Mathias Dus

Francesco Ferrante

Christophe Prieur

ON L ∞ STABILIZATION OF DIAGONAL SEMILINEAR HYPERBOLIC SYSTEMS BY SATURATED BOUNDARY CONTROL

Keywords: 1991 Mathematics Subject Classification. 93D05, 93D15, 93D20. 2018 Diagonal semilinear hyperbolic systems, Saturation, Lyapunov theory

This paper considers a diagonal semilinear system of hyperbolic partial differential equations with positive and constant velocities coupled with a nonlinear source term. The boundary condition is composed of an unstable linear term and a saturated feedback control. Weak solutions with initial data in L 2 ([0, 1]) are considered and well-posedness of the system is proven using nonlinear semigroup techniques. Local L ∞ exponential stability is tackled by a Lyapunov analysis and convergence of semigroups. Moreover, an explicit estimation of the region of attraction is given.

Introduction

Literature Review

The problem of saturated control has gained interest in the last few years because of the increasing need of precision for modeling real actuators. Physical controllers cannot provide infinite energy and sometimes, they saturate rendering classical unsaturated models obsolete. To avoid such situations, engineers choose controllers powerful enough to avoid saturation when the system operates in standard conditions. However, over-dimensioning actuators is not optimal in term of mass and cost of operation for many sophisticated systems as satellites for example. Moreover, in some exceptional configurations, actuators could saturate and lead to very dangerous situations; unpredictable via linear theory.

In fact, disregarding nonlinearities coming from saturation in the input can be source of undesirable and even catastrophic behaviors for the closed-loop system. In [START_REF] Campo | Robust control of processes subject to saturation nonlinearities[END_REF], authors showed that in presence of magnitude saturation, the closed-loop system can become unstable if the initial data is too "large" for a certain norm. As a consequence, it is important to determine a precise estimation of the region of attraction.

In this article, we are interested in infinite-dimensional systems involving hyperbolic 1-D partial differential equations (PDEs). To the best of our knowledge, the first work analyzing the effect of saturation in infinite-dimensional systems is [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF]. In particular, in [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF] the author focuses on the case of compact and bounded control operators, with an a priori constraint. Recently, in [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] the case of distributed saturating control has been considered. The results in [START_REF] Lasiecka | Strong stability of elastic control systems with dissipative saturating feedback[END_REF] suggest the use of an observability condition for the analysis of systems modeled by PDEs controlled via closed-loop saturating controllers. In particular, the contraction semigroup obtained from the saturating closed-loop system is compared with the corresponding saturation-free semigroup.

In this paper, we focus on feedback boundary control of a diagonal system of semilinear hyperbolic PDEs. The literature on unsaturated linear boundary control for semilinear hyperbolic systems is rich; see, e.g., [START_REF] Bastin | Stability And Boundary Stabilization Of 1-D Hyperbolic Systems[END_REF] or [START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF], just to mention a few. However, when input saturation comes into play, the inherent nonlinear nature of the problem renders the analysis much harder. As such, only a few papers focused on saturated boundary control of systems modeled via PDEs. For example, in [START_REF] Li | Semi-global C 1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems[END_REF], the authors prove that two-dimensional quasilinear hyperbolic systems with opposite velocities are stabilizable with bounded C1 boundary control inputs. Nevertheless, the method mainly relies on characteristics and does not seem to be generalizable to system with a larger number of PDEs. The results in [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] are tailored to the wave equation z tt = z xx subject to a nonlinear saturated boundary condition; which is a special second-order hyperbolic PDE. In particular, inspired by [START_REF] Ball | Feedback stabilization of distributed semilinear control systems[END_REF], the authors in [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] relies on the theory of nonlinear semigroups to prove well-posedness and global H 1 exponential stability for the wave equation, in the presence of distributed or boundary saturated controllers. The main idea consists of using a sector bounded approach inspired by the literature of finite dimensional systems [START_REF] Tarbouriech | Stability and Stabilization of Linear Systems with Saturating Actuators[END_REF], to ensure exponential decay of an H 1 -Lyapunov functional.

In this manuscript, as opposed to [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] and [START_REF] Li | Semi-global C 1 solution and exact boundary controllability for reducible quasilinear hyperbolic systems[END_REF], we directly consider the following system of semilinear hyperbolic PDEs of arbitrary dimension

d ∈ N    R t + ΛR x =g • R R(0, t) =HR(1, t) + Bu(t) R(., 0) =R 0
where Λ is a diagonal positive definite matrix, H and B are d × d real matrices, and g ∈ C 1 (R d ) is a globally Lipschitz function, with Lipschitz constant L g , such that g(0) = 0. Moreover, g is diagonal in the sense that for all

R ∈ R d , g i (R) = g i (R i ).
The open-loop system may turn out to be unstable if the matrix H is too "large" (in the sense of a certain norm). The source term g has also its impact on the stability. According to its form, it could make the open-loop system more or less stable. In [START_REF] Bastin | Stability And Boundary Stabilization Of 1-D Hyperbolic Systems[END_REF] chapter 1, typical examples of systems modeled by hyperbolic PDEs with feedback boundary conditions are cited; the telegrapher equations for electrical lines, the shallow water (Saint-Venant) equations for open channels, the isothermal Euler equations for gas flow in pipelines or even the Aw-Rascle equations for road traffic.

In [START_REF] Bastin | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF], authors found a sufficient condition on matrix K ∈ M d (R) such that the linear control u(t) = KR(1, t) ensures C 1 exponential stability of the following "unsaturated" closed-loop system:

   R t + ΛR x =0 R(0, .) =(H + BK)R(1, .) R(., 0) =R 0 ∈ C 1 ([0, 1]). (1)
In particular, defining for all matrices

M ∈ M d (R), R ∞ (M ) = max i=1..d d j=1 |M i,j |, [4, Theorem 3.3] established that if:
ρ ∞ (H + BK) := inf ∆∈D + d (R) R ∞ ∆(H + BK)∆ -1 < 1 then the unsaturated system (1) is C 1 exponentially stable for the canonical norm of C 1 ([0, 1]). Note that [START_REF] Bastin | Dissipative boundary conditions for one-dimensional quasilinear hyperbolic systems: Lyapunov stability for the C 1 norm[END_REF]Theorem 3.3] was proven for small initial data and for quasilinear systems.

Definition of the system and contribution

In this paper, it is assumed that there exists a matrix gain K such that ρ ∞ (H + BK) < 1 and we will study the L ∞ stability of the saturated closed-loop system:

   R t + ΛR x =g • R R(0, .) =HR(1, .) + Bσ(KR(1, .)) R(., 0) =R 0 ∈ L ∞ ([0, 1]) (2)
with σ defined as a saturation by component i.e. there exists a σ s > 0 such that for all i ∈ 1, d , x ∈ R,

σ i (x) = x if |x| ≤ σ s σ i (x) = sign(x)σ s otherwise.
Remark 1. For simplicity, we take σ such that the value of the saturation level σ s is identical for each component. This is not a restriction with respect to the general case

σ i (x) = x if |x| ≤ σ s,i σ i (x) = sign(x)σ s,i otherwise where (σ s,i ) i∈ 1,d ∈ (R + ) d , under a suitable change of variables.
The initial data R 0 being in L ∞ ([0, 1]), solutions of (2) has to be understood in a weak sense. The main contribution of this paper is to answer the two following problems: Problem 1. Define the sense of a weak solution to system (2) and prove a well-posedness theorem.

Problem 2. Prove the L ∞ local exponential stability of this system with an estimation of the region of attraction. Problem 1 will be solved using a smooth approximation of the system (2) coming from a smoothed sequence of saturations. Convergence of semigroups allows to define weak solutions to (2) and prove the well-posedness. Problem 2 will be tackled using an approximation of the L ∞ ([0, 1]) norm by L p ([0, 1]) norms (p ∈ N).

The rest of this paper is organized as follows. In Section 2, all main results are formulated by two theorems; the first one states the well-posedness and the other, the exponential stability. In the same section, an estimation of the region of attraction is given. In Section 3, the estimated region of attraction for systems taken from the literature is compared with the region of non-saturation. Some concluding remarks and further orientations are given in Section 4.

Notation: For any integers n and m, the set m, n := {m, m + 1, . . . , n}. Unless specified, spaces of vector valued functions in

L p ([0, 1]), C p ([0, 1]) (p ∈ 1, ∞ ) are equipped, respectively, with the canonical norms || • || L p ([0,1]) and || • || C p ([0,1]) . The symbol D + d (R) designates the set of d×d positive definite diagonal matrices. Let R = (R 1 , R 2 , . . . , R d ) ∈ R d , |R| denotes the Euclidean norm of R while |R| max = max i∈ 1,d |R i |. Given M ∈ M d (R), we denote |M | = sup |R|∈R d , |R|=1 |M R|, R ∞ (M ) = max i=1..d d j=1 |M i,j | , ρ ∞ (M ) = inf ∆∈D + d (R) R ∞ ∆M ∆ -1 . Given a function (x, t) → f (x, t), f t and f x denote,
respectively, the partial derivative of f with respect to t and x. When unspecified, T stands for an arbitrary positive real used to define spaces of the form C q ([0, T ], X) where q ∈ N and X is a Banach space.

Main results

In this section, results for well-posedness and exponential stability are stated.

Problem 1

To properly define a weak solution to system (2), we need to give a precise sense to the trace of this solution on the lines s → (x = 0, t = s) and s → (x = 1, t = s). To do so, smoothed solutions subject to smoothed saturations are used. Such smoothed saturations approximate σ in the sense of Definition 1.

Definition 1. (σ n ) n is a smooth approximation of σ if it is in C 1 (R)
and converges uniformly to σ on R. Remark 2. An example of smoothed saturation (σ n ) n (with n an integer) approximation of σ is defined by:

       σ ′ n,i (x) = 1 if x ∈ [-σ s , σ s ] σ ′ n,i (x) = 1 2 + cos(n(x-σs)) 2 if x ∈ [σ s , σ s + π/n] σ ′ n,i (x) = 1 2 + cos(n(x+σs)) 2 if x ∈ [-σ s , -σ s -π/n] σ ′ n,i (x) = 0 otherwise and σ n which is a primitive of σ ′ n , is chosen as:        σ n,i (x) = x if x ∈ [-σ s , σ s ] σ n,i (x) = x+σs 2 + sin(n(x-σs)) 2n if x ∈ [σ s , σ s + π/n] σ n,i (x) = x-σs 2 + sin(n(x+σs)) 2n if x ∈ [-σ s , -σ s -π/n] σ n,i (x) = σ s + π n otherwise . It is easy to show that (σ n ) n tends uniformly towards σ on R. We represent both sequences (σ n ) n and (σ ′ n ) n in Figure 1.
Remark 3. In this paper and for every approximation of σ, the maximum of σ n will be denoted σ s,n for all integers n. Now, taking a smooth approximation (σ n ) n of σ and an integer n, we define another system "smoother" than (2). The system subject to the saturation σ n is defined by:

   R n,t + ΛR n,x =g • R n R n (0, .) =HR n (1, .) + Bσ n (KR n (1, .)) R n (., 0) =R 0,n ∈ H 2 ([0, 1]). (3)
Thanks to Theorem A.1 given in Appendix, we will show that if the initial data is H 2 ([0, 1]) and satisfies compatibility conditions of order 1 (4) then the previous system of PDEs has a unique solution in

C 0 ([0, T ], H 1 ([0, 1])) ∩ C 1 ([0, T ], L 2 ([0, 1]
)) for any T > 0. Hence, traces of this unique solution R n (0, .) and R n (1, .) are well-defined on almost everywhere sense. Remark 4. Note that compatibility conditions of order 1 depends on the chosen σ n . They are expressed as follows

R 0 (0) = HR 0 (1) + Bσ n (KR 0 (1)) R 0,x (0) = Λ -1 [H + Bσ ′ n (R 0 (1))](ΛR 0,x (1) -g(R 0 (1))) + g(R 0 (0)) . (4)
The definition of weak solutions is given here.

Definition 2. For all T > 0, R ∈ C 0 ([0, T ], L 2 ([0, 1]
)) is a weak solution to the problem:

   R t + ΛR x =g • R R(0, .) =HR(1, .) + Bσ(KR(1, .)) R(., 0) =R 0 ∈ L 2 ([0, 1]), (5)
if there exists a sequence (σ n ) n , smooth approximation of σ, and a sequence (R 0,n ) n in H 2 ([0, 1]) satisfying the compatibility conditions of order 1 (4) (which depend on the saturation σ n chosen) tending towards R 0 in L 2 ([0, 1]), such that the sequence of solutions (R n ) n to (3) converges towards R in C 0 ([0, T ], L 2 ([0, 1])).

Remark 5. This definition is different from the common definition of a weak L 2 solution [3, Definition A.3]. The adjoint (for the usual L 2 ([0, T ]) scalar product) of the boundary operator f → Hf + Bσ(Kf ) may not exist. As a consequence, it is impossible to define a boundary condition on test functions. Therefore, we cannot use the common notion of weak solutions. In [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF], authors proved the well-posedness of quasilinear scalar problems subject to L ∞ boundary conditions. They used the method of vanishing viscosity to prove the existence and the uniqueness of the weak solution. The method consists of using a regularized system with additional viscosity and pass to the limit in the weak formulation of the PDE considered. Here, we use the same idea: a regularized system is considered and by a passage to the limit, the weak solution is defined.

It turns out that this problem is well-posed in the sense of Hadamard:

Theorem 1 (Well-Posedness). There exists a unique weak solution to problem [START_REF] Blandin | Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws[END_REF]. Moreover, the flow operator defined by:

U T : L 2 ([0, 1]) → C 0 ([0, T ], L 2 ([0, 1])) R 0 → R is continuous for all T > 0.
This theorem is proven in Appendix A. Remark 6. Theorem 1 holds for whatever σ bounded and continuous such that there exists a smooth approximation of σ in the sense of Definition 1. Moreover, Theorem 1 is also valid for a nondiagonal (but Lipschitz) source term.

•

Problem 2

For exponential stability, we introduce the Lipschitz constants L g,i such that

∀R ∈ R d , i ∈ 1, d , |g i (R i )| ≤ L g,i |R i |. (6)
Hence, for all integers i, the scalar function

R → g i (R) is a scalar function from R to R which is L g,i Lipschitz. Then defining ∀f ∈ L ∞ ([0, 1]), V (f ) := max i∈ 1,d |δ i f i e -µx | L ∞ ([0,1]) (7)
where ∆ = diag(δ i ) is selected such that R ∞ (∆(H + BK)∆ -1 ) ≤ 1 (possible because ρ ∞ (H + BK) < 1 is assumed all along this article). One gets the following result:

Theorem 2 (Exponential Stability). Suppose ρ ∞ (H +BK) < 1. For all ∆ ∈ D + d (R) satisfying R ∞ (∆(H +BK)∆ -1 ) < 1, all positive µ < -log(R ∞ (∆(H + BK)∆ -1 )) and for all initial data R 0 ∈ L ∞ ([0, 1]), if µλ min -L g,max ≥ 0 where L g,max := max i∈ 1,d L g,i ,
and

V (R 0 ) < e -µ R ∞ (∆B∆ -1 )σ s |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -e -µ | , (8)
then the weak solution to (5) verifies:

∀t ≥ 0, V (R(., t)) ≤ e -(µλmin-Lg,max)t V (R 0 ).
This theorem is proven in Appendix B.

Remark 7.

As

∀R ∈ L ∞ ([0, 1]), min i {δ i } e -µ ||R|| L ∞ ([0,1]) ≤ V (R) ≤ max i {δ i } ||R|| L ∞ ([0,1]) ,
Theorem 2 gives the classical local exponential stability of system (5) with respect to the usual L ∞ ([0, 1]) norm.

Remark 8. Concerning the estimation of the region of attraction, we will see in the proof of Theorem 2 (Remark 11) that such an estimation (8) comes from a sector bounded condition imposed on the dead-zone function; the difference between the linear and the saturated control.

Numerical example

In this section, we analyze a typical example of diagonal semilinear systems taken from [START_REF] Prieur | Boundary control design for linear conservation laws in the presence of energy-bounded measurement noise[END_REF]; an article considering the same kind of systems plus a disturbance; discarded here for our purposes. Matrices are defined as:

Λ = 1 0 0 √ 2 , H = 0 1.1 1 0 , B = I 2 .
In [START_REF] Prieur | Boundary control design for linear conservation laws in the presence of energy-bounded measurement noise[END_REF], authors consider a system of transport PDEs with positive velocities and without source term. They give a method to find a gain matrix K such that the equivalent linear system subject to the control R(0, t) = (H + BK)R(1, t) be L 2 exponentially stable and robust. Three gain matrices K were compared with an increasing rate (or at least an estimation of this rate) of exponential decay for the L 2 norm:

K 1 = 0 -0.1050 -0.1045 0 K 2 = 0 -0.4777 -0.4651 0 .
The last gain is taken from [START_REF] Espitia | Stabilization of boundary controlled hyperbolic PDEs via Lyapunov-based event triggered sampling and quantization[END_REF]:

K 3 = 0 -0.7 -1 0 .
For all these matrix gains, we evaluate the region of attraction thanks to the estimation given in Theorem 2 and evaluate if it is larger than the domain where the saturation does not apply.

We take µ = 0 in [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and approximate the stability region of Theorem 2 by:

V (R 0 ) < R ∞ (∆B∆ -1 )σ s |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -1| (9)
equivalent to:

∀i ∈ 1, d , |R 0,i | L ∞ ([0,1]) < |R 0,i | L ∞ ,lim where |R 0,i | L ∞ ,lim := 1 δi R∞(∆B∆ -1 )σs |R∞(∆(H+BK)∆ -1 )+R∞(∆B∆ -1 )R∞(∆K∆ -1 )-1| . Taking ∆ ∈ D + d (R) minimizing R ∞ (∆(H + BK)∆ -1
) and a saturation such that σ s = 1, one gets the numerical results from Table 1 for the estimation of the region of attraction:

K K 1 K 2 K 3 ρ ∞ (H + BK) 0.95 0.58 ≃ 0 |R 0,1 | L ∞ ,lim 17.5 11.0 ≃ 0 |R 0,2 | L ∞ ,lim 16.6 10.1 ≃ 0 |(KR 0 ) 1 | lim 1.84 5.61 ≃ 0 |(KR 0 ) 2 | lim 1.93 5.91 ≃ 0 Table 1. The estimation of the region of attraction where |(KR 0 ) 1 | lim := sup |(KR) 1 | | |R i | < |R 0,i | L ∞ ,lim , ∀i ∈ 1, 2 and |(KR 0 ) 2 | lim := sup |(KR) 2 | | |R i | < |R 0,i | L ∞ ,lim , ∀i ∈ 1, 2 .
We added a row giving the values of ρ ∞ (H + BK) as it gives an estimation of the rate of convergence µλ min -L g,max of the L ∞ norm of the solution R. This can be seen from the condition "µ

< -log(R ∞ (∆(H + BK)∆ -1 ))" of Theorem 2.
We remark that for K 1 and K 2 , |(KR 0 ) 1 | lim and |(KR 0 ) 2 | lim are both larger than the saturation σ s = 1 and hence, the estimated region of attraction is larger than the linear unsaturated region. Then, we also remark that there exists a balance between the rate of convergence of the saturated system estimated by ρ ∞ (H + BK) and the region of attraction. Keep in mind that the smaller ρ ∞ (H + BK), the larger the estimation of the rate of exponential convergence of the saturated system. From this and results presented in Table 1, if one wants a vast region of attraction, the estimation of the rate of convergence will not be important. On the contrary, if one wants a strong rate of convergence, then the region of attraction will be limited.

Another comment has to be made on the case K = K 3 . Here the matrix gain K 3 is chosen such that ρ ∞ (H + BK) ≃ 0 which means that the system is exponentially stable with a very large rate of convergence. From Table 1, the estimation of the region of attraction gives bad results. This is mainly because R ∞ (∆B + K + ∆ -1 ) ≃ +∞. This last analysis tends to confirm the link between the estimation of the rate of convergence and the estimation of the region of attraction underlined earlier.

Conclusion

The well-posedness and the local L ∞ exponential stability of a wide class of diagonal semilinear systems was established. The PDEs under consideration resulted from a transport with constant velocities coupled with a nonlinear source term and a nonlinear boundary condition. The saturated control was applied at the boundary in order to stabilize the openloop system. The well-posedness was tackled using nonlinear semigroup techniques. The stability has been proven using convergence of semigroups and Lyapunov theory. This work let some questions open. The case of mixed positive and negative velocities is not treated; the method of [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] which differentiates the Lyapunov functional for components with positive and negative velocities seems to be a promising idea. The case of space varying velocities (with constant sign) is also interesting and already solved in [START_REF] Bastin | Stability And Boundary Stabilization Of 1-D Hyperbolic Systems[END_REF]Chapter 3.5] for unsaturated systems. The generalization to non-diagonal source terms would be an important improvement. The article [START_REF] Hayat | Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions[END_REF], where a specific space dependent Lyapunov functional is introduced, would be a good starting point to tackle the problem. Finally, the L ∞ (or even L p ) stability for systems of nonlinear d scalar conservation laws remains an open question even for unsaturated controllers. In [START_REF] Blandin | Regularity and Lyapunov stabilization of weak entropy solutions to scalar conservation laws[END_REF], a feedback control was found for a single scalar conservation law whose flux is either convex or concave. Additionally, authors of [START_REF] Coron | Output feedback stabilization for a scalar conservation law with a nonlocal velocity[END_REF] study the stabilization of a nonlocal one-dimensional conservation law. Starting from a linearized system, they find a sufficient condition for stability and adapt the proof to the full nonlinear PDE. However, to our knowledge, nothing seems to be generalizable to conservation laws of arbitrary dimension.

Appendix A. Proof of Theorem 1

Let X = L 2 ([0, 1]) be the base space; the scalar product on X was introduced by [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and is defined by:

∀u, v ∈ X, (u, v) := 1 0 u T ve ν(x-1) dx. (10)
A.1. Existence and uniqueness of solution with a smoothed saturation Take an arbitrary smooth approximation (σ n ) n of σ in the sense of Definition 1. For all integers n, we define the operator A n by:

A n R = -ΛR x D(A n ) = R ∈ H 1 ([0, 1]); R(0) = HR(1) + Bσ n (KR(1)) .
Moreover, the operator G can be defined as follows:

GR = g • R D(G) = L 2 ([0, 1])
The following theorem states the well-posedness for the closed-loop system whose control is smoothly saturated.

Theorem A.1. There exists ζ > 0 dependent on σ s , H, B and K such that for all integers n, the operator

A n + G is ζ dissipative. Moreover, A n + G generates a semigroup T n of type ζ and for all R 0,n ∈ D(A n ), T n (.)R 0,n is the C 0 ([0, T ], L 2 ([0, 1]
)) solution to the Cauchy problem:

   R n,t = -ΛR n,x + g • R n R n (t = 0) = R 0,n R n (t) ∈ D(A n ) (11)
where R n,t is defined as the Fréchet derivative with respect to t in the L 2 space:

R n (., t + dt) -R n (., t) dt -R n,t (., t) L 2 ([0,1]) → dt→0 0 Finally, if R 0,n is H 2 ([0, 1]
) and satisfies compatibility conditions of order 1 (4) then the solution R n belongs to

C 1 ([0, T ], L 2 ([0, 1])) ∩ C 0 ([0, T ], H 1 ([0, 1])).
Remark 9. A definition of the ζ dissipativity can be found in [START_REF] Miyadera | Nonlinear Semigroups. Translations of mathematical monographs[END_REF]. Moreover, in the following proof, we do not use the form of σ n but the fact that it is bounded in C 1 (n fixed). Hence, conclusions of Theorem A.1 are valid for whatever bounded σ n ∈ C 1 (R).

Let n be an integer. To get the conclusions of Theorem A.1, we need to prove some properties on the operators A n + G. They are listed below:

• A n + G is ζ dissipative with ζ independent on n. • It satisfies the range condition: Rg(I -ρ(A n + G)) ⊃ D(A n + G) for all positive ρ sufficiently small. • A n + G is closed
Having done that, we use [15, Theorem 5.12] to prove that A n + G generates the expected semigroup.

Proof. Let n be an integer fixed all along the proof. Constants may depend on n but, in the following, this dependence is skipped when it is useless.

(1) A n + G is ζ dissipative: Let u and v in D(A n + G) = D(A n ).
Recalling the definition of the scalar product [START_REF] Espitia | Stabilization of boundary controlled hyperbolic PDEs via Lyapunov-based event triggered sampling and quantization[END_REF], one has

((A n + G)u -(A n + G)v, u -v) = - 1 0 (u x -v x ) T Λ(u -v)e ν(x-1) dx + 1 0 (g • u -g • v) T (u -v)e ν(x-1) dx (12)
By an integration by parts, one has:

((A n + G)u -(A n + G)v, u -v) = -[(u -v) T Λ(u -v)e ν(x-1) ] 1 0 + 1 0 (u -v) T Λ(u x -v x )e ν(x-1) dx + ν 1 0 (u -v) T Λ(u -v)e ν(x-1) dx + 1 0 (g • u -g • v) T (u -v)e ν(x-1) dx.
From [START_REF] Krstic | Boundary Control of PDEs: A Course on Backstepping Designs[END_REF], one gets :

((A n + G)u -(A n + G)v, u -v) = -[(u -v) T Λ(u -v)e ν(x-1) ] 1 0 -((A n + G)u -(A n + G)v, u -v) + ν 1 0 (u -v) T Λ(u -v)e ν(x-1) dx + 2Re 1 0 (g • u -g • v) T (u -v)e ν(x-1) dx.
It implies necessarily that:

2Re((A n + G)u -(A n + G)v, u -v) = -[(u -v) T Λ(u -v)e ν(x-1) ] 1 0 + ν 1 0 (u -v) T Λ(u -v)e ν(x-1) dx + 2Re 1 0 (g • u -g • v) T (u -v)e ν(x-1) dx
and taking the real part in last equation:

2Re((A n + G)u -(A n + G)v, u -v) = -Re [(u -v) T Λ(u -v)e ν(x-1) ] 1 0 + ν Re 1 0 (u -v) T Λ(u -v)e ν(x-1) dx + 2Re 1 0 (g • u -g • v) T (u -v)e ν(x-1) dx.
Using the fact that velocities are bounded from above, 2Re((

A n + G)u -(A n + G)v, u -v) ≤ -Re [(u -v) T Λ(u -v)e ν(x-1) ] 1 0 + νλ max Re(u -v, u -v) + 2Re 1 0 (g • u -g • v) T (u -v)e ν(x-1) dx.
Using the fact that g is Lipschitz, there exists a constant ς > 0 depending on L g such that:

2Re

((A n + G)u -(A n + G)v, u -v) ≤ -Re[(u -v) T Λ(u -v)e ν(x-1) ] 1 0 + (νλ max + ς)Re(u -v, u -v). (13)
To simplify the notation, u(1) and v(1) will be denoted, respectively, by u 1 and v 1 in following computations. Boundary terms can be rewritten as follows:

Re [(u -v) T Λ(u -v)e ν(x-1) ] 1 0 = Re(u 1 -v 1 ) T ΛRe(u 1 -v 1 ) -e -ν Re[H(u 1 -v 1 ) + B(σ n (Ku 1 ) -σ n (Kv 1 ))] T ΛRe[H(u 1 -v 1 ) + B(σ n (Ku 1 ) -σ n (Kv 1 ))] +Im(u 1 -v 1 ) T ΛIm(u 1 -v 1 ) -e -ν Im[H(u 1 -v 1 ) + B(σ n (Ku 1 ) -σ n (Kv 1 ))] T ΛIm[H(u 1 -v 1 ) + B(σ n (Ku 1 ) -σ n (Kv 1 ))].
For ν = ν n large enough compared to the norm of H, B and σ s,n , we deduce that

Re [(u -v) T Λ(u -v)e νn(x-1) ] 1 0 ≥ 1 2 (Re(u 1 -v 1 ) T ΛRe(u 1 -v 1 ) + Im(u 1 -v 1 ) T ΛIm(u 1 -v 1 )) ≥ 0
which implies with (13) that:

2Re (A n + G - λ max ν n + ς 2 I)u -(A n + G - λ max ν n + ς 2 I)v, u -v ≤ -Re[(u -v) T Λ(u -v)e νn(x-1) ] 1 0 ≤ 0 and therefore A n + G is λmaxνn+ς 2 dissipative.
Recall that, ν n was taken large enough compared to the norms of H, B and σ s,n . As (σ s,n ) n is a bounded sequence, we can take a sequence (ν n ) n bounded from above by a positive real ν.

As A n + G is νnλmax+ς 2 dissipative and νnλmax+ς 2 ≤ νλmax+ς 2 , A n + G is νλmax+ς 2 = ζ dissipative with ζ independent on n.
(2) A n + G satisfies the range condition:

Let us now prove the following range condition:

∃ρ sup > 0; ∀ρ ∈ (0, ρ sup ), Rg(I -ρ(A n + G)) ⊃ D(A n ). ( 14
)
It is equivalent to prove that for all v in D(A n ), there exists an element u in D(A n ) such that: 1)) This property is the most difficult to prove. It consists of proving the existence of a solution to a nonlinear ODE with a nonlinear boundary condition. To prove the existence, first, we will deal with the nonlinear boundary condition and then, using a fixed point theorem, the nonlinear source term will be taken into account; the method being inspired from [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] and [START_REF] Tisdell | Existence of solutions to first-order periodic boundary value problems[END_REF].

u + ρΛu x -ρg(u)=v u(0) =Hu(1) + Bσ n (Ku(
(2.

1) Taking into account the nonlinear boundary condition

Let us now prove the following range condition:

∀(v 1 , v 2 ) ∈ D(A n + G) = D(A n ), v 1 + ρGv 2 ∈ Rg(I -ρA n ).
(15) To do so, take v 1 , v 2 both in C 0 ([0, 1]). To prove assertion [START_REF] Miyadera | Nonlinear Semigroups. Translations of mathematical monographs[END_REF], we have to find an element u in D(A n ) solution of:

u + ρΛu x =v 1 + ρ(g • v 2 ) u(0)
=Hu(1) + Bσ n (Ku(1)) equivalent to:

u x + Λ -1 ρ u=Λ -1 v1 ρ + g • v 2 u(0) =Hu(1) + Bσ n (Ku(1)). ( 16
)
We define T :

C 0 ([0, 1]) → C 1 ([0, 1]) T : C 0 ([0, 1]) → C 1 ([0, 1]) y → u solution of the following system u x + Λ -1 ρ u=Λ -1 v1 ρ + g • v 2 u(0) =Hu(1) + Bσ n (Ky(1)). ( 17
)
If we prove that T is well-defined and admits a fixed point in D(A n ), then assertion ( 15) is proven.

For all y in C 0 ([0, 1]), solutions u to the ODE in (17

) are C 1 ([0, 1]) (because v 1 , g(v 2 ) ∈ C 0 ([0, 1]
)) and can be expressed as follow:

∀x ∈ [0, 1], u(x) = e -Λ -1 x ρ Z(y) + x 0 e -Λ -1 ρ (x-s) × Λ -1 v 1 ρ + g • v 2 ds
where Z(y) is a constant of R d . Thus, for all y in C 0 ([0, 1]), u is a solution to system [START_REF] Prieur | Wave equation with cone-bounded control laws[END_REF] if and only if Z(y) satisfies the following equation:

Z(y) = He -Λ -1 ρ Z(y) + H 1 0 e -Λ -1 ρ (1-s) × Λ -1 v 1 ρ + g • v 2 ds + Bσ n (Ky(1))
equivalent to:

(I d -He -Λ -1 ρ )Z(y) = H 1 0 e -Λ -1 ρ (1-s) × Λ -1 v 1 ρ + g • v 2 ds + Bσ n (Ky(1))
and for ρ sufficiently small, say ρ < δ(H, Λ) with δ(H, Λ) > 0, one can invert the last relation:

∀y ∈ C 0 ([0, 1]), Z(y) = (I d -He -Λ -1 ρ ) -1 H 1 0 e -Λ -1 ρ (1-s) × Λ -1 v 1 ρ + g • v 2 ds + Bσ n (Ky(1)) . (18)
Last equation ensures that for ρ < δ(H, Λ), T is well defined and

∀x ∈ [0, 1], T (y)(x) = e -Λ -1 x ρ Z(y) + x 0 e -Λ -1 ρ (x-s) × Λ -1 v 1 ρ + g • v 2 ds
with Z(y) defined in [START_REF] Slemrod | Feedback stabilization of a linear control system in Hilbert space with an a priori bounded control[END_REF].

The operator T being well-defined, we can focus on the fixed point argument. As σ n is bounded, Z(y) is bounded when y scans C 0 ([0, 1]). Hence, T (C 0 ([0, 1])) is a bounded set of C 1 ([0, 1]) and there exists a set

K = w ∈ C 1 ([0, 1]); ||w|| C 0 ([0,1]) ≤ M and ||w ′ || C 0 ([0,1]) ≤ M where M is a constant such that T (C 0 ([0, 1])) ⊂ K. As K is bounded in C 1 ([0, 1]), K is compact in C 0 ([0, 1]
) by Ascoli-Arzela's theorem. Moreover, T is continuous and K is closed and convex allowing to use Schauder fixed point theorem to conclude that:

∃u ∈ K : T (u) = u.
Hence, u ∈ D(A n ) = D(A n + G) and the assertion ( 15) is proven.

(2.2) Taking into account the source term

Let v be in D(A n + G) = D(A n ), ρ ≤ δ(H, Λ) and H : C 0 ([0, 1]) → C 0 ([0, 1]) be such that for all w in C 0 ([0, 1]), H(w) is solution of: u x + Λ -1 ρ u=Λ -1 ( v ρ + g • w) u(0) =Hu(1) + Bσ n (Ku(1)).
By assertion [START_REF] Miyadera | Nonlinear Semigroups. Translations of mathematical monographs[END_REF], H is well defined (take v 1 ← v and v 2 ← w). We will prove that H has a fixed point in D(A n ) which implies that the range condition ( 14) is verified.

To do so, we will prove that there exists a ball of C 0 ; B r of radius r such that B r is invariant under H and H(B r ) precompact in C 0 ([0, 1]).

(2.2.1) There exists a ball of C 0 ([0, 1]), B r , invariant under H Let w be in C 0 ([0, 1]) and let recall the definition of the usual C 0 ([0, 1]) norm:

||w|| C 0 ([0,1]) := max x∈[0,1] |w(x)| = max x∈[0,1] d i=1 w i (x) 2 . ( 19
)
The continuous function H(w) can be expressed as:

∀w ∈ C 0 ([0, 1]), ∀x ∈ [0, 1], H(w)(x) = e -Λ -1 x ρ Z(w) + x 0 e -Λ -1 ρ (x-s) × Λ -1 v ρ + g • w ds (20)
where:

∀w ∈ C 0 ([0, 1]), Z(w) = (I d -He -Λ -1 ρ ) -1 H 1 0 e -Λ -1 ρ (1-s) × Λ -1 v ρ + g • w ds + Bσ n (KH(w)(1)) . (21)
Fixing an w in C 0 ([0, 1]), we focus on a term present in both expressions of Z(w) and of H(w):

∀x ∈ [0, 1],

x

0 e -Λ -1 ρ (x-s) × Λ -1 ρ v ds = v(x) -e -Λ -1 ρ x v(0) - x 0 e -Λ -1 ρ (x-s) v x ds.
Hence,

∀x ∈ [0, 1], x 0 e -Λ -1 ρ (x-s) × Λ -1 ρ v ds = d i=1 x 0 e -λ -1 i ρ (x-s) × λ -1 i ρ v i ds 2 = d i=1 v i (x) -e -λ -1 i ρ x v i (0) - x 0 e -λ -1 i ρ (x-s) v i,x ds 2 ≤ d i=1 v i (x) 2 + d i=1 e -2λ -1 i ρ x v 2 i (0) + d i=1 x 0 e -λ -1 i ρ (x-s) v i,x ds 2
and because e -Λ -1 ρ (x-s) ≤ I d when s ≤ x:

∀x ∈ [0, 1], x 0 e -Λ -1 ρ (x-s) × Λ -1 ρ v ds ≤ 2||v|| C 0 ([0,1]) + ||v|| H 1 ([0,1]) =: C(v). (22)
Remark that C(v) is independent on ρ, w.

The second term to study is:

∀x ∈ [0, 1], x 0 e -Λ -1 ρ (x-s) × Λ -1 (g • w) ds 2 = d i=1 x 0 e -λ -1 i ρ (x-s) × λ -1 i (g i • w) ds 2 ≤ d i=1 |g i • w| 2 L 2 ([0,1]) ρ 2λ i (1 -e -2 λ -1 i ρ x ) ≤ d i=1 ρ 2λ i |g i • w| 2 L 2 ([0,1]) ≤ ρ 2 min i λ i ||g • w|| 2 C 0 ([0,1])
where we have used Cauchy-Schwarz inequality to get first inequality.

Using the fact that g(0) = 0 and g Lipschitz as a function from R d → R d for the canonical R d norm, one gets:

∀w ∈ C 0 ([0, 1]), ||g • w|| C 0 ([0,1]) ≤ max x∈[0,1] |(g • w)(x)) -g(0)| + |g(0)| ≤ L g max x∈[0,1] |w(x)|. Hence, ∀w ∈ C 0 ([0, 1]), ||g • w|| C 0 ([0,1]) ≤ L g ||w|| C 0 ([0,1])
and as a consequence:

∀w ∈ C 0 ([0, 1]), x ∈ [0, 1], x 0 e -Λ -1 ρ (x-s) × Λ -1 (g • w) ds ≤ ρ 2 min i λ i L g ||w|| C 0 ([0,1]) . (23)
Finally, using the boundedness of σ n , one gets the existence of a constant C 3 (only dependent on σ s , H, B) such that ∀ρ > 0 :

|(I d -He -Λ -1 ρ ) -1 Bσ n (KH(w)(1))| ≤ C 3 . (24)
Injecting inequalities ( 22), ( 23) and (24) in the definition of Z(w) (21), one gets the existence of C 1 (v), C 2 , C 3 independent on ρ, w such that:

∀w ∈ C 0 ([0, 1]), |Z(w)| ≤ C 1 (v) + C 2 √ ρ||w|| C 0 ([0,1]) + C 3 . (25)
Putting all together inequalities (22), ( 23) and (25) in the expression of H(w) [START_REF] Tisdell | Existence of solutions to first-order periodic boundary value problems[END_REF], there exists three positive constants C1 (v), C2 , C3 independent on ρ, w such that:

∀w ∈ C 0 ([0, 1]), ||H(w)|| C 0 ([0,1]) ≤ C1 (v) + C2 √ ρ||w|| C 0 ([0,1]) + C3 . ( 26
)
Taking ρ < 1 C2 2 (note that this bound is independent on v), it is possible to choose a strict positive r such that C1 (v) + C2 √ ρr + C3 ≤ r and:

∀w ∈ B r , ||H(w)|| C 0 ([0,1]) ≤ r.
which is no more than B r is invariant under H.

(2.2.

2) The set H(B r ) is precompact in C 0 ([0, 1])

The following claim allows to get compactness:

Claim 1. The set H(B r ) is uniformly bounded in C 1 ([0, 1]).
Proof of Claim 1. Let w ∈ B r . Denoting u = H(w), we have

u x = - Λ -1 ρ u + Λ -1 v ρ + g • w .
As u, v, w are all continuous, u x is also continuous. By previous section, u ∈ B r and as a consequence

||u x || C 0 ([0,1]) ≤ C
where C may depend on (r, ρ, λ, L g , v) but not on w.

This ends the proof of Claim 1.

To conclude, using Ascoli-Arzela's theorem, H(B r ) is relatively compact in C 0 ([0, 1]) and by Schauder fixed point theorem, H admits a fixed point which ends the proof of the range condition ( 14).

(3) A n + G is a closed operator:

To prove this, we first prove that A n is closed. Then, using the continuity of G, we conclude on the closedness of the operator A n + G. ). We have to show that u belongs to D(A n ) and that ũ = A n u. Let us define for all integers k, ũk := A n u k → L 2 ([0,1]) ũ which can also be written as:

∀k ∈ N, u k,x = -Λ -1 ũk → L 2 ([0,1]) -Λ -1 ũ. ( 27
)
As a consequence,

∀k ∈ N, ∀x ∈ [0, 1], u k (x) = u k (0) - x 0 Λ -1 ũk (s)ds = Hu k (1) + Bσ n (Ku k (1)) - x 0 Λ -1 ũk (s)ds. Let x be in [0, 1], we have | x 0 Λ -1 (ũ k (s) -ũ(s))ds| ≤ C × ||ũ k -ũ|| L 2 ([0,1]
) which tends to zero as k tends towards infinity. Hence,

∀x ∈ [0, 1], x 0 Λ -1 ũk (s)ds ----→ k→∞ x 0 Λ -1 ũ(s)ds. ( 28
)
Then as

H 1 ([0, 1]) ⊂ C 0 ([0, 1]) continuously : ∀k, m ∈ N, |u k (1) -u m (1)| ≤ C × ||u k -u m || H 1 ([0,1])
where C is the constant of the continuous injection

H 1 ([0, 1]) ⊂ C 0 ([0, 1]). Moreover, by (27), ||u k -u m || 2 H 1 ([0,1]) = ||u k -u m || 2 L 2 ([0,1]) + ||Λ -1 ũk -Λ -1 ũm || 2 L 2 ([0,1]
) which tends to zero as (k, m) tends towards infinity. Hence, (u k (1)) k is Cauchy and tends towards a real u 1 . From this and the pointwise convergence stated in (28), Hu k (1) + Bσ n (Ku k (1)) -x 0 Λ -1 ũk (s)ds converges pointwise towards a function denoted w (which can be different from u because the convergence of w is just pointwise) and:

∀x ∈ [0, 1], w(x) = Hu 1 + Bσ n (Ku 1 ) - x 0 Λ -1 ũ(s)ds. ( 29
)
Obviously, w is in H 1 ([0, 1]) and w x = Λ -1 ũ. The convergence of (u k ) k towards w is also in L 2 ([0, 1]):

1 0 (w(x) -u k (x)) 2 dx = 1 0 H(u 1 -u k (1)) + Bσ n (Ku 1 ) -Bσ n (Ku k (1)) + x 0 Λ -1 (ũ(s) -ũk (s))ds 2 dx ≤ 2 × (H(u 1 -u k (1)) + Bσ n (Ku 1 ) -Bσ n (Ku k (1))) 2 + 2 × 1 0 x 0 Λ -1 (ũ(s) -ũk (s))ds 2 dx ≤ C × (|u k (1) -u 1 | 2 + ||ũ k -ũ|| 2 L 2 ([0,1]) )
where we have used the identity (a+b) 2 ≤ 2a 2 +2b 2 to get first inequality and C is a constant depending on H, B, σ s,n and Λ.

The right-hand side of last equation tending towards zero as k tends to infinity, we have proven that (u k ) k tends towards w in L 2 ([0, 1]) and by the uniqueness of the limit w = u in L 2 ([0, 1]). Note that as w is continuous, we can take w = u in the sense of C 0 ([0, 1]). Moreover, as u k (1) tends towards w(1) = u(1) (because (u k ) k tends towards w pointwise) and towards u 1 (by definition of u 1 ) at the same time, we have that u(1) = u 1 . Injecting this last equality in (29), we have:

∀x ∈ [0, 1], u(x) = Hu(1) + Bσ n (Ku(1)) - x 0 Λ -1 ũ(s)ds.
Thus, u ∈ D(A n ) and ũ = A n u.

(3.2) A n + G is a closed operator

As A n is closed by previous paragraph and G is continuous as an operator from

L 2 ([0, 1]) into L 2 ([0, 1]) (because g is Lipschitz), A n + G is closed.
(4) First conclusions on the proof of Theorem A.1

As A n + G satisfies the range condition and is ζ dissipative; by [15, Theorem 5.12], A n + G generates a unique semigroup T n of type ζ. By Remark 2 p. 148 (and Theorem 4.10 (ii)) of the same book, the additional facts that A n + G is closed and L 2 ([0, 1]) is reflexive; if R 0,n ∈ D(A n ) then t → T n (t)R 0,n is the unique solution of the Cauchy problem:

   R n,t = (A n + G)R n R n (t = 0) = R 0,n R n (t) ∈ D(A n )
in the sense that for almost every t ≥ 0, T n (t)R 0,n is in D(A n ), time-Fréchet differentiable in L 2 ([0, 1]) and verifies the system presented just above.

As T n is a semigroup of type ζ in L 2 ([0, 1]), we have:

t → T n (t)R 0,n ∈ C 0 ([0, T ], L 2 ([0, 1])).
(

5) Regularity of the solution

Here we use the C 1 regularity assumptions on g and σ n . The augmented system satisfied by

U n := (R n , R n,x ) writes      U n,t + diag(Λ, Λ)U n,x = g 2 • U R n (0) = HR n (1) + Bσ n (KR n (1)) R n,x (0) = Λ -1 [H + Bσ ′ n (R n (1))](ΛR n,x (1) -g(R n (1))) + g(R n (0)) (30)
where

g 2 : R 2d → R 2d U := (U 1 , U 2 ) → (g(U 1 ), g ′ (U 1 )U 2 )
In substance, this augmented system is very similar to system [START_REF] Hayat | Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions[END_REF]. The transport part is identical and the source term is Lipschitz (g ′ is bounded by assumption). For the boundary condition of R n,x , it is linear in the variable R n,x . Knowing this, we can easily adapt the reasoning used for the system [START_REF] Hayat | Exponential stability of general 1-D quasilinear systems with source terms for the C 1 norm under boundary conditions[END_REF] for the variable R n alone to the system (30), the unbounded operator being now defined as

     D(A 2,n ) = (U 1 , U 2 ) ∈ H 1 ([0, 1]) 2 | U 1 (0) = HU 1 (1) + Bσ n (KU 1 (1)), U 2 (0) = Λ -1 [H + Bσ ′ n (U 1 (1))](ΛU 2 (1) -g(U 1 (1))) + g(U 1 (0)) A 2,n (U 1 , U 2 ) = -(ΛU 1,x , ΛU 2,x )
and the source term

D(G 2 ) = (L 2 ([0, 1])) 2 G 2 (U 1 , U 2 ) = g 2 (U 1 , U 2 ).
Recall that to apply the theory of nonlinear semigroups from [START_REF] Miyadera | Nonlinear Semigroups. Translations of mathematical monographs[END_REF], we need to prove the three following statements; A 2,n + G 2 is ζ dissipative for some ζ ∈ R, satisfies the range condition and is closed. As these proofs are very similar to what was done in the parts (1) ( 2) and (3) for the operator A n + G, we will just give a sketch of the proof and insist on crucial hypothesis.

• For ζ dissipativity, it is a comparison between boundary terms. The fact that σ ′ n is bounded (n fixed) is the key hypothesis to show this ζ dissipativity.

• For range condition, we need to solve (I -

ρ(A 2,n + G 2 ))(u 1 , u 2 ) = (v 1 , v 2 ) where (u 1 , u 2 ) is the unknown, (v 1 , v 2 ) ∈ D(A 2,n
) and ρ belonging to [0, ρ 0 ] with ρ 0 independent on (v 1 , v 2 ) to determine. We find u 1 using the fact that A n + G satisfies the range condition. For u 2 , it suffices to remark that the ODE to solve is linear in u 2 (even for the boundary condition) with a bounded linear source term (g ′ is bounded as g is Lispchitz).

• Finally, for the closedness one can proceed using same techniques as in previous sections and the closedness of

A n + G. Hence (R n , R n,x ) is C 0 ([0, T ], L 2 ([0, 1])). As a consequence, R n ∈ C 0 ([0, T ], H 1 ([0, 1])) and as R n,t = g(R n ) -ΛR n,x , we also have R n,t ∈ C 0 ([0, T ], L 2 ([0, 1])). To conclude, R n ∈ C 0 ([0, T ], H 1 ([0, 1])) ∩ C 1 ([0, T ], L 2 ([0, 1])).
All points of Theorem A.1 are now proven.

A.2. Convergence of solutions with smoothed saturations in L 2 ([0, 1])

We define the operator A by:

AR = -ΛR x D(A) = R ∈ H 1 ([0, 1]); R(0) = HR(1) + Bσ(KR(1)) . Note that D(A) = L 2 ([0, 1]) because C ∞ c ((0, 1)) ⊆ D(A) and C ∞ c ((0, 1)) is a dense subset of L 2 ([0, 1]
) by [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]Corollary 4.23]1 . The following lemma will be useful to prove the convergence of semigroups (T n ) n .

Lemma A.2. It holds A ⊂ lim n→∞ A n which means that every element of the graph of A is the limit of a sequence

{(x n , A n x n )} n (in L 2 ([0, 1]) × L 2 ([0, 1])) where x n ∈ D(A n ). Proof. Let R ∈ D(A). Let us define the sequence (R n ) n in H 1 ([0, 1]) by: ∀n ∈ N, ∀x ∈ [0, 1], R n (x) := (1 -x 2 )B[σ n (KR(1)) -σ(KR(1))] + R(x).
For all integers n, we have R n (0) = HR(1) + Bσ n (KR( 1)) (use the fact that R ∈ D(A)) and R n (1) = R(1). Hence, for all integers n, R n ∈ D(A n ) and by the pointwise convergence of (σ n ) n , we have

R n → H 1 ([0,1]) R which is equivalent to (R n , A n R n ) → L 2 ([0,1])×L 2 ([0,1]) (R, AR).
The convergence of semigroups (T n ) n is given by the following theorem:

Theorem A.3. The operator A + G generates a semigroup T of type ζ. Moreover, for all initial data R 0 in D(A) = L2 ([0, 1]) and sequence (R 0,n ) n such that R 0,n ∈ D(A n ) for all integers n and converging to R 0 in L 2 ([0, 1]):

∀T > 0, lim n→∞ T n (•)R 0,n = T (•)R 0 ∈ C 0 ([0, T ], L 2 ([0, 1]). ( 31
)
Proof. After Lemma A.1, all the A n + G are ζ dissipative. Moreover, A + G satisfies the range condition, the proof being identical to the proof of "A n + G satisfies the range condition ( 14)" (because the argument only uses the boundedness and the continuity of the saturation operator). Hence, it satisfies the distance condition:

∀Q ∈ D(A), lim inf ρ→0 + ρ -1 d(Rg(I -ρ(A + G)), Q) = 0. (32)
Finally, A ⊂ lim n→∞ A n by previous lemma. As a consequence, by the continuity of G as an operator from

L 2 ([0, 1]) to L 2 ([0, 1]), A + G ⊂ lim n→∞ A n + G.
Take R 0 ∈ D(A) and a sequence (R 0,n ) n of elements from D(A n ) for all n converging to R 0 in L 2 ([0, 1]). From [START_REF] Miyadera | Nonlinear Semigroups. Translations of mathematical monographs[END_REF]Theorem 6.8], A + G generates a semigroup T and

lim n→+∞ T n (t)R 0,n = T (t)R 0 in L 2 ([0, 1])
for all time t ≥ 0.

Moreover, the equality above holds uniformly on every bounded interval of [0, ∞) which proves Theorem A.3.

A.3. Conclusion of the proof of Theorem 1

(1) Existence Let R 0 be the initial data in D(A) = L 2 ([0, 1]). Take a smooth approximation (σ n ) n of σ and (R 0,n ) n a sequence of C ∞ ((0, 1)) satisfying compatibility conditions of order 1 converging towards R 0 in L 2 ([0, 1]). Note that there exists at least one sequence (R 0,n ) n of such initial data. Indeed, C ∞ c ((0, 1)) is dense in L 2 ([0, 1]) and functions of C ∞ c ((0, 1)) obviously satisfy compatibility conditions of order 1.

By Theorem A.1, for all integers n, the operator A n + G generates a semigroup T n and R n := T n (.)R 0 is the unique solution to [START_REF] Bastin | Stability And Boundary Stabilization Of 1-D Hyperbolic Systems[END_REF].

After Theorem A.3, (R n ) n converges in C 0 ([0, T ], L 2 ([0, 1])) towards an element R ∈ C 0 ([0, T ], L 2 ([0, 1]))
. R satisfies all requirements for being a weak solution to system (5).

(2) Uniqueness The limit R neither depends on the smooth approximation (σ n ) n nor on the sequence of initial data (R 0,n ) n chosen. To prove this, we take an arbitrary sequence of smooth approximations (σ n ) n and define the corresponding sequence of operators ( Ãn ) n . Then, take an arbitrary sequence ( R0,n ) n of elements of D( Ãn )∩H 2 ([0, 1]) converging to R 0 in L 2 ([0, 1]). For all integers n, we define Rn as the solution of:

   R n,t + ΛR n,x =g • R n R n (0, .) =HR n (1, .) + B σn (KR n (1, .)) R n (., 0) = R0,n .
We note R, the limit in C 0 ([0, T ], L 2 ([0, 1])) of ( Rn ) n . We define (σ n ) n as equal to σ n if n is even and equal to σn otherwise. In the same way, we define (R 0,n ) n as equal to R 0,n if n is even and equal to R0,n otherwise. Obviously, (σ n ) n is a smooth approximation of σ and (R 0,n ) n is a sequence of H 2 function satisfying compatibility conditions of order 1 and converging to R 0 in L 2 ([0, 1]). Hence, we can apply what was done before to prove that there exists a limit R in

C 0 ([0, T ], L 2 ([0, 1])) of the sequence (R n ) n .
As a consequence R = R = R and the limit is unique.

(3) Continuity of U T Let R 0 , R0 be two initial data in L 2 ([0, 1]) and R, R the associated weak solutions.

Take an arbitrary smooth approximation (σ n ) n of σ and sequences of approximation (R 0,n ) n and ( R0,n ) n of R 0 and R0 respectively. Using the ζ dissipativity of the A n + G proven in Theorem A.1 with ζ independent on n, one gets:

∀n ∈ N, ∀0 ≤ t ≤ T , ||T n (t)R 0,n -T n (t) R0,n || L 2 ([0,1]) ≤ e ζt ||R 0,n -R0,n || L 2 ([0,1]) .
Taking the limit for all time t ≤ T (see Theorem A.3):

∀0 ≤ t ≤ T, ||T (t)R 0 -T (t) R0 || L 2 ([0,1]) ≤ e ζt ||R 0 -R0 || L 2 ([0,1]) which implies that: ∀0 ≤ t ≤ T, ||T (t)R 0 -T (t) R0 || L 2 ([0,1]) ≤ e ζT ||R 0 -R0 || L 2 ([0,1]) .
It shows that U T is Lipschitz and hence continuous. This concludes the proof of Theorem 1.

Appendix B. Proof of Theorem 2

Take a ∆ ∈ D + d (R) satisfying R ∞ (∆(H + BK)∆ -1 ) < 1 and a positive µ < -log(R ∞ (∆(H + BK)∆ -1 )).
Take the smooth approximation (σ n ) n of σ from Remark 2. Notation A n , D(A n )... are the same as in Appendix A.1.

Let R 0 be an initial data in [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]) and as

D(A) = L 2 ([0, 1]) (for the moment we do not suppose that R 0 is in L ∞ ([0, 1])) . As C ∞ c ((0, 1)) is dense in L 2 ([0, 1]) (see
C ∞ c ((0, 1)) ⊂ D(A n ) for all integers n, one can construct a sequence (R 0,n ) n belonging to C ∞ c ((0, 1)) ⊂ D(A n ) converging to R 0 in L 2 ([0, 1]).
In what follows, we will consider such a sequence of initial data.

B.1. Local L ∞ stability of C 1 ([0, T ], L 2 ([0, 1])) ∩ C 0 ([0, T ], H 1 ([0, 1])) solutions All along section B.1, n is a fixed integer.
R n is the solution to the smooth problem:

   R n,t + ΛR n,x =g • R n R n (0, .) =HR n (1, .) + Bσ n (KR n (1, .)) R n (., 0) =R n,0 ∈ D(A n ) ∩ H 2 ([0, 1]).
As the initial data is H 2 ([0, 1]), satisfies compatibility conditions of order 1, the solution belongs to

C 1 ([0, T ], L 2 ([0, 1])) ∩ C 0 ([0, T ], H 1 ([0, 1])) for all T > 0 by Theorem A.1. Recall the definition of V (.), ∀f ∈ L ∞ ([0, 1]), V (f ) := max i∈ 1,d |δ i f i e -µx | L ∞ ([0,1]) (33)
where diag i∈ 1,d (δ i ) = ∆.

For all integers p, the functional V 2p is defined by:

∀f ∈ L ∞ ([0, 1]), V 2p (f ) := d i=1 1 0 q 2p p,i f 2p i e -2pµx dx 1/2p
where Q p = diag i∈ 1,d (q p,i ) will be chosen wisely later.

Claim 2. We can differentiate V 2p (R n (., t)) with respect to time and

∀t ≥ 0, dV 2p 2p (R n (., t)) dt = 2p d i=1 1 0 q 2p p,i R n,i,t R 2p-1 n,i e -2pµx dx. ( 34
)
Proof of Claim 2. Take t ≥ 0, dt > 0 and T > t.

V 2p 2p (Rn(.,t+dt))-V 2p 2p (Rn(.,t)) dt -2p d i=1 1 0 q 2p p,i R n,i,t R 2p-1 n,i e -2pµx dx ≤ d i=1 1 0 q 2p p,i R 2p n,i (x,t+dt)-R 2p n,i (x,t) dt -2pR n,i,t (x, t)R 2p-1 n,i (x, t) e -2pµx dx.
Thus it is sufficient to prove that at the limit when dt tends to zero

∀i ∈ 1, d , R 2p n,i (., t + dt) -R 2p n,i (., t) dt -2pR n,i,t (., t)R 2p-1 n,i (., t) → L 1 ([0,1]) 0. Let i be in 1, d , 1 0 R 2p n,i (.,t+dt)-R 2p n,i (.,t) dt -2pR n,i,t (., t)R 2p-1 n,i (., t) dx = 1 0
Rn,i(.,t+dt)-Rn,i(.,t) dt

(R n,i (., t + dt) 2p-1 + • • • + R n,i (., t) 2p-1 ) -2pR n,i,t (., t)R 2p-1 n,i (., t) dx ≤ || Rn,i(.,t+dt)-Rn,i(.,t) dt -R n,i,t || L 2 ([0,1]) × 1 0 |R n,i (., t + dt) 2p-1 + • • • + R n,i (., t) 2p-1 | 2 dx + 1 0 |R n,i,t | R n,i (., t + dt) 2p-1 + • • • + R n,i (., t) 2p-1 -2pR n,i (., t) 2p-1 dx
The term Rn,i(.,t+dt)-Rn,i(.,t) dt

-R n,t L 2 ([0,1]) converges towards zero because R n,i is in C 1 ([0, T ], L 2 ([0, 1])) while the term 1 0 R n,i (., t + dt) 2p-1 + • • • + R n,i (., t) 2p-1 2 dx is bounded because R n,i ∈ C 0 ([0, T ], H 1 ([0, 1])) ⊆ C 0 ([0, T ] × [0, 1]
). Finally, the term

1 0 R n,i,t ||R n,i (., t + dt) 2p-1 + • • • + R n,i (., t) 2p-1 -2pR n,i (., t) 2p-1 dx is bounded by ||R n,i,t || L 2 ([0,1]) ||R n,i (., t+ dt) 2p-1 +• • •+R n,i (., t) 2p-1 -2pR n,i (., t) 2p-1 || L 2 ([0,1]) . The term ||R n,i,t || L 2 ([0,1]) is bounded as R n is in C 1 ([0, T ], L 2 ([0, 1]))
whereas the other term converges towards zero by the dominated convergence theorem. Therefore, Claim 2 is proven.

Next, as R n verifies R n,t + ΛR n,x = g • R n , one gets:

dV 2p 2p (R n (., t)) dt = 2p d i=1 1 0 q 2p p,i (-λ i R n,i,x + g i • R) R 2p-1 n,i e -2pµx dx. ( 35
)
There are two terms in (35) whose origins are different:

• the transport term:

W 2p (R n (., t)) := -2p d i=1 1 0 q 2p p,i λ i R n,i,x R 2p-1 n,i e -2pµx dx (36)
• and the source term:

Z 2p (R n (., t)) := 2p d i=1 1 0 q 2p p,i (g i • R)R 2p-1 n,i e -2pµx dx (37)
so that:

dV 2p 2p (R n (., t)) dt = W 2p (R n (., t)) + Z 2p (R n (., t)). (38)
Define:

∀t ≥ 0, ξ n (t) := ∆R n (1, t).

(39) In the rest of the proof, we will sometimes denote ξ n (t) as ξ n for readability.

Study of the term transport term W 2p

Lemma B.1. For all µ < -log(R ∞ (∆(H + BK)∆ -1 )) and ε > 0 there exists p = p(µ, H, B, K, ∆, ε) such that for all initial data R 0,n ∈ H 2 ([0, 1]) satisfying compatibility conditions of order 1, the solution R n = T n (.)R 0,n to (3) verifies the following assertion.

If at time t ≥ 0,

|ξ n (t)| max < R ∞ (∆B∆ -1 )σ s,n |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -e -µ-ε | . ( 40
)
Then:

∀p > p, W 2p (R n (., t)) ≤ -2pµλ min V 2p 2p (R n (., t)). (41)
Remark 10. The positive real ε is introduced to prevent an eventual dependence of p with respect to t. Indeed, further in the paper, we integrate (41) with respect to time. This would not be possible if p = p( t).

•

Proof. In this proof we will sometimes drop the time dependence notation ξ(t) or R n (t) for readability.

Using an integration by parts in (36), we compute :

W 2p (R n (., t)) = - d i=1 q 2p p,i λ i R 2p n,i e -2pµx 1 0 -2pµ d i=1 1 0 q 2p p,i λ i R 2p n,i e -2pµx dx
and therefore

W 2p (R n (., t)) ≤ -W 2p,1 (R n (., t)) -2pµλ min V 2p 2p (R n (., t)) (42) where W 2p,1 (R n (., t)) := d i=1 q 2p p,i λ i R 2p n,i e -2pµx 1 0 .
Using the fact that R n (., t) is in D(A n ), one gets:

W 2p,1 (R n (., t)) = d i=1 q 2p p,i λ i R 2p n,i (1, t)e -2pµ -q 2p p,i λ i HR n (1, t) + Bσ n (KR n (1, t)) 2p i .
We have,

W 2p,1 (R n (., t)) = W 2p,11 + W 2p,12 (43) with:
W 2p,11 := d i=1 q 2p p,i λ i R 2p n,i (1, t)e -2pµ -q 2p p,i λ i (H + BK)R n (1, t) 2p i (44)
and

W 2p,12 := d i=1 q 2p p,i λ i (H + BK)R n (1, t) 2p i -q 2p p,i λ i HR n (1, t) + B(σ n (KR n (1, t))) 2p i . (45)
Note that if ξ n = 0, W 2p,1 is zero and the conclusion of Lemma B.1 holds. In what follows, we suppose that ξ n = 0.

Study of W 2p,11

For what follows, we choose Q p such that:

∀i ∈ 1, d , ∀p ∈ N, q p,i = λ -1/2p i δ i (46)
Inspired by [3, p. 123], we get the following estimates for W 2p,11

W 2p,11 = d i=1 q 2p p,i λ i R 2p n,i (1, t)e -2pµ -q 2p p,i λ i (H + BK)R n (1, t) 2p i ≥ |ξ n | 2p max e -2pµ - d i=1 q 2p p,i λ i (H + BK)R n (1, t) 2p i = |ξ n | 2p max e -2pµ - d i=1 δ 2p i d j=1 (H + BK) i,j ξn,j δj 2p ≥ |ξ n | 2p max e -2pµ -d × R ∞ (∆(H + BK)∆ -1 ) 2p
where we have used the equations q 2p p,i

λ i = δ 2p i , |ξ n | 2p max ≤ d i=1 |ξ i | 2p , ( 39
) and the definition of R ∞ .

For µ < -log(R ∞ (∆(H + BK)∆ -1 )) and p > -

log(d)+log(2)
2(µ+log(R∞(∆(H+BK)∆ -1 ))) , one has:

W 2p,11 |ξ n | 2p max ≥ e -2pµ 2 ≥ 0. ( 47
)
Study of W 2p,12 :

The term W 2p,12 is deeply related to the effect of saturation; it corresponds to the difference between the action of a linear controller and a saturated one. We recall the definition of W 2p,12 in (45):

W 2p,12 = d i=1 q 2p p,i λ i ( (H + BK)R n (1, t) 2p i -HR n (1, t) + B(σ n (KR n (1, t))) 2p i )
equivalent to:

W 2p,12 = d i=1 δ 2p i (χ 2p i -υ 2p i ) (48)
with:

χ = (H + BK)R n (1, t) and υ = HR n (1, t) + Bσ n (KR n (1, t)).
The following claim allows to neglect the term W 2p,12 relatively to W 2p,11 at the limit when p tends to infinity: Claim 3. Under the conditions of Lemma B.1 and if ξ n = 0, then there exists a p = p(µ, H, B, K, ∆, ε) such that:

∀p > p, |W 2p,12 | ≤ e -2pµ |ξ n | 2p max 4 . ( 49
)
Proof of Claim 3. Let i be in 1, d :

δ i |χ i | = | d j=1 (H + BK) i,j δ i R n,j (1, t)| = | d j=1 (H + BK) i,j δ i δ j ξ n,j | ≤ R ∞ (∆(H + BK)∆ -1 )|ξ n | max (50)
Then, denoting R n,j (1, t) by R n,j (j ∈ 1, d ) and Sat i the set

Sat i = {j ∈ 1, d | |(KR n ) j | > σ s,n and B i,j = 0} , one has δ i |υ i | =δ i d j=1 (H + BK) i,j R n,j + d j=1 B i,j (σ n,j ([KR n ] j ) -[KR n ] j ) =δ i d j=1 (H + BK) i,j R n,j + j∈Sati B i,j (σ n,j ([KR n ] j ) -[KR n ] j ) ≤δ i d j=1 |(H + BK) i,j R n,j | + δ i j∈Sati B i,j σ n,j ([KR n ] j ) - d k=1 K j,k R n,k ≤δ i d j=1 |(H + BK) i,j R n,j | + δ i j∈Sati |B i,j | | d k=1 K j,k R n,k | -σ s,n ≤R ∞ (∆(H + BK)∆ -1 )|ξ n | max + R ∞ (∆B∆ -1 )(R ∞ (∆K∆ -1 )|ξ n | max -σ s,n )
If Sat i is empty ie if the saturation does not act on the ith coordinate, then υ i = χ i and

δ i |υ i | ≤ R ∞ (∆(H + BK)∆ -1 )|ξ n | max (51)
by (50).

Otherwise, Sat i is non empty and

δ i |υ i | ≤R ∞ (∆(H + BK)∆ -1 )|ξ n | max + R ∞ (∆B∆ -1 )(R ∞ (∆K∆ -1 )|ξ n | max -σ s,n ). ( 52
)
Moreover as µ < -log(R ∞ (∆(H + BK)∆ -1 )) by hypothesis, there exists α ∈ (0, 1) such that

R ∞ (∆(H + BK)∆ -1 ) = αe -µ .
Additionally by (40),

R ∞ (∆(H + BK)∆ -1 )|ξ n | max + R ∞ (∆B∆ -1 )(R ∞ (∆K∆ -1 )|ξ n | max -σ s,n ) < e -µ-ε |ξ n | max .
Injecting the last two equations in (50)-(52), ). This condition recalls a local sector bounded condition [19, Section 1.7.2]. Indeed, the term W 2p,12 is induced by the difference between the linear control law and the its saturated version, i.e., this term arises from a deadzone nonlinearity. The term on the right-hand side of (54), e -µ |ξ n | max , represents the state of the system. Thus formally speaking, the condition

∀i ∈ 1, d , δ i |χ i | ≤ αe -µ |ξ n | max δ i |υ i | ≤ max(αe -µ , e -µ-ε )|ξ n | max . (53)
W 2p,12 = p→+∞ o(|ξ n | 2p max e -2pµ ), (54)
is somewhat equivalent to a regional sector condition [19, Section 1.7.2].

Remark 12. Conditions (40) are less restrictive when the saturation σ s,n is weaker and when the exponential decay rate µ decreases.

Analysis of the term Z 2p

Lemma B.2. For all integers p and for all time t ≥ 0,

|Z 2p (R n (., t))| ≤ 2pL g,max V 2p 2p (R n (., t)).
Proof. Recall the definition of Z 2p (R n (., t)) (37):

Z 2p (R n (., t)) = 2p d i=1 1 0 q 2p p,i (g i • R n )R 2p-1 n,i e -2pµx dx.
Using the hypothesis on g in ( 6):

|Z 2p (R n (., t))| ≤ 2p d i=1 1 0 q 2p p,i L g,i R 2p n,i e -2pµx dx ≤ 2pL g,max V 2p 2p (R n (., t)).
Conclusion on the L ∞ stability of regular solutions Lemma B.3. Under conditions of Lemma B.1, the solution R n = T n (.)R 0,n to problem (3) satisfies the following statement.

If at a time t ≥ 0, condition (40) is satisfied then:

dV 2p (R n (., t)) dt ≤ -(µλ min -L g,max )V 2p (R n (., t))
for all p > p where p defined in Lemma B.1.

Proof. Let p > p. By (38), we have:

dV 2p 2p dt = 2p dV 2p dt V 2p-1 2p = W 2p + Z 2p . (55)
dV 2p dt ≤ -(µλ min -L g,max )V 2p .
Before going further into the proof, we need the following lemma which may be useful in future works :

Lemma B.4. ∀R ∈ L ∞ ([0, 1]), V 2p (R) → p→+∞ V (R). Moreover, the convergence is uniform on all bounded sets of H 1 ([0, 1]). Proof. Let R be in L ∞ ([0, 1]) ∀p ∈ N, V 2p (R) = d i=1 1 0 δ 2p i λ i R 2p i e -2pµx dx 1/2p
.

As a consequence,

λ 1/2p min ||x → ∆R(x)e -µx || L 2p ([0,1]) ≤ V 2p (R) ≤ λ 1/2p max ||x → ∆R(x)e -µx || L 2p ([0,1]) . By a classic result of analysis ||Q|| L 2p ([0,1]) converges towards ||Q|| L ∞ ([0,1]) for all Q ∈ L ∞ ([0, 1]
). Hence, passing to the limit in last inequalities:

||x → ∆R(x)e -µx || L ∞ ([0,1]) ≤ lim p→+∞ V 2p (R) ≤ ||x → ∆R(x)e -µx || L ∞ ([0,1]) and: lim p→+∞ V 2p (R) = ||x → ∆R(x)e -µx || L ∞ ([0,1]) = V (R).
We also need to prove the uniform convergence for all bounded sets of H 1 ([0, 1]). We will prove it for the case of scalar functions; the case of vector valued function being similar. Take an r > 0. Let B r be the ball of radius r in H 1 ([0, 1]).

Define for ω > 0 and

f in B r , S f,ω := {x ∈ [0, 1]; |f (x)| ≥ |f | L ∞ ([0,1]) -ω}.
For all ω > 0 and f in B r , there exists an x in S f,ω/2 . For all y in [0, 1], we have:

f (y) = f (x) + y x f ′ (z)dz.
Using Cauchy-Schwartz inequality for the L 2 canonical scalar product, one gets

|f (y)| ≥ |f (x)| -r|x -y|. As x is in S ω/2 : |f (y)| ≥ |f | L ∞ ([0,1]) - ω 2 -r|x -y|
Thus, for all y in [0, 1] such that |x -y| ≤ ω 2r :

|f (y)| ≥ |f | L ∞ ([0,1]) -ω which implies that: ∀y ∈ [0, 1]; |x -y| ≤ ω 2r =⇒ y ∈ S f,ω .
As a consequence:

∀f ∈ B r , ∀0 < ω < r, µ(S f,ω ) ≥ ω r (56)
where µ designates the usual Lebesgue measure. Now using the definition of S f,ω and (56), one has: Now take an 0 < ε < 1, f in B r and pose ω = ε 2 ;

∀f ∈ B r , ∀ω > 0, ∀p ∈ N, ω r 1/2p (|f | L ∞ ([0,1]) -ω) ≤ µ(S f,ω ) 1/2p (|f | L ∞ ([0,1]) -ω) = S f,ω (|f | L ∞ ([0,1]) -ω) 2p dx 1/2p ≤ |f | L 2p ([0,1]) . Moreover, ∀f ∈ L ∞ ([0, 1]), |f | L p ([0,1]) ≤ |f | L ∞ ([0,1]) . Hence: ∀f ∈ B r , ∀ω > 0, ∀p ∈ N, ω r 1/2p (|f | L ∞ ([0,1]) -ω) ≤ |f | L 2p ([0, 1
ω r

1/2p (|f | L ∞ ([0,1]) -ω) -|f | L ∞ ([0,1]) ≤ ω r 1/2p -1 |f | L ∞ ([0,1]) -ω + |f | L ∞ ([0,1]) -ω -|f | L ∞ ([0,1]) ≤ ω r 1/2p -1 (r + ω) + ω = ε 2r 1/2p -1 (r + ε 2 ) + ε 2 ≤ ε 2r 1/2p
-1 (r + 1) + ε 2 .

There exists a P (r, ε) (independent on f ) such that for p > P (r, ε):

ε 2r 1/2p
-1 (r + 1) ≤ ε 2 so that:

∀f ∈ B r , |f | L 2p ([0,1]) -|f | L ∞ ([0,1]) ≤ ε.
This finishes the proof of Lemma B.4.

Lemma B.5. For all initial data R 0,n ∈ H 2 ([0, 1]) satisfying compatibility conditions of order 1 and µ < -log(R ∞ (∆(H + BK)∆ -1 )), if    µλ min -L g,max ≥ 0 V (R 0,n ) < e -µ R ∞ (∆B∆ -1 )σ s,n |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -e -µ | then, the solution R n = T n (.)R 0,n to (3) verifies: ∀t ≥ 0, V (R n (., t)) ≤ e -(µλmin-Lg,max)t V (R 0,n ).

Proof. Using a continuity argument, we can take a ε > 0 such that

V (R 0,n ) < e -µ R ∞ (∆B∆ -1 )σ s,n |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -e -µ-ε | . ( 57
)
Define for all integers p, T p := sup T ≥ 0; ∀t ∈ [0, T ], V 2p (R n (., t)) ≤ e -(µλmin-Lg,max)t V 2p (R 0,n ) .

(1) (T p ) p is not a bounded sequence

We will prove that the sequence (T p ) p is not bounded by contradiction. Suppose (T p ) p bounded and take a subsequence still denoted (T p ) p converging towards a limit denoted T ∞ .

(1.1) T ∞ is strictly positive For all integers i in 1, d ξ n,i (0)e -µ = δ i R n,i (1, 0)e -µ ≤ V (R n (., 0)) = V (R 0,n ). By (57), |ξ n | max (0) < R ∞ (∆B∆ -1 )σ s,n |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -e -µ-ε | .

By continuity of t → ξ n (t) = ∆R n (1, t), there exists a dt 0 > 0 independent on p such that:

∀t ∈ [0, dt 0 ], |ξ n | max (t) < R ∞ (∆B∆ -1 )σ s,n |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -e -µ-ε | .

Applying Lemma B.3 and noting that condition (40) is satisfied for t ← t ∈ [0, dt 0 ], there exists a p(µ, H, B, K, Λ, ε) ∈ N (independent on time):

∀p > p, ∀t ∈ [0, dt 0 ], dV 2p (R n (., t)) dt ≤ -(µλ min -L g,max )V 2p (R n (., t)).

Integrating on [0, dt 0 ], it holds ∀p > p, ∀t ∈ [0, dt 0 ], V 2p (R n (., t)) ≤ e -(µλmin-Lg,max)t V 2p (R 0,n ). This allows to assert that: ∀p > p, T p ≥ dt 0 and passing to the limit, T ∞ ≥ dt 0 > 0.

(1.2) Proof of the contradiction As R n ∈ C 0 ([0, 1] × [0, T ]) for all T > 0 and by definition of T p : ∀p ∈ N, V 2p (R n (., T p )) = e -(µλmin-Lg,max)Tp V 2p (R 0,n ).

As (R n (T p )) p is a bounded sequence of H 1 ([0, 1]) (because R n is in C 0 ([0, T ], H 1 ([0, 1])) for all T > 0), we can use the uniform convergence proven in Lemma B.4 to pass to the limit as p tends towards infinity in last equation:

V (R n (., T ∞ )) = e -(µλmin-Lg,max)T∞ V (R 0,n ) ≤ V (R 0,n ). As a consequence, for all integers i in 1, d ξ n,i (T ∞ )e -µ = δ i R n,i (1, T ∞ )e -µ ≤ V (R n (., T ∞ )) ≤ V (R 0,n ) and by (57), we have:

|ξ n | max (T ∞ ) <
R ∞ (∆B∆ -1 )σ s,n |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1 ) -e -µ-ε | .

As t → ξ n (t) = ∆R n (1, t) is a continuous function (remember that R n is in C 0 ([0, 1] × [0, T ]) for all T > 0), there exists a 0 < dt < T ∞ independent on p (but dependent on n) such that:

∀t ∈ [T ∞ -dt, T ∞ + dt], |ξ n | max (t) < R ∞ (∆B∆ -1 )σ s,n |R ∞ (∆(H + BK)∆ -1 ) + R ∞ (∆B∆ -1 )R ∞ (∆K∆ -1
) -e -µ-ε | .

Applying Lemma B.3 and noting that condition (40) is satisfied for t ← t ∈ [T ∞ -dt, T ∞ + dt], there exists p(µ, H, B, K, Λ, ε) ∈ N such that :

∀p > p, ∀t ∈ [T ∞ -dt, T ∞ + dt],
dV 2p (R n (., t)) dt ≤ -(µλ min -L g,max )V 2p (R n (., t))

which is no more than:

∀p > p, ∀t ∈ [T ∞ -dt, T ∞ + dt], d[V 2p (R n (., t
))e (µλmin-Lg,max)t ] dt ≤ 0.

Integrating last statement on [T ∞ -dt, t] for T ∞ -dt ≤ t ≤ T ∞ + dt: ∀p > p, ∀t ∈ [T ∞ -dt, T ∞ + dt], V 2p (R n (., t)) ≤ e -(µλmin-Lg,max)(t+dt-T∞) V 2p (R n (., T ∞ -dt)) (58) Moreover, as T ∞ -dt < T ∞ and lim p→∞ T p = T ∞ , there exists a P (dt) such that for all p > P (dt), T ∞ -dt < T p . It implies that: ∀p > P (dt), ∀t ≤ T ∞ -dt, V 2p (R n (., t)) ≤ e -(µλmin-Lg,max)t V 2p (R 0,n ). Using last inequality and (58), one has: ∀p > max(P (dt), p), ∀t ≤ T ∞ + dt, V 2p (R n (., t)) ≤ e -(µλmin-Lg,max)t V 2p (R 0,n ). As a consequence and by the definition of T p : ∀p > max(P (dt), p), T p ≥ T ∞ + dt passing to the limit as p tends towards infinity, one gets:

T ∞ ≥ T ∞ + dt implying that dt ≤ 0 which is a contradiction.

(2) Proof of exponential convergence Hence, (T p ) p is not a bounded sequence and there exists a non decreasing subsequence (T φ(p) ) p of (T p ) p such that: lim p→∞ T φ(p) = +∞. Take an arbitrary time t ≥ 0, there exists a P such that for p > P then T φ(p) ≥ t. It implies that: ∀p > P, V 2φ(p) (R n (., t)) ≤ e -(µλmin-Lg,max)t V 2φ(p) (R 0,n ) Passing to the limit as p tends towards infinity, we get:

V (R n (., t)) ≤ e -(µλmin-Lg,max)t V (R 0,n ) and Lemma B.5 is proven.

Figure 1 .

 1 Figure 1. Functions σ ′ n and σ n for n = 20 and σ s = 1

( 3 . 1 )

 31 A n is a closed operator: Take a sequence (u k ) k of elements of D(A n ) such that lim k→∞ u k =: u in L 2 ([0, 1]) and lim k→∞ A n u k =: ũ in L 2 ([0, 1]

  ]) ≤ |f | L ∞ ([0,1])which implies:∀f ∈ B r , ∀ω > 0, ∀p ∈ N, 0 ≤ |f | L 2p ([0,1]) -|f | L ∞ ([0,1]) ≤ ω r 1/2p (|f | L ∞ ([0,1]) -ω) -|f | L ∞ ([0,1]) .

  This ends the proof of Claim 3.As a consequence, by (47), (49) and for p > p,W 2p,1 = W 2p,11 + W 2p,22 ≥ W 2p (R n (., t)) ≤ -2pµλ min V 2p 2p (R n (.,t)). This finishes the proof of Lemma B.1. Remark 11. In the proof of the above result, we enforce W 2p,12 = p→+∞ o(|ξ n | 2p max e -2pµ

	Finally, from (53) and (48), we have
	∀p ∈ N, Hence, there exists an integer p > -2(µ+log(R∞(∆(H+BK)∆ -1 ))) (depending on (ε, α) but not on |ξ n | max ) such that W 2p,12 max ≤ 2d × max e -2pε , α 2p e -2pµ . |ξ n | 2p log(d)+log(2)
	∀p ≥ p,	W 2p,12 |ξ n | 2p ≤	e -2pµ 4	.
	(42), one gets:			|ξn| 2 max pe -2p 4	≥ 0. Injecting last statement in

Univ. Paul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse Cedex 9, mathias.dus@math.univtoulouse.fr

[START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] Univ. Grenoble Alpes, CNRS, Grenoble INP, GIPSA-

lab, 38000 Grenoble, France. Research by F. Ferrante has been partially supported by the CNRS-INS2I under the JCJC grant CoBrA and by the Grenoble Institute of Technology under the grant CrYStAL. c EDP Sciences, SMAI 1999

In fact, it is shown that C ∞ c (Ω) is dense in L

(Ω) for Ω open of R N (N ∈ N)) but the proof can be easily adapted to our context.

B.2. Local L ∞ ([0, 1]) stability for weak solutions Finally, we prove Theorem 2. The integer n is not fixed anymore and we will pass to the limit in the exponential stability of the sequence (R n ) n .

Proof. By [START_REF] Brezis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF]Corollary 4.23], for all R ∈ L ∞ ([0, 1]) there exists a sequence of elements of C 2 c ((0, 1)), (R n ) n , such that:

and

where the convergence holds in

Applying the Corollary stated above to x → ∆R 0 (x)e -µx , there exists a subsequence (S 0,n ) n of elements of C 2 c ((0, 1)) such that:

and

Defining, for all integers n, R 0,n = x → ∆ -1 S 0,n (x)e µx , one gets:

This concludes the proof of Lemma B.6.

Take the sequence (R 0,n

As for all integers n, V (R 0,n ) ≤ V (R 0 ), σ s,n ≥ σ s and (8):

we can apply Lemma B.5 for all integers n,

Now let t ≥ 0 and p ∈ N. By Fatou's lemma and the fact that (R n (., t)) n converges up to a subsequence towards R(., t) in the almost everywhere sense (because it converges in L 2 ([0, 1]) by Theorem A.3): 46) and the definition of V and V 2p ), we have:

where we have used the fact that V (R 0,n ) ≤ V (R 0 ) by construction.

Passing to the limit when p goes to infinity, one gets:

V (R(., t)) ≤ e -(µλmin-Lg,max)t V (R 0 ). Theorem 2 is proven.


Sylvain Achelle
email: sylvain.achelle@univ-rennes1.fr

Julián Rodríguez-López

Filip Bureš

Françoise Robin-Le Guen

Julián Rodríguez López

Tuning the Photophysical Properties of Push-Pull Azaheterocyclic Chromophores by Protonation: A Brief Overview of a French-Spanish-Czech Project

Keywords: white-light emission, push-pull chromophores, nitrogen heterocycles, protonation, photoluminescence

An account is provided of our studies on the effect that the addition of acid has on the photophysical properties of π-extended push-pull chromophores that incorporate azaheterocyclic rings as electron-withdrawing moieties. The work carried out to afford white light-emitting materials based on an equilibrium between neutral and protonated forms that emit complementary colors is described.

Introduction

The design of push-pull chromophores has been the subject of intensive research during the past few decades. [1] In this context, π-deficient nitrogen heterocycles such as pyridine, [2] quinoline, [3] diazines (pyridazine, [4] pyrimidine, [5] and pyrazine [6] ), benzodiazines (quinoxaline, [ 3 , 6 ] quinazoline, [ 3 , 5 c] phthalazine, [7] etc.), phenanthroline, [START_REF] Bonaccorso | [END_REF] triazines, [9] and tetrazines [10] are considered as moderate-to-strong electron withdrawing groups. Some pushpull chromophores that bear these heterocycles as the electron-attracting part exhibit intense luminescence properties. In such cases an intramolecular charge transfer (ICT) occurs and this can be easily tuned by adjusting the donor/attractor couple, the π-conjugated linker or by developing multipolar structures. [ 1 ,11] An enhanced ICT leads to a bathochromic effect on the absorption spectrum whereas the effect on the emission becomes less predictable: either an increase or decrease in the intensity accompanied by a red shift, along with total quenching, can be observed depending on the structure. Engaging the electron lone pair of the nitrogen R e v i s e d m a n u s c r i p t atoms of azaheterocycles provides a way to increase their electron-withdrawing character. In this respect, it is feasible to induce substantial changes in the photophysical properties of push-pull molecules bearing such fragments by complexation, [12] alkylation, [13] or reversible protonation. [14] This latter case is called halochromism or acidochromism. The general mechanism for the halochromic response is the redistribution of the charge density due to protonation, which promotes changes in the electronic structure. As a consequence, azaheterocyclic chromophores can be used as pH sensors. [15] Likewise, a series of pyridinebased molecules were reported to be acid-triggered reversible luminescent and nonlinear optical (NLO) switches both in solution and in the solid state. [16] The linear and NLO properties could also be tuned by coordination to metal centers. [16c,d] Moreover, tuning of the photo-and electroluminescence was achieved by an in situ controlled non-reversible thermal deprotonation, which can be exploited to fabricate smart thermal sensors and antifraud devices. [17] The controlled protonation of some blue-emitting azaheterocyclic derivatives affords white photo-and electroluminescence by the formation of orange-emissive acidified species. [18] Over the past few decades our groups have intensively studied the photophysical properties of push-pull chromophores based on nitrogen heterocycles. In particular, we recently focused our investigations on the structure-dependent effect of protonation on the luminescence properties of such chromophores. One of our goals was to obtain efficient white light-emitting materials based on neutral and protonated forms that emit complementary colors. In this personal account, we would like to provide a brief overview of our 'French-Spanish-Czech' collaborative efforts made in this direction. The pyrimidine nucleus generally acts as a monobasic compound with pK a ≈ 1.1.

Pyrimidine chromophores

Thus, it is a weaker base than pyridine (pK a ≈ 5.2). Protonation of the first nitrogen atom leads to a marked decrease in the basicity of the second nitrogen atom due to the attractive inductive effect of the resulting quaternary nitrogen (second pK a ≈ -6.3). [START_REF] Milcent | Chimie Organique Hétérocyclique[END_REF] Although some of us described some basic information on the pH sensitivity of arylpyrimidine derivatives in 2008, [START_REF] Achelle | [END_REF] the first comprehensive study on the effects of the protonation concerns 4,6-bis(arylvinyl)pyrimidines 1-8 (Figure 1). [21] With the exception of compound 3, which has a trifluoromethyl electron-withdrawing substituent, the addition of trifluoroacetic acid (TFA) to a CH 2 Cl 2 solution of compounds 1-8 induced a dramatic color change (Figure 1). This color change was fully reversible by neutralization with a base such as Et 3 N or KBu t O. As far as emission is concerned, compounds substituted with moderate electron-donating groups (1, 4, 5 and 7) or CF 3 fragments (3) remained luminescent upon protonation and this was associated with a red-shifted emission band, a significant decrease in the Stokes shifts and, in some cases, an increase in the fluorescence quantum yield (1, 3 and 4). For stronger electron-donating groups such as NPh 2 (chromophore 2) and for the dendritic structure 8, the emission was completely quenched after protonation, whereas ferrocene derivative 6 remained non-emissive in acidic media. Similarly, an acetonitrile solution of the arylvinylpyrimidine 9, which contains a dipicolylamine substituent, showed red-shifted absorption and emission quenching upon addition of TFA (Figure 1). [22] R e v i s e d m a n u s c r i p t 4-Arylvinyl-2,6-di(pyridin-2-yl)pyrimidines [23] 10, 11, 13 and 14, 4arylvinylpyrimidines 15-17, 19-22, 24, 27 and 28, and 2-arylvinylpyrimidines 29-33 (Figure 2) [24,25,26,27] exhibited similar behavior to that found for 4,6-bis(arylvinyl)pyrimidine derivatives 1-8. Namely, THF or CH 2 Cl 2 solutions of these compounds underwent a significant color change upon addition of acid (Figure 3). Whereas a red shift of the charge transfer absorption band was observed, the emission was either completely quenched for the compounds substituted with strong electron-donating amino groups (11, 14, 16, 17, and 24) or red-shifted for chromophores with weaker methyl, methoxy, and thiomethyl electron donors (10, 13, 15, 19, 21, and 22). As an example, the change in the absorption and emission spectra of a THF solution of 10 upon gradual addition of p-toluensulfonic acid (p-TSA) is shown in unit. [25] The most favorable site for protonation in the ground state was found at N1. It should be noted that for dimethylamino derivatives 12 and 23, as well as piperidinyl compounds 18

and 25, the protonation led to a blue-shifted absorption due to protonation of the basic amino groups. Nevertheless, this phenomenon was not observed for 2-arylvinylpyrimidines 31 and 33. [26] Surprisingly, in contrast to monosubstituted triphenylamine 16 and 2,6-di(pyridin-2yl)pyrimidine analog 14, the protonated form of 26 remained luminescent in CH 2 Cl 2 with an emission band at 570 nm. [28] This fact can be explained by a slightly reduced ICT in this case.

Color change and protonation for compounds 27 and 28, with an extended π-conjugated linker, was observed at higher concentrations of TFA (10 -2 -10 -1 M) than for other diazine derivatives with shorter π-conjugated scaffolds. [27] 4-Arylvinylpyrimidine chromophores have also been incorporated into polymethyl methacrylate-based copolymers 34-36 (Figure 5). [29] The macromolecules displayed similar photophysical properties in CH 2 Cl 2 solution when compared to the free chromophores. Thus, upon adding acid, a red shift was obtained both in the absorption and emission spectra. The emission was partially (34) or totally (35) quenched for the amino-substituted compounds. The behavior upon acidic treatment of pyranylidene derivatives 37-39 (Figure 6)

depends on the π-conjugated linker. The addition of acid to compounds 37 and 38 led to a significant red shift in the absorption and this is attributed to protonation of the pyrimidine ring, with the solution remaining non-emissive. [30] Compound 39, with a phenylene unit in the linker, showed different behavior: a blue-shifted absorption band appeared progressively upon addition of TFA and the protonated species exhibited an intense blue emission at 450 nm (Φ F = 0.13) attributed to the formation of a pyrylium cation. Chromophores 11 (Figure 2) and 40 (Figure 7) have been incorporated into pluronic F68 nanoparticles and stable aqueous colloidal solutions were obtained. [31] The position of the absorption band of the aqueous emulsions was virtually the same as that of the chloroform solutions. On decreasing the pH, these emulsions underwent an immediate and fully reversible color change from yellow to purple. The spectra showed the decrease of the initial The change in the color emission of a CH 2 Cl 2 solution of the tristyrylpyrimidine 41 upon addition of (1S)-(+)-10-camphorsulfonic acid (CSA) is illustrated in Figure 8. [32] The photoluminescence spectra are characterized by the progressive disappearance of the band of the neutral form at 441 nm, which is accompanied by the progressive appearance of a new red-shifted band at 540 nm that corresponds to the monoprotonated species. The neutral form emits dark blue light whereas the protonated form emits green light. A mixture of the two produces cyan light. The pictures were taken in the dark upon irradiation with a hand-held UV lamp (λ exc = 366 nm). Adapted with permission from reference [32] (copyright 2018 American Chemical Society).

As previously shown, the emission of amino-substituted pyrimidines is generally (partially) quenched upon protonation. Nevertheless, the presence of weaker electrondonating groups occasionally results in a moderate quenching or even an enhancement of the fluorescence intensity. The fluorescence quantum yields of these compounds are mostly low, but the extension of the π-conjugated scaffold leads to a significant enhancement of the fluorescence intensity without modifying the solubility properties of the chromophores. Blue emitting neutral forms can be combined with red/orange protonated species to obtain white emitting materials. With this goal, a series of seven methoxy-substituted pyrimidine derivatives 5 (Figure 1), 19 (Figure 2), and 42-46 (Figure 9) with extended π-conjugated backbones was studied. [25,33] In CH for 44 with 100 equivalents of TFA (λ exc = 355 nm). Fluorophore 5 also exhibited white photoluminescence by controlled protonation both in solution and the solid state when incorporated into a polymer matrix. [33] The emission spectra and the color changes of thin films of polystyrene doped with 1 wt% of 5 in the absence and the presence of TFA, respectively, are shown in Figure 11. Photographs were taken upon irradiation with a hand-held UV-lamp (λ exc = 366 nm). Adapted with permission from reference [33] (copyright 2015 Royal Chemical Society).

A series of carbazole-and triphenylamine-substituted pyrimidines 47-52 (Figure 12)

was also designed for white light emission. [34] In this case, the shortening of the π-conjugated scaffold is compensated by the stronger electron-donating character of the carbazolyl and

Other diazine and benzodiazine chromophores

Other diazines such as pyrazine (pK a ≈ 0.65) and pyridazine (pK a ≈ 2.3) and benzodiazines such as quinoxaline (pK a ≈ 0.56) and quinazoline (pK a ≈ 3.5) are also monobases that can be protonated. [START_REF] Milcent | Chimie Organique Hétérocyclique[END_REF] Compounds 53-55 (Figure 14) were compared to their pyrimidine analog 19 (Figure 2). [25] All compounds displayed red-shifted absorption upon protonation by TFA. As far as the emission was concerned, the effect of protonation varied according to the nature of the diazine ring. Whereas the progressive addition of acid prompted a color change from blue to green in the emission of pyrimidine 19, the luminescence of pyrazine 54 and quinoxaline 55 was attenuated and ultimately quenched, as illustrated in Figure 15 for a CH 2 Cl 2 solution of compound 54. In contrast, pyridazine 53 was not luminescent and remained non-emissive after protonation. Theoretical calculations predicted that protonation was favored on the nitrogen atom farthest from the vinyl moiety. The protonated pyridazine derivative 53 remained non-fluorescent because vanishing overlap occurs between the HOMO located on the methoxynaphthylvinyl moiety and the LUMO centered on the diazine ring. For pyrazine and quinoxaline derivatives 54 and 55, the emission was computed to be considerably redshifted to the near IR region where nonradiative decays are well known to be more efficient.

Moreover, given that the bright excited state for quinoxaline 55 in the gas phase was envisaged to be destabilized by more than 11 kcal/mol, such nonradiative channels probably explain the fluorescence quenching of this protonated quinoxaline derivative. Nevertheless, some quinoxaline chromophores exhibit a more intense and red-shifted emission after protonation. [24] On the other hand, dimethylamino derivatives 56-58 (Figure 14) displayed blue-shifted absorption due to the protonation of the amino group. [24] R e v i s e d m a n u s c r i p t Figure 15. Changes in the emission spectra of a CH 2 Cl 2 solution of 54 upon addition of TFA (10-30000 equiv), λ exc = 352 nm. Reproduced with permission from reference [25] (copyright 2016 American Chemical Society).

The effect of protonation on the photophysical properties of a series of aminosubstituted quinazoline chromophores 59-63 (Figure 16) has been also studied. [35] In all cases, a red-shifted absorption, as evidenced by the significant color change experienced by their CH 2 Cl 2 solutions (Figure 17), associated with total emission quenching was observed. It would be interesting to study the effect of protonation with other quinazolines substituted by weaker electron-donating fragments, such as methoxy or carbazolyl groups.

Pyridine chromophores

Pyridine (pK a ≈ 5.2) is a stronger base than (benzo)diazines. The use of this basic heterocyclic compound allows easy control of the acid-base equilibrium with the aim of obtaining white luminescence. As the pyridine ring has a weaker electron-withdrawing character than pyrimidine, the methoxy or carbazole substituents were replaced by stronger electron-donating triphenylamino groups. Thus, a series of triphenylamine derivatives bearing pyridine units 64-70 was designed (Figure 18). [36] Likewise, the corresponding trifluoroacetate salts were isolated and characterized. For all series of compounds, the gradual addition of TFA to CHCl 3 solutions led to the progressive disappearance of both of the absorption and emission bands and the appearance of a new red-shifted absorption/emission band. As an example, the changes for compound 70 are represented in Figure 19. Timeresolved emission studies performed on compounds 67-69 revealed an energy transfer from the neutral to the protonated species on the 1 ps time scale. [37] As expected, a substantial equivalents of TFA (from left to right). Photographs were taken upon irradiation with a handheld UV-lamp (λ exc = 366 nm). Adapted with permission from reference [36] (copyright 2017 Elsevier).

In order to overcome this problem, the reinforcement of the electron-donating character of the triphenylamine was tested by the introduction of methoxy, carbazolyl or methoxythienyl fragments at the triphenylamine core (compounds 71-76, Figure 21).

Unfortunately, the protonated forms of these chromophores were not emissive in any case. [38] R e v i s e d m a n u s c r i p t

Conclusions

The extraordinary optical properties of π-extended push-pull molecules that incorporate azaheterocyclic rings as electron-withdrawing moieties provide an important tool to develop new sensing and luminescent materials. In the last year, we have prepared and studied a large library of such chromophores through a fruitful collaborative 'French-Spanish-Czech' project. In this review we have illustrated the effect that the addition of acid has on the photophysical properties of these materials. White light photoluminescence can be obtained by controlled protonation, which involves an equilibrium between two different forms of complementary emitting colors. Nevertheless, major efforts are still required to harness the full potential of these compounds.

3 Filip

 3 Bureš studied chemistry at the University of Pardubice, where he also obtained his PhD in organic chemistry in 2005. He pursued (post)doctoral studies with Professor P. Knochel (LMU, Munich) and Professor F. Diederich (ETH, Zurich). In 2007 he established a working group at the Institute of Organic Chemistry, University of Pardubice, which focuses on the design and synthesis of π-conjugated molecules for miscellaneous (opto)electronic applications. Françoise Robin-le Guen received her PhD in organometallic chemistry from the University of Bretagne Occidentale (France). She is currently full professor at IUT Lannion, University of Rennes 1 and head of the heterocyclic organometallic teams at the Chemical Sciences Institute of Rennes. Her main research topics are focused on the design of π-conjugated heterocyclic and organometallic systems for dye-sensitized solar cells, luminescence and nonlinear optics.

Figure 1 .

 1 Figure 1. Color change of CH 2 Cl 2 solutions of compounds 1-8 (acetonitrile solution for 9) after treatment with TFA.

Figure 4 .

 4 Figure 4. Compound 20 remained non-emissive upon protonation. Theoretical calculations

Figure 2 .

 2 Figure 2. Chemical structures of compounds 10-33.

Figure 3 .

 3 Figure 3. Color change of CH 2 Cl 2 solutions of compounds 16, 20, and 17 (from left to right) (c = 10 -3 M in the presence of 10 -2 M TFA).Adapted with permission from reference[24] (copyright 2012 American Chemical Society).

Figure 4 .

 4 Figure 4. Changes in the absorption (left) and emission (right) spectra of 10 (2.0 × 10 -5 M in THF) upon addition of p-TSA (7.5 × 10 -5 M to a large excess). Reproduced with permission from reference [23] (copyright 2011 American Chemical Society).

Figure 5 .

 5 Figure 5. Chemical structures of copolymers 34-36.

Figure 6 .

 6 Figure 6. Chemical structures of pyranylidenes 37-39.

  around 420 nm for both chromophores, together with the development of a new red-shifted band located at 480 nm for 11 and 530 nm for 40 (Figure7). The pK a values were estimated to be 1.1 and 1.8 for 11 and 40, respectively. The 'pH switch' value between the yellow and purple forms is very low, but these nanoparticle assemblies behave as colorimetric pH sensors in aqueous media. It should be noted that all aqueous emulsions were non-emissive -probably due to hydrogen bonding interactions between the hydrophilic pluronic segments and/or water protons with the nitrogen atoms of the pyrimidine ring.

Figure 7 .

 7 Figure 7. Absorption spectra of dilute pluronic F-68 nanoparticles incorporating chromophore 40 at acidic, basic and intermediate pH (dotted line). Inset: color change upon addition of HCl. Adapted with permission from reference [31] (copyright 2012 Springer).

Figure 8 .

 8 Figure 8. Left: changes in the emission spectra of a CH 2 Cl 2 solution of 41 (c = 9.64 × 10 -6 M) upon addition of CSA (0.1-50 equiv). λ exc = 380 nm. Right: changes in the color after addition of 0.8 equiv (middle) and 50 equiv of CSA (right). The pictures were taken in the dark upon irradiation with a hand-held UV lamp (λ exc = 366 nm). Adapted with permission from reference[32] (copyright 2018 American Chemical Society).

Figure 9 .

 9 Figure 9. Chemical structures of methoxy derivatives 42-46.

Figure 10 .Figure 11 .

 1011 Figure 10. a) Changes in the color of a solution of 42 (c = 1.6 × 10 -4 M in CH 2 Cl 2 , left) after addition of 10 equivalents (middle) and 1000 equivalents (right) of TFA. b) Changes in the color of a solution of 44 (c = 1.2 × 10 -5 M in CH 2 Cl 2 , left) after addition of 100 equivalents (middle) and 1000 equivalents (right) of TFA.Photographs were taken upon irradiation with a hand-held UV-lamp (λ exc = 366 nm). Reproduced with permission from reference[25] (copyright 2016 American Chemical Society).

  diphenylamino groups with respect to the methoxy group. All of these compounds exhibited an intense purple/blue luminescence in CH 2 Cl 2 solution with high fluorescence quantum yields (up to 0.86 for 49). Triphenylamine derivatives 47, 49, and 51 as well as 2-carbazolyl pyrimidine 48 showed almost no luminescence upon addition of camphorsulfonic acid, although 50 and 52 led to intense yellow luminescence under the same conditions (Φ F = 0.63 for 50 and Φ F = 0.45 for 52). White light emission both in solution and in solid state was observed by a combination of neutral and protonated forms of 52 (Figure13).

Figure 12 .

 12 Figure 12. Chemical structures of compounds 47-52.

Figure 13 .

 13 Figure 13. Top: Changes in the color of a solution of 52 (c = 1.25 × 10 -5 M in CH 2 Cl 2 , left) after addition of 45 equivalents (middle) and in 10 -2 M CSA (right). Bottom: Changes in the color of filter paper samples after immersion in a CH 2 Cl 2 solution of polystyrene doped with 52 (1 wt%) in the absence (left) and presence of 0.1 equivalents (middle) and 80 equivalents (right) of camphorsulfonic acid. Photographs were taken upon irradiation with a hand-held UV-lamp (λ exc = 366 nm).

Figure 14 .

 14 Figure 14. Chemical structures of compounds 53-58.

Figure 16 .

 16 Figure 16. Chemical structures of compounds 59-63.

Figure 17 .

 17 Figure 17. Color change of CH 2 Cl 2 solutions of several quinazoline derivatives in the presence of TFA. Adapted with permission from reference [35] (copyright 2014 American Chemical Society).

  positive emission solvatochromic behavior for 64 was observed when protonated, which is consistent with a significant enhancement of the ICT in the acidified form. For all compounds, the controlled protonation of the blue emitting pyridine derivatives led to white photoluminescence in solution due to the formation of yellow/orange emissive pyridinium species(Figure 20, top). The chromaticity coordinates were close to those of pure white light (0.33, 0.33). Doped polystyrene thin films of compounds 64 and 70 were also prepared using different ratios of neutral and protonated forms. Unfortunately, although the films remained luminescent, it was not possible to obtain white light in the solid state. When compound 64 was used both the absorption and the emission band experienced a significant blue shift with regard to the CHCl 3 solution, and the chromaticity coordinates were far from those of pure white light. Multi-color photoluminescence was achieved by tuning the ratio of the pyridine/pyridinium forms in the case of fluorophore 70(Figure 20, bottom).

Figure 18 . 20 Figure 19 .

 182019 Figure 18. Chemical structures of pyridine derivatives 64-70.

Figure 20 .

 20 Figure 20. Top: changes in the color of a CHCl 3 solution of 70 (c = 1.86 × 10 -5 M) after the addition of 0.8 equivalents (middle) and 5 equivalents of TFA (right). Bottom: colors of polystyrene thin films of 70 (1 wt%) in the absence and the presence of 0.4, 0.6, and 5equivalents of TFA (from left to right). Photographs were taken upon irradiation with a handheld UV-lamp (λ exc = 366 nm). Adapted with permission from reference[36] (copyright 2017 Elsevier).

Figure 21 .

 21 Figure 21. Chemical structures of compounds 71-76.

  44, and 46 are highly emissive in both the neutral (Φ F < 0.14-0.90) and protonated forms (Φ F < 0.15-0.72). The color changes of CH 2 Cl 2 solutions of 42 and 44 upon gradual addition of TFA are shown in Figure10. White light emission was observed with chromaticity

	coordinates of (0.32, 0.36) for 42 with 20 equivalents of TFA (λ exc = 346 nm) and (0.33, 0.33)

2 Cl 2 solution, compound 19 and thiophene derivatives 43 and 45 exhibited low emission quantum yields (Φ F < 0.04). Nevertheless, compounds 5, 42, R e v i s e d m a n u s c r i p t

Acknowledgments

The authors wish to thank all those individuals who have been involved in the different stages of the work described in the manuscript. Funding by the Junta de Comunidades de Castilla-La Mancha/FEDER (project SBPLY/17/180501/000214) and the Ministerio de Economía y Competitividad/Agencia Estatal de Investigación/FEDER (project CTQ2017-84561-P) is also gratefully acknowledged. F.B. is particularly indebted to the

After two post-doctoral stays at the the University of Texas at Austin for one year, under the supervision of Professor Philip Magnus.