Guillaume Aupy

Anne Benoit

Thomas Hérault

Yves Robert

Jack Dongarra

Optimal Checkpointing Period: Time vs. Energy

This short paper deals with parallel scientific applications using non-blocking and periodic coordinated checkpointing to enforce resilience. We provide a model and detailed formulas for total execution time and consumed energy. We characterize the optimal period for both objectives, and we assess the range of time/energy trade-offs to be made by instantiating the model with a set of realistic scenarios for Exascale systems. We give a particular emphasis to I/O transfers, because the relative cost of communication is expected to dramatically increase, both in terms of latency and consumed energy, for future Exascale platforms.

I. INTRODUCTION

A significant research effort is focusing on the characteristics, features, and challenges of High Performance Computing (HPC) systems capable of reaching the Exaflop performance mark [START_REF] Dongarra | The international exascale software project: a call to cooperative action by the global high-performance community[END_REF], [START_REF] Sarkar | Exascale software study: Software challenges in extreme scale systems[END_REF]. The portrayed Exascale systems will necessitate billion way parallelism, resulting not only in a massive increase in the number of processing units (cores), but also in terms of computing nodes. Considering the relative slopes describing the evolution of the reliability of individual components on one side, and the evolution of the number of components on the other side, the reliability of the entire platform is expected to decrease, due to probabilistic amplification. Even if each independent component is quite reliable, the Mean Time Between Failures (MTBF) is expected to drop drastically. Executions of large parallel applications on these systems will have to tolerate a higher degree of errors and failures than in current systems. The de-facto general-purpose error recovery technique in high performance computing is checkpoint and rollback recovery. Such protocols employ checkpoints to periodically save the state of a parallel application, so that when an error strikes some process, the application can be restored into one of its former states. The most widely used protocol is coordinated checkpointing, where all processes periodically stop computing and synchronize to write critical application data onto stable storage. Coordinated checkpointing is well understood, at least in its blocking form (when no computing activity takes place during checkpoints), and good approximations of the optimal checkpoint interval exist; they are known as Young's and Daly's formula [START_REF] Young | A first order approximation to the optimum checkpoint interval[END_REF], [START_REF] Daly | A higher order estimate of the optimum checkpoint interval for restart dumps[END_REF].

While reliability is a major concern for Exascale, another key challenge is to minimize energy consumption, both for This work was supported in part by the ANR RESCUE project. A. Benoit and Y. Robert are with the Institut Universitaire de France. economic and environmental reasons. One of the most powerconsuming components of today's systems is the processor: even when idle, it dissipates a significant fraction of the total power. However, for future Exascale systems, the power dissipated to execute I/O transfers is likely to play an even more important role, because the relative cost of communication is expected to dramatically increase, both in terms of latency and consumed energy [START_REF] Shalf | Exascale computing technology challenges[END_REF].

In this short paper, we investigate trade-offs between execution time and energy consumption for the execution of parallel applications on future Exascale systems. The optimal period T opt Time given by Young's and Daly's formula [START_REF] Young | A first order approximation to the optimum checkpoint interval[END_REF], [START_REF] Daly | A higher order estimate of the optimum checkpoint interval for restart dumps[END_REF] will minimize (expected) execution time. However, this period T opt Time will not minimize energy consumption, mainly because the fraction of power P Cal spent when computing (by the CPUs) is not the same as the fraction of power P I/O spent when checkpointing. In particular, we revisit the work of Meneses, Sarood and Kalé [START_REF] Meneses | Assessing Energy Efficiency of Fault Tolerance Protocols for HPC Systems[END_REF] for checkpoint/restart, where formulas are given to compute the time-optimum and energy-optimum periods. However, our model is more precise: (i) we carefully assess the impact of the power consumption required for I/O activity, which is likely to play a key role at the Exascale; (ii) we consider non-blocking checkpointing that can be partially overlapped with computations; (iii) we give a more accurate analysis of the consumed energy.

Altogether, this short paper provides the following main contributions:

• We provide a refined analytical model to compute both the execution time and the consumed energy with a given checkpoint period. The model handles the case where checkpointing activity can be non-blocking, i.e., partially overlapped with computations. • We provide analytical formulas to approximate the optimal period for time T opt Time as well as the optimal period for energy T opt Energy , thereby refining and extending Daly [START_REF] Daly | A higher order estimate of the optimum checkpoint interval for restart dumps[END_REF] and Meneses, Sarood and Kalé [START_REF] Meneses | Assessing Energy Efficiency of Fault Tolerance Protocols for HPC Systems[END_REF] results to non-blocking checkpoints.

• We assess the range of time/energy trade-offs to be made by instantiating the model with a set of realistic scenarios for Exascale systems.

II. MODEL

In this section, we introduce all the model parameters. We start with parameters related to resilience (checkpointing) before moving to parameters related to energy consumption.

A. Checkpointing

We model coordinated checkpointing [START_REF] Chandy | Distributed snapshots: Determining global states of distributed systems[END_REF] where checkpoints are taken at regular intervals, after some fixed amount of work units have been performed. This corresponds to an execution partitioned into periods of duration T . Every period, a checkpoint of length C is taken.

An important question is whether checkpoints are blocking or not. On some architectures, we may have to stop executing the application before writing to the stable storage where the checkpoint data is saved; in that case checkpoint is fully blocking. On other architectures, checkpoint data can be saved on the fly into a local memory before the checkpoint is sent to the stable storage, while computation can resume progress; in that case, checkpoints can be fully overlapped with computations. To deal with all situations, we introduce a slowdown factor ω: during a checkpoint of duration C, the work that is performed is ωC work units. In other words, (1 -ω)C work units are wasted due to checkpoint jitter disrupting the progress of computation. Here, 0 ≤ ω ≤ 1 is an arbitrary parameter. The case ω = 0 corresponds to a fully blocking checkpoint, while ω = 1 corresponds to a checkpoint totally overlapped with computations. All intermediate situations can be represented.

Next we have to account for failures. During t time units of execution, the expectation of the number of failures is t µ , where µ is the MTBF (Mean Time Between Failures) of the platform. Note that if the platform if made of N identical resources whose individual mean time between failures is µ ind , then µ = µind N . This relation is agnostic of the granularity of the resources, which can be anything from a single CPU to a complex multi-core socket. When a failure strikes, there is a downtime of length D (time to reboot the resource or set up a spare), and then a recovery of length R (time to read the last stored checkpoint). The work executed by the application since the last checkpoint and before the failure needs to be reexecuted. Clearly, the shorter the period T , the less work to re-execute, but also the more overhead due to frequent checkpoints in a failure-free execution. The best trade-off when ω = 0 (blocking checkpoint) is achieved for T = √ 2Cµ + C (Young's formula [START_REF] Young | A first order approximation to the optimum checkpoint interval[END_REF]) or T = 2C(µ + D + R)+C (Daly's formula [START_REF] Daly | A higher order estimate of the optimum checkpoint interval for restart dumps[END_REF]). Both formulas are first-order approximations and valid only if all checkpoint parameters C, D and R are small in front of µ (and these formulas collapse if they become negligible). In Section III, we show how to extend these formulas to the case of non-blocking checkpoints (see also [START_REF] Bosilca | Unified model for assessing checkpointing protocols at extreme-scale[END_REF] for more details).

B. Energy

To compute the energy consumption of the application, we need to consider the energy consumption of the different phases, and hence the power consumption at each time-step. To this purpose, we define:

• P Static : this is the base power consumed when the platform is switched on. • P Cal : when the platform is active, we have to consider the CPU overhead in addition to the static power P Static .

• P I/O : similarly, this is the power overhead due to file I/O. This supplementary power consumption is induced by checkpointing, or when recovering from a failure. • P Down : for coordinated checkpointing, when one processor fails, the rest of the machine stays idle. P Down is the power consumption overhead when one machine is down, that may be incurred for instance by rebooting the machine. In general, we let P Down = 0. Meneses, Sarood and Kalé [START_REF] Meneses | Assessing Energy Efficiency of Fault Tolerance Protocols for HPC Systems[END_REF] have a simpler model with two parameters, namely L, the base power (corresponding to P Static with our notations), and H, the maximum power (corresponding to P Static + P Cal with our notations). They use

P I/O = P Down = 0.
In Section III, we show how to compute the optimal period that minimizes the energy consumption. In Section IV, we instantiate the model with expected values for power consumption of Exascale platforms.

III. OPTIMAL CHECKPOINTING PERIOD

We consider a parallel application whose execution time is T base without any overhead due to the resilience method or the occurrence of failures. We compute the expectation T final of the total execution time (accounting both for checkpointing and for failures) in Section III-A, and the expectation E final of the total energy consumed during this execution of length T final in Section III-B. We will compute the optimal period T that minimizes the objective, either T final or E final .

A. Execution time

The total execution time T final of the application depends on two sources of overhead. We first compute T ff , the time taken by a fault-free execution, thereby accounting only for the overhead due to periodic checkpointing. Then we compute T fails , the time lost due to failures. Finally, T final = T ff + T fails . We detail here both computations:

T ff = T base T T -(1 -ω)C
.

• The reasoning to compute T fails is the following. Since the mean time between two failures is µ, the average number of failures during execution is Tfinal µ . For each failure, the time lost is expressed as:

-D + R for downtime and recovery; a time ωC for the work that was done during the previous checkpoint and that has to be redone because it was not checkpointed (because of the failure);

with probability T -C T , the failure happens while we are not checkpointing, and the time lost is on average A = T -C 2 ; otherwise, with probability C T , the failure happens while we are checkpointing, and the time lost is on average

B = T -C + C 2 = T -C 2 .
The time lost for each failure is

D + R + ωC + T -C T A + C T B = D + R + ωC + T 2 .
Finally,

T fails = T final µ D + R + ωC + T 2 .
We are now ready to express the total execution time:

T final = T ff + T fails = T base T T -(1 -ω)C + T final µ D + R + ωC + T 2 = T (T -(1 -ω)C) 1 -D+R+ωC+T /2 µ T base = T (T -a) b -T 2µ T base , where a = (1 -ω)C and b = 1 -D+R+ωC µ .
This equation is minimized for

T opt Time = 2(1 -ω)C(µ -(D + R + ωC)). (1)
When ω = 0, we obtain an expression close to that of Young and Daly, but slightly different because they have less accurately approximated the total execution time. In the following, we let ALGOT be the checkpointing strategy that checkpoints with period T opt Time .

B. Energy consumption

In order to compute the total energy consumption of the execution, we consider the different phases during which the different powers introduced in Section II-B are used:

• First, we consume P Static during each time-step of the execution. Indeed, even when a node fails and is shutdown, we still pay for the power of all the other nodes, for the cooling system, etc. The corresponding energy cost is T final P Static .

• Next, let T Cal be the time during which the CPU is used, inducing a power overhead P Cal . T Cal includes the base work T base , and T re-exec , the work that must be re-executed after each failure (which we multiply by the number of failures T final /µ):

with probability T -C T , the failure does not happen during a checkpoint, and the work to re-execute is A = ωC + T -C 2 ; with probability C T , the failure happens during the execution of a checkpoint, and the work to re-execute is

B = ωC + T -C + ωC 2 . We derive T re-exec = T -C T A + C T B, hence T re-exec = ωC + T 2 -C 2 2T + ωC 2 2T .
Finally, we have:

T Cal = T base + T final µ ωC + T 2 -C 2 2T + ωC 2 2T .
The corresponding energy consumption is T Cal P Cal .

• Let T I/O be the time during which the I/O system is used, inducing a power overhead P I/O . This time corresponds to checkpointing and recovery from failures.

-The total number of checkpoints that are taken in a fault-free execution is equal to the number of periods, T , the failure does not happen during a checkpoint, and there is no additional I/O overhead; 3) however, with probability C T , the failure happens during a checkpoint, and the I/O time wasted is (in average) C 2 . Altogether, we obtain

T I/O = T base C T -(1 -ω)C + T final µ R + C 2 2T .
The corresponding energy consumption is T I/O P I/O .

• Finally, let T Down be the total down time, incurring a power overhead P Down . We have

T Down = T final µ D,
and the corresponding energy cost is T Down P Down . This term is only included for full generality, as we expect to have P Down = 0 in most scenarios.

The final expression for the total energy consumed is

E final = T Cal P Cal + T I/O P I/O + T Down P Down + T final P Static = T base + T final µ ωC + T 2 -C 2 2T + ωC 2 2T P Cal + T final µ R + C 2 2T + C T base T -(1 -ω)C P I/O + T final µ DP Down + T final P Static .
It is important to understand that T final = T Cal +T I/O +T Down , unless ω = 0. Indeed, CPU and I/O activities are overlapped (and both consumed) when checkpointing. To ease the derivation of the optimal period that minimizes E final , we introduce some notations and let P Cal = αP Static , P I/O = βP Static , and P Down = γP Static . Re-using parameters a = (1 -ω)C and b = 1 -D+R+ωC µ from Section III-A, we obtain:

T ′ final T base = -ab + T 2 2µ (T -a) 2 b -T 2µ 2
, and

E ′ final P Static = T ′ final µ αωC +βR+γD+ αT 2 - α(1-ω)C 2 2T + βC 2 2T +µ + T final 2µ α+ α(1-ω)C 2 T 2 -βC 2 T 2 -βCT base (T-(1-ω)C) 2 .
Then, letting

K = (T -a) 2 (b-T 2µ )
2

PStaticTbase

, we have:

KE ′ final = -ab+ T 2 2µ µ (αωC +βR+γD+µ)+ αT 2 + α(1-ω)C 2 2T + βC 2 2T + (T-a)(b-T 2µ ) 2µ α+ α(1-ω)C 2 -βC 2 T -βC b-T 2µ 2 = T 3 1 4µ -1 4µ +T 2 αωC+βR+γD 2µ 2 + b+ a 2µ 2µ -βC 4µ 2 + 1 2µ +T -ab 2µ -ab 2µ + βCb µ -2 (α(1-ω)-β)C 2 4µ 2 -βCb 2 - ab(αωC+βR+γD+µ) µ -b 2µ -a 4µ 2 (α(1 -ω)-β)C 2 + 1 T (α(1 -ω)-β) C 2µ -(α(1 -ω)-β) C 2µ = T 2 αωC+βR+γD 2µ 2 + b 2µ + a-βC 4µ 2 + 1 2µ +T (βC-a)b µ -2 (α(1-ω)-β)C 2 4µ 2 - ab(αωC+βR+γD+µ) µ -βCb 2 + b 2µ + a 4µ 2 (α(1 -ω)-β)C 2 .
Let T opt Energy be the only positive root of this quadratic polynomial in T : T opt Energy is the value that minimizes E final . In the following, we let ALGOE be the checkpointing strategy that checkpoints with period T opt Energy . As a side note, let us emphasize the differences with the approach of Meneses, Sarood and Kalé [START_REF] Meneses | Assessing Energy Efficiency of Fault Tolerance Protocols for HPC Systems[END_REF] when restricting to the case ω = 0 (because they only consider the blocking variant). For each failure, they consider that: 2T P I/O . Theses differences come from our more detailed analysis of the impact of the failure location, which can strike either during the computation phase, or during the checkpointing phase, of the whole period.

IV. EXPERIMENTS

In this section, we instantiate the previous model with scenarios taken from current projections for Exascale platforms [START_REF] Dongarra | The international exascale software project: a call to cooperative action by the global high-performance community[END_REF], [START_REF] Sarkar | Exascale software study: Software challenges in extreme scale systems[END_REF], [START_REF] Shalf | Exascale computing technology challenges[END_REF], [START_REF] Ferreira | Evaluating the Viability of Process Replication Reliability for Exascale Systems[END_REF]. We choose realistic values for all model parameters: this includes all types of power consumption (P Static , P Cal , P I/O and P Down ), all checkpoint parameters (C, R, D and ω), and the platform MTBF µ. We start with a word of caution: our choices for these parameters may be somewhat arbitrary, and do not cover the whole range of scenarios that can be investigated. However, a key feature of our model is its robustness: as long as µ is reasonably large in front of checkpoint times, the model is able to accurately predict the best period for execution time and for energy consumption.

The power consumption of an Exascale machine is capped to 20 Mega-watts. With 10 6 nodes, this represents a nominal power of 20 watts per node. Let us express all power values in watts. A reasonable scenario is to assume that half this power is used for operating the platform, hence to let P Static = 10. The overhead due to computing would represent the other half, hence P Cal = 10. As for communications and I/Os, which are expected to cost an order of magnitude more than computing [START_REF] Shalf | Exascale computing technology challenges[END_REF], we take an overhead of 100, hence P I/O = 100.

A key parameter for the experimental study is the ratio

ρ = P Static + P I/O P Static + P Cal = 1 + β 1 + α . (2)
With our values, we get ρ = 5.5. Note that if we used P Static = 5 and kept the same overheads 10 and 100 for computing and I/O respectively, we would get P Cal = 10, P I/O = 100, and ρ = 7. These two representative values of ρ (ρ = 5.5 and ρ = 7) are emphasized by vertical arrows in the plots below on Figure 1. As for P Down , the power during downtime, we use P Down = 0, meaning that during downtime we only account for the static power P Static of the processors that are idle.

The Jaguar platform, with N = 45, 208 processors, is reported to have experienced about one fault per day [START_REF] Zheng | A scalable double in-memory checkpoint and restart scheme towards exascale[END_REF], which leads to an individual (processor) MTBF µ ind equal to 45,208 365 ≈ 125 years. Therefore, we set the individual (processor) MTBF to µ ind = 125 years. Letting the total number of processors N vary from N = 219, 150 to N = 2, 191, 500 (future exascale platforms), the platform MTBF µ varies from µ = 300 min (5 hours) down to µ = 30 min. The experiments use resilience parameters that are representative of current and forthcoming large-scale platforms [START_REF] Ferreira | Evaluating the Viability of Process Replication Reliability for Exascale Systems[END_REF], [START_REF] Cappello | Preventive migration vs. preventive checkpointing for extreme scale supercomputers[END_REF]. We take C = R = 10 min, D = 1 min, and ω = 1/2.

On Figures 1 and2, we evaluate the impact of the ratio ρ (see Equation ( 2)) on the gain in energy and loss in time of ALGOE with respect to ALGOT. The general trend is that using ALGOE can lead to significant gains in energy at the price of a small increase in execution time.

We then study in Figure 3 the scalability of the approach on forthcoming platforms. We set the duration of the complete checkpoint and rollback (C and R, respectively) to 1 minute, independently of the number of processors, and we let the downtime D equal to 0.1 minutes. It is reasonable to consider that checkpoint storage time will not increase with the number of nodes in the future, but on the contrary will remain constant. Indeed, system designers are studying a couple of alternative approaches. One consists of providing each computing node with local storage capability, ensuring through hardware mechanisms that this storage will remain available during a failure of the node. Another approach consists iof using the memory of the other processors to store the checkpoint, pairing nodes as "buddies", thus allowing to take advantage of the high bandwidth capability of the high speed network to design a scalable checkpoint storage mechanism [START_REF] Zheng | FTC-Charm++: an in-memory checkpoint-based fault tolerant runtime for Charm++ and MPI[END_REF], [START_REF] Ni | Hiding checkpoint overhead in HPC applications with a semi-blocking algorithm[END_REF], [START_REF] Dongarra | Revisiting the double checkpointing algorithm[END_REF], [START_REF] Rajachandrasekar | A 1 PB/s file system to checkpoint three million MPI tasks[END_REF]. The MTBF for 10 6 nodes is set to 2 hours, and this value scales linearly with the number of components. Given these parameters, Figures 3a and3b shows (i) the execution time ratio of ALGOE over ALGOT, and (ii) the energy consumption ratio of ALGOT over ALGOE, both as a function of the number of nodes. Figures 3a and3b confirm the important gain in energy that can be achieved, namely up to 30% for a time overhead of only 12%. When the number of nodes gets very high (up to 10 8 ), then we observe that both energy and time ratios converge to 1. Indeed, when C becomes of the order of magnitude of the MTBF, then both periods T opt Time and T opt Energy become close to C to account for the higher failure rate.

V. CONCLUSION

In this short paper, we have provided a detailed analysis to compute the optimal checkpointing period, when the checkpointing activity can be partially overlapped with computations. We have considered two distinct objectives: either the goal is to minimize the total execution time, or it is to minimize the total energy consumption. Because of the different power consumption overheads due to computations and I/Os, we obtain different optimal periods.

We have instantiated the formulas with values derived from current and future Exascale platforms, and we have studied the impact of the power overhead due to I/O activity on the gains in time and energy. With current values, we can save more than 20% of energy with an MTBF of 300 min, at the price of an increase of 10% in the execution time. The maximum gains are expected for a platform with between 10 6 and 10 7 processors (up to 30% energy savings).

Our analytical model is quite flexible and can easily be instantiated to investigate scenarios that involve a variety of resilience and power consumption parameters.

TbaseT

  -(1-ω)C , and the time taken by checkpoints is therefore TbaseC T -(1-ω)C . -For each failure, there is an additional overhead: 1) the system needs to recover, which lasts R timesteps; 2) with probability T -C

•

  energy lost due to re-execution is T -2C 2 P Cal , while we have T -C T T -C 2 + C T (T -C) P Cal = T 2 -C 2 2T P Cal ; • energy lost due to I/O is CP I/O , while we have C 2

Figure 1 :

 1 Figure 1: Time and energy ratios as a function of ρ, with C = R = 10 min, D = 1 min, γ = 0, ω = 1/2, and various values for µ.

Figure 2 :

 2 Figure 2: Ratios of the different strategies with C = R = 10 min, D = 1 min, γ = 0, ω = 1/2 as a function of µ and ρ.

  Time and energy ratios, as a function of the number of nodes, when ρ = 5Time and energy ratios, as a function of the number of nodes, when ρ = 7

Figure 3 :

 3 Figure 3: Ratios of total energy and time for the two period strategies, as a function of the number of nodes, with µ = 120 min for 10 6 nodes, C = R = 1 min, D = 0.1 min, γ = 0, ω = 1/2.

• The reasoning to derive T ff is simple. We need to execute a total amount of work equal to T base . During each period of length T , there is an amount of time T -C where only computations take place, and an amount of time C of checkpointing, where only a work ωC is done. Therefore, the total number of work units executed during a period of length T isT -C + ωC = T -(1 -ω)C, and


Ying Tang

Christophe Prieur

Antoine Girard

A new H 2 -norm Lyapunov function for the stability of a singularly perturbed system of two conservation laws

In this paper a class of singularly perturbed system of conservation laws is considered. The partial differential equations are equipped with boundary conditions which may be studied to derive the exponential stability. Lyapunov stability technique is used to derive sufficient conditions for the exponential stability of this system. A Lyapunov function in H 2 -norm for a singularly perturbed system of conservation laws is constructed. It is based on the Lyapunov functions of two subsystems in L 2 -norm.

I. INTRODUCTION

The singular perturbation techniques occurred at the beginning of the 20th century. A great deal of the early motivation in this area arose from the studies of physical problems exhibiting both fast and slow dynamics, for instance DC-motor model, voltage regulator in [START_REF] Kokotovic | Singular pertrubation methods in control: analysis and design[END_REF] and semiconducting diode in [START_REF] Smith | Singular-perturbation theory: an introduction with application[END_REF]. The development of this method led to the efficient use in various fields in mathematical physics and engineering, for example, fluid mechanics, fluid dynamics, elasticity, quantum mechanics, chemical-reactor, aerodynamics etc. (see [START_REF] Kadalbajoo | Singularly perturbed problems in partial differential equations: a survey[END_REF] for a survey).

The stability properties of singularly perturbed systems have been investigated by several authors. In the papers [START_REF] Habets | Stabilité asymptotique pour des problèmes de perturbations singulières[END_REF], [START_REF] Chow | Asymptotic stability of a class of nonlinear singularly perturbed systems[END_REF], [START_REF] Grujic | Uniform asymptotic stability of nonlinear singularly perturbed and large scale systems[END_REF], [START_REF] Chow | A two-stage Lyapunov-Bellman feedback design of a class of nonlinear systems[END_REF] and [START_REF] Saberi | Quadratic-type Lyapunov Functions for Singularly Perturbed Systems[END_REF], Lyapunov method, which is commonly used for stability analysis of dynamical systems, has been employed. The main idea is to consider two subsystems: the reduced system representing the slow dynamic and the boundary-layer system standing for the fast dynamic. Assuming that each of the two subsystems is stable and has a Lyapunov function, the stability of the singularly perturbed system can be established by a Lyapunov function which is obtained by the composition of the Lyapunov function of the reduced and boundary-layer systems, for a sufficiently small perturbation parameter. As in [START_REF] Saberi | Quadratic-type Lyapunov Functions for Singularly Perturbed Systems[END_REF], a composite Lyapunov function has been investigated for the asymptotic stability of a singularly perturbed finite-dimensional nonlinear system.

In our previous work [START_REF] Tang | Lyapunov stability of a singularly perturbed system of two conservation laws[END_REF], a class of singularly perturbed system of two conservation laws with a small perturbation parameter introduced to both the dynamics and the boundary conditions has been studied. As soon as the two subsystems, the reduced and boundary-layer systems, are stable, the stability of this system can be obtained by a Lyapunov ying.tang@gipsa-lab.fr, christophe.prieur@gipsa-lab.fr.

A. GIRARD is with Laboratoire Jean Kuntzmann, Université de Grenoble, B.P. 53, 38041 Grenoble, France. antoine.girard@imag.fr. function in L 2 -norm. The present paper focuses on the stability problem of the singularly perturbed system of two conservation laws with a small perturbation parameter ε introduced only to the dynamics. The first problem under our consideration is the stability of the two subsystems. Some necessary and sufficient conditions are stated for exponential stability of these subsystems. Each of the two subsystems has a strict Lyapunov function in L 2 -norm. Furthermore, we consider the exponential stability of the whole singularly perturbed system. For a sufficiently small perturbation parameter, with some additional conditions on the boundary conditions, the stability of the whole singularly perturbed system can be established via strict Lyapunov function in H 2 -norm. The new element here is the use of Lyapunov function in H 2 -norm. This kind of Lyapunov function has been studied in several papers. In [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] a strict H 2 -norm Lyapunov function has been constructed to analyze the stability of solutions to a system of two hyperbolic conservation laws around equilibrium. The stability of one-dimensional n × n nonlinear hyperbolic systems has also been considered in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]. And in the work of [START_REF] Gügat | H 2 stabilization of the isothermal Euler equations: a Lyapunov function approach[END_REF], it is concerned with H 2 -stabilization of the Isothermal Euler equations.

The paper is organized as follows. The class of singularly perturbed system of conservation laws under consideration in this paper is given in Section 2. Section 3 states the exponential stability of the reduced and boundary-layer systems via Lyapunov functions in L 2 -norm. Section 4 analyzes the stability for the overall singularly perturbed system. A strict Lyapunov function in H 2 -norm is constructed. In Section 5, an illustrative example is provided to show the main result. Finally, concluding remarks end the paper. Some proofs have been omitted due to space limitation.

Notation. For a partitioned symmetric matrix P , the symbol stands for symmetric block, P 0 means that P is positive semidefinite. Given a matrix A, the transpose matrix of A is denoted as A T . The associate norm in H 2 (0, 1) space is denoted by || || H 2 , defined for all functions f ∈ H 2 (0, 1), by

||f || H 2 = 1 0 f 2 + f 2 x + f 2 xx dx 1/2 . (1)
Following [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], we introduce the notation, for all matrices

M ∈ R n×n , ρ 1 (M ) = inf{ ∆M ∆ -1 , ∆ ∈ D n,+ }, (2)
where || || denotes the usual matrix 2-norm and ∆ ∈ D n,+ denotes the set of diagonal positive matrix in R n×n .

According to [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF], for all matrices M ∈ R n×n , ρ(M ) is defined as

ρ(M ) = inf{|||∆M ∆ -1 |||, ∆ ∈ D n,+ }; (3)
where

|||M ||| = max    n j=1 |M ij |; i ∈ 1, . . . , n    . (4)

II. PROBLEM FORMULATION

We consider the following singularly perturbed system of conservation laws for a small positive perturbation parameter ε:

y t (x, t) + y x (x, t) = 0, εz t (x, t) + z x (x, t) = 0, (5)
where

x ∈ [0, 1], t ∈ [0, +∞), y : [0, 1] × [0, +∞) → R, z : [0, 1] × [0, +∞) → R, with the boundary conditions y(0, t) z(0, t) = G y(1, t) z(1, t) , (6)
where

G = G 11 G 12 G 21 G 22 is a 2 × 2 constant matrix.
Given two continuous functions y 0 : [0, 1] → R and z 0 : [0, 1] → R, the initial conditions are:

y(x, 0) z(x, 0) = y 0 z 0 . (7)
Let us compute the reduced and boundary-layer systems following the approach of [START_REF] Kokotovic | Singular pertrubation methods in control: analysis and design[END_REF]. By setting ε = 0 in system (5), the reduced system is defined as:

y t (x, t) + y x (x, t) = 0, (8a) z x (x, t) = 0. (8b)
Substitute (8b) into the boundary condition (6) yields:

y(0, t) = G 11 + G12G21 1-G22 y(1, t), z(., t) = G21 1-G22 y(1, t). (9)
The reduced system is rewritten as

ȳt (x, t) + ȳx (x, t) = 0, (10)
with the boundary condition

ȳ(0, t) = G 11 + G 12 G 21 1 -G 22 ȳ(1, t). (11)
To define the boundary-layer system, the variable y(x, t) is seen as a constant with respect to time, which yields:

zτ (x, τ ) + zx (x, τ ) = 0, (12)
with the boundary condition:

z(0, τ ) = G 21 ȳ(1) + G 22 z(1, τ ), (13)
where τ = t ε is a stretching time scale. ȳ(1) in ( 13) is handled as a fixed parameter for the boundary-layer system.

Remark. According to Proposition 2.1 in [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], for every y 0 z 0 ∈ H 2 (0, 1) satisfying the following compatibility conditions:

y 0 (0) z 0 (0) = G y 0 (1) z 0 (1) , (14)
y 0 x (0) 1 ε z 0 x (0) = G y 0 x (1) 1 ε z 0 x (1) , (15)
the system ( 5) and ( 6) has a unique maximal classical solution y z ∈ C 0 ([0, +∞), H 2 (0, 1)).

III. STABILITY ANALYSIS OF REDUCED AND BOUNDARY-LAYER SYSTEMS

The first problem under consideration is the stability analysis of the reduced system ( 10) and ( 11) and the boundarylayer system [START_REF] Gügat | H 2 stabilization of the isothermal Euler equations: a Lyapunov function approach[END_REF] and [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF]. Our objective in this section is to establish the stability properties of the reduced and boundarylayer systems via strict Lyapunov function in L 2 -norm.

Definition 1: The reduced system ( 10) and ( 11) is exponentially stable in L 2 -norm if there exist α > 0 and C > 0 such that, for every ȳ0 ∈ L 2 (0, 1), the solution to the reduced system [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF]

satisfies ||ȳ(., t)|| L 2 Ce -αt ||ȳ 0 || L 2 .
Proposition 1: The reduced system [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and ( 11) is exponentially stable in L 2 -norm to 0 if and only if the boundary condition satisfies

G 11 + G 12 G 21 1 -G 22 < 1. ( 16
)
Under this condition, a strict Lyapunov function is defined as:

V (ȳ) = 1 0 e -µx ȳ2 dx, (17)
for all ȳ ∈ L 2 (0, 1), where µ > 0 satisfies

e -µ G 11 + G 12 G 21 1 -G 22 2 . ( 18
)
Proposition 2: Given any ȳ(1) the boundary-layer system (12) and ( 13) is exponentially stable in L 2 -norm to the equilibrium point G21 1-G22 ȳ(1) if and only if the boundary condition satisfies

|G 22 | < 1. ( 19
)
Under this condition, a strict Lyapunov function is defined as:

W (z) = 1 0 e -νx z - G 21 1 -G 22 ȳ(1) 2 dx, (20)
for all z ∈ L 2 (0, 1), where ν > 0 satisfies

e -ν G 2 22 . (21)

IV. STABILITY ANALYSIS OF SINGULARLY PERTURBED SYSTEM OF CONSERVATION LAWS

The aim of this section is to state the exponential stability of the singularly perturbed system of conservation laws from that of the reduced and boundary-layer systems, for small ε > 0.

In the previous section, the exponential stability has been established via Lyapunov functions in L 2 -norm for the reduced and boundary-layer systems. As we will see later, in order to do the stability analysis of the overall singularly perturbed system of conservation laws, it is needed to consider a strict Lyapunov function in H 2 -norm.

Definition 2: The singularly perturbed system of conservation laws ( 5) and ( 6) is exponentially stable to the origin in H 2 -norm if there exist γ > 0 and C 1 > 0, for every y 0 z 0 ∈ H 2 (0, 1), the solution to the system ( 5) and ( 6) satisfies

(||y(., t)|| H 2 + ||z(., t)|| H 2 ) C 1 e -γt ||y 0 || H 2 + ||z 0 || H 2 .
Our result shows that as soon as the reduced system ( 10) and ( 11) and the boundary-layer system ( 12) and ( 13) are exponentially stable in L 2 -norm, with some additional conditions on the boundary conditions, for sufficiently small ε, the exponential stability of singularly perturbed system of conservation laws ( 5) and ( 6) holds in H 2 -norm.

Let us start by stating our assumptions. Assumption 1: The reduced system [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] with boundary condition [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF] is exponentially stable in L 2 -norm.

Assumption 2: The boundary-layer system [START_REF] Gügat | H 2 stabilization of the isothermal Euler equations: a Lyapunov function approach[END_REF] with boundary condition [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF] is exponentially stable in L 2 -norm. The last assumption (Assumption 3) deals with technical properties on matrix G in the boundary condition [START_REF] Grujic | Uniform asymptotic stability of nonlinear singularly perturbed and large scale systems[END_REF].

Assumption 3: For a given d such that 0 < d < 1, µ > 0 (resp. ν > 0) such that e -µ > G 2 11 and e -µ > G

11 + G12G21 1-G22 2 (resp. e -ν > G 2 22
), the following three inequalities hold: a)

(1 -d)R 1 -dG 2 21 0, (22) b
) dR 2 -(1 -d)G 2 12 0, (23) c
)

(1 -d)R 1 -dG 2 21 dR 2 -(1 -d)G 2 12 -((1 -d)R 3 + dR 4 ) 2 0.
(24) where:

R 1 = e -µ -G 2 11 , R 2 = e -ν -G 2 22 , R 3 = G 11 G 12 , R 4 = G 21 G 22 .
Remark. 1) Due to Proposition 1, Assumption 1 is equivalent to G 11 + G12G21 1-G22 < 1, which guarantees that y = 0 is an exponentially stable equilibrium point of the reduced system [START_REF] Coron | A strict Lyapunov function for boundary control of hyperbolic systems of conservation laws[END_REF] and [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF].

2) Similarly, due to Proposition 2, Assumption 2 is equivalent to |G 22 | < 1, which plays the same role for the boundarylayer system (12) and ( 13) which has an equilibrium point z = G21

1-G22 y(1). It is important to notice that, for the boundary-layer system, y(1) is treated as a constant with respect to time. On the other hand it is reasonable to consider y(1) as a variable of time for the overall system ( 5) and ( 6).

3) Assumption 3 stands for the additional conditions on the boundary conditions, which ensures the solutions converge in H 2 -norm. We can state the following theorem:

Theorem 1: Under Assumptions 1, 2 and 3, let the positive value ε * = min(ε 1 , ε 2 ), ε 1 , ε 2 ∈ (0, +∞] be given by: Case 1:

If G 12 = 0 or G 21 = 0 ε 1 = dK 1 R 2 (1 -d)G 2 12 K 1 + (1 -d)K 2 2 ;
(25)

ε 2 = (1 -d)µe -µ νe -ν (1 -G 22 ) 2 2dG 2 21 . ( 26
)
where:

K 1 = e -µ - G 11 + G12G21 1-G22 2
and

K 2 = G 11 G 12 + G 2 12 G21
1-G22 .

Case 2: If G 12 = 0 and G 21 = 0

ε 1 = ε 2 = +∞, (27)
Then, for all 0 < ε < ε * and 0 < ε < +∞, the singularly perturbed system of conservation laws ( 5) and ( 6) is exponentially stable in H 2 -norm to the origin and it has a strict Lyapunov function:

L(y, z) = (1 -d) 1 0 e -µx (y 2 + y 2 x + y 2 xx )dx + d 1 0 e -νx z - G 21 1 -G 22 y(1) 2 + η 1 (ε)z 2 x + η 2 (ε)z 2 xx ( 28
)
where η 1 , η 2 are positive values which depend on ε.

Remark. ε * can be computed by ε 1 and ε 2 , moreover, to ensure the stability of system ( 5) and ( 6) in H 2 -norm for all ε < ε * , ε * should be taken as the minimum value between ε 1 and ε 2 . Sketch of the proof: First, let us decompose L(y, z) in the following way:

L(y, z) = L 1 + L 2 + L 3 , (29)
with L 1 , L 2 and L 3 selected respectively to the zero-th, first and second space-derivative of the solutions, that is:

L 1 = (1 -d) 1 0 e -µx y 2 dx +d 1 0 e -νx z - G 21 1 -G 22 y(1)
2 dx, (30)

L 2 = (1 -d) 1 0 e -µx y 2 x dx + dη 1 (ε) 1 0 e -νx z 2 x dx, (31)
L 3 = (1 -d) 1 0 e -µx y 2 xx dx + dη 2 (ε) 1 0 e -νx z 2 xx dx. (32)
Next, we use 4 steps to demonstrate the proof of Theorem 1.

Step 1: Compute the time derivative of the first term L 1 along the solutions to ( 5) and ( 6)

L1 = L 11 + L 12 ( 33
)
with

L 11 = -(1 -d) e -µx y 2 x=1 x=0 - d ε e -νx z - G 21 1 -G 22 y(1) 2 x=1 x=0 , (34)
and

L 12 = -(1 -d)µ 1 0 e -µx y 2 dx + 2dG 21 1 -G 22 1 0 e -νx z - G 21 1 -G 22 y(1) y x (1)dx - d ε ν 1 0 e -νx z - G 21 1 -G 22 y(1) 2 dx. (35)
Under the boundary conditions ( 6) and replacing z(1) by the right-hand side of the following equation

z(1) = z(1) - G 21 1 -G 22 y(1) + G 21 1 -G 22 y(1),
the term L 11 follows

L 11 = - y(1) z(1) -G 21 1-G 22 y(1) T M 11 y(1) z(1) -G 21 1-G 22 y(1)
with

M 11 = (1 -d)K 1 -(1 -d)K 2 d ε R 2 -(1 -d)G 2 12 ,
where K 1 , K 2 are defined in Theorem 1 and R 2 is defined in Assumption 3.

To prove L 11 is non positive, which is equivalent to prove the matrix M 11 0, it is sufficient to require that

0 < ε dK 1 R 2 (1 -d)G 2 12 K 1 + (1 -d)K 2 2 , ( 36
) if G 12 = 0. And pick any ε > 0 if G 12 = 0. Due to Assumption 1 (resp. Assumption 2), K 1 (resp. R 2 )
is always positive. The right-hand side of (36) gives ε 1 .

Step 2: Differentiating (5) with respect to x, we have

y xt (x, t) + y xx (x, t) = 0, εz xt (x, t) + z xx (x, t) = 0, (37)
with the boundary conditions

y x (0, t) = G 11 y x (1, t) + 1 ε G 12 z x (1, t), z x (0, t) = εG 21 y x (1, t) + G 22 z x (1, t). (38)
Compute the time derivative of the second term L 2 along the solutions to (37) and (38)

L2 = L 21 + L 22 (39)
with

L 21 = -(1 -d)[e -µx y 2 x ] x=1 x=0 - dη 1 (ε) ε [e -νx z 2 x ] x=1 x=0 , (40)
and

L 22 = -(1 -d)µ 1 0 e -µx y 2 x dx - dνη 1 (ε) ε 1 0 e -νx z 2 x dx. (41)
Take η 1 (ε) = 1 ε , under the boundary conditions (38), the term L 21 follows

L 21 = - y x (1) z x (1) T M 21 y x (1) z x (1) , with:
M 21 = (1 -d)R 1 -dG 2 21 -( (1-d)R3 ε + dR4 ε ) d ε 2 R 2 - (1-d)G 2 12 ε 2
, where R 1 , R 2 , R 3 and R 4 are defined in Assumption 3.

To prove L 21 is non positive, which is equivalent to prove the matrix M 21 0, it is sufficient to require the following conditions are satisfied:

(1 -d)R 1 -dG 2 21 0; (42)
it is equivalent to the first inequality of Assumption 3.

d ε 2 R 2 - (1 -d)G 2 12 ε 2 0; (43)
it is equivalent to the second inequality of Assumption 3.

(

-d)R 1 -dG 2 21 dR 2 -(1 -d)G 2 12 ε 2 - ((1 -d)R 3 + dR 4 ) 2 ε 2 0; 1
it is equivalent to the third inequality of Assumption 3.

Step 3: Differentiating (37) with respect to x, we have:

y xxt (x, t) + y xxx (x, t) = 0, εz xxt (x, t) + z xxx (x, t) = 0, (45)
with boundary conditions:

y xx (0, t) = G 11 y xx (1, t) + 1 ε 2 G 12 z xx (1, t), z xx (0, t) = ε 2 G 21 y xx (1, t) + G 22 z xx (1, t). (46)
Compute the time derivative of the third term L 3 along the solutions to (45) and ( 46)

L3 = L 31 + L 32 (47)
with

L 31 = -(1 -d)[e -µx y 2 xx ] x=1 x=0 - dη 2 (ε) ε [e -νx z 2 xx ] x=1 x=0 , (48)
and

L 32 = -(1 -d)µ 1 0 e -µx y 2 xx dx - dνη 2 (ε) ε 1 0 e -νx z 2 xx dx. (49)
Take η 2 (ε) = 1 ε 3 , under the boundary conditions (46), the term L 31 follows

L 31 = - y xx (1) zxx(1) ε T M 21 y xx (1)
zxx( 1) ε

.

As soon as the matrix M 21 0, L 31 is non positive.

Step 4: Combining all the integral terms (35), ( 41) and (49), with η

1 (ε) = 1 ε , η 2 (ε) = 1 ε 3 , we get L 12 + L 22 + L 32 = -(1 -d)µ 1 0 e -µx (y 2 + y 2 x + y 2 xx )dx - dν ε 1 0 e -νx z - G 21 1 -G 22 y(1) 2 + z 2 x ε + z 2 xx ε 3 + 2dG 21 1 -G 22 1 0 e -νx z - G 21 1 -G 22 y(1) y x (1)dx. (50)
The following inequalities hold

-(1 -d)µ 1 0 e -µx (y 2 + y 2 x + y 2 xx )dx -(1 -d)µe -µ y 2 H 2 . (51)
Moreover

- dν ε 1 0 e -νx z - G 21 1 -G 22 y(1) 2 + z 2 x ε + z 2 xx ε 3 dx - d ε νe -ν z - G 21 1 -G 22 y(1) 2 H 2 . (52)
Furthermore

2dG 21 1 -G 22 1 0 e -νx z - G 21 1 -G 22 y(1) y x (1)dx 2 √ 2dG 21 1 -G 22 z - G 21 1 -G 22 y (1)
H 2 y H 2 . ( 53
)
According to (51), ( 52) and (53), L 12 + L 22 + L 32 follows

L 12 + L 22 + L 32 - y H 2 (z -G 21 1-G 22 y(1) H 2 T M 4 y H 2 (z -G 21 1-G 22 y(1) H 2 , with M 4 = (1 -d)µe -µ -| √ 2dG21 1-G22 | dν ε e -ν .
To prove M 4 > 0, it is sufficient to require that

0 < ε < (1 -d)µe -µ νe -ν (1 -G 22 ) 2 2dG 2 21 . ( 54
)
Let ε 2 be given by the right-hand side of (54). Under Assumptions 1, 2 and 3, pick ε * = min(ε 1 , ε 2 ), where ε 1 and ε 2 are given by ( 25) and (26) in Case 1 and (27) in Case 2. λ is the minimal eigenvalue of M 4 , it is obtained:

L -λ z - G 21 1 -G 22 y(1) 2 H 2 + y 2 H 2 . ( 55
)
Therefore the Lyapunov function ( 28) is a strict Lyapunov function for singularly perturbed system of conservation laws ( 5) and ( 6). This concludes the proof of Theorem 1.

V. NUMERICAL EXAMPLE

In this section, we consider the following boundary condition for the singularly perturbed system (5):

G = 0.7 2 0.1 -0.5 , (56)
where Assumptions 1 and 2 hold. Let take d = 0.95, µ = ν = 0.1, it is computed: R 1 = 0.4148, R 2 = 0.6548, R 3 = 1.4, R 4 = -0.05, the three inequalities of Assumption 3 hold. Following the statement of Theorem 1 it is computed: K 1 = 0.2104, K 2 = 1.6666. According to (25) and ( 26), ε 1 and ε 2 are computed:

ε 1 = 0.7233, ε 2 = 0.0485.
The admissible perturbation parameter ε is chosen as 0.045.

Applying Theorem 1, the system (5) and ( 56) is exponentially stable. Considering a diagonal positive definite matrix ∆ = 1 0 0 4 , the inequality ||∆G∆ -1 || < 1 holds and thus ρ 1 (G) < 1. Therefore with [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF], we recover the stability of system ( 5) and (56). In other words, the stability condition of [START_REF] Coron | Dissipative boundary conditions for one-dimensional nonlinear hyperbolic systems[END_REF] applies for system (5) and (56). However ρ(|G|) > 1, and thus the stability condition of [START_REF] Li | Global classical solutions for quasilinear hyperbolic systems[END_REF] does not apply.

Let us check the stability in numerical simulation of ( 5) and (56). Discretize the equation ( 5) using a two-step variant of the Lax-Wendroff method which is presented in [START_REF] Shampine | Two-step Lax-Friedrichs method[END_REF] and the solver on Matlab in [START_REF] Shampine | Solving hyperbolic PDEs in Matlab[END_REF]. More precisely, we divide the space domain [0, 1] into 100 intervals of identical length, and 15 as final time. We choose a time-step that satisfies the CFL condition for the stability and select the following initial functions:

y(x, 0) = sin(4πx) z(x, 0) = sin(5πx) for all x ∈ [0, 1].
The time evolutions of the solutions y z and of square of H 2 -norm

||y|| 2 H 2 z -G21 1-G22 y(1)

VI. CONCLUSIONS

In this paper, the stability for a class of singularly perturbed system of conservation laws has been studied. Necessary and sufficient condition has been derived for the stability of the reduced and boundary-layer systems. For a perturbation parameter sufficiently small, under suitable assumptions on the boundary conditions, the exponential stability of the singularly perturbed system of conservation laws has been established in H 2 -norm. This result was proved by means of a Lyapunov function approach. This work was restricted to analyze singularly perturbed system of two conservation laws. It is natural to extend to the system of balanced laws and to systems of higher dimension. Another interesting point is to consider some physical application, like gas flow through pipelines in [START_REF] Dick | Classical solutions and feedback stabilization for the gas flow in a sequence of pipes[END_REF], [START_REF] Colombo | Optimal control in networks of pipes and canals[END_REF] and open channels in [START_REF] Santos | Boundary control of open channels with numerical and expermental validations[END_REF].

2 H 2 Fig. 1 .Fig. 2 . 2 H 2 (

 221222 Fig. 1. Time evolutions of the first component y (top) and of the second component z (bottom) of the solution of the system (5) and (56)