Gilles Enee
email: enee@i3s.unice.fr

Cathy Escazut
email: escazut@i3s.unice.fr

Classifier systems evolving multi-agent system with distributed elitism

Keywords: Genetic Algorithm, Classifier System, Multi-Agent System, Genetics Based Machine Learning, Knowledge Sharing

niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Solving complex problems with classical algorithms has shown its limits. In our work, the solutions space of complex problems is explored thanks to genetic algorithm [Goldberg, 1989]. Genetic algorithms are not dedicated to machine learning. Then [Holland, 1992] has introduced Classifier Systems which are an extension of genetic algorithm as it is described in the next sections. These systems are specialized in learning. But even with these tools, we are obliged to split complex problems into simple ones [Bull, 1995]. These simple problems are considered as agents co-evolving in a multi-agent system. But we still encounter difficulties to answer to complex problems due to the actual lack of general conceptual approach. In this paper, we will first describe Pittsburgh-style classifier systems and multi-agent systems. Section 3 will present our experiments. In that section, we will first explain the traffic signal controller we used for the tests, second will be the Elitism experiment, third will be the Distributed Elitism experiment, fourth will be the mixed one and last we show robustness of the chosen classifier system through three experiments. Finally, last section will attempt to show new conceptual openings.

Multi-Agent System

Before explaining what is a multi-agent system we must first describe what a genetic algorithm is.

Genetic Algorithm in Classifier Systems

Genetic algorithms are usually used for function optimization. The problem is encoded on a bit 'fragment' called chromosome referring to DNA. They work on a population of chromosomes called individuals. Genetic algorithms use the implementation of DNA's manipulators like crossover and mutation on this population. They implement natural selection of good solutions based upon Darwin's model. The initial set of chromosomes is randomly initialized over a ternary alphabet 0, 1, #✁ where '#' is a wildcard matching for a 1 or a 0. Crossover consists in randomly taking two chromosomes (parents) and crossing them at a random position exchanging part of their fragments giving two children. For example, let us describe a crossover between individuals 0011 and 1100. If we suppose that the crossover site is situated on the middle of the chromosome, the two children 0000 and 1111 are obtained.

Mutation consists in changing a bit value into one of the two other possible values of the alphabet. Mutation and crossover are applied with a probability of occurring. Genetic algorithms are also used in machine learning system.

Pittsburgh-style Classifier Systems

Classifier systems are genetics based machine learning systems. They only use genetic based operators to find problem solution. They are made of sets (individuals) of rules (chromosomes), the whole consists in a population. Classifier systems exist in two forms : Michigan-style ones and Pittsburghstyle ones. In the first one, one individual is a single rule, in the second one, one individual is a set of rules. Let us detail Pittsburgh-style classifiers [Smith, 1980], since it is the approach we have chosen. The basic cycle of such a classifier system is the following one :

Initialize(Pop); While(conditions not satisfied) { Evaluate(Pop); GA(Pop); }
The population is a set of individuals, each individual being a set of rules. Rules encode the problem as a production rule i.e with a condition part and an action part. The final population should contain many individuals representing the solution to execute the task to be learnt. The evaluation consists in steps during which each individual of the population will be evaluated many times to measure its performance to accomplish the asked task. The performance gives a reward to the individual in the form of strength. This strength will be used by the genetic algorithm and is essential to the selection. The more there are trials, the more the strength is relevant. Once the evaluation done, individuals need to be evolved. This is done by the genetic algorithm. The highly fit individuals (best strength) should 'reproduce' each other because the random selection is made following a scheme that favors the more fit members. Then crossover and mutation mechanisms are applied to recombine individuals. A new population is thus created, containing a higher proportion of the characteristics possessed by the good members of the previous population. Each passing through the loop corresponds to a generation.

Multi-agent Classifier System

Complex problems need to be split into simple ones [START_REF] Bull | Evolving cooperative communicating classifier systems[END_REF], expecting these simple problems might be coordinated to answer the main one. This is what we do in a multi-agent system. An agent, in our work, is a classifier system that must solve a simple problem. Thus a multi-agent system is a system composed of agents (classifier systems) that solve together the complex problem.

Experiments

Let us now describe the application of our multi-agent classifier system.

Adaptation to a traffic signal

To implement our system, we have chosen the traffic signal controller which is a good example of homogeneous environment [Bull, Carse and Fogarty, 1995], [START_REF] Mikami | Self-organized control of traffic signals through genetic reinforcement learning[END_REF] and [START_REF] Mikami | Genetic reinforcement learning for cooperative traffic signal control[END_REF].

This problem has been widely treated in related areas [Escazut and Forgarty, 1997] and [START_REF] Montana | Evolving control laws for a network of traffic signals[END_REF]. Each agent manages one junction, the aim is to control a map of crossroads that implies cooperation between junctions.

The grid map we used is a 3 ✂ 3 one. A junction is made with four roads at each cardinal point (North, East, South, West). Each road has a traffic light which colors are opposed vertically and horizontally. The traffic light is green or red. There is also two sensors giving informations on incoming cars. The nearest one of the traffic light is detecting if a car is waiting at the crossroad, the other one detects a car approaching (see figure 1). Cars have a constant speed or are waiting for the signal to turn green (no speed), in other words, there is no acceleration. Each road has a traffic flow which is the probability that a car may appear on it. The number of cars by road is limited to 10. The controller shall decide whether the signal must change or not. A cycle is the time between each decision of the controller. Each cycle starts with the execution of the action dictated by the controller. Then, vehicles that can move forward, achieve a one-step move. A vehicle that crosses the junction can go straight or turn left or right using a turning probability of 10%. When a car turns, cutting the opposed road, there is no collision. Roads are 'double-sided'. Roads of junctions that are connected to an other junction only receive cars from that neighbour junction i.e. cars cannot be created on that road.

The main problem is to control a 3 ✂ 3 grid map of junctions. Then according to our choices, we say that the multiagent system is the map and that an agent is attributed to a junction. Thus we have here nine agents. Now let us see how an agent will rule the junction in our particular case.

There are four input roads per junction and each road has two sensors returning true or false. Thus, the condition part of a rule has two bits by road representing the state of the two sensors on it and a bit to represent the traffic light on the north road that gives the color of other signals. So the condition part of a rule is made of nine bits. The action part consists in changing whether or not the traffic light which is represented using one bit. For instance, the rule 11 00 11 00 1:1 says that when there is no cars on the East-West roads (xx 00 xx 00 x:x) and there is a North-South important flow (11 xx 11 xx x:x) and when the vertical traffic lights are red (xx xx xx xx 1:x) then the controller must change the state of the traffic lights (xx xx xx xx x:1). Figure 2 represents an individual where A is the Arriving sensor and J the Junction sensor of the indicated road.

Rule n Rule 1 Rule

i A A A A J J J J North Red Light North East South West Action Figure 2: An Individual.
At each tic (i.e. time interval), a car should advance or wait. A cycle corresponds to 5 tics. Each cycle an individual takes a decision, choosing randomly one rule among the matching rules. Each junction is ruled by an agent also called population. The evaluation of populations is made measuring the performance of all individuals filling it. Each individual ✄ is evaluated regarding its performance on its junction. At the end of an evaluation, ✄ is given a strength which is the fluctuation between the average delay obtained at the end of the first generation by the whole of the individuals and the one obtained by ✄ for the current generation. Strength is given using this function :

☎ ✝✆ ✟✞ ✝✠ ☛✡ ☞ ✍✌ ✏✎ ✒✑ ✓✆ ✕✔ ✝✌ ✗✖ ✙✘ ✛✚ ✢✜ ✤✣ ✦✥ ★✧ ✪✩ ✬✫ ✤✭ ✮ ✙✯ ✱✰ ✳✲ ✎ ✴✎ ✵✑ ✶✆ ✕✌ ✷ ✙✑ ✶✸ ✺✹ ✼✻ ✽ ✾✠ ☛✎ ✒✿ ❀✌ ❁ ❂✡ ✼✑ ✓✎ ✵✹ ✝✔ ✤✑ ✷ ✙✑ ✶✸ ✺✹ ✼✻ ❄❃ ❅ ❇❆ ❉❈ ❋❊ ✼• ✒❍ ■ ❑❏ ▼▲ ✵❊ ✝◆ ❖ ✕▲ ◗ ✦▲ ✒| ❙◆ ❯❚ is the mean of the average delays the 900 individuals (i.e. 9 junctions of 100 individuals) have obtained at the end of the first generation upon their junction.

❅ ❲❱ (❳ ✍❊ )❊ )▲ ✒❬ ❭❍ ◗ (▲ )| ❙◆ ❚ is the average delay the evaluated individual has obtained at the end of the current generation upon its junction.

The ❆ ❉❈ ❋❊ ✼• ✒❍ ■ ❑❏ ▼▲ ✵❊ ✝◆ ❖ ✕▲ ◗ ✦▲ ✒| ❙◆ ❯❚ is the reference performance for the next generations. Figures 3 shows how an individual will be rewarded when it is evaluated comparing its performance to the mean of the first average delays obtained by the 900 individuals. First generation mean values proposed here are 20, 30 or 50 tics. The starting mean strength is 0.7 for the 900 individuals. We use ▲ ✒) ❴ function as a delay minimization function. It allows an important enough slope to strengthen or to weaken an evaluated individual. Thus, the more delay decreases through generations, the more strength increases. The parameters of our experiment for the genetic algorithm are :

❅ Mutation probability : 0.01

❅ Crossover probability : 0.6

❅ Crossover : single point ❅ Selection method : Roulette wheel population is composed of 20 individuals, each of them being the concatenation of 15 randomly initialized rules. Individuals performance is averaged after 100 evaluations. Three different flows for roads have been chosen. High flow represents 9 cars per 60 tics, medium flow corresponds to 6 cars per 60 tics, low flow corresponds to 3 cars per 60 tics arriving on a road. The results we present here are averaged over 20 runs with different random seed values. In order to have a global vision of the performance of the whole multi-agent system, the curves we present in the next experiments gives each generation the mean of the average delays for all the 9 agents. In the same way, we always give a reference curve which is the result of an experiment without any enhancement.

Elitism

Inspired by works made by [START_REF] Jong | [END_REF] on Genetic Algorithms, the aim of the Elitism is to keep the best individuals deleting the worst ones in order to strengthen reinforcement.

To implement Elitism, the ❵ best individuals of the last generation are to be placed in the new generation. They replace the ❵ worst individuals. The remaining individuals are created as usually, i.e applying crossover and mutation operators.

In this experiment, North flow is high, South flow is medium and East-West flows are low. Roads that are connected have an empty flow because cars come from the neighbor junctions. We have chosen different percentages of Elitism. So ❵ is dependent of this percentage. We have tested the multi-agent classifier system with different values for ❵ : 3 (15% of the size of an agent, i.e 15% of 20), 6 (30%), 10 (50%) and 12 (60%). Reference curve corresponds We can see that the simple evolution (reference) does not learn quickly the solution to the main problem. We notice the quick and relatively efficient convergence of Elitism. Toward generation 10, the 30%, 50% and 60% curves have converged to a stable average delay of about 4.2 tics and then decreased slowly to an average delay of about 1.9 tics at generation 50 for a car to wait before crossing junction. Differences between 30%, 50% and 60% curves are slight. The 15% curve has more difficulties to converge. It seems that a 50% rate of Elitism on population is here the best rate to enhance classifier systems convergence. This curve shows a gain of almost 96% on the first average delay. The 60% rate is too high and its curve is higher than 50% one. Now that we have seen the already proved efficiency of Elitism, we should extend that method.

Distributed Elitism

Pittsburgh-style classifier systems can be enhanced like Michigan-style one with some features. Elitism is a good approach that we can apply to a multi-agent Pittsburgh-style system. In an homogeneous environment, where agents learn locally a similar problem, we should share knowledge. Distributed Elitism consists in exchanging the best individuals between these identical agents, trying to reinforce the whole as quickly as possible, through a pool that will receive for transit those individuals. We conceive Distributed Elitism equally sharing individuals between agents. Distribution is made after the evaluation of whole population : The distribution of the best individuals of each agent is made through a pool classifier. This pool is not a classifier system so it does not evolve. The pool is filled with the best individuals of each agent. To have a better sharing of the pool, the global strength of each agent is used to know how many individuals an agent will pool. The more it has a high strength, the more it will place individuals in the pool :

❛ ❝❜ ❞ ✾❡ ✴❡ ✵✸ ❢✑ ✓✞ (✘ ❄❞ ✾❡ ✴❡ ✒✸ ☞ ❣✠ ✐❤ )✑ ✯ ✛• ☞ •❦ ☛❁ ❂✔ ✤✑ ✓✆ ✕✌ ♥♠ • ☞ •❦ ☛❁ ✳✸ ❢✸ ❢♠
❅ ❇♦ q♣ r ✙s ✝s ✝| ❙▲ )t is the number of individuals pooled.

❅ ❇r ✙s ✝s ✝| ✉ ★❈ ❋✈ ❯▲ is the size of the pool.

❇✇ ①✉ a function that calculates the Global Strength of an agent or more.

❅ ❇■ ❑❖ ▼▲ ✒❬ ❭❍ is the considered agent.

❅ ❇■ ②| ✐| represents all the agents to get the whole strength of our multi-agent system.

the pool is filled with the best individuals of all agents, these individuals have to be distributed only once, replacing the worst individuals of each agent. The redistribution is ordered : the worst agents take the best individuals from the pool classifier and the best agents take the 'worst of the best' individuals from the pool. To know how many individuals will be taken by an agent, we use also an equal share method based upon the strength where the worst the strength is, the greatest the number of taken individuals is :

❛ ✱❜ ③ ❂✹ ✤④ ✑ ✓✆ ✱✘ ❞ ✾❡ ✵❡ ✵✸ ☞ ⑤✠ ✐❤ )✑ ✯ ❦ • ☞ •❦ ☛❁ ⑥✸ ❢✸ ⑦♠ ❣⑧ • ☞ •❦ ☛❁ ❂✔ ✤✑ ✓✆ ✕✌ ♥♠ ✗♠ • ☞ •❦ ☛❁ ✳✸ ✺✸ ⑦♠ ✯ ❦ ☛❛ ✱❜ ❁ ✬✔ ✤✑ ✶✆ ✕✌ ✗✿ ★⑧ ⑩⑨ ✶♠ ❅ ❇♦ q♣ ❶
②◆ ▼❷ ✟▲ ✵❬ is the number of individuals taken from the pool.

❅ ❇♦ q♣ ■ ❑❖ ▼▲ ✒❬ ❭❍ ♥• is the number of agents of our multi-agent system.

In this experiment, North flow is high, South flow is medium and East-West flows are low. Roads that are connected have an empty flow because cars come from the neighbor junctions. We compare an alone evolution (reference) and different percentages of the whole population using distributed elitism. The different sizes of the pool, that reflect these percentages, are : 27 (15% the whole population, i.e 3 ✂ 3 ✂ 20), 54 (30%) and 90 (50%). Figure 5 shows the average delays that cars wait on the map to cross junctions.

Distributed Elitism gives good results indicating that each junction is efficient and that the whole map is also very efficient at the end of the learning. Toward generation 15, the 3 curves have converged to a stable average delay of about 3 tics and converge slowly to an average delay of 1.1 at generation 50. The best curve is 30% of Distributed Elitism. In term of convergence, it is more efficient than simple Elitism.

The gain is about 98% on the first average delay. But it needs 3 generations more to converge to less than 4.2 tics. So why should we not mix the both methods ?

Mixed experiment

The efficiency of both methods in their domain have been shown (quick convergence for Elitism and better convergence for Distributed Elitism). We have made a last experiment to see if combining these two methods can more improve the results.

The implementation of the combined methods is simple because Elitism and Distributed Elitism are separated in the algorithm :

Initialize(Pop); While(conditions not satisfied) { Evaluate(Pop); Distributed_Elitism(Pop); GA(Pop); /* Elitism */ }
Thus in this experiment, Elitism is performed within each independent agent, and Distributed Elitism is applied between all the agents of the system. So the mechanism favoring the best individual is applied at both levels of our multiagent system. The reference curve used in this experiment is free of any form of elitism. Figure 6 also represents the best Elitism curve (50% of the population of an agent) and the best Distributed Elitism curve (30% of the whole population). The fourth curve displays the average delay vs. the generations for the system using both Elitism at 50% and Distributed Elitism at a 30% rate.

Results are impressive because we should think that further gain was nearly impossible. On the fifth generation, average delay is about 3.4 tics, better than the best Elitism. And as final result at generation 50, we have an average delay of 0.47 tic. It is a gain of more than 99% from the first average delay. To figure out this results and to test robustness, we have made three more experiments. Traffic flows on each roads are changed at regular intervals to verify that there is no convergence toward a local optimum and that our mixed solution is a good one.

Changing flow : Robustness

To test the robustness of our multi-agent classifier systems using the mixed solution, we have done two more experiments during which flows change every 10 generations. In the first test, we will show how the system reacts when faced to a low traffic progressively becoming a very high one. The second experiment is the opposite : the system has to learn how to manage with a very high traffic progressively becoming a low one. The notion of very high flow is represented by a probability of 0.5 for a car to appear on a road at each tic. In other words, with this probability there is every expectation that 30 cars will arrive in 60 tics. On both experiments, flows are randomly set for the ten last generations, i.e. from generation 40 to 50.

The aim of the first experiment is to show that incremental difficulty is not too tough to handle. At the beginning, all incoming flows are low, then they increase to very high : Figure 7 represents the reference curve with no Elitism nor Distributed Elitism, and the mixed solution one. The reference experiment seems to have no problems with the changing flows. But the global behavior of the system is not suitable since at generation 50 the delay is about 20. Considering the mixed solution test, we can notice that at each change of flow the system has to recover from the new flow difficulty. But in all cases, it has the expected behavior since it converges quickly even with a sharp difficulty at generation 30 when flows become very high ones. Each 10 generations, production rules that are needed to handle a harder problem are available in the population and are quickly used to lower the average delay. Then, random flows present no difficulties at all to the system, since all cases have already been learnt. Thus, we can say the population keeps enough diversity in order to ensure the robustness of the system.

Let us now detail the second experiment. Here, the system has to learn how to manage with very high flows progressively becoming low ones. Figure 8 plot with no Elitism nor Distributed Elitism, and the mixed solution one. We can notice that with this test, the reference system has more difficulties to manage the changing of flows that in the previous experiment. But globally, the average delay is the same (about 20). But, considering the mixed solution plot, it is very impressive to see that when the difficulty is higher, the system learns even faster to handle the map of junctions. With a difficulty lowering each 10 generations, it is able to handle at a very good rate the average delay. We can deduce from this experiment that rules needed to handle easier flow problems are contained in the population of the harder one. A third experiment has been done : during 50 generations, flows are randomly set and change every 5 generations. Experiment is shown on figure 9. Reference curve uses no Elitism nor Distributed Elitism. It converges slowly as before toward an average delay of 20 encountering slight difficulties each 5 generations. Mixed solution curve learns quickly with a little stop on generation 5 at the first changing flow. This test , we noticed that our system is robust since the average delay is about 0.57 tics at the end of the experiment. After generation 15, when the classifier system has learnt, the variation of the average delay is only of 0.2 tics each 5 generations while flows change. Those experiments have shown the good reactivity of the classifier system using a mixed Distributed Elitism and Elitism solution : Robustness is shown.

Conclusion and further works

This work on multi-agent classifier systems has shown that simplest solutions can be good ones. There is a lot of work to do after this, because generalization of classifiers is far away. We think that there is a way to explore in order to extend conceptual approaches of complex problems to solve them with classifier systems. It seems that in an homogeneous environment, Distributed Elitism mixed with Elitism is both efficient and robust, now we shall integrate it to classifier systems for learning in such an environment. We shall use a classifier system to rule percentages of Distributed Elitism and Elitism. That classifier will allow to set rates without knowledge of the best ones for the problem it will handle.

Heterogeneous systems are not as easy to generalize because agents have different goals to achieve, but they are correlated to solve a single complex problem. Sensors of agents are different and indicate different measures that pooled together and evaluated, give an answer to the problem. These systems need to exchange information between agents introducing the notion of communication and language. They cannot exchange individuals as in an homogeneous environment, disabling the Elitism solution as we present it. Evolution as Artificial-Life of such a communication is a big interest point. Pitssburgh-style classifier systems are interesting and efficient but they need 'evolutions' as Michigan-style classifier systems did. Putting them in an heterogeneous situation and studying their comportment while communicating should be a good answer to rule more and more complex problems.

Figure 1 :

 1 Figure 1: The Simulator.

  Classifier Systems (Based upon the mean of the first average delays = 20, 30, 50 tics)

Figure 4 :

 4 Figure 4: Elitism.

  Figure 5: Distributed Elitism.

  Figure 6: Mixed experiment.

❅

  from generation 0 to 9 flows are low ones ❅ from generation 10 to 19 flows are medium ones ❅ from generation 20 to 29 flows are high ones ❅ from generation 30 to 39 flows are very high ones.

Figure 7 :

 7 Figure7represents the reference curve with no Elitism nor Distributed Elitism, and the mixed solution one. The reference experiment seems to have no problems with the changing flows. But the global behavior of the system is not suitable since at generation 50 the delay is about 20. Considering the mixed solution test, we can notice that at each change of flow the system has to recover from the new flow difficulty. But in all cases, it has the expected behavior since it converges quickly even with a sharp difficulty at generation 30 when flows become very high ones. Each 10 generations, production rules that are needed to handle a harder problem are available in the population and are quickly used to lower the average delay. Then, random flows present no difficulties at all to the system, since all cases have already been learnt. Thus, we can say the population keeps enough diversity in order to ensure the robustness of the system.Let us now detail the second experiment. Here, the system has to learn how to manage with very high flows progressively becoming low ones. Figure8represents the reference

  Figure 8: Changing flow from very high to easy.

Figure 9 :

 9 Figure 9: Changing flow each five generations.


Clément Mifsud

Bruno Després

Nicolas Seguin

Dissipative formulation of initial boundary value problems for Friedrichs' systems

In this article we present a dissipative definition of a solution for initial boundary value problems for Friedrichs' systems posed in the space L 2 t,x . We study the information contained in this definition and prove an existence and uniqueness theorem in the non-characteristic case and with constant coefficients. Finally, we compare our choice of boundary condition to previous works, especially on the wave equation.

Introduction

The aim of this paper is to present a notion of solution for initial boundary value problems in the case of linear hyperbolic systems. This problem has been studied by many different authors [START_REF] Friedrichs | Symmetric Positive Linear Differential Equations[END_REF][START_REF] Kreiss | Initial-Boundary Value Problems for Hyperbolic Equations with Uniformly Characteristic Boundary[END_REF][START_REF] Massey | Differentiability of solutions to hyperbolic initial-boundary value problems[END_REF][START_REF] Madja | Initial-Boundary Value Problems for Hyperbolic Equations with Uniformly Characteristic Boundary[END_REF][START_REF] Bardos | Maximal Positive Boundary Value Problems as Limits of Singular Perturbation Problems[END_REF][START_REF] Rauch | Symmetric positive systems with boundary characteristic constant multiplicity[END_REF][START_REF] Ern | Discontinuous Galerkin Methods for Friedrichs' systems. I. General Theory[END_REF] and there are still a lot of open questions for this kind of problem (in particular in the case of variable multiplicities : see [START_REF] Métivier | Hyperbolic boundary value problems for symmetric systems with variable multiplicities[END_REF]). In addition, many physical systems can recast in a Friedrichs' framework : linear elasticity theory [START_REF] Serre | On the L 2 -well posedness of an initial boundary value problem for the 3D linear elasticity[END_REF], isotropic Maxwell equations [START_REF] Falk | Explicit Finite Element Methods for Symmetric Hyperbolic Equations[END_REF], linearized Euler equations, MHD [START_REF] Métivier | Hyperbolic boundary value problems for symmetric systems with variable multiplicities[END_REF].

Having in mind dissipative problems such as viscoplastic problems for which boundary conditions play a important mechanical role and their associated limit plastic problems, which can develop singularities and discontinuities, our initial goal was to give a new definition of solution that requires minimal regularity and in particular, the solution does not need a trace on the boundary. In [START_REF] Després | Weak solutions to friedrichs systems with convex constraints[END_REF], a new formulation for Friedrichs' systems on the whole space is proposed, very similar to the Kruzhkov's definition of entropy solutions of scalar quasilinear equations [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] and to dissipative solutions introduced by Lions for incompressible Euler equations [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF]. In [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF], Otto extends the concept of entropy solutions to bounded domains with Dirichlet conditions, without requiring any assumption on the existence of traces of the solutions (contrary to the theory developed in [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF]). We aim to do the same for linear systems. In order to verify that this original formalism contains enough information, we prove in the classical case of constant coefficients in the half-plane that this formulation leads to a well-posed theory in L 2 t,x and that our formulation is equivalent to the usual strong formulation for sufficiently smooth solutions. Finally, since different theories of boundary conditions are available in the literature, we compare our formulation to other formulations on a simple example. This work is organized as follows. In the first section, we explain formally where the definition comes from, how this definition can be extended in order to take account of non-homogeneous boundary conditions and variable coefficients, and then we show in the case of constant coefficients when the domain is a half-plane that a H 1 solution of our weak formulation is a solution of the problem in the strong sense. Then we prove that there exists a unique solution (in the sense of our definition) in the non-characteristic setting for H 1 initial data and L 2 source term, again for constant coefficients and if the spatial domain is a half-plane (the regularity of the initial data is necessary for our proof, but in this case, the classical theory is valid for L 2 initial data and L 2 source term). Finally we compare our approach of the boundary conditions to the literature (namely to [START_REF] Friedrichs | Symmetric Positive Linear Differential Equations[END_REF], [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF] and [START_REF] Dubois | Boundary Conditions for Nonlinear Hyperbolic Systems of Conservation Laws[END_REF]) and we examine these boundary conditions on the simple example of the wave equation in 1D with Dirichlet condition.

Definition and basic properties 2.1 Motivations and definitions

We are interested here in the following problem

         ∂ t U + n i=1 A i ∂ xi U = f, on (0, T ) × Ω, U (0, x) = U 0 (x),
on Ω, (A(ν(x)) -M (ν(x)))U (t, x) = 0, on (0, T ) × ∂Ω, [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] where Ω is a open domain (not necessarily bounded) of R n with (smooth) boundary, denoted by ∂Ω, U is a vector of size m, f ∈ L 2 ((0, T ) × Ω) m , U 0 ∈ L 2 (Ω) m (in the following L 2 ((0, T ) × Ω) m will be simply denoted by L 2 t,x and L 2 (Ω) m by L 2 x ), the matrices A i are symmetric, A(ν(x)) = n i=1 ν i A i and ν(x) = (ν 1 , ν 2 , . . . , ν n ) is the exterior unit normal of Ω at the point x ∈ ∂Ω, M (ν(x)) is a symmetric non-negative matrix such that ker

A(ν(x)) ⊂ ker M (ν(x)) (2)
and

R m = ker(A(ν(x)) -M (ν(x))) + ker(A(ν(x)) + M (ν(x))). ( 3
)
The symmetry of the matrix M is an important hypothesis in our approach. In the work of Friedrichs [START_REF] Friedrichs | Symmetric Positive Linear Differential Equations[END_REF] the matrix M does not need to be symmetric, but its symmetric part has to be non-negative (see section 4). In his work, Friedrichs only considers matrices M such that the previous sum is direct (this hypothesis is too restrictive, especially in the case where det(A(ν(x)) = 0), but one can see (using lemma 1) that in the non-characteristic case (i.e. det(A(ν(x)) = 0) the sum is always direct. In the following, the matrices A i are independent of the space and time variables but our formulation can be extended to matrices with variables coefficients (see remark 5).

Remark 1 To design a matrix M (ν(x)), we could consider an orthogonal matrix P (ν(x)) such that t P (ν(x))A(ν(x))P (ν(x)) is a diagonal matrix and its diagonal terms are

λ 1,1 ≤ • • • ≤ λ p,p < λ p+1,p+1 = 0 = • • • = λ q,q < • • • ≤ λ m,m , then define M (ν(x)) = t P (ν(x))diag(|λ 1,1 |, • • • , |λ p,p |, 0, • • • , 0, λ q+1,q+1 , • • • , λ m,m )P (ν(x)).
It means there is no information coming from the incoming characteristic (since we consider here an homogeneous boundary condition). The hypothesis that M is symmetric still allows us to use a large class of boundary conditions coming from the theory of characteristics.

To shorten notation, we write A ν instead of A(ν(x)) and M instead of M (ν(x)).

Remark 2 Here we denote by . | . the canonical scalar product on R m .

Lemma 1

Let A ν and M be symmetric matrices verifying the properties (2) and (3). The following statements are true 1. We have the decomposition

R m = ker A ν ⊕ ((ker(A ν -M )) ∩ ImA ν ) ⊕ ((ker(A ν + M )) ∩ ImA ν ) ,
and we denote k = k 0 + k -+ k + the decomposition of k ∈ R m over this sum.

2. M is positive on (ker

(A ν + M )) ∩ ImA ν . 3. For all k, κ ∈ R m , k | A ν κ = k -| A ν κ -+ k + | A ν κ + . 4. For all k ∈ R m , 0 ≤ k -| M k -≤ k | M k and 0 ≤ k + | M k + ≤ k | M k .
The proof of this lemma uses classical tools of linear algebra. It allows us to make some algebraic manipulations (see below) to derive the following definition Definition 1 We say that U ∈ L 2 t,x is a dissipative solution of the system (1) if for all ϕ ∈ W 1,∞ c,+ , the space of W 1,∞ ((0, T ) × Ω) functions with non-negative values and compact supports (in R n+1 ), and for all k ∈ R m the following inequality holds

T 0 Ω |U (t, x) -k| 2 ∂ t ϕ(t, x)dxdt + T 0 Ω n i=1 U (t, x) -k | A i (U (t, x) -k) ∂ xi ϕ(t, x)dxdt + Ω |U 0 (x) -k| 2 ϕ(0, x)dx + T 0 Ω 2 f (t, x) | U (t, x) -k ϕ(t, x)dxdt + T 0 ∂Ω k + | M (ν(x))k + ϕ(t, x)dxdt ≥ 0, (4)
where k + stands for the projection on (ker(A ν + M )) ∩ ImA ν with respect to the decomposition of R m presented in lemma 1.

Remark 3 As A ν and M depend on x ∈ ∂Ω, k + also depends on ∂Ω. In the case where A ν and M do not depend on x, k + is independent of x ∈ ∂Ω (see remark 8 for more details).

It is worth pointing out that the formulation does not require much regularity on the solution, in particular we do not use any traces of the solution either at the initial time or at the boundary. This formulation is inspired by the one exposed in [START_REF] Després | Weak solutions to friedrichs systems with convex constraints[END_REF] for Friedrichs' systems under constraints in the whole space. Furthermore, this definition can be used in the case of variable coefficients (see remark 5) and when the number of non-negative eigenvalues of A ν is not constant along ∂Ω, this is the case of variable multiplicities (as long as we can find a matrix M that satisfies the hypotheses (2) and (3) and belongs to L ∞ ), these cases are of deep interest for applications (the condition of constant multiplicities is difficult to insure for a general domain Ω, in particular for a bounded domain ∂Ω) and that is still an active domain of research (see for example [START_REF] Métivier | Hyperbolic boundary value problems for symmetric systems with variable multiplicities[END_REF]).

We call this formulation dissipative since taking k = 0 and ϕ(t, x) = T -t in (4), we get the L 2 -norm of a solution is non-increasing

T 0 Ω |U | 2 (t, x)dtdx ≤ T Ω |U 0 | 2 (x)dx.
This possible loss of energy of the solution allows us to call the solution dissipative (as P-L. Lions did for Euler equations in [START_REF] Lions | Mathematical topics in fluid mechanics[END_REF]). The word dissipative also refers in the literature to special kind of boundary conditions, our boundary condition can be called dissipative in that sense (see section 4).

Let us briefly explain where this formulation comes from. If we assume that the solution U is regular and we use the fact that the derivatives of k cancel, we go from (1) to

T 0 Ω |U (t, x) -k| 2 ∂ t ϕ(t, x)dxdt + T 0 Ω n i=1 U (t, x) -k | A i (U (t, x) -k) ∂ xi ϕ(t, x)dxdt + T 0 Ω 2 f (t, x) | U (t, x) -k ϕ(t, x)dxdt - T 0 ∂Ω U (t, x) -k | A ν (U (t, x) -k) ϕ(t, x)dσ(x)dt + Ω |U 0 (x) -k| 2 ϕ(0, x)dx - Ω |U (T, x) -k| 2 ϕ(T, x)dx = 0,
by taking the scalar product with U -k and then integrating by parts. Then we write the decomposition of k = k 0 + k -+ k + and observe that since U ∈ ker(A ν -M ) at the boundary we know that U + = 0 and it follows that

T 0 ∂Ω U (t, x) -k | A ν (U (t, x) -k) ϕ(t, x)dσ(x)dt = T 0 ∂Ω U (t, x) -k -| A ν (U (t, x) -k -) ϕ(t, x)dσ(x)dt + T 0 ∂Ω k + | A ν k + ϕ(t, x)dσ(x)dt = T 0 ∂Ω U (t, x) -k -| M (U (t, x) -k -) ϕ(t, x)dσ(x)dt - T 0 ∂Ω k + | M k + ϕ(t, x)dσ(x)dt, thanks to lemma 1. Consequently, T 0 Ω |U (t, x) -k| 2 ∂ t ϕ(t, x)dxdt + T 0 Ω n i=1 U (t, x) -k | A i (U (t, x) -k) ∂ xi ϕ(t, x)dxdt + T 0 Ω 2 f (t, x) | U (t, x) -k ϕ(t, x)dxdt + Ω |U 0 (x) -k| 2 ϕ(0, x)dx + T 0 ∂Ω k + | M k + ϕ(t, x)dσ(x)dt = T 0 ∂Ω U (t, x) -k -| M (U (t, x) -k -) ϕ(t, x)dσ(x) + Ω |U (T, x) -k| 2 ϕ(T, x)dxdt ≥ 0.
Remark 4 Here we explain how to take account of an inhomogeneous boundary condition of the form

(A ν -M )(U (t, x) -U b (t, x)) = 0, on (0, T ) × ∂Ω,
where U b ∈ L 2 ((0, T ) × ∂Ω) m . From the previous formal calculations, we want to eliminate the trace of U in the term

T 0 ∂Ω U (t, x) -k | A ν (U (t, x) -k) ϕ(t, x)dσ(x)dt.
Since we know that, formally and using lemma 1,

U -k | A ν (U -k) = U b -k | A ν (U b -k) + U -U b | A ν (U -U b ) + 2 U -U b | A ν (U b -k) = (U b -k) -| M (U b -k) --(U b -k) + | M (U b -k) + + U -U b | M (U -U b ) + 2 (U -U b ) | M (U b -k) - ≥ -(U b -k) + | M (U b -k) + ,
we just replace the term

T 0 ∂Ω k + | M k + ϕ(t, x)dtdσ(x), by T 0 ∂Ω (k -U b ) + | M (k -U b ) + ϕ(t, x)dtdσ(x),
in the formulation.

Remark 5

We also can write a definition of solution for symmetric matrices A i that belong to the space W 1,∞ ((0, T ) × Ω) m×m (assuming that one can find a matrix M that satisfies the conditions (2) and (3) and belongs to L ∞ ((0, T ) × ∂Ω) m×m ). One just need to add the term

T 0 Ω n i=1 U (t, x) -k | ∂ xi [A i ] (U (t, x) -k) ϕ(t, x)dxdt,
to the left-hand side of the inequality (4).

Remark 6 One can see in the functions parametrized by k ∈ R m

F (U ) = |U -k| 2 (5)
the analogue of the entropy functions of Kružkov [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] in the scalar case of quasilinear initial value problems.

Example 1 In the scalar case (m = 1), the theory of initial boundary value problems is well known ( [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF], [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF]). Using our theory, we see that if A ν ≥ 0 then M = A ν is the only choice for the boundary matrix and it leads to the fact that there is no condition at that point (this corresponds exactly to the case when the BLN condition [START_REF] Bardos | First order quasilinear equations with boundary conditions[END_REF] is automatically verified whatever the trace is at that point). In the case A ν < 0, the choice for M is -A ν , it implies (at least formally) that U = 0 at this point of the boundary (again this corresponds to the BLN condition).

What is the information contained in our formulation?

Our goal is here to show that dissipative solutions (which satisfy inequalities) satisfy the original equation, initial condition and boundary condition in a weak sense (that are equalities). For technical reasons we will nevertheless make the hypothesis that the spatial domain is a half-plane for the study of the boundary condition.

First, we are going to show that if U ∈ L 2 t,x is a solution of (1) in the sense of definition 1 then U is a solution of the equation

∂ t U + n i=1 A i ∂ xi U = f,
in the sense of the distributions.

Lemma 2

A solution U in the sense of definition 1 verifies

∂ t U + n i=1 A i ∂ xi U = 0, in D ((0, T ) × Ω).
Proof : Let ϕ be a function of D((0, T ) × Ω) with non-negative values. Then, from definition 1, we know that

T 0 Ω |U (t, x) -k| 2 ∂tϕ(t, x)dxdt + T 0 Ω n i=1 U (t, x) -k | Ai(U (t, x) -k) ∂x i ϕ(t, x)dxdt ≥ 0.
Expanding the terms |U (t, x) -k| 2 and U (t, x) -k | Ai(U (t, x) -k) and using the fact that the partial derivatives of k vanish leads us to the following inequality

T 0 Ω |U (t, x)| 2 -2 k | U (t, x) ∂tϕ(t, x) + n i=1 ( U (t, x) | AiU (t, x) -2 U (t, x) | Aik ) ∂x i ϕ(t, x) dxdt ≥ 0.
Since k is independent of t and x, we rewrite the previous inequality as

T 0 Ω |U (t, x)| 2 ∂tϕ(t, x) + n i=1 U (t, x) | AiU (t, x) ∂x i ϕ(t, x) dxdt ≥ 2 k T 0 Ω U (t, x)∂tϕ(t, x) + n i=1 AiU (t, x)∂x i ϕ(t, x)dxdt .
The left-hand side of the inequality is independent of k, thus the previous inequality implies that

T 0 Ω U (t, x)∂tϕ(t, x) + n i=1 AiU (t, x)∂x i ϕ(t, x)dxdt = 0. (6)
It remains to show that equation ( 6) is true for all ϕ ∈ D((0, T ) × Ω). Let ϕ be a function in D((0, T ) × Ω), then ϕ = ϕ + -ϕ -where ϕ + (resp. ϕ -) is the positive part (resp. negative part) of ϕ and ϕ + (resp. ϕ -) is an element of W 1,∞ c,+ ((0, T ) × Ω). Equation ( 6) is true for ϕ + (resp. ϕ -) and thus the lemma is proven.

Remark 7

The previous and following lemmas are also true in the case of variable coefficients and when Ω is a general domain with smooth boundary. Now, we focus our attention on the initial condition and the boundary condition.

Lemma 3 Let ξ ∈ W 1,∞ c,+ (Ω). We have lim τ →0 lim α→0 1 α τ τ -α Ω |U (t, x) -U 0 (x)| 2 ξ(x)dxdt = 0.
Proof : For convenience, we take f = 0. The same conclusion can be drawn for a general f ∈ L 2 t,x . Let ξ ∈ W 1,∞ c,+ (Ω) and ρ n be the classical positive n-dimensional mollifier with > 0. Let α > 0 and τ ∈ (α, T ). We define the function ϕ :

[0, T ] × Ω × Ω → R + by ϕ(t, x, y) = ξ(x)ηα(t)ρ n (x -y),
where ηα(t) = max(0, min(1, (τ -t)/α)). Observe that η α (t) ≤ 0 and η α -→ α→0 + -δτ in the distributional sense. Using the Cauchy-Schwarz inequality, we have

|U (t, x) -U0(y)| 2 = |U (t, x) -U0(x)| 2 + |U0(x) -U0(y)| 2 + 2 U (t, x) -U0(x) | U0(x) -U0(y) ≥ |U (t, x) -U0(x)| 2 + |U0(x) -U0(y)| 2 -2|U (t, x) -U0(x)||U0(x) -U0(y)|.
From the inequality ∂tϕ ≤ 0, we get for almost every (t, x, y),

|U (t, x) -U0(y)| 2 ∂tϕ(t, x, y) ≤ |U (t, x) -U0(x)| 2 ∂tϕ(t, x, y) + |U0(x) -U0(y)| 2 ∂tϕ(t, x, y) -2|U (t, x) -U0(x)||U0(x) -U0(y)|∂tϕ(t, x, y).
We integrate this inequality over [0, T ] × R n × R n (we extend U and U0 by zero outside of Ω) to get, thanks to Fubini's theorem,

T 0 R n R n |U (t, x) -U0(y)| 2 ∂tϕ(t, x, y)dxdydt ≤ T 0 R n R n |U (t, x) -U0(x)| 2 ∂tϕ(t, x, y)dxdydt - R n R n |U0(x) -U0(y)| 2 ξ(x)ρ n (x -y)dxdy -2 R n R n |U0(x) -U0(y)|ξ(x)ρ n (x -y) T 0 |U (t, x) -U0(x)|η α (t)dtdxdy.
We take k = U0(y) and use the above-defined test function ϕ(t, x, y) in the definition of a dissipative solution and integrate with respect to y to obtain

T 0 R n R n |U (t, x) -U0(y)| 2 ∂tϕ(t, x, y)dxdydt ≥ - T 0 Rα, (t)dt - R n R n |U0(x) -U0(y)| 2 ξ(x)ρ n (x -y)dxdy,
where Rα, belongs to L 1 (0, T ) and is for almost every time t defined by

Rα, (t) = R n R n n i=1 U (t, x) -U0(y) | Ai(U (t, x) -U0(y)) ηα(t)∂x i (ξ(x)ρ n (x -y)) dxdy + R n ∂Ω U0(y)+ | M U0(y)+ ξ(x)ηα(t)ρ n (x -y)dσ(x)dy. (7)
Using these two inequalities, we get

- T 0 Rα, (t)dt ≤ T 0 R n R n |U (t, x) -U0(x)| 2 ∂tϕ(t, x, y)dxdydt -2 R n R n |U0(x) -U0(y)|ξ(x)ρ n (x -y) T 0 |U (t, x) -U0(x)|η α (t)dtdxdy.
Using the fact that for (a, b) ∈ R 2 , ab ≤ b 2 + a 2 4 and, since ∂tϕ ≤ 0,

-|U0(x) -U0(y)||U (t, x) -U0(x)|ξ(x)ρ n (x -y)η α (t) = |U0(x) -U0(y)||U (t, x) -U0(x)||∂tϕ(t, x, y)| ≤ |U0(x) -U0(y)| 2 ξ(x)ρ n (x -y)|η α (t)| - |U (t, x) -U0(x)| 2 4 ∂tϕ(t, x, y).
By integration of this inequality over

[0, T ] × R n × R n , it follows - T 0 Rα, (t)dt ≤ 3 4 T 0 R n R n |U (t, x) -U0(x)| 2 ∂tϕ(t, x, y)dxdydt + R n R n |U0(x) -U0(y)| 2 ξ(x)ρ n (x -y)dxdy.
We are going to send α to 0. By the definition of ηα (and since η0 = 1 [0,τ ] ) and equation ( 7), we have

T 0 [Rα, (t) -R0, (t)] dt ≤ τ τ -α R n R n n i=1 | U (t, x) -U0(y) | Ai(U (t, x) -U0(y)) ∂x i ϕ(t, x, y)| dxdydt +2 τ τ -α R n ∂Ω U0(y)+ | M U0(y)+ ϕ(t, x, y)dσ(x)dydt.
We now use the fact that, if It follows that, when α → 0,

k = k 0 + k-+ k+ with k± ∈ ker(A ± M ) ∩ ImA
lim α→0 1 α τ τ -α R n R n |U (t, x) -U0(x)| 2 ξ(x)ρ n (x -y)dxdydt ≤ 4 3 T 0 R0, (t)dt + 4 R n R n |U0(x) -U0(y)| 2 ξ(x)ρ n (x -y)dxdy.
But, by Fubini's theorem, we remark that for every > 0,

R n R n |U (t, x) -U0(x)| 2 ξ(x)ρ n (x -y)dxdy = R n |U (t, x) -U0(x)| 2 ξ(x)dx.
Then, we let τ going to 0 in the last inequality, and we use that for all , we know that

T 0 R0, (t)dt ≤ τ 0 R n R n n i=1 Ci(|U0| 2 (x) + |U0| 2 (y))∂x i [ξ(x)ρ n (x -y)] dxdydt + τ 0 R n ∂Ω C|U0| 2 (y) ξ ∞ dσ(x)dydt,
where Ci and C respectively depend on Ai and M . By the dominated convergence theorem, we get lim

T →0 lim α→0 1 α τ τ -α Ω |U (t, x) -U0(x)| 2 ξ(x)dxdt = lim T →0 lim α→0 1 α τ τ -α R n |U (t, x) -U0(x)| 2 ξ(x)dxdt ≤ 4 R n R n |U0(x) -U0(y)| 2 ξ(x)ρ n (x -y)dxdy.
The left-hand side is independent of and the right-hand side goes to 0 as goes to 0.

We now discuss the boundary condition in the special case of the n-dimensional half-place

Ω = R n -= {(x 1 , x 2 , • • • , x n ) = (x , x n ) , x n < 0} . (8)
In the following, we assume that Ω = R n -and that the matrices A i and M are independent of the space and time variables (see remark 8 for more details).

Lemma 4

Assume Ω is a half-space [START_REF] Falk | Explicit Finite Element Methods for Symmetric Hyperbolic Equations[END_REF].

If U ∈ L 2 t,
x is a solution of (1) in (0, T )×R n -in the sense of definition 1, then U satisfies the following condition at the boundary

lim →0 1 R n-1 0 - T 0 U (t, x) + ψ(t)λ(x )dtdx n dx = 0, for every ψ ∈ W 1,∞ ((0, T )) and λ ∈ W 1,∞ (R n-1
) with compact supports (resp. in R and in R n-1 ).

Proof : Using the fact that any function of W 1,∞ with compact support is the sum of its positive part and the opposite of its negative part (that belong to W 1,∞ c,+ ), we may suppose that

ψ ∈ W 1,∞ c,+ ((0, T )) and λ ∈ W 1,∞ c,+ (R n-1
). We take in the formulation of a solution in definition 1 the function φ(x, t) = a (xn)ψ(t)λ(x ) where a (xn) = max(0, (xn + )/ ). We get

R n-1 0 - T 0 |U (t, x) -k| 2 a (xn)λ(x )ψ (t)dtdxndx + 1 R n-1 0 - T 0 U -k | An(U -k) ψ(t)λ(x )dtdxndx + R n-1 0 - T 0 n-1 i=1 U -k | Ai(U -k) ψ(t)a (xn)∂x i λ(x )dtdxndx + T 0 R n-1 k+ | M k+ ψ(t)λ(x )dtdx ≥ 0,
then we use the fact that for all vector v ∈ R n we know that

v | Av = v-| M v--v+ | M v+ ,
and we apply this identity in the second integral to get

Ω T 0 |U (t, x) -k| 2 a (x)ψ (t)λ(x )dtdxndx + Ω T 0 n-1 i=1 U -k | Ai(U -k) ψ(t)a (xn)∂x i λ(x )dtdxndx + 1 Ω T 0 (U (t, x) -k)-| M (U (t, x) -k)-ψ(t)λ(x )dtdxndx - 1 Ω T 0 (U (t, x) -k)+ | M (U (t, x) -k)+ ψ(t)λ(x )dtdxndx + T 0 R n-1 k+ | M k+ ψ(t)λ(x )dtdx ≥ 0,
where Ω = R n-1 × (-, 0). The linearity of the projector and the fact that k+ and M are independent of x leads us to

Ω T 0 |U (t, x) -k| 2 a (x)ψ (t)λ(x )dtdxndx + 1 Ω T 0 (U (t, x) -k)-| M (U (t, x) -k)-ψ(t)λ(x )dtdxndx + Ω T 0 n-1 i=1 U -k | Ai(U -k) a (x)ψ(t)∂x i λ(x )dtdxndx ≥ 1 Ω T 0 U (t, x)+ | M U (t, x)+ ψ(t)λ(x )dtdxndx - 2 Ω T 0 U (t, x)+ | M k+ ψ(t)λ(x )dtdxndx . Now, we set k = -αk0 where k0 = 1 Ω T 0 U (t,
x)+ψ(t)λ(x )dtdxndx and α ∈ R is to be chosen later.

Observe that k0 ∈ ker(A + M ) ∩ ImA and consequently k-= 0, using that k in the previous inequality we obtain

2α k0 | M k0 ≤ 2α 2 C1 0 - R n-1 ∩supp λ T 0 |k0| 2 a (x)dtdxndx +2C2 0 - R n-1 ∩supp λ T 0 |U (t, x)| 2 a (x)dtdxndx + 1 Ω T 0 U (t, x)-| M U (t, x)-ψ(t)λ(x )dtdxndx ,
where C1 and C2 are positive number depending on Ai, λ and ψ. We now choose α to be equal to

α = k0 | M k0 2C1 0 - R n-1 ∩supp λ T 0 |k0| 2 a (x)dtdxn , (9)
and consequently we get that

2α k0 | M k0 -2α 2 C1 0 - R n-1 ∩supp λ T 0 |k0| 2 a (x)dtdxndx = α k0 | M k0 ,
and we obtain

α k0 | M k0 ≤ 2C2 0 - R n-1 ∩supp λ T 0 |U (t, x)| 2 a (x)dtdxndx + 1 Ω T 0 U (t, x)-| M U (t, x)-ψ(t)λ(x )dtdxndx .
Since M is coercive on ker(A + M ) ∩ ImA (we denote by γ the constant of coercivity of M ) , equation ( 9) implies

α ≥ γ|k0| 2 2|k0| 2 C1 0 - R n-1 ∩supp λ T 0 a (x)dtdxn
.

We see that α and are related by the following inequality

α ≥ C ,
where C = C(ψ, M, Ai, λ). Finally,

k0 | M k0 ≤ 2 C1 C 0 - R n-1 ∩supp λ T 0 |U (t, x)| 2 a (x)dtdxndx + 1 C Ω T 0 U (t, x)-| M U (t, x)-ψ(t)λ(x )dtdxndx .
Since U ∈ L 2 t,x , the first term of the right hand side of the inequality goes to 0 as goes to 0. The dominated convergence and the fact that 2 show us that the second term also goes to 0 as goes to 0. Observe that M is coercive on ker(A + M ) ∩ ImA, so by the definition of k0 we get the claim.

U (t, x)-| M U (t, x)-≤ U (t, x) | M U (t, x) ≤ C|U (t, x)|
Remark 8 In the proof of the previous lemma, we crucially use the fact that we were in the case of constant coefficients to get some information on the boundary conditions when we say that k 0 ∈ ker(A ν + M ) ∩ ImA ν . For the initial conditions, we always have some information since the boundary at the time t = 0 is flat and the boundary matrix for t = 0 is the identity matrix. On the contrary, recovering some information in the case where the matrices A ν and M depending on x and t seems more tricky and is left for further studies.

Using the previous lemmas, we will show that a regular solution in the sense of definition 1 is a solution of the problem 1:

Theorem 1

Assume that Ω is a half-plane [START_REF] Falk | Explicit Finite Element Methods for Symmetric Hyperbolic Equations[END_REF]. Assume that U is a dissipative solution and U is a H 1 t,x . Then U is a solution of 1 in the classical sense, i.e.

∂

t U + n i=1 A i ∂ xi U = f in L 2 t,x , 2. U (0, x) = U 0 (x) in L 2 x , 3. (A ν -M )U (t, x) = 0 in L 2 ((0, T ) × ∂Ω).
The proof of this theorem follows from the discussion below. First, observe that lemmas 2 and (3) imply respectively the first and the second point of this lemma. Secondly, we need two lemmas to prove that the third point of the previous lemma is true.

Lemma 5

Assume Ω is a half-space [START_REF] Falk | Explicit Finite Element Methods for Symmetric Hyperbolic Equations[END_REF]. Let U ∈ W 1,p ((0, T )×Ω) (1 < p < ∞), λ ∈ D((0, T )) and β ∈ D(R n-1 ). Then the limit when → 0 of

1 0 - T 0 R n-1 U (t, x , x n )λ(t)β(x )dx dtdx, is T 0 R n-1 U (t, x , 0)λ(t)β(x )dx dt.
Proof : First, observe that the modulus of the quantity

1 0 - T 0 R n-1 U (t, x , xn)λ(t)β(x )dx dtdxn - T 0 R n-1 U (t, x , 0)λ(t)β(x )dx dt, is bounded above by 1 0 - T 0 R n-1 U (t, x , xn) -U (t, x , 0) λ(t)β(x )dx dtdxn ≤ 0 - T 0 R n-1 U (t, x , xn) -U (t, x , 0) xn λ(t)β(x )dx dtdxn ≤ λ ∞ β ∞ T 0 U (t, x , xn) -U (t, y, 0) xn L 2 ((-,0)×R n-1 )
dx dtdxn, But using Hardy's inequality (proved in [START_REF] Tartar | An introduction to Sobolev spaces and interpolation spaces[END_REF]), we know that for all t ∈ (0, T )

U (t) -U (t, xn = 0) xn L p (R n -) ≤ p p -1 ∂x n U (t) L p (R n -) ,
and consequently (t, x , xn) → U (t, x , xn) -U (t, x , 0) xn belongs to L 2 t,x . The lemma follows from the dominated convergence theorem.

To apply the previous result to U + , we need the following lemma

Lemma 6

Assume Ω is a half-space [START_REF] Falk | Explicit Finite Element Methods for Symmetric Hyperbolic Equations[END_REF]. Suppose that A ν U ∈ H 1 ((0, T ) × Ω). Then U + ∈ H 1 ((0, T ) × Ω).

Proof : Since ker Aν ⊂ ker M , there exists a matrix B such that M = BAν and consequently M U also belongs to H 1 ((0, T ) × Ω). But

(Aν -M )U = (Aν -M )(U 0 + U-+ U+) = (Aν -M )U+ = -2M U+.
We know that M is non-singular on ker(Aν + M ) ∩ ImAν , so U+ belongs to H 1 .

Since U ∈ H 1 t,x , A ν U also belongs to H 1 t,x and thanks to the previous lemma, U + ∈ H 1 t,x . Now using lemmas 5 and 4, we deduce that

T 0 R n-1 U (t, x , 0) + λ(t)β(x )dx dt = 0,
for every λ and β test-functions. Consequently, U + = 0, a.e on (0, T ) × {x n = 0}, the boundary condition is satisfied in the strong sense. Similar arguments apply to the case of the initial condition thanks to lemma 3.

Existence and uniqueness result in the half-plane

This section is devoted to prove the following theorem Theorem 2

Suppose that Ω = R n -, the matrices A i and M are supposed to be constant and A n is non singular. For every f ∈ L 2 t,x and U 0 ∈ L 2

x , there exists a dissipative solution of (1). If U 0 ∈ H 1 x , the dissipative solution is unique.

Remark 9

In the previous theorem, we need the H 1 regularity for the initial condition to satisfy the hypothesis of the theorem 6 but also to know that the function g, defined by [START_REF] Otto | Initial-boundary value problem for a scalar conservation law[END_REF], is L 2 . But, if one can obtain a comparison principle for L 2 data, as in the case developed in [START_REF] Després | Weak solutions to friedrichs systems with convex constraints[END_REF] for the whole space, the uniqueness would be easy to obtain. For now, the H 1 regularity allows us to avoid technical problem about traces on the boundary.

The existence is proven thanks to the classical theory of Friedrichs' systems for regular data and the uniqueness result makes use of mollification in the tangential directions (i.e in the time direction and in the x -direction). In order to use the regularity theory (developed in [START_REF] Massey | Differentiability of solutions to hyperbolic initial-boundary value problems[END_REF]), we assume that

A ν = A n is non singular. ( 10
)
In order to prove the existence result, one can use, for example, the theory developed in [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF] and [START_REF] Massey | Differentiability of solutions to hyperbolic initial-boundary value problems[END_REF] for Friedrichs' systems. To do so, we need a matrix B ∈ M p×n (R) where p is the number of negative eigenvalues of A ν and examine the following initial boundary value problem

         ∂ t U + n i=1 A i ∂ xi U = f, on (0, T ) × Ω, U (0, x) = U 0 (x), on Ω, BU (t, x) = 0, on (0, T ) × ∂Ω, (11)
where f ∈ H 1 ([0, T ] × Ω) and U 0 ∈ H 1 (Ω). We suppose that U 0 obeys to the following compatibility

condition ∀x ∈ ∂Ω, BU 0 (x) = 0. ( 12
)
To connect our boundary condition and the boundary condition using a matrix B, we need that ker B = ker(A ν -M ). For example, since ker(A ν -M ) ⊕ ker(A ν + M ) = R n , one can take the matrix B that gives the coordinates in a basis of ker(A ν + M ) of the projection into the space ker(A ν + M ). Since here we are dealing with constant matrices (A ν and M ), B is also constant.

Under all these assumptions, we can say that there exists a unique solution to the problem [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] and this solution belongs to the space C 1 ([0, T ], L 2

x ) ∩ C 0 ([0, T ], H 1 x ). Consequently, the initial condition is satisfied in a strong sense and the partial differential equation in [START_REF] Kružkov | First order quasilinear equations with several independent variables[END_REF] has to be understood as an equality in C 0 ([0, T ], L 2

x ). The boundary condition is satisfied in C 0 ([0, T ], L 2 (∂Ω) m ). This result can be found in [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF] (theorem 9.16). The normality condition and the uniform Kreiss-Lopatinski ȋ condition (cf [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF]) are automatically fulfilled since B is a strictly dissipative and we are dealing with a symmetric system (cf proposition 4.4 in [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF]).

The following theorem tells us that the regular solution of the problem ( 11) is in fact a regular dissipative solution of (1).

Theorem 3

Assuming that f ∈ H 1 ([0, T ] × Ω), U 0 ∈ H 1 (Ω) and condition [START_REF] Leoni | A First Course in Sobolev Spaces[END_REF], there exists a solution to problem (1) in the sense of definition 1.

Proof : As we said before, we have

∂tU + n i=1 Ai∂x i U = f, in C 0 ((0, T ), L 2
x ), ( 13)

BU (t, x) = 0, in C 0 ((0, T ), L 2 x ). ( 14
)
If k is a vector of R n independent of t and x, then we also have the following equality

∂t(U -k) + n i=1 Ai∂x i (U -k) = f, in C 0 ((0, T ), L 2 x ). ( 15
)
Consequently, if

ϕ ∈ W 1,∞ c,+ ((0, T ) × Ω), T 0 Ω ∂t(U -k) | U -k (t, x)ϕ(t, x)dxdt + T 0 Ω n i=1 U -k | Ai∂x i (U -k) (t, x)ϕ(t, x)dxdt = T 0 Ω f | U -k (t, x)ϕ(t, x)dxdt.
Since for almost every x ∈ Ω, the function t → ∂t(U -k) | U -k (t, x) belongs to C 1 (0, T ), the Green formula and Fubini's theorem give us

T 0 Ω ∂t(U -k) | U -k (t, x)ϕ(t, x)dxdt = 1 2 Ω |U (T, x) -k| 2 ϕ(T, x)dx - Ω |U0(x) -k| 2 ϕ(0, x)dx - T 0 Ω |U -k)| 2 (t, x)∂tϕ(t, x)dxdt .
Since the function x → U (t, x) belongs to H 1 (Ω) for every t ∈ [0, T ], it implies that the function x → U | AiU belongs to W 1,1 (Ω, R) (with ∂x i 1/2 U | AiU = U | Ai∂x i U and consequently admits a trace on ∂Ω (in L 1 (∂Ω)). The Green formula (applied to a W 1,1 (Ω, R) function and a W 1,∞ (Ω, R+) function) tells us that

Ω U -k | Ai∂x i (U -k) (t, x)ϕ(t, x)dx = - 1 2 Ω U -k | Ai(U -k) (t, x)∂x i ϕ(t, x)dx + 1 2 ∂Ω νiT1,1 U -k | Ai(U -k) (t, x)ϕ(t, x)dσ(x).
Here T1,1 (resp. T1,2) is the classical trace operator from W 1,1 (Ω) to L 1 (∂Ω) (resp. from W 1,2 (Ω) to L 2 (∂Ω)). By definition of the trace operator and by the density of C 1 (Ω) in W 1,1 (Ω) (resp. W 1,2 (Ω)), we know that for almost every

x ∈ ∂Ω T1,1 U -k | Ai(U -k) (t, x) = T1,2U -k | Ai(T1,2U -k) .
Consequently,

n i=1 T 0 Ω U -k | Ai∂x i (U -k) (t, x)ϕ(t, x)dxdt = - 1 2 n i=1 T 0 Ω U -k | Ai(U -k) (t, x)∂x i ϕ(t, x)dxdt + 1 2 T 0 ∂Ω T1,2U -k | Aν (T1,2U -k) (t, x)ϕ(t, x)dσ(x)dt.
Since we know that ker B = ker(Aν -M ), we can conclude that T1,2U (x, t) ∈ ker(Aν -M ) almost everywhere on (0, T ) × ∂Ω. Lemma 1 tells us that

T 0 ∂Ω T1,2U -k | Aν (T1,2U -k) (t, x)ϕ(t, x)dσ(x)dt = T 0 ∂Ω T1,2U -k-| M (T1,2U -k-) (t, x)dσ(x)dt - T 0 ∂Ω k+ | M k+ (t, x)dσ(x)dt.
Finally we deduce from the facts that

T 0 ∂Ω T1,2U -k-| M (T1,2U -k-) (t, x)dσ(x)dt ≥ 0,
and

Ω |U (T, x) -k| 2 ϕ(T, x)dx ≥ 0,
that we have a dissipative solution to the problem (1) i.e. ∀k ∈ R n , ∀ϕ ∈ W 1,∞ c,+ ((0, T ) × Ω) we have the following inequality

T 0 Ω |U (t, x) -k| 2 ∂tϕ(t, x) + n i=1 U -k | Ai(U -k)) (t, x)∂x i ϕ(t, x)dxdt + T 0 Ω 2 f | U -k (t, x)ϕ(t, x)dxdt + Ω |U0(x) -k| 2 ϕ(0, x)dx + T 0 ∂Ω k+ | M k+ (t, x)dσ(x)dt ≥ 0.
Using this result, we can compare regular solutions.

Theorem 4

Let U and Ũ be two solutions (in the sens of definition 1) associated with (f, U 0 ) (resp. ( f , Ũ0 )). We suppose that U and Ũ are H 1 t,x . Then there exists a constant C > 0 (depending only on T ) such that

U -Ũ 2 L 2 t,x ≤ C f -f 2 L 2 t,x + U 0 -Ũ0 2 L 2
x .

Proof : Using the results of the section 2, we know that U and Ũ are solutions of the problem (1) in the classical sense. Consequently, one has

T 0 Ω ∂t(U -Ũ ) U -Ũ (t, x)ϕ(t, x)dxdt + T 0 Ω n i=1 Ai∂x i (U -Ũ ) U -Ũ (t, x)ϕ(t, x)dxdt = T 0 Ω f -f U -Ũ (t, x)ϕ(t, x)dxdt,
Taking the function ϕ : (t, x) → (T -t)/T , we get

1 2T T 0 Ω U -Ũ 2 (t, x)dxdt + 1 2 T 0 U -Ũ Aν (U -Ũ ) (t, 0)ϕ(t, 0)dt = T 0 Ω f -f U -Ũ (t, x)ϕ(t, x)dxdt + 1 2 Ω U -Ũ U -Ũ (0, x)dx,
Using the fact that Aν (U -Ũ ) = M (U -Ũ ) on the boundary xn = 0, we obtain thanks to the fact that M is non-negative 2

Ω T f -f U -Ũ (t, x)ϕ(t, x)dxdt + Ω |U0 -Ũ0| 2 (x)dx ≥ 1 T Ω T |U -Ũ | 2 (t, x)dxdt.
The Cauchy-Schwarz inequality leads us to

1 T U -Ũ 2 L 2 t,x ≤ 2 f -f L 2 t,x U -Ũ L 2 t,x + U0 -Ũ0 2 L 2 x . Finally, 1 2T U -Ũ 2 L 2 t,x ≤ 2T f -f 2 L 2 t,x + U0 -Ũ0 2 L 2 x .
The assertion of the lemma follows easily.

This contraction principle leads to the existence theorem.

Theorem 5

If f ∈ L 2 t,x and U 0 ∈ L 2 x , then there exists a solution of system 1 in the sense of definition 1.

Proof : We regularize the problem by taking fn ∈ D((0, T )×Ω) and U0,n ∈ D(Ω) (remark that automatically we have BU0,n = 0 on ∂Ω) such that fn (resp. U0,n) tends to f in L 2 t,x (resp. to U0 in L 2

x ), we denote by Un the solution given by theorem 3. Then the previous lemma ensures that (Un) n∈N is a Cauchy sequence in L 2 t,x hence it converges to a function U ∈ L 2 t,x . Passing to the limit when n goes to infinity in the inequality (4) leads to the existence of a solution. Now, we are going to show a uniqueness result in the case where U 0 ∈ H 1

x and f ∈ L 2 t,x . Here, we take a solution U ∈ L 2 t,x a solution of (1) associated with the function f ∈ L 2 t,x and U 0 ∈ H 1 x . We are going to regularize U ∈ L 2 t,x in the tangential variables (i.e. the variable (t, x )), so that we will be able to use the tools of theorem 6 which needs more regular solutions. Now we use convolution in time (with support in [-, -/2] and 0 < < 1) and classical convolution in the x = (x 1 , • • • , x n-1 ) variables. We denote by ρ ,l the time convolution kernel, ρ ,x the x -convolution kernel. First we extend U for negative times and we denote by V the function U 1 (0,T )×Ω + U 0 1 (-∞,0)×Ω , then we define the function V as

V (t, x , x n ) = V t,x ρ (t, x) = V t ρ ,l x ρ (t, x) = V t ρ ,l x ρ (t, x) = R n V (s, y, x n )ρ ,l (s -t)ρ (y -x )dsdy,
where f is the function defined by f (x) = f (-x).

Lemma 7

The function V verifies the following properties 1. V goes to U in L 2 t,x when → 0.

2. V verifies (in the sense of distribution)

∂ t V + n i=1 A i ∂ xi V = f .
3. for all , V ∈ H 1 ((0, T ) × Ω).

4. for all , V (t = 0) = U 0, = U 0 x ρ (straightforward with the support of ρ ,l ).

5. V + = 0 on (0, T ) × ∂Ω.

Proof : Let ϕ be in W 1,∞ c,+ ((-∞, T ) × Ω).
The key observation is the following one : if U is a dissipative solution of (1) with data f and U0 then the function V verifies the following inequality

T -∞ Ω |V | 2 -2 V | k (t, x)∂tϕ(t, x)dxdt + T -∞ Ω n i=1 ( V | AiV -2 V | Aik ) (t, x)∂x i ϕ(t, x)dxdt + T -∞ Ω 2 g | V -k (t, x)ϕ(t, x)dxdt + 0 -∞ ∂Ω U0 | An(U0 -k) (t, x , 0)ϕ(t, x , 0)dx dt + T 0 ∂Ω k-| M k-(t, x , 0)ϕ(t, x , 0)dx dt ≥ 0, (16)

Lemma 8

Assume that A n is non singular. Let U be in L 2 t,x such that ∂ t U , ∂ 1 U ,...,∂ n-1 U belong to L 2 t,x and we suppose that the following inequality holds for every k ∈ R m and ϕ ∈ W 1,∞ c,+ ((0, T ) × Ω),

T 0 Ω |U (t, x) -k| 2 ∂ t ϕ(t, x)dxdt + T 0 Ω n i=1 U (t, x) -k | A i (U (t, x) -k) ∂ xi ϕ(t, x)dxdt +2 T 0 Ω f | U -k ϕ(t, x)dxdt + Ω |U 0 (x) -k| 2 ϕ(0, x)dx + T 0 ∂Ω k + | M (ν(x))k + ϕ(t, x)dxdt ≥ T 0 Ω g t (U )∂ t ϕ(t, x) + n i=1 g i (U )∂ xi ϕ(t, x)dxdt + Ω g 0 (U )ϕ(0, x)dx.
where g t (U ), g i (U ) and g 0 (U ) belong to L 2 t,x and are independent of k. Then U ∈ H 1 ((0, T ) × Ω).

Proof : Following the proof of the fact that a solution in the sense of definition 1 is a weak solution, we get that U is a weak solution to the PDE

∂tU + n i=1 Ai∂x i U = f.
Consequently, since Aν is non-singular by hypothesis, we get that

∂x n U = A -1 n f -∂tU + n-1 i=1 Ai∂x i U ∈ L 2 t,x .
This implies that U ∈ H 1 ((0, T ) × Ω).

Remark 10

The hypothesis that A ν is invertible is important. In fact in the singular case, there is a loss of regularity for the solution (see [START_REF] Rauch | Symmetric positive systems with boundary characteristic constant multiplicity[END_REF]).

Theorem 6

Let U and Ũ be two L 2 t,x solutions (in the sense of definition 1) associated with (f, U 0 ) ∈ L 2 t,x × H 1

x and ( f , Ũ0 ) ∈ L 2 t,x × H 1 x respectively. Then there exists a constant C > 0 (depending only on T ) such that

U -Ũ 2 L 2 t,x ≤ C f -f 2 L 2 t,x + U 0 -Ũ0 2 L 2 x .
Remark 11 From the previous theorem, we deduce that there exists a unique solution in the sense of definition 1 of the problem (1) for f ∈ L 2 t,x and U 0 ∈ H 1 x .

Proof : From U and Ũ , we construct their approximations V and Ṽ , lemma 7 allows us to mimic the proof of the theorem 6 to get

V -Ṽ 2 L 2 t,x ≤ C g -g 2 L 2 t,x + U0, -Ũ0 2 L 2 x .
Passing to the limit, since V and g are in L 2 loc ((-∞, T ) × Ω) m then V and g tends to V and g in L 2 ((0, T ) × Ω) m (see for example [START_REF] Leoni | A First Course in Sobolev Spaces[END_REF]), we get the result.

Comparison with other formulations

In the literature, there are several ways for taking into account the boundary condition for linear hyperbolic system, for example the initial approach of Friedrichs [START_REF] Friedrichs | Symmetric Positive Linear Differential Equations[END_REF], the formulation developed in [START_REF] Dubois | Boundary Conditions for Nonlinear Hyperbolic Systems of Conservation Laws[END_REF] by Dubois and Le Floch and the maximal and strictly dissipative theories (summarized in the book of Benzoni-Gavage and Serre [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF]). We first describe these different approaches which are slightly different from the algebraic point of view, and then compare these formulations on a simple example: the wave equation in 1D. We will see on this example that, except for the maximal dissipative theory, the theory must be modified to take into account the classical Dirichlet boundary condition. This modification can be performed at least formally with the use of a small parameter such that the homogeneous Dirichlet condition is recovered at the limit of a family of dissipative solutions in the sense of definition 1.

Friedrichs' approach

In the seminal work of Friedrichs [START_REF] Friedrichs | Symmetric Positive Linear Differential Equations[END_REF], the boundary condition takes the following form : first he assumes a decomposition of A ν = β -+ β + , M is now the matrix β + -β -and the boundary condition is (A ν -M )U = 0. Friedrichs also assumes that three conditions (denoted by III 0 , III 1 and III 2 in [START_REF] Friedrichs | Symmetric Positive Linear Differential Equations[END_REF]) are fulfilled by the matrices β -and β

+ 1. ker(β -) ⊕ ker(β + ) = R m . 2. Im(β -) ∩ Im(β + ) = {0}. 3. t M + M ≥ 0.
From lemma 1 and the hypothesis that we made on the matrix M , we see that all Friedrichs' hypothesis are satisfied in the non-characteristic case (we have already said that the first hypothesis was too restrictive in the characteristic case for example take all the matrices A i equal to zero). As far as assumption 3 is concerned, we have made a restrictive hypothesis : we suppose that M is a symmetric matrix.

Dubois-Le Floch's formulation

Now we describe Dubois-Le Floch's formulation. In order to define admissible boundary condition for nonlinear hyperbolic systems, Dubois and Le Floch (in [START_REF] Dubois | Boundary Conditions for Nonlinear Hyperbolic Systems of Conservation Laws[END_REF]) also define admissible boundary condition for linear strictly hyperbolic systems. To take account of inhomogeneous boundary condition, they say that the space of admissible condition (for the problem in (0,

T ) × R n -) is U b + Vect {r k , k = p + 1, • • • , m} , (18)
where r k is an eigenvector associated with the eigenvalue λ k (for k ≥ p + 1, λ k ≥ 0 and for k < p + 1, λ k < 0) of the boundary matrix A ν . Using our formalism, we can rewrite this space of admissible boundary condition as

U -U b ∈ ker(A ν -M ),
where M is the matrix constructed in the remark 1 with the matrix P associated with the eigenvectors chosen in (18) (recall that these eigenvectors are orthogonal and we may assume without loss of generality that they are orthonormal and consequently that the matrix P is orthogonal). Therefore the type of boundary condition discussed by Dubois and Le Floch is contained in our formalism of boundary condition (in fact strictly contained, see the example of the wave equation).

Maximal dissipative and strictly dissipative theories

We have already used the fact that our formalism can be transformed to use strictly dissipative theory. In these two theories, the boundary condition is taking into account thanks to a matrix B ∈ M p×n (R) where p is the number of negative eigenvalues of A ν . We say that the matrix B is maximal dissipative (in the non-characteristic case) if

U ∈ ker B ⇒ A ν U | U ≥ 0, ( 19
)
and B is maximal for that property, i.e. ker B is not a proper subspace of vector space V such that

U ∈ V ⇒ A ν U | U ≥ 0,
to get L 2 -well-posedness (theorem 3.2 of [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF]). This boundary condition (or equivalently the boundary condition used in the work of Rauch, for example in [START_REF] Rauch | Symmetric positive systems with boundary characteristic constant multiplicity[END_REF]) contains more possibilities than the one we use in this paper (in fact, we'll see in the next example that the formalism of [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF] allows more boundary conditions than the Friedrichs' approach).

There is also a more stronger condition (see for example [START_REF] Benzoni-Gavage | Multi-dimensional Hyperbolic Partial Differential Equations[END_REF]): the notion of strictly dissipative boundary condition. A matrix B is strictly dissipative (in the non-characteristic case) if

1. U ∈ ker B ⇒ A ν U | U > 0, if U = 0.
2. ker B is maximal for the previous property.

3. B is onto.

A simple example: the wave equation in 1D

We consider here the wave equation problem

∂ tt w -∂ xx w = 0, on (0, T ) × R -,
that we rewrite as an hyperbolic problem in dimension m = 2 as

∂ t u p -A∂ x u p = 0, (20)
where u = ∂ t w, p = ∂ x w and A = 0 1 1 0 . It is classical (see for example [START_REF] Brézis | Functional Analysis, Sobolev Spaces and Partial Differential Equations[END_REF] section 10.3) that the wave equation is well-posed with the natural following conditions w(x, 0) = f (x), on {t = 0} × R -, ∂ t w(x, 0) = g(x), on {t = 0} × R -, w(0, t) = 0, on (0, T ) × {x = 0}. Now that the unknowns in ( 20) are u = ∂ t w and p = ∂ x w, it seems natural to know if one can obtain the boundary condition u(0, t) = 0, on (0, T ) × {x = 0}

We are going to show that this kind of boundary condition is not contained in our formalism, Friedrichs' formalism or Dubois-Le Floch's formalism but it can be obtain with the use of a maximal dissipative boundary condition. First, we are going to show that one can not find two matrices β -and β + such that ker β -= 0 p , p ∈ R and the conditions 1,2,3 are satisfied. Let β -be the matrix a b c d .

In fact, the condition ker β -= 0 p , p ∈ R implies that b = d = 0. The condition 3 rewrites as

t M + M = β + -β -+ t β + -t β -= -2a -2c -2 -2c -2 0 ≥ 0,
and consequently, it imposes that c = 1, but then β + = -a -1 -2 0 is invertible and consequently, the condition 1 can not be fulfilled.

It means that such the boundary condition (21) can not be directly use in the Friedrichs' framework. Since our framework and Dubois-Le Floch's formalism are contained in Friedrichs' framework, one can not expect to use this boundary condition in these theories.

In this particular example, one can see that our framework contains more possibility for the boundary condition than the theory of characteristics of Dubois-Le Floch. Indeed, if we take the example of homogeneous boundary condition, the only possibility from Dubois-Le Floch is that u + p = 0 on the boundary x = 0.

Let's determine all the possible matrices M for the wave equation in 1D that fulfill the conditions of section 2. Necessarily, we have dim ker(A ± M ) = 1 otherwise, either A -M = 0 or A + M = 0, these two equalities can not happen since A is neither non-positive nor non-negative. It allows an infinite number of boundary condition for the wave equation in one dimension, parameterized by c. Finally, we examine the maximal and strictly dissipative boundary conditions in the case of the wave equation. First to use a maximal dissipative boundary condition, one only need to find a matrix B ∈ M 1×2 (R) it simply means that dim ker B = 1. In fact, any matrix B of the form a 0 is maximal dissipative and give the boundary condition (21). One can easily see that the boundary condition u = 0 can not be strictly dissipative and that in fact the boundary condition (in the general case) that we are considering is such that the space ker(A ν -M ) ∩ ImA ν is strictly dissipative in the sense that the first two previous condition are verified.

Nevertheless, one way to get a formulation that is valid for a L 2 t,x solution in this particular example would be to send c to +∞ in the dissipative formulation to get a non-dissipative weak formulation. This is very similar to penalization procedure.

One could also modify the definition 1 by restricting the test vectors k to the ones satisfying k + = 0 if the support of ϕ touches ∂Ω. This formulation seems nevertheless less general than the one studied in this work.

0

  and k 0 ∈ ker A, then k-| M k+ = 0 and consequently 0 ≤ k+ | M k+ ≤ k | M k . Therefore we know that the right-hand side of the previous inequality belongs to L 1 (0, T ), so the dominated convergence theorem implies that lim α→0 T Rα, (t)dt = T 0 R0, (t)dt.

,

  In particular, det(A ± M ) = 0 = ac -(-1 ± b) 2 , subtracting this two equalities tells us that b = 0. If b = 0, to insure that det(A ± M ) = 0, we get that a = 1 c). In our formalism, all the matrices M possible areM c = with c > 0.The matrix M c is associated with the following boundary condition u + p c = 0, on (0, T ) × {x = 0}.

Acknowledgments

The authors wish to express their thanks to Jean-François Babadjian for many stimulating conversations.

where

Why ( 16) is true? We know that U is a solution and consequently, the following inequality is true by definition:

But since U0 is regular, independent of t, we have

Adding this two inequalities and using the definition of V and g gives us [START_REF] Métivier | Hyperbolic boundary value problems for symmetric systems with variable multiplicities[END_REF]. Then the tangential regularization gives us a function V satisfying the following inequality

The following lemma and its proof ensure that points 2 and 3 are true. As in the proof of lemma 4, we obtain that V + = 0 on (0, T ) × ∂Ω.

The following lemma explains why we only need to regularize in the (t, x )-directions to get the full H 1 -regularity.