Mich Ele Sebag
email: michele.sebag@polytechnique.frmarc.schoenauer@polytechnique.fr

Marc Schoenauer

Mutation by Imitation in Boolean Evolution Strategies

Adaptive heuristics have been developed in the Evolution

Strategy (ES) frame regarding the mutation of real-valued variables. But these heuristics poorly extend to discrete variables: when the rate or variance of mutation gets too small, mutation has no e ect any more.

To overcome this problem, we propose two mutation operators, that use the worst individuals of the current population as beacons indicating the limits of the current promising region: Mutation by di erentiation drives individuals away from the beaconindividuals. Mutation by imitation inversely assumes that beacon-individuals still contain relevant informations, and aims at moving the individual at hand nearer to the beacons. Mutation by imitation produces o spring that share the features of several \parents"; but in contrast with classical crossover, it allows one to control the distance between the o spring and the main parent, by xing the number of bits to mutate. Mutation by imitation thus permits a tunable exchange of informations among individuals. Both operators have been implemented in a boolean ( + ) ES framework, and experimented on four problems: the Royal Road problem, a GA-deceptive problem, the combinatorial multiple knapsack optimization problem and the Long Path problem. Comparative validation is presented and discussed.

Introduction

The three historical roots of evolutionary computation are known to be evolutionary programming (EP), evolution strategies (ES) and genetic algorithms (GA). Original ES were developed in the mid sixties [START_REF] Schwefel | Numerical Optimization of Computer Models[END_REF][START_REF] Rechenberg | Evolutionstrategie: Optimierung Technisher Systeme nach Prinzipien des Biologischen Evolution[END_REF] to deal with real parameter optimization. EP was initially developed in the context of Finite State Machines 7], but was later modi ed and tuned to handle di erent representations, including real parameters. Last, early GAs heavily relied on the bitstring representation 10, 9], and were later extended to other representations 22].

No optimization method can accurately handle all optimization problems, according to the \No Free Lunch Theorem " 28]. The key question thus remains to choose the good evolutionary algorithm depending on the current problem. To address this issue, comparative validations have been conducted on many test-beds, most of them involving real-valued tness functions (e.g. the DeJong test-suite 3]); see 2, 5, 6] among many others. Except in the GA literature, little attention has been paid to boolean problems, despite the fact that real-world problems often involve discrete features. The evolutionary landscape thus appears divided: GAs are to be preferred for discrete problems only [START_REF] Radcli E | Equivalence class analysis of genetic algorithms[END_REF][START_REF] Janikow | An experimental comparison of binary and oating point representations in genetic algorithms[END_REF]; In the meanwhile, the e cient strategies developed in EP and ES to evolve real-valued features seem to transpose poorly to the discrete case 1]. This paper is concerned with the mutation of discrete features in the framework of evolution strategies. The idea is to use the current population as a reservoir of information. More precisely, the worst individuals in the current population are considered as beacons signaling the limits of the present promising region, which adaptively move as the population is driven toward regions of increasingly high tness. These beacons give rise to two mutation operators: mutation by imitation moves the individual at hand toward the beacons, whereas mutation by di erentiation rather moves the individual at hand away from the beacons. These mutation operators allow individuals to exchange information and directly in uence each other. This was only possible so far through recombination. The advantage of the proposed mechanism is that it allows one to control the distance between o spring and parents by xing the number of bits to mutate, whereas classical recombination o ers no guarantee as to the distance between o spring and parents. This paper is organized as follows. Section 2 brie y reviews the main trends regarding the mutation of real-valued and discrete features in EP and ES. Mutation by imitation and mutation by di erentiation are described in section 3. These operators are experimented on four problems: the Royal Road 18, 19, 8], the Ugly (GA-deceptive) problem 27], the combinatorial multiple knapsack optimization problem 21, 14], and the Long Path problem 11]. Comparative results are discussed in section 4. We conclude with some avenues for further research.

State of the art

The general real-valued optimization problem is stated as: F being a real-valued function de ned on IR n , nd x 2 E IR n , such that F(x) = MaxfF(y); y 2 Eg Real-valued evolutionary algorithms now exist in all three communities (see for instance 16] for an example of real-valued GA). Real-valued crossover is based on the linear combination of two or several parents 16, 2], or on classical n-point recombination. Real-valued mutation involves the addition of a Gaussian random variable:

x i := x i + N(0; i ) where N(0; ) represents the Gaussian random variable of standard deviation . The main di culty remains to adjust parameters i .

Most real-valued GAs use xed i , or i decreasing along generations according to a xed schedule. Early ESs use the well-known 1=5 rule 23], where a global is modi ed according to the recent e ects of mutation: if more than one-fth of the mutations have been successful in the last generation (i.e. lead to an o spring more t than the parent), increase , otherwise decrease . The schedule for increasing and decreasing is geometrical schedule (the factor suggested by Schwefel 24] is 1.1). Last, in early EP 4], the standard deviation of mutation was determined on the basis of the tness of the individual at hand, in such way that most t individuals undergo mutation with proportionally smaller standard deviations.

Both ES and EP nally turned to adaptive mutation: the standard deviations i are encoded in the individuals, and themselves undergo evolution. The standard deviations are therefore adjusted \for free" along evolution 2].

This mechanism encounters several di culties in the GA framework, which uses mutation very sparingly (mutation rate per bit is often around 10 3 ): this gives little chance for the adaptive mutation of real-valued features to actually adapt. Further, Gaussian-like mutation seems to poorly evolve discrete features 1]: when the standard deviation gets very small, mutation comes to having no e ect at all on discrete features. This violates the strong causality principle, stating that \a small change in the parameters should result in a small change in the tness" 23]; as a matter of fact, no \small change" below the one-bit threshold is possible in the discrete case.

Evolution under In uence

This section presents three population-driven mechanisms to mutate discrete features in the ES framework.

Evolution and Induction

Our approach is loosely inspired from previous work devoted to the inductive control of evolution 26, 25]: evolution is considered a sequence of events (operators giving birth to new individuals), classi ed good or bad depending on whether the o spring are more or less t than the parents. These events, gathered through either spying evolution or experimenting on the current population, are processed by Inductive Learning algorithms [START_REF] Michalski | A theory and methodology of inductive learning[END_REF][START_REF] Kodrato | Introduction to Machine Learning[END_REF]. Induction can thus construct online some \law of bad events", which may then be used in a variety of ways to guide the next generations (e.g. to control the disruptiveness of evolution operators, or prevent the loss of genetic diversity). The coupling of Evolution and Induction allows one to continually create, use and update knowledge about evolution.

The present paper is similarly based on the assumption that one can get valuable hints by observing evolution, and use them to guide the further steps of evolution. The di erence is that what was previously observed was the evolution operators, whereas the present work directly considers individuals.

Imitation and Di erentiation

Induction needs positive and negative examples; these are respectively set to the most t and less t o spring 1 .

Indeed, the di erence of tness between a positive and a negative example is due to the di erence of their gene values. From this fact, one may infer that mutation should preserve these di erences, and preferably modify those bits which take same value for the good individual at hand, and the negative examples. Practically, for each individual Ex, the pro le (p 1 ; : : : ; p N ) of its di erences with the negative examples is computed, where p i is the fraction of negative examples di ering from Ex on bit x i . Mutation by di erentiation then mutates bit x i in individual Ex with a probability proportional to 1 p i ; the resulting o spring will be still farther from the negative examples, than the parent was.

The number K of bits to mutate is set by the user, and it is presently constant along evolution and over the population. Mutation by di erentiation expectedly leads to highly diversi ed populations: if most o spring share a given bit value, this bit will be mutated with high probability. This covers as a particular case strategies devised to restore genetic diversity (see 16] among others).

One could also consider that negative examples contain valuable, if not rstrate, information: after all, these individuals or their parents have survived up to now. This leads to mutation by imitation, where bit x i in individual Ex is mutated with a probability proportional to p i . Mutation by imitation tends to \repair" explorers whenever they have lost some information shared by the previous population; it favors uniformity.

Last, and since the two above operators have opposite virtues, a third possibility consists in alternating mutation by imitation and mutation by di erentiation: this hopefully allies the conservative properties of mutation by imitation and the high rate of exploration of mutation by di erentiation.

We nally propose three kinds of evolution \under in uence": evolution by imitation, evolution by di erentiation and alternate evolution. The two rst schemes respectively use mutation by imitation and mutation by di erentiation as single evolution operator. In the third scheme, even-numbered generations undergo mutation by imitation whereas odd-numbered generations undergo mutation by di erentiation.

Experimental Validation

Evolution under in uence has been implemented in a standard ( + ) ES frame, and compared to canonical GA, canonical ES (based on the 1=5 rule), and standard adaptive ES, on four problems.

The problems

The Royal Road problem was conceived by Holland, Mitchell and Forrest 18] to study into details the interaction of features most adapted to GA search. An analysis of the unexpected di culties of this problem is found in [START_REF] Mitchell | When will a genetic algorithm outperform hillclimbing ?[END_REF][START_REF] Forrest | What makes a problem hard for a genetic algorithms : Some anomalous results and their explanation[END_REF].

The Ugly problem is a GA-deceptive problem 27], built by concatenation of 10 instances of the elementary problem, de ned on = f0; 1g 3 , by F(x) = 3 if x = 111; F(x) = 2 for x in 0 ? ?, and F(x) = 0 otherwise.

The combinatorial multiple knapsack optimization problem 14] is de ned as follows:

Given P knapsacks having respective capacities c 1 ::c P , Given N objects having respective costs p 1 ::; p N , Given the overall dimension w i;j of object i regarding knapsack j, Determine a subset of objects noted X = x 1 ; ::x N , with x i boolean, that is feasible, i.e. satis es the constraints relative to the maximal capacities of all knapsacks, and maximizes the overall pro t: Max f P N i=1 p i :x i ; 8j = 1::P; P N i=1 w i;j x i < c j :g A usual heuristics in evolutionary constrained optimization 16] consists in reducing the tness of non feasible individuals:

F(X) = 8 < : P N i=1 p i x i if X is feasible r 2 P N i=1 p i x i if

r is the percentage of satis ed constraints

The Long Path problem was conceived by Horn and Goldberg 11] as a unimodal problem hard for local searchers, that is, hill-climbers and the GAs mutation operator. The tness landscape is composed of a large low-tness plateau, within which a path of slowly increasing tness leads to the unique optimum. This path is of length 2 N=2 (N is the length of the bitstring). Two successive individuals on the path di er by one bit, while all other individuals on the path are at Hamming distance of at least 2.

Experimental settings

All results are averaged on 15 independent runs. The dynamics of evolution is visualized by plotting the (averaged) best tness obtained for a number of tness calculations. We compared six evolutionary schemes:

A canonical GA (CGA) serves as reference 9]: the parents are selected based on their tnesses with rank-based selection, 2-point crossover is applied at the rate of 0.6; the mutation is applied at the rate of 0:2 N , where N denotes the length of the bitstring.

A \traditional ES" (TES) is a boolean ( + ) evolution strategy involving a single mutation rate per bit ; is modi ed according to the 1/5 rule of Rechenberg 23]. The geometrical factor used to increase ranges from 1.1 to 2. An \adaptive ES" (AES) is a boolean ( + ) evolution strategy that transposes the adaptive mutation described

Comparative results

The Royal Road problem. The canonical GA evolves a population of 60 individuals; all other algorithms follow a (30 + 60) ES scheme. The imitation scheme outstandingly outperforms all other schemes. This can be explained from the resemblance between mutation by imitation and crossover, and the fact that this problem explicitly relies on the building block hypothesis. In the meanwhile, the elitist ( + ) scheme prevents evolution from the eeting discovery phenomenon 8], which was considered the main cause for the di culties of standard GAs on this problem.

Nb of generations

The Ugly problem. The canonical GA evolves a population of 40 individuals; all other algorithms follow a (20 + 40) ES scheme.

The AES algorithm (ES with adaptive mutation) slightly supersedes the TES algorithm (ES following the 1/5 rule), and both signi cantly outperform the canonical GA algorithm on this GA-deceptive problem (Fig. 2(a)). The di erentiation scheme appears the best one on this problem, though the imitation scheme soon catches up; this can be interpreted as di erentiation allows rejecting suboptimal solutions previously found (the deceptive schemata). Again, this is possible only because the ( + ) scheme counterbalances the disruptive e ects of mutation by di erentiation, which basically escapes any good schema the current population is settled in.

The Knapsack problem. The canonical GA evolves a population of 40 individuals; all other algorithms follow a (20 + 40) ES scheme. Figure 4 shows average results for 1000 generations. Note that the optimum will always be found by all methods in the (very) long run, by climbing up the path through mutations. What is measured here is the ability to nd short-cuts in spite of the low-tness plateau between di erent components of the path. The long path problem presents intriguing particularities. First, the simple, traditional ES signi cantly supersedes CGA, whereas this problem was meant to be hard for mutation alone. Second, it demonstrates that alternate evolution can behave quite di erently (and in fact, much better) than evolution by imitation or evolution by di erentiation stand-alone. A tentative explanation is related to the accordion-like behavior of alternate evolution: mutation by imitation reduces the diversity and moves an individual toward the previous local optimum closest to this individual; then, mutation by di erentiation re-increases the diversity and spreads individuals away. The combination of these two phases seemingly eases the discovery of shortcuts on the long and sneaky path toward the optimum.

Discussion and Perspectives

The central characteristics of evolution under in uence is to allow individuals to exchange information along evolution. Such exchange is usually ensured by recombination, does it concern two parents or more 2]. And, according to the building blocks principle 10, 9], this exchange is the essential factor of biological and arti cial evolution.

Another analysis 13] emphasizes that recombination can, in some contexts, be advantageously replaced by macro-mutation | equivalent to recombination with a random parent. It is noteworthy that macro-mutation and recombination o er di erent kinds of control on the di erence between o spring and parents. Macro-mutation ensures that the o spring will be signi cantly di erent from the parent | but does not set any bias on the di erence. In opposition, recombination is heavily biased: the localization, and hence the amount, of the di erence between an o spring and a parent depends on the current population.

To sum up, macro-mutation controls how much o spring are di erent from parents, but does not care about where, that is, which bits are modi ed. Inversely, recombination tends to modify bits so as to copy other individuals in the population, which implies that it hardly has any e ect when the population is on the verge of converging.

Mutation by imitation combines the e ects of recombination and macromutation in that it allows for a quantitative and qualitative control of the difference between o spring and parents: The amount of di erence (the number of mutated bits) is set by the user; The localization of the di erence (which bits are modi ed) depends on the other individuals. Mutation by di erentiation and alternate mutation similarly enable an exchange of information between individuals, that is tunable by the user.

The main originality of this work, in our opinion, is to make clear the distinction between the amount and the nature of modi cations done by evolution. Further, evolution under in uence determines the nature (that is, the localization) of the modi cations in a way which depends on both the current population, and the individual at hand. Many improvements can be brought to evolution under in uence. Further research is concerned with adaptively adjusting the number of bits to mutate, and the kind of mutation (by imitation, di erentiation, or alternate) to apply on a given individual. Mutation could also take into account, besides the di erence between the current individual and the negative examples, its di erence with other positive examples.

Last, this scheme will be extended to multi-modal tness landscapes; the idea would be to evolve an individual depending on the negative examples nearest to this individual, in order to separately follow several tracks.

  in 1] with minor di erences: Each individual is attached one mutation rate p (the probability of mutation of any single bit) which itself undergoes mutation according to Obalek's rule 20, bound of 1=N on p to guarantee e ective mutation. Mutations under in uence have been implemented in the frame of a ( + ) ES. Positive and negative examples respectively are the most t and less t individuals among the o spring. Imitation-and di erentiation-based evolutions respectively use mutation by imitation and mutation by di erentiation as single evolution operators. Alternate evolution alternatively evolves populations through mutation by imitation and mutation by di erentiation. Unless otherwise speci ed, the number K of bits to mutate is set to 1.

Figure 1 :

 1 The Royal Road, Dynamics of evolution (beware of the Y-scale).

Figure 2 :

 2 The Ugly problem, Dynamics of evolution.

Figure 3 :

 3 The Knapsack problem, Dynamics of evolution. This problem includes a great many of local optima, and a strong mutation is needed to avoid premature convergence; mutating more than just one bit (K = 2) demonstrates clearly bene cial in the Imitation and Di erentiation schemes.The Long Path problem. The canonical GA evolves a population of 21 individuals; all other algorithms follow a (7+21) ES scheme. The size of the problem (number of bits of individuals) is 91. Parameter K is set to 2, the minimum jump for short-cuts on the path.

Figure 4 :

 4 The Long Path problem, Dynamics of evolution.

Care must be exercised in discarding individuals with intermediate tnesses: otherwise hazardous conclusions can be drawn from making distinctions between almost equally t individuals.

Nb of generationsMax. Fitness

Max. Fitness


Amaury Hayat

Global exponential stability and Input-to-State Stability of semilinear hyperbolic systems for the L 2 norm

Keywords: Global stabilization, exponential stability, Lyapunov, hyperbolic systems; nonlinear, nonlocal, inhomogeneous 2010 MSC: 35F60, 35F61, 93D09, 93D15, 93D20, 93D30

In this paper we study the global exponential stability in the L 2 norm of semilinear 1-d hyperbolic systems on a bounded domain, when the source term and the nonlinear boundary conditions are Lipschitz. We exhibit two sufficient stability conditions: an internal condition and a boundary condition. This result holds also when the source term is nonlocal. Finally, we show its robustness by extending it to global Input-to State Stability in the L 2 norm with respect to both interior and boundary disturbances.

Introduction

Hyperbolic systems can be found everywhere in sciences and nature: from biology [START_REF] Filbet | Derivation of hyperbolic models for chemosensitive movement[END_REF], to fluid mechanics, population dynamics [START_REF] Tadmor | Critical thresholds in flocking hydrodynamics with non-local alignment[END_REF], electromagnetism, networks [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF][START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF][START_REF] Espitia | Fluid-flow modeling and stability analysis of communication networks[END_REF] etc. For this reason, they are of large importance for practical applications and the question of their stability and stabilization is paramount.

For linear 1-d systems, studying the exponential stability or the stabilization can Email address: amaury.hayat@enpc.fr (Amaury Hayat)

Preprint submitted to Journal of L A T E X Templates November 22, 2020 be achieved by looking at the eigenvalues and using spectral mapping theorems [START_REF] Lichtner | Spectral mapping theorem for linear hyperbolic systems[END_REF][START_REF] Renardy | On the type of certain C 0 -semigroups[END_REF]. For nonlinear systems, the situation is much more tricky. For nonlinear systems, the situation is much more tricky. In general the stabilities in different norms are not equivalent [START_REF] Coron | Dissipative boundary conditions for nonlinear 1-d hyperbolic systems: sharp conditions through an approach via timedelay systems[END_REF]. Indeed, for the same system, stabilities in different norms can require different criteria. For semilinear systems the spectral tools may still work (in contrast with quasilinear systems), but the resulting exponential stability may only hold locally, meaning for small enough perturbations.

Worse, most of the time spectral tools are hard to use when the system is inhomogeneous. Several tools were developed to deal with this situation and obtain local exponential stability results. A first method is the characteristic analysis, which was originally used by Li and Greenberg in 1984 in [START_REF] Greenberg | The effect of boundary damping for the quasilinear wave equation[END_REF] then generalized in [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF][START_REF] Qin | Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems[END_REF][START_REF] Zhao | The boundary value problem for systems of first-order quasilinear hyperbolic equations[END_REF][START_REF] Wang | Exact controllability for nonautonomous first order quasilinear hyperbolic systems[END_REF] for quasilinear homogeneous hyperbolic systems in the C1 norm.

A second method is the use of basic Lyapunov functions 1 . This method was, for instance, applied in [START_REF] Bastin | Stability and Boundary Stabilisation of 1-D Hyperbolic Systems[END_REF]Chapter 6] for general semilinear systems in the H 1 norm and quasilinear systems in the H2 norm, but also in many particular cases [START_REF] Bastin | A quadratic Lyapunov function for hyperbolic density-velocity systems with nonuniform steady states[END_REF][START_REF] Gugat | Neumann boundary feedback stabilization for a nonlinear wave equation: A strict H 2 -Lyapunov function[END_REF][START_REF] Hayat | A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope[END_REF][START_REF] Hayat | Exponential stability of density-velocity systems with boundary conditions and source term for the H 2 norm[END_REF][START_REF] Dick | A strict H 1 -Lyapunov function and feedback stabilization for the isothermal Euler equations with friction[END_REF]. This will be our approach in this article. A third method is the backstepping method, a very powerful tool originally designed for finite-dimensional systems, modified for PDEs using a Volterra transform in [START_REF] Krstic | Boundary control of PDEs[END_REF], 2 and then used in [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF][START_REF] Hu | Boundary exponential stabilization of 1-d inhomogeneous quasilinear hyperbolic systems[END_REF] for quasilinear hyperbolic systems in the H 2 norm. Such backstepping approach was also used to derive controllability [START_REF] Coron | Optimal time for the controllability of linear hyperbolic systems in one-dimensional space[END_REF][START_REF] Coron | Null-controllability of linear hyperbolic systems in one dimensional space[END_REF] or finite-time stabilization [START_REF] Coron | Null controllability and finite time stabilization for the heat equations with variable coefficients in space in one dimension via backstepping approach[END_REF][START_REF] Coron | Finite-time stabilization in optimal time of homogeneous quasilinear hyperbolic systems in one dimensional space[END_REF][START_REF] Coron | Lyapunov functions and finite time stabilization in optimal time for homogeneous linear and quasilinear hyperbolic systems[END_REF] in both parabolic and hyperbolic settings. Other results using a more general transform were then introduced [START_REF] Zhang | Internal rapid stabilization of a 1-D linear transport equation with a scalar feedbackWorking paper or preprint[END_REF][START_REF] Coron | Rapid stabilization of a linearized bilinear 1-D Schrödinger equation[END_REF]. The main drawback of this method is that it involves controls that are usually using full-state measurements and cannot take the simple form of output feedback controllers (see ( 3)). Therefore these controls might be less convenient for practical implementation. Although sometimes observers can be designed to tackle this issue [START_REF] Di Meglio | A backstepping boundary observer for a class of linear first-order hyperbolic systems[END_REF]. Other methods exist, as for instance the study of stability based on time delay systems introduced in [START_REF] Coron | Dissipative boundary conditions for nonlinear 1-d hyperbolic systems: sharp conditions through an approach via timedelay systems[END_REF] where the authors give criteria for exponential stability in the W 2,p norms for any p ≥ 1 (see also [START_REF] Chitour | Approximate and exact controllability of linear difference equations[END_REF]).

So far, the nonlinear stability results for hyperbolic systems have been obtained in the H 1 norm for semilinear systems and for the H 2 norm for quasilinear systems. The H 1 and H 2 norm enabling to bound the nonlinear terms of the source term and of the transport term respectively, using the Sobolev embed-

dings H p ([0, L]; R) ⊂ C p-1 ([0, L]; R), for p ≥ 1.
Other results have been shown

for the C 0 and C 1 norm [START_REF] Hayat | Boundary Stability of 1-D Nonlinear Inhomogeneous Hyperbolic Systems for the C 1 Norm[END_REF][START_REF] Hayat | On boundary stability of inhomogeneous 2× 2 1-D hyperbolic systems for the C 1 norm[END_REF]. For weaker norms, such as the L 2 norm, one is usually unable to derive any exponential stability result when the system is nonlinear. However, in this paper we show that having a Lipschitz source term, with some condition on the size of the source, is enough to obtain the exponential stability in the L 2 norm for semilinear systems. Besides, in contrast with most of the previous analyses cited above, this result holds for a nonlocal source term. Nonlocal source terms are found in many important phenomena as population dynamics, material sciences, flocking, traffic flow [START_REF] Tadmor | Critical thresholds in flocking hydrodynamics with non-local alignment[END_REF][START_REF] Berestycki | The non-local Fisher-KPP equation: travelling waves and steady states[END_REF][START_REF] Bayen | Modeling multi-lane traffic with moving obstacles by nonlocal balance laws[END_REF], and open the door to many potential applications. Moreover, while all the above previous approaches were dealing with local exponential stability, we obtain here global exponential stability. Concerning semilinear systems with Lipschitz source terms, one should highlight the work of [START_REF] Dus | On L ∞ Stabilization of Diagonal Semilinear Hyperbolic Systems by Saturated Boundary Control[END_REF] where the authors study the exponential stability in C 0 norm of a semilinear system with a diagonal and Lipschitz source term, and saturating boundary conditions. They give a potentially large explicit bound on the basin of attraction, and they prove in addition the well-posedness in L 2 .

Finally, we show that these results can be extended to a wider notion: the Input-to-State Stability (ISS). The ISS measures the resilience of the stability of a system when adding disturbances in the boundary conditions or in the source term [START_REF] Sontag | Smooth stabilization implies coprime factorization[END_REF][START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF]. These disturbances could have many origins such as actuator errors, quantized measurments, uncertainties of model parameters, etc. The ISS is therefore a more relevant notion from an application perspective, and is also paramount for designing observers. While exponential stability of nonlinear hyperbolic systems has been studied for several decades now, fewer results are known concerning this wider notion of ISS. Until recently, the most up-to-date results were given in [40, Part II], for L p norms, p ∈ N * ∪ {+∞} (see also [START_REF] Prieur | ISS-Lyapunov functions for time-varying hyperbolic systems of balance laws[END_REF] for instance for nonautonomous systems), and recently several works have been providing quite good conditions by extending exponential stabilization results obtained through Lyapunov approach to ISS results under the same conditions [START_REF] Ferrante | Boundary Control Design for Conservation Laws in the Presence of Measurement Noise[END_REF][START_REF] Weldegiyorgis | An analysis of the input-to-state stabilization of linear hyperbolic systems of balance laws with boundary disturbances[END_REF][START_REF] Bastin | Input-to-State Stability in sup norms for hyperbolic systems with boundary disturbances[END_REF]. These results suffer however the same limitations as the exponential stabilization results they are generalizing: local validity and strong norms. One can also refers to [START_REF] Dashkovskiy | Robust stability of a perturbed nonlinear wave equation[END_REF][START_REF] Dashkovskiy | Input-to-state stability of infinitedimensional control systems[END_REF][START_REF] Lhachemi | PI Regulation of a Reaction-Diffusion Equation with Delayed Boundary Control[END_REF][START_REF] Hayat | PI controllers for the general Saint-Venant equationsWorking paper or preprint[END_REF] for other ISS results on hyperbolic systems in particular cases, and to [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF] for a more detailed review on ISS results for PDEs in general. This paper is organized as follows: in Section 2 we state some definitions and our main result, which is proven in Section 3 using a Lyapunov approach. The well-posedness and the extension to ISS are dealt with in the Appendix.

Statement of the problem and main results

A semilinear hyperbolic system can always be written in the following way [START_REF] Li | Boundary value problems for quasilinear hyperbolic systems[END_REF]:

∂ t u + Λ(x)∂ x u + B(u, x) = 0, (1)
where u(t, x) ∈ R n , Λ(x) is a diagonal matrix with non vanishing eigenvalues,

Λ : x → Λ(x) belongs to C 1 ([0, L]) and B ∈ C 0 (L 2 (0, L) × [0, L], L 2 (0, L))
is the nonlinear source term, with B(0, x) = 0. Note that B could be potentially nonlocal at it takes a function as argument, thus B(u, x) refers here to B(u(t, •), x).

Throrough the article we will assume that B(•, x) is Lipschitz with respect to u with a Lipschitz constant C B in the following sense: for u and v two functions

of L 2 (0, L), B(u, •) -B(v, •) L 2 ≤ C B u -v L 2 . ( 2
)
Of course, this assumption is satisfied if

B is local, takes argument in R n × [0, L]
and is Lipschitz with respect to the first argument, with a Lipschitz constant that might depend on x but as a L 2 function. We will come back to this special case later on in Remark 2.4. When the system is equipped with a control static and exerted at the boundaries, the boundary conditions can be written in the following way:

  u + (t, 0) u -(t, L)   = G   u + (t, L) u -(t, 0)   , (3)
where G is a continuous and Lipschitz function such that G(0) = 0. The notation u + is used to refer to the components of u corresponding to positive propagation speeds Λ i > 0, whereas the notation u -is used to refer to the components corresponding to negative propagation speeds. In the following, we assume without loss of generality that Λ i > 0 for i ∈ {1, ..., m} and Λ i < 0 for i ∈ {m + 1, ..., n}. Note that the boundary conditions (3) are nonlinear. As G is Lipschitz, all of its components are Lipschitz, which implies that there exists a matrix K such that for any i ∈ {1, ..., n}, System (1), ( 3) with ( 2), ( 4) is well posed in L 2 in the following sense:

G i   u + (t, L) u -(t, 0)   ≤ m j=1 K ij |u j (t, L)| + n j=m+1 K ij |u j (t, 0)|. (4)
Theorem 2.1 (Well posedness). For any T > 0 and any

u 0 ∈ L 2 (0, L) the Cauchy problem (1)-(3), with initial condition u(0, •) = u 0 has a unique solution u ∈ C 0 ([0, T ], L 2 (0, L)). Moreover, u(t, •) L 2 ≤ C(T ) u 0 L 2 , ∀ t ∈ [0, T ], (5)
where C(T ) is a constant depending only on T .

This theorem is shown in the Appendix. Most of the proof is a subcase of a remarkable result in [START_REF] Dus | On L ∞ Stabilization of Diagonal Semilinear Hyperbolic Systems by Saturated Boundary Control[END_REF]Theorem A.1], where the authors study the framework of saturating boundary conditions. The only differences are some slight changes in the estimates to deal with a nonlocal functional and a density argument.

These changes are indicated in Appendix A, together with a proper definition of the notion of weak solution to System (1), (3).

Remark 2.2. As it could be expected, the well posedness also holds for more regular solutions. In particular for any u 0 ∈ H 1 (0, L) satisfying the compatibility conditions given by ( 3), the Cauchy problem ( 1), ( 3) with initial condition

u(0, •) = u 0 has a unique solution u ∈ C 0 ([0, T ], H 1 (0, L)) ∩ C 1 ([0, T ], L 2 (0, L)).
This is also shown in Appendix A.

Before stating our main result, we recall the definition of exponential stability for the L 2 norm.

Definition 2.1 (Exponential stability). We say that System (1)-( 3) is exponentially stable for the L 2 norm with decay rate γ and gain C if there exists constants δ > 0, γ > 0, and C > 0 such that for any T > 0 and

u 0 ∈ L 2 (0, L) such that u 0 L 2 ≤ δ, the Cauchy problem (1)-(3) with initial condition u(0, •) = u 0 has a unique solution u ∈ C 0 ([0, T ], L 2 (0, L)) and u(t, •) L 2 ≤ Ce -γt u 0 L 2 . ( 6
)
Moreover, if

δ = +∞, (7)
then the system is said globally exponentially stable.

We can now state our main result.

Theorem 2.2. Let a system be of the form (1), [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF], where

Λ ∈ C 1 ([0, L])
and B is Lipschitz with respect to u. If there exist K ∈ M n (R) satisfying (4),

J ∈ C 1 ([0, L]; M n (R))
where J(x) is a diagonal matrix with positive coefficients, and M ∈ C 0 ([0, L]; M n (R)), such that the following conditions are satisfied

1. (Interior condition) -(J 2 Λ) + J 2 M + M J 2 (8)
is positive definite and there exists D ∈ C 1 ([0, L]; M n (R)) where D(x) is a diagonal matrix with positive coefficients, such that

C g < λ m 2 max i,x (D i ) max i,x (D i J 2 i ) , (9)
where C g is the Lipschitz constant of g :=B -M and λ m denotes the smallest eigenvalue of

-D(J 2 Λ) D + DJ 2 M D + DM J 2 D, (10)
2. (Boundary condition) the matrix

  J 2 + (L)Λ + (L) 0 0 J 2 -(0)|Λ -(0)|   -K   J 2 + (0)Λ + (0) 0 0 J 2 -(L)|Λ -(L)|   K (11)
is positive semidefinite, then the system is globally exponentially stable for the L 2 norm. Moreover the

gain is J -1 L ∞ J L ∞ and an admissible decay rate is λ m (2 max i,x (D i J 2 i )) -1 - C g max i,x (D i )
We prove this theorem in Section 3. Note that (9) does not involve directly the Lipschitz constant of B but the Lipschitz constant of g= B -M , which is B minus a linear part that can be chosen. Of course, the Lipschitz constant of B would be suitable by setting M = 0, but other choices of M could lead to less restrictive conditions. Let us note that the apparent complexity of the interior condition aims at giving a good explicit computable bound on C g for practical applications: indeed finding the values of λ m can be numerically solved.

Besides, choosing D = Id or K = C G I would also give a sufficient condition that is simpler to write, but the sufficient condition would be more restrictive.

Remark 2.3 (Linear case). When B is a local and linear operator we recover the result found in [16, Proposition 5.1] (see also [START_REF] Diagne | Lyapunov exponential stability of 1-D linear hyperbolic systems of balance laws[END_REF] when B is in addition marginally diagonally stable). Indeed, we can choose M = B, then g = 0 and the interior condition is reduced to the existence of J, diagonal matrix with positive coefficients such that -(ΛJ 2 ) + J 2 M + M J 2 is positive definite.

Remark 2.4 (Local case). In the special case where the system is local, i.e. B is a function on R × [0, L] and B(u, x) = B(u(t, x), x), the condition (9) of the previous theorem can be slightly improved as follows: assume that B is Lipschitz with respect to the first variable with a Lipschitz constant C(x) ∈ L 2 (0, L), then for any matrix M , g = B -M is also Lipschitz with respect to the first variable and we can denote again its Lipschitz constant by C g (x) ∈ L 2 (0, L). Then, the interior condition (9) in Theorem 2.2 can be replaced by

C g < λ m (x) max i (J 2 i )(x) or C g < µ m (x) max i (J i )(x) inf i (J i )(x) , (12)
where λ m (x) and µ m (x) are the smallest eigenvalues at a given x of the matrix given by ( 8) and ( 10) respectively.

Input-to-State Stability

In fact, this result can be extended to a more general notion: the Inputto-State Stability (ISS). This notion is more relevant when looking at practical implications as it takes into account the external disturbances that can arise.

When such disturbance arise, System (1), ( 3) is replaced by

∂ t u + Λ(x)∂ x u + B(u, x) + d 1 (t, x) = 0,   u + (t, 0) u -(t, L)   = G   u + (t, L) u -(t, 0)   + d 2 (t), (13)
where d 1 and d 2 are respectively the distributed and boundary disturbances.

We define the ISS as follows:

Definition 2.2 (Input-to-State Stability). We say that System ( 13) is strongly Input-to-State stable (or ISS) with fading memory for the L 2 norm if there exists positive constants δ > 0, C 1 > 0, C 2 > 0, γ > 0, such that for any T > 0 and any u 0 ∈ L 2 (0, L) with u 0 L 2 ≤ δ and

d 1 L 2 + d 2 L 2 ≤ δ, there exists a unique solution u ∈ C 0 ([0, T ], L 2 ([0, L]))
to System (1), (3), and

u(t, •) L 2 ≤ C 1 e -γt u 0 L 2 + C 2 e -γ(t-s) d 1 (s, x) L 2 ((0,t)×(0,L)) + e -γ(t-s) d 2 (s) L 2 (0,t) , for any t ∈ [0, T ]. (14)
Moreover, if δ = +∞, then the system is said to be globally strongly ISS with fading memory.

This defines a strong notion of ISS with an exponentially fading memory.

The fading memory comes from the e -γ(t-s) in the L 2 norms of d 1 and d 2 .

It means that the influence of the disturbances at a given time s decreases exponentially with time. One could have chosen other and less restrictive fading factors (see [START_REF] Karafyllis | Input-to-state stability for PDEs[END_REF]Chapter 7] for a more complete description of ISS estimates with fading memory). The constants C 1 and C 2 are called the gains of the ISS estimate. When such notion of ISS cannot be achieved, weaker notions exist and can be found for instance in [START_REF] Mironchenko | Characterizations of input-to-state stability for infinite-dimensional systems[END_REF]. We have the following result, analogous to Theorem 2.2

Theorem 2.3. Let a system be of the form [START_REF] Wang | Exact controllability for nonautonomous first order quasilinear hyperbolic systems[END_REF] where

Λ ∈ C 1 ([0, L]), d 1 ∈ L 2 ((0, T ) × (0, L)), d 2 ∈ H 1 ([0, T ]
) and B is Lipschitz with respect to u. If the condition (9) is satisfied and the matrix defined by (11) is positive definite, then the system is globally strongly ISS with fading memory for the L 2 norm.

The proof of this theorem is very similar to the proof of Theorem 2.2. The only difference being that the assumption on [START_REF] Qin | Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems[END_REF] has to be slightly stronger than in Theorem 2.2 (positive definite instead of positive semidefinite). A way to adapt the proof of Theorem 2.2 is given in 4. Besides, the gains can again be computed explicitly as a function of K, B and Λ (see [START_REF] Dus | On L ∞ Stabilization of Diagonal Semilinear Hyperbolic Systems by Saturated Boundary Control[END_REF]).

Exponential stability in the L 2 norm

Proof of Theorem 2.2. Let a semilinear system be of the form (1), (3) with Λ ∈ C 1 ([0, L], M n (R)) and B being L 2 with respect to u with Lipschitz con-stant C B . We will first show Theorem 2.2 for H 1 solutions and then recover it for L 2 solutions using a density argument. Let T > 0, and let u 0 ∈ H 1 (0, L). From Theorem 2.1 and Remark 2.2, there exists a unique solution u ∈ C 0 ([0, T ], H 1 (0, L))∩C 1 ([0, T ], L 2 (0, L)) associated to this initial condition.

Let us now define the following Lyapunov function candidate:

V (u) = L 0 (J(x)u(t, x)) J(x)u(t, x)dx, (15)
where

J = diag(J 1 , ..., J n ) ∈ C 1 ([0, L], D + n (R n ))
, where D + n is the space of diagonal matrices with positive coefficients. The function V is well defined on

L 2 (0, L) and equivalent to u(t, •) 2 L 2 , as u(t, •) 2 L 2 J -1 -2 L ∞ ≤ V (u) ≤ J 2 L ∞ u(t, •) 2 L 2 . ( 16
)
We would like to show that V decreases exponentially quickly along u. Before has the form (15) with J(x) = diag(q i e -µsix ) where s i = 1 if Λ i > 0 and s i = -1 if Λ i < 0 and q i and µ are positive constants to be chosen. In our case however, such function might not work. This is due to the inhomogeneity and this a phenomena that can be seen in balance laws in general [START_REF] Bastin | On boundary feedback stabilization of non-uniform linear 2× 2 hyperbolic systems over a bounded interval[END_REF]. For instance, in [START_REF] Hayat | A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope[END_REF] is found a basic quadratic Lyapunov function that exists for any length L > 0 provided good boundary conditions, while this could not happen with a basic quadratic Lyapunov function made of exponential weights.

As u ∈ C 1 ([0, T ], L 2 (0, L)), V (u(t,
•)) can be differentiated with time, and we have

dV (u(t, •)) dt = L 0 2u J 2 ∂ t udx = - L 0 2u J 2 Λ∂ x udx -2 L 0 u J 2 B(u, x)dx = -u J 2 Λu L 0 + L 0 u (J 2 Λ) udx -2 L 0 u J 2 B(u, x)dx. ( 17
)
We used here that J and Λ commute as they are both diagonal. Now, let M ∈ C 0 ([0, L], M n (R)) to be selected later on and set g(u, x) = B(u, x)-M (x)u(t, x)

which is again Lipschitz in u in the sense of (2). We have

dV (u(t, •)) dt = -u J 2 Λu L 0 + L 0 u (J 2 Λ) udx -2 L 0 u J 2 M udx -2 L 0 u J 2 g(u, x)dx = -u J 2 Λu L 0 - L 0 u [-(J 2 Λ) + J 2 M + M J 2 ]udx -2 L 0 u J 2 g(u, x)dx (18)
where we used that u J 2 M u = u M J 2 u, as it is a scalar. Now, we set

I 2 : = u J 2 Λu L 0 , I 3 : = L 0 u [-(J 2 Λ) + J 2 M + M J 2 ]udx + 2 L 0 u J 2 g(u, x)dx (19)
We would like to show that under assumptions 1. and 2. of Theorem 2.2, I 2 is a nonnegative definite quadratic form with respect to the boundary conditions, and

I 3 ≥ µ u L 2
where µ is a positive constant. We will show that this is exactly the point of Assumptions 1. and 2.. Let us start with I 2 . From (3),

I 2 =   u + (t, L) u -(t, L)   J 2 (L)Λ(L)   u + (t, L) u -(t, L)   -   u + (t, 0) u -(t, 0)   J 2 (0)Λ(0)   u + (t, 0) u -(t, 0)   = m i=1 J 2 i (L)Λ i (L)u 2 i (L) - n i=m+1 J 2 i (0)Λ i (0)u i (0) 2 + n i=m+1 J 2 i (L)Λ i (L)   G i   u + (t, L) u -(t, 0)     2 - m i=1 J 2 i (0)Λ i (0)   G i   u + (t, L) u -(t, 0)     2 . ( 20
)
We set x i := 0 if i ∈ {1, ..., m} and x i := L if i ∈ {m + 1, ..., n}. Then using that Λ i > 0 for i ∈ {1, ..., m} and Λ i < 0 otherwise, and using (4),

I 2 = n i=1 J 2 i (L -x i )|Λ i (L -x i )|u 2 i (L -x i ) - n i=1 J 2 i (x i )|Λ i (x i )|   G i   u + (t, L -x i ) u -(t, L -x i )     2 ≥ n i=1 J 2 i (L -x i )|Λ i (L -x i )|u 2 i (L -x i ) - n i=1 J 2 i (x i )|Λ i (x i )|   n j=1 K ij |u j (t, L -x j )|   2 , (21)
This can be rewritten as

I 2 ≥ Y N Y, (22)
where Y is a vector with components Y i = |u i (t, L -x i )| and N is given by

N = J 2 + (L)|Λ+(L)| 0 0 J 2 -(0)|Λ-(0)| -K J 2 + (0)|Λ+(0)| 0 0 J 2 -(L)|Λ-(L)| K. (23)
From ( 11) the matrix N is positive semidefinite, thus

I 2 ≥ 0. (24)
Let us now deal with I 3 . Assume that the condition (9) holds. Then there exists

D ∈ C 1 ([0, L], M n (R)) such that D(x) is diagonal with positive coefficients for any x ∈ [0, L]. Thus -D(J 2 Λ) D + DJ 2 M D + DM J 2 D ( 25
)
is a symmetric and definite positive matrix and we denote by λ m its smallest eigenvalue on [0, L]. We have from ( 19), using Cauchy-Schwarz inequality and using the fact that g is Lipschitz with u and the fact that g(0, x) = B(0, x) = 0,

I 3 ≥ L 0 (D -1 u) [-D(J 2 Λ) D + DJ 2 M D + DM J 2 D](D -1 u)dx -2 L 0 |D -1 u| 2 dx 1/2 L 0 |DJ 2 g(u, x)| 2 dx 1/2 ≥ L 0 (D -1 u) [-D(J 2 Λ) D + DJ 2 M D + DM J 2 D](D -1 u)dx -2 max i,x (D i J 2 i (x)) L 0 |D -1 u| 2 dx 1/2 L 0 |g(u, x)| 2 dx 1/2 ≥ L 0 (D -1 u) [-D(J 2 Λ) D + DJ 2 M D + DM J 2 D](D -1 u)dx -2 max i,x (D i J 2 i (x)) L 0 |D -1 u| 2 dx 1/2 C g L 0 |u| 2 dx 1/2 ≥ L 0 (D -1 u) [-D(J 2 Λ) D + DJ 2 M D + DM J 2 D](D -1 u)dx -2C g max i,x (D i J 2 i (x)) max i,x (D i (x)) L 0 |D -1 u| 2 dx 1/2 ≥ λ m D -1 u 2 L 2 -2C g max i,x (D i J 2 i (x)) max i,x (D i (x)) D -1 u 2 L 2 . (26)
Therefore if

C g < λ m /(2 max i,x (D i (x)) max i,x (D i (x)J 2 i (x))) then I 3 ≥ µ D -1 u L 2 , (27)
with µ = λ m -2C g max i,x

(D i J 2 i (x)) max i,x
(D i (x)) > 0. Thus from ( 16), the positive definiteness of D, hence D -1 , ( 18), [START_REF] Hu | Boundary exponential stabilization of 1-d inhomogeneous quasilinear hyperbolic systems[END_REF], and ( 27) we can set γ = µ(max(D i J 2 i )) -1 > 0 such that such that for any t ∈ [0, T ].

dV (u(t, •) dt ≤ -γV, (28)
and therefore

V (u(t, •)) ≤ V (u(s, •))e -γ(t-s) , ∀ 0 ≤ s ≤ t ≤ T. (29)
From ( 16), this implies that

u(t, •) L 2 ≤ J -1 L ∞ J L ∞ e -γ 2 (t-s) u 0 L 2 , (30)
which is exactly the estimate wanted with decay rate γ/2. So far this estimate is only true for H 1 solutions. However, it only involves the L 2 norm. Thus, as the system is well-posed in C 0 ([0, T ], L 2 (0, L)) and • L ∞ ((0,T );L 2 (0,L)) is lower semicontinuous, the estimate (30) also hold for L 2 solutions by density (more details on this argument can be found in the proof of [START_REF] Bastin | Exponential boundary feedback stabilization of a shock steady state for the inviscid Burgers equation[END_REF]Lemma 4.2]).

Adapting the proof in the ISS case

In this section we show how to adapt the proof of Theorem 2.2 to get Theorem 2.3.

Proof. Let us consider System (13) and let T > 0. Let u 0 ∈ H 1 (0, L) and u ∈ C 1 ([0, T ], H 1 (0, L)) the associated solution. Then, defining V as in [START_REF] Hayat | A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope[END_REF], and differentiating along u, we obtain as previously

dV (u(t, •)) dt = -I 2 -I 3 -2 L 0 u J 2 d 1 dx, (31)
where I 2 and I 3 are given by [START_REF] Hayat | Exponential stability of density-velocity systems with boundary conditions and source term for the H 2 norm[END_REF]. Thus, using Young's inequality

dV (u(t, •)) dt = -I 2 -I 3 + ε 0 V + J 2 L ∞ ε 0 d 1 (t, •) 2 L 2 , (32)
where ε 0 > 0 and can be chosen. As previously, from (9), I 3 ≥ µV where µ > 0.

Therefore, choosing ε 0 = µ/2, we have

-I 3 + ε 0 V ≤ - µ 2 V. (33)
Concerning I 2 , if we denote by I 2,0 the quantity in the absence of disturbances (i.e. the quantity given by the first equality of ( 21)) we get

I 2 = I 2,0 - n i=1 J 2 i (x i )|Λ i (x i )|   d 2 2,i + 2d 2,i G i   u + (t, L) u -(t, 0)     ≥ Y N Y - n i=1 J 2 i (x i )|Λ i (x i )| 1 + 1 ε d 2 2,i -εYK   J 2 + (0)|Λ + (0)| 0 0 J 2 -(L)|Λ -(L)|   KY, (34)
where we used Young's inequality and where N is the matrix given in [START_REF] Coron | Local exponential H 2 stabilization of a 2 × 2 quasilinear hyperbolic system using backstepping[END_REF], Y is defined as in [START_REF] Xiang | Null controllability of a linearized Korteweg-de Vries equation by backstepping approach[END_REF], and ε > 0 is to be chosen. Using the definition of N and the fact that N is positive definite (and not positive semidefinite in contrast with Theorem 2.2), we get by continuity that there exists ε > 0 such that

N -εK   J 2 + (0)|Λ + (0)| 0 0 J 2 -(L)|Λ -(L)|   K is semipositive definite. ( 35
)
Therefore,

I 2 ≥ -(1 + ε -1 ) J 2 ∞ Λ ∞ |d 2 (t)| 2 and (32) becomes dV (u(t, •)) dt ≤ - µ 2 V + 2 J 2 L ∞ µ d 1 (t, •) 2 L 2 + (1 + ε -1 ) J 2 ∞ Λ ∞ |d 2 (s)| 2 , (36)
thus, using Gronwall's Lemma,

V (u(t, •)) ≤ V (u 0 )e -µt 2 + 2 J 2 L ∞ µ t 0 e -µ 2 (t-s) d 1 (s, •) 2 L 2 + µ 2 (1 + ε -1 ) Λ ∞ |d 2 (t)| 2 ds, (37)
which, together with ( 16) and the concavity of the square root function gives

u(t, •) L 2 ≤ J -1 L ∞ J L ∞ u 0 L 2 e -µt 4 + J -1 L ∞ J L ∞ 2 µ max 1, µ 2 (1 + ε -1 ) Λ L ∞ e -µ 2 (t-s) d 1 (s, x) L 2 ((0,t)×(0,L)) + + d 2 (t) L 2 (0,t) , (38)
which is the ISS estimate wanted and this holds for any H 1 solutions. And, by density, this holds also for any L 2 solutions. Note that the gains of the estimate can again be computed explicitly. This ends the proof of Theorem 2.3

Numerical simulations

In this section we present a numerical illustration of the previous result on a simple example. We consider a system inspired from [16, Section 5.6] and given as

∂ t u 1 + ∂ x u 1 = cL -1 sin L 0 u 2 (t, x)dx ∂ t u 2 -∂ x u 2 = cL -1 sin L 0 u 1 (t, x)dx u 1 (t, 0) -u 2 (t, 0) = 0 u 1 (t, L) -u 2 (t, L) = ku 1 (t, L) (39)
where one boundary condition can be imposed through a design parameter k while the other one is imposed. Note first that in open-loop, i.e. k = 0, the null steady-state is an unstable steady-states for any c ∈ R and any length of the domain L > 0. Indeed, there is a continuum of travelling wave solutions: for

any ε > 0    u 1 (x) = εe 2πi L (t-x) u 2 (x) = εe -2πi L (t-x) (40)
is a solution of (39) with k = 0. Nevertheless, Theorem 2.2 can be applied to find a feedback in closed loop as long as |c|L < 1/2: set M = 0, D = Id, ε > 0 to be defined, and k = 1/(1 + 2Lε -1 ). Set also, J

= diag( √ L + ε -x, √ L + ε + x),
one has Λ = diag(1, -1) therefore -(J 2 Λ) = I d and therefore is positive definite with smallest eigenvalue 1. Besides max i,x (J 2 i ) = ε + 2L and

g(U ) -g(V ) 2 L 2 = 1 L 2 L 0   c sin L 0 U 2 (x)dx c sin L 0 U 1 (x)dx -   -   c sin L 0 V 2 (x)dx c sin L 0 C 1 (x)dx -   2 dx ≤ |c|L -1   L 0 |U 2 -V 2 |dx 2 + L 0 |U 1 -V 1 |dx 2   ≤ |c| U -V 2 L 2 . (41)
Hence, condition (9) becomes |c| < (ε + 2L) -1 . Now, as |c| < (2L) -1 , one can choose ε = 3(|c| -1 -2L)/4 such that condition ( 9) is satisfied. Finally, one can easily check that condition [START_REF] Qin | Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems[END_REF] becomes

(1 -k) 2 ≤ ε ε + 2L , (42)
which is also satisfied from our definition of k. Thus Theorem 2.2 applies and the system is globally stable for the L 2 norm. On Figure 1

Conclusion and perspective

We derived sufficient conditions for the global stability in the L 2 norm of semilinear systems with Lipschitz boundary conditions and source term (potentially nonlocal). We also showed that a strong ISS property with respect to boundary and internal disturbances holds globally under the same conditions.

This result could have many applications in practice. Knowing whether such conditions are optimal for the existence of a basic quadratic Lyapunov function, at least for n = 2 as it is in the linear and local case, is an open question. Another interesting direction for future works would be to try to extend, at least partially, these results to quasilinear but Lipschitz nonlocal systems. and therefore u

(1) n → u in C 0 ([0, T ], L 2 ), which implies that u = u (1) in C 0 ([0, T ], L 2 ). This holds for any T > 0, and ends the proof.

Well-posedness in the ISS case

The well-posedness of system ( 13) is again a consequence of the three properties of the operator (A + B) and the results in [START_REF] Miyadera | Translations of Mathematical Monographs[END_REF]. In particular, let T > 0 and u 0 ∈ L 2 (0, L), and assume (d with boundary conditions (3), and where B is seen again as an operator on L 2 (0, L). The interest of this reformulation is that the boundary conditions of v are now again (3) and do not depend on time (contrary to the boundary conditions of ( 13)) except through v (see also [START_REF] Ferrante | Boundary Control Design for Conservation Laws in the Presence of Measurement Noise[END_REF]). Using the fact that (A + B)

satisfies

Remark 2 . 1 (

 21 Choice of K). Of course the matrix K = C G I, where I is the identity matrix and C G the Lipschitz constant of G would work. However, there might be other matrices K satisfying (4) and some could lead to potentially less restrictive conditions in Theorem 2.2 than the matrix C G I (see (11) below).

  going any further, let us comment on the choice of the form of this Lyapunov function candidate. Functions of this type are sometimes called basic quadratic Lyapunov function or basic Lyapunov function for the L 2 norm because they can be seen as the simplest functional equivalent of the L 2 norm. A commonly used Lyapunov function candidate for hyperbolic systems of conservation laws

we represent the L 2

 2 norm of the solution for various values of k when c = 1/4 and L = 1. In blue is represented the open-loop situation (i.e. k = 0), in green the closed-loop situation with k = 3/4, and in red k = 1/2.

Figure 1 :

 1 Figure 1: Stability of the system (39) in open-loop (blue) and closed-loop with k = 3/4 (green) and k = 1/2 (red). horizontal axis represents time, and vertical axis represents the L 2 norm of the solution with initial condition u 1,0 (x) = √ 2πx and u 2,0 (x) = e -2πx .

1 , d 2 )

 12 ∈ L 2 ((0, T ) × (0, L)) × H 1 (0, T ), and setL : x → diag(L i (x)) where L i (x) = (L -x)/L if i ∈ {1, ..., m} and L i = x/L if i ∈ {m + 1, ..., n}. A function u ∈ C 0 ([0, T ], L 2 (0, L)) is an L 2 solution to (13), u(0, •) = u 0 , if and only if v(t, x) := u(t, x) -L(x)d 2 (t) is an L 2 solution to ∂ t v = -(A + B)v + d 1 + [(Λ∂ x + B)(Ld 2 ) -L ḋ2 ] v(0, •) = u 0 -L(x)d 2 (0) =: v 0 , (A.[START_REF] Qin | Global smooth solutions of dissipative boundary value problems for first order quasilinear hyperbolic systems[END_REF]

  the range condition and is dissipative of type ζ, from [54, Theorem 5.18 and Remark 2°], there exists a unique integral solution u ∈ C 0 ([0, T ]; L 2 (0, L)), satisfying a definition analogous to Appendix A.1 (the precise definition is omitted here due to the lack of space).

see[START_REF] Hayat | Stabilization of 1D nonlinear hyperbolic systems by boundary controls[END_REF] Definition 1.4.3] for a proper definition and[START_REF] Hayat | A quadratic Lyapunov function for Saint-Venant equations with arbitrary friction and space-varying slope[END_REF] for an overview of this method

see[START_REF] Xiang | Null controllability of a linearized Korteweg-de Vries equation by backstepping approach[END_REF] for more details

Acknowledgments

The author would like to thank Jean-Michel Coron for his advice and many interesting discussions. The author would also like to thank Benedetto Piccoli for interesting discussions, the NSF for support via the CPS Synergy project "Smoothing Traffic via Energy-efficient Autonomous Driving" (STEAD) CNS 1837481, the French Corps des IPEF, the ANR Finite4Sos, and the INRIA CAGE team.

Appendix A. Well-posedness of the system

In this section we deal with the well-posedness of the system and extend [38, Theorem A.1] to get Theorem 2.1. But first, we give the definition of a weak L 2 solution for System (1), [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF]. Definition Appendix A.1. Let u 0 ∈ L 2 (0, L). We say that u ∈ C 0 ([0, +∞); L 2 (0, L)) is an L 2 solution of the Cauchy problem (1), (3), u(0, •) = u 0 , if for every T > 0 there exists a sequence of functions u 0,n ∈ H 1 (0, L) satisfying [START_REF] Leugering | On the modelling and stabilization of flows in networks of open canals[END_REF] and such that

where u n ∈ C 0 ([0, T ], H 1 (0, L)) is a weak solution of (1), (3) with initial condition u 0,n , i.e. u n satisfies (3) and for any φ

Remark Appendix A.1. As noted in [START_REF] Dus | On L ∞ Stabilization of Diagonal Semilinear Hyperbolic Systems by Saturated Boundary Control[END_REF], this definition is slightly different from the definition given in [START_REF] Bastin | Stability and Boundary Stabilisation of 1-D Hyperbolic Systems[END_REF]Definition A.3] when looking at linear systems.

The reason comes from the nonlinear boundary conditions which may prevent the adjoint of the boundary operator from existing. Of course, in the linear case, a solution in the sense of [START_REF] Bastin | Stability and Boundary Stabilisation of 1-D Hyperbolic Systems[END_REF]Definition A.3] is also a solution in the sense of Definition Appendix A.1.

With this definition in mind, we prove Theorem 2.1, by slightly adapting the proof of [START_REF] Dus | On L ∞ Stabilization of Diagonal Semilinear Hyperbolic Systems by Saturated Boundary Control[END_REF]Theorem A.1] Proof of Theorem 2.1. Let T > 0. We define the operator A = -Λ(x)∂ x on the domain D(A) defined by

We also consider B as an operator on the domain D(B) = L 2 (0, L), and in the

First of all, we can restrict ourselves to the case where Λ has only positive components. Indeed, if not, we define v = (v i ) i∈{1,...,n} by

and

Clearly, u is a L 2 solution to the system (1), (3) if and only if v is an L 2 solution to a system of the form ( 1 or equivalently that for any v ∈ D(A + B), there exists u ∈ H 1 (0, L) such that

The difficulty comes from the nonlinearity of the equation and this was the main point shown in [START_REF] Dus | On L ∞ Stabilization of Diagonal Semilinear Hyperbolic Systems by Saturated Boundary Control[END_REF]Theorem A.1]. In our case, all we need to do is to change slightly their proof to take into account the nonlocal operator and the fact that Λ depends on x. The latter is easy to take into account by replacing e -Λ -1 x/ρ by e -x 0 (Λ -1 (s)/ρ)ds when integrating, which has a similar behavior (this holds as Λ is diagonal). To take into account the nonlocal operator, we need to get the estimate [38, (26)] while replacing the estimations in [38, 2.2.1], which hold

where C 2 is a constant that depends only on the parameters of the system.

This enables to recover the estimate [38, (26)] which is then used to apply

which implies the convergence of u n to u in C 0 ([0, T ], L 2 (0, L)). To conclude we only need to show that this is the unique solution in the sense of Definition Appendix A.1. Let assume that there is another solution u (1) with initial condition u 0 . Let T > 0. By assumption there exists a sequence u

n → u (1) in C 0 ([0, T ], L 2 (0, L)). For any n ∈ N, u

n ∈ C 1 ([0, T ], L 2 (0, L)), therefore u