 (0, x) = R 0 2 (x) W (0) = W 0 .
(14) Remark 1: According to Proposition 1 and the first line of [START_REF] Levaggi | Sliding modes in banach spaces[END_REF], Ṡ(t) = 0 for any t > t r . Then, the solution (R 1 , R 2 ) of ( 14) reaches the sliding surface Σ in finite time t r and remains on it. Since W (t) = 0 for any t > t r , then according to [START_REF] Levaggi | Infinite dimensional systems' sliding motions[END_REF], one has v(t) + d(t) = 0 for any t > t r . As a consequence, the system ( 14) can be rewritten as ( 5) on the sliding surface, which is exponentially stable from [1, Theorem 2.11].

Our proof strategy is based on semigroup theory. For this, it is interesting to mention the scalar product which will be used in this paper.

Since λ 1 (•) and λ 2 (•) are positive functions, define a scalar product on X as follows: for all

z 1 w 1 , z 2 w 2 ∈ X, z 1 w 1 , z 2 w 2 := ℓ 0 1 λ 1 (x) z 1 (x)z 2 (x)dx + ℓ 0 1 λ 2 (x) w 1 (x)w 2 (x)dx. (15)
Now, consider the following system

d dt Y (•, t) = AY (•, t) + B d(t), (16)
where Y = (Y 1 , Y 2 ), and d ∈ L 2 (0, T ; IR). The operator A is defined as

     AY = (-λ 1 Y ′ 1 , λ 2 Y ′ 2 ), D(A) = Y ∈ (H 1 (0, ℓ)) 2 | Y 1 (0) = k 1 Y 2 (0), Y 2 (ℓ) = k 2 Y 1 (ℓ)} , (17)
and according to the proof of [1, Theorem A.1], it generates a C 0 -semigroup (T(t)) t≥0 of contractions in X. The operator B is the delta function at x = 0 in L(R 2 , D(A * ) ′ ) i.e ⟨φ, By⟩ D(A * ),D(A * ) ′ = φ(0)y for all y ∈ IR 2 and φ ∈ D(A * ) where A * is the adjoint operator of A, D(A * ) its domain and ⟨•, •⟩ D(A * ),D(A * ) ′ is the dual product.

In this paper, we consider the mild solution of ( 16) in the sense of the next definition.

Definition 1: Let T > 0, d ∈ L 2 (0, T ; IR). Then for every (R 0 1 , R 0 2 ), we say that the map

(Y 1 , Y 2 ) : [0, T ] × (0, ℓ) → R 2 is a mild solution of (16), if (Y 1 , Y 2 ) ∈ C(0, T ; X) ∩ H 1 (0, T ; D(A * ) ′ ) such that for all t ∈ [0, T ] Y 1 (t, •) Y 2 (t, •) = T(t) Y 0 1 Y 0 2 + t 0 T(t -s)B d(s)ds (18)
where (T(t)) t≥0 is the C 0 -semigroup generated by the operator A define in [START_REF] Orlov | Sliding mode control in indefinitedimensional systems[END_REF].

Next, the solutions of ( 14) are understood in the sense of the following definition.

Definition 2: Let T > 0 and (R 0 1 , R 0 2 , W 0 ) ∈ X × IR. We say that the map (R 1 , R 2 ) : [0, T ] × (0, ℓ) → R 2 and W : [0, T ] → R is a mild solution of the Cauchy problem (14) if (R 1 , R 2 ) ∈ C(0, T ; X) such that for all t ∈ [0, T ], (R 1 (t, •), R 2 (t, •)) satisfies (18) with d(t) = -α|S(t)| 1 2 sign(S(t)) + W (t) + νS(t) 0 ( 19
)
and W is absolutely continuous and satisfies

Ẇ (t) ∈ ḋ(t) -βsign(S(t)) (20)
for a.e t ∈ [0, T ] where S is given in [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF].

The main results of this paper can be formulated as follows:

Theorem 1 (Well-posedness): Assume that (13) holds. Then, for all T > 0 and for all (R 0 1 , R 0 2 , W 0 ) ∈ X × IR, the closed-loop system (14) admits a mild solution (R 1 , R 2 , W ).

Theorem 2 (Global asymptotic stability): Assume that (13) holds. Then, for any (R 0 1 , R 0 2 , W 0 ) ∈ X × IR, 0 is globally asymptotically stable for [START_REF] Liard | Boundary sliding mode control of a system of linear hyperbolic equations: a lyapunov approach[END_REF]. In other words, there exists a KL-function τ such that for any (R 0 1 , R 0 2 , W 0 ) ∈ X × R and for any t ≥ 0:

∥(R 1 (t, •), R 2 (t, •)∥ X +|W (t)| ≤ τ (∥(R 0 1 , R 0 2 )∥ X +|W 0 |, t). (21)
III. PROOF OF THEOREM 1 AND THEOREM 2

A. Proof of Theorem 1

This section provides a proof of Theorem 1. More precisely, the aim consists in proving the well-posedness of the closed-loop system [START_REF] Liard | Boundary sliding mode control of a system of linear hyperbolic equations: a lyapunov approach[END_REF] and the regularity of the function S defined by [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF].

Let (R 0 1 , R 0 2 , W 0 ) ∈ X × IR and consider the following ODE      γ(t) = -α|γ(t)| 1 2 sign(γ(t)) + η(t), t ∈ IR + , η(t) ∈ ḋ(t) -βsign(γ(t)), γ(0) = S 0 , η(0) = W 0 . ( 22
)
where

S 0 = ℓ 0 q 1 (x)R 0 1 (x) + q 2 (x)R 0 2 (x) dx.
The system [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] is understood in the sense of Filippov [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF].

In the next lemma, we state that there exists a solution to [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF].

Lemma 1: Assume that (13) holds. There exists an absolutely continuous map (γ, η) that satisfies [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] for almost t ≥ 0.

Proof: We consider the function f : IR 2 → IR 2 defined by

f (γ, η) = f + (γ, η) = (-α √ γ + η, -β) if γ > 0, f -(γ, η) = (α √ -γ + η, β) if γ < 0 (23) and let F d : (γ, η) ∈ IR 2 → F d (γ, η) be the set-valued map defined by F d (γ, η) = B(0, C)+ {f (γ, η)} if γ ̸ = 0, conv{f + (γ, η), f -(γ, η)} if γ = 0 (24)
where B(0, C) is a closed ball of IR 2 centered at 0 and of radius C. Since f is continuous on IR \ {0} × IR, then the function F d is non-empty, compact, convex and upper semi-continuous. Then according to [2, Theorem 3.6], there exists at least one solution of the differential inclusion

ζ ∈ F d (ζ) (25)
where ζ = (γ, η). Since F d is the Filippov's construction associated to [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF], then, there exists an absolutely continuous map that satisfies [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF] for almost t ≥ 0, concluding therefore the proof. Since γ and η are continuous then, according to the first line of ( 22), we deduce that γ is also continuous. Next, we show the following well-posedness result for the system [START_REF] Orlov | Discontinuous unit feedback control of uncertain infinite-dimensional systems[END_REF].

Lemma 2: Let (γ, η) be a solution of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]. Then, for all (Y 0 1 , Y 0 2 ) ∈ X and for all T > 0, the system (16) with

d(t) = νγ(t) + γ(t) 0 (26) admits a mild solution (Y 1 , Y 2 ) ∈ C([0, T ]; X).
Proof: Let T > 0 and (γ, η) be a solution of [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]. Then according to Lemma 1, γ and γ are continuous. Therefore, d ∈ L 2 (0, T ). As a consequence, since A generates a C 0 -semigroup (T(t)) t≥0 of contractions in X and if one proves that the operator B is an admissible operator (see e.g [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Chapter 4]) for the C 0 -semigroup (T(t)) t≥0 , one can apply the result provided by [9, Theorem 2.2], and conclude that the statement of Lemma 2 holds. Since X is a Hilbert space, proving that the operator B is an admissible operator for the C 0 -semigroup (T(t)) t≥0 is equivalent to prove that the adjoint operator B * of B is an admissible observation operator for the adjoint of the C 0 -semigroup (T(t)) t≥0 . Then, we consider the dual system of ( 16):

       d dt φ 1 φ 2 = A * φ 1 φ 2 y * = B * φ 1 φ 2 (27)
where A * and B * are given by

                             A * φ 1 φ 2 = λ 1 φ ′ 1 -λ 2 φ ′ 1 , D(A * ) = φ 1 φ 2 ∈ (H 1 (0, ℓ)) 2 | φ 2 (0) = k 1 φ 1 (0), φ 1 (ℓ) = k 2 φ 2 (ℓ) , B * : φ 1 φ 2 ∈ D(A * ) → φ 1 (0) 0 .
(28) For all (φ 0 1 , φ 0 2 ) ∈ D(A * ), the function

φ 1 (t) φ 2 (t) = T * (t) φ 0 1 φ 0 2 (29)
defines the unique classical solution of (27) where T * (t) is a C 0 -semigroup with infinitesimal generator A * on X. Now, consider the following function

E(t) = ℓ 0 Q 1 (x)φ 2 1 (t, x) + Q 2 (x)φ 2 2 (t, x) dx, (30)
Q 2 (x) = a 2 λ 2 (x) exp - x 0 ν λ 2 (σ) dσ , Q 1 (x) = a 1 λ 1 (x) exp x 0 ν λ 1 (σ) dσ , (31)
where ν, a 1 , a 2 > 0 will be chosen later. If we select ν, a 1 and a 2 as in [1, Proof of Theorem 2.11], then one deduces that for all t ≥ 0

|φ 1 (t, 0)| 2 ≤ -1 a 1 -a 2 k 2 1 Ė(φ 1 (t, •), φ 2 (t, •)). (32)
Therefore, for all T > 0 and for all φ 1 (0, •)

φ 2 (0, •) ∈ D(A * ) T 0 |y * (t)| 2 dt ≤ - 1 a 1 -a 2 k 2 1 T 0 Ė(φ 1 (t, •), φ 2 (t, •))dt ≤ 1 a 1 -a 2 k 2 1 E(φ 1 (0, •), φ 2 (0, •)) (33) - 1 a 1 -a 2 k 2 1 E(φ 1 (T, •), φ 2 (T, •)) ≤ 1 a 1 -a 2 k 2 1 E(φ 1 (0, •), φ 2 (0, •)).
Since E is equivalent to the usual norm, there exists a positive constant C such that, for all T > 0 and for all φ 1 (0, •)

φ 2 (0, •) ∈ D(A * ) T 0 |y * (t)| 2 dt ≤ C∥(φ 1 (0, •), φ 2 (0, •))∥ 2 X . (34)
This proves that B is admissible for the C 0 -semigroup (T(t)) t≥0 and concludes the proof of Lemma 2.

The aim is now to prove that, for any Filippov solution (γ, η) of ( 22) with initial condition (S 0 , W 0 ), the solution (Y 1 (•), Y 2 (•), η) of ( 16)-( 26)-( 22) is a mild solution of [START_REF] Liard | Boundary sliding mode control of a system of linear hyperbolic equations: a lyapunov approach[END_REF].

To that end, we will show that the following function

σ(t) = S(Y 1 (t, •), Y 2 (t, •)), (35)
with S defined in ( 7) and (Y 1 , Y 2 ) the solution of ( 16), is equal to γ for any t > 0.

Lemma 3: For any T > 0, σ is a Carathéodory1 solution of

σ(t) = -νσ(t) + γ(t) + νγ(t), t ∈ [0, T ], σ(0) = S 0 . (36)
Proof: Let T > 0. According to ( 4) and ( 8), we have λ 1 q 1 λ 2 q 2 ∈ D(A * ). Then, taking the inner product with λ 1 q 1 λ 2 q 2 on both sides of ( 16), we obtain for almost every

t ∈ [0, T ] d dt Y 1 (t, •) Y 2 (t, •) , λ 1 q 1 λ 2 q 2 = Y 1 (t, •) Y 2 (t, •) , A * λ 1 q 1 λ 2 q 2 + B * λ 1 q 1 λ 2 q 2 d(t) (37)
where ⟨•, •⟩ is defined in [START_REF] Moreno | Strict lyapunov functions for the super-twisting algorithm[END_REF]. This implies that for a.e t ∈ [0, T ]

d dt ℓ 0 (q 1 (x)Y 1 (t, x) + q 2 (x)Y 2 (t, x)) dx = -ν ℓ 0 (q 1 (x)Y 1 (t, x) + q 2 (x)Y 2 (t, x)) dx (38) + 1 0 d(t)
Then, after integration by parts, one gets for a.e t ∈ [0, T ] Since the couple (R 1 , R 2 ) is continuous on [0, t r ] , then, according to [START_REF] Filippov | Differential equations with discontinuous righthand sides: control systems[END_REF], S is also continuous. Therefore, S is bounded on [0, t r ]. Moreover, d is bounded on [0, t r ]. As a consequence, W is bounded on [0, t r ] according to [START_REF] Landet | Modeling and control of heave-induced pressure fluctuations in managed pressure drilling[END_REF] and [START_REF] Levaggi | Infinite dimensional systems' sliding motions[END_REF]. Then, the function -α|S(•)| As a consequence, according to (45), ( 46), (47) and (48), exists C 1 > 0 (independent of t r ) such that for all t ∈ [0, t r ],

ℓ 0 (q 1 (x)Y 1 (T, x) + q 2 (x)Y 2 (T, x)) dx - ℓ 0 (q 1 (x)Y 1 (0, x) + q 2 (x)Y 2 (0, x)) dx = -ν T 0 ℓ 0 (q 1 (x)Y 1 (t, x) + q 2 (x)Y 2 (t, x)) dxdt (
∥(R 1 (t, •), R 2 (t, •)) T ∥ X + |W (t)| ≤ ψ ∥(R 0 1 (•), R 0 2 (•)) T ∥ X + |W 0 | . ( 49
)
where is given by ψ :

x ∈ IR + → C 1 (x + √ x).
This concludes the proof of Theorem 2.

IV. CONCLUSION

A new approach for sliding mode control (precisely supertwisting control) has been proposed for a class of PDEs, namely a system of two transport equations. It is a Lyapunov approach, since the sliding variable is based on the gradient of the classical Lyapunov function given in [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]. The existence of solutions of the closed-loop system has been proved as well as the disturbance rejection and the asymptotic stability of the closed-loop control system.

  , (26), (35) one has for all T > 0 σ(T ) -σ(0) = -ν T 0 σ(t)dt + T 0 νγ(t) + γ(t)dt. (40) This concludes the proof of Lemma 3.

1 2 1 2

 11 sign(S(•)) + W (•) + νS is also bounded on [0, t r ]. Therefore, there existsK 2 > 0 such that ∥ -α|S(•)| sign(S(•)) + W (•) + νS(•)∥ L 2 ((0,tr),I R) ≤ K 2 t to [15,Theorem 2], there are positive constants K 3 , K 4 (dependent on the bound of ḋ) such thatt r < K 3 (|S(0)| + |W (0)|) , |W (t)| ≤ K 4 |W (0)|. (47)Using Holder's inequality there exists C > 0 such that|S(0)| ≤ ∥q 1 (•), q 2 (•)∥ L ∞ ((0,L),I R 2 ) ∥R 0 1 (•), R 0 2 (•)∥ L 1 ((0,L),IR 2 ) (48) ≤ C∥R 0 1 (•), R 0 2 (•)∥ X .

A Carathéodory solution of (36) is an absolutely continuous map that satisfies (36) for almost every t.

Lemma 4: For all (R 0 1 , R 0 2 ) ∈ X, for all W 0 ∈ IR and for all t ∈ IR + , σ(t) = γ(t) and W (t) = η(t).

Proof: Note that it is enough to prove σ(t) = γ(t) to be able to conclude W (t) = η(t). For this proof, we refer to [START_REF] Liard | Boundary sliding mode control of a system of linear hyperbolic equations: a lyapunov approach[END_REF]Section 4.1]. This concludes the proof of Lemma 4.

Then, according to Lemma 2, Lemma 3 and Lemma 4, one concludes that, for any Filippov solution (γ, η) of ( 22) with initial condition γ(0) = S 0 , η(0) = W 0 , d satisfies [START_REF] Schunk | Transport equations for aeronomy[END_REF] and the associated solutions (Y 1 , Y 2 , η) is a mild solution of [START_REF] Liard | Boundary sliding mode control of a system of linear hyperbolic equations: a lyapunov approach[END_REF] in the sense of Definition 2. This concludes the proof of Theorem 1.

B. Proof of Theorem 2

Let us start proving [START_REF] Levaggi | Sliding modes in banach spaces[END_REF]. Let T > 0 and (R 0 1 , R 0 2 , W 0 ) ∈ X×IR. We consider (R 1 , R 2 , W ) a mild solution of ( 14) with initial condition (R 0 1 , R 0 2 , W 0 ). Then, according to Definition 2, there exists 18) with d which satisfies [START_REF] Schunk | Transport equations for aeronomy[END_REF]. As a consequence, (R 1 , R 2 ) satisfies ( 16). Then, by replacing 39) and using ( 7), [START_REF] Schunk | Transport equations for aeronomy[END_REF] we obtain that, for all T > 0

Then, according to (41), we obtain for a.e t ∈ [0, T ]

Since w(t) ∈ sign(S(t)), then (S, W ) is a Filippov solution of [START_REF] Levaggi | Sliding modes in banach spaces[END_REF] with initial condition (S(R 0 1 , R 0 2 ), W 0 ). Now, we are going to prove the Theorem 2. Let (R 0 1 , R 0 2 , W 0 ) ∈ X × IR. Then, according to Proposition 1 and Remark 1, there exists a finite time t r such that, for all t > t r , the system ( 14) is equivalent to the system (5) and hence is exponentially in X from [START_REF] Bastin | Stability and boundary stabilization of 1-D hyperbolic systems[END_REF]Theorem 2.11]. Therefore, to conclude the proof of Theorem 2, it is just necessary to prove the Lyapunov stability of the system ( 14) over the time interval [0, t r ]. It is stated in the following Lemma.

Lemma 5: There exists a K-function ψ such that for all

for all mild solutions (R 1 (t, •), R 2 (t, •), W (•)) of ( 14).

Proof: Let (R 0 1 , R 0 2 , W 0 ) ∈ X × IR and we consider (R 1 , R 2 , W ) a mild solution of ( 14) associated (R 0 1 , R 0 2 , W 0 ). Then, using the Definition 2, there exists C > 0 such that, for all t ∈ [0, t r ], we have

As a consequence, since (T(t)) t≥0 is exponentially stable and B is admissible operator for (T(t)) t≥0 , then according to [START_REF] Tucsnak | Observation and control for operator semigroups[END_REF]Proposition 4.3.3], there exists K 1 > 0 independent of t r such that