Olivier Ley
email: olivier.ley@insa-rennes.fr

Vinh Duc Nguyen
email: vinh.nguyen@math.cnrs.fr

GRADIENT BOUNDS FOR NONLINEAR DEGENERATE PARABOLIC EQUATIONS AND APPLICATION TO LARGE TIME BEHAVIOR OF SYSTEMS

Keywords: May 15, 2015. 2010 Mathematics Subject Classification. Primary 35J70; Secondary 35K40, 35B40, 35B45, 35B50 35F21, 49L25 Nonlinear degenerate parabolic equations, Nonlinear degenerate elliptic equations, Hamilton-Jacobi equations, monotone systems, gradient bounds, oscillation, strong maximum principle, ergodic problem, asymptotic behavior, viscosity solutions

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Gradient bounds for nonlinear degenerate parabolic equations and application to large time behavior of systems

Introduction

One of the main result of this work is to obtain new results about the large time behavior of the solution u(x, t) = (u 1 (x, t), • • • , u m (x, t)) of the fully nonlinear parabolic system

           ∂u i ∂t + sup θ∈Θ -trace(A θi (x)D 2 u i ) + H θi (x, Du i ) + m j=1
d ij u j = 0, (x, t) ∈ T N × (0, +∞),

u i (x, 0) = u 0i (x), x ∈ T N , 1 ≤ i ≤ m, (1.1)
in the periodic setting (T N denotes the flat torus R N /Z N ), where the equations are linearly coupled through a matrix D = (d ij ) ij which is assumed to be monotone and irreducible. The set Θ is a metric space, the diffusion matrices can be written A θi (x) = σ θi (x)σ θi (x) T with σ θi bounded Lipschitz continuous and H i , u 0i are continuous.

To simplify the presentation, we present our results in the simplest case without dependence with respect to θ and for m = 2. See Section 3.5 for some discussions about the general case. We then consider

         ∂u 1 ∂t -trace(A 1 (x)D 2 u 1 ) + H 1 (x, Du 1 ) + u 1 -u 2 = 0, ∂u 2 ∂t -trace(A 2 (x)D 2 u 2 ) + H 2 (x, Du 2 ) + u 2 -u 1 = 0, (x, t) ∈ T N × (0, +∞), u 1 (x, 0) = u 01 (x), u 2 (x, 0) = u 02 (x) x ∈ T N , (1.2)
The precise assumptions on the Hamiltonians H i will be explained below. We prove the following asymptotic behavior of the solution, u i (x, t) + c i t → v i (x) uniformly in T N as t → ∞, for i = 1, 2, (1.3) where c = (c 1 , c 2 ) ∈ R 2 and v = (v 1 , v 2 ) ∈ W 1,∞ (T N ) 2 are solutions of the so-called ergodic problem

-trace(A 1 (x)D 2 v 1 ) + H 1 (x, Dv 1 ) + v 1 -v 2 = c 1 , -trace(A 2 (x)D 2 v 2 ) + H 2 (x, Dv 2 ) + v 2 -v 1 = c 2 , x ∈ T N . (1.4)
Let us give immediately one of the most striking application of our result. We are able to prove the convergence for asymmetric systems like

     ∂u 1 ∂t -trace(A 1 (x)D 2 u 1 ) + b 1 (x), Du 1 + ℓ 1 (x) + u 1 -u 2 = 0, ∂u 2 ∂t -trace(A 2 (x)D 2 u 2 ) + |Du 2 | 2 + ℓ 2 (x) -u 1 + u 2 = 0, (1.5)
where A 1 is uniformly elliptic and A 2 may be degenerate. The name asymmetric means that different equations can have different natures as above: the first one contains a sublinear Hamiltonian and is uniformly elliptic whereas the second one contains a superlinear one and may be degenerate. The general framework is presented below. To explain the main difficulty to prove this result, let us recall the related results for scalar equations.

To study the large time behavior for parabolic nonlinear equations ∂u ∂t -trace(A(x)D 2 u) + H(x, Du) = 0, (x, t) ∈ T N × (0, +∞), (1.6) one has first to establish uniform gradient bounds for the stationary equation ǫφ ǫ -trace(A(x)D 2 φ ǫ ) + H(x, Dφ ǫ ) = 0, x ∈ T N , ǫ > 0.

(1.7) By uniform gradient bounds, we mean |Dφ ǫ | ∞ ≤ K, where K is independent of ǫ. (1.8) This condition is crucial to be able to send ǫ to 0 in (1.7) in order to solve the ergodic problem (1.4), which is a first step when trying to prove (1.3). Barles and Souganidis [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] obtained the first results concerning both estimates like (1.8) and the asymptotic behavior (1.3) for scalar equations (m = 1) with A(x) = I in two contexts. The first one is for sublinear Hamiltonians, i.e., for Hamiltonians with a sublinear growth with respect to the gradient. A typical example is

H(x, p) = b(x), p + ℓ(x), b ∈ C(T N ; R N ), ℓ ∈ C(T N ).
The second context is for superlinear Hamiltonians. The precise assumption is more involved (see (2.42)) and designed to allow the use of weak Bernstein-type arguments ( [START_REF] Barles | A weak Bernstein method for fully nonlinear elliptic equations[END_REF]). The most important example is Hamiltonians with a superlinear growth with respect to the gradient H(x, p) = a(x)|p| 1+α + ℓ(x), α > 0, a, ℓ ∈ C(T N ) and a > 0.

(1.9)

As a consequence of these bounds together with the strong maximum principle, they obtain the convergence for the solutions of (1.6) when either H is sublinear or H is superlinear.

Using the extension of viscosity solutions to monotone systems of parabolic equations by Ishii and Koike [START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF], it is not difficult to adapt the results of [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] to the case of sublinear systems (1.2) (i.e., systems for which all the Hamiltonians are sublinear in the sense of [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF]) on the one hand, and to the case of superlinear systems on the other hand. In this work, we focus on the more delicate issue of asymmetric systems like (1.5) containing both sublinear and superlinear Hamiltonians. The main difficulty is that the proofs of (1.8) in [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] are completely different in the two contexts. In the case of sublinear Hamiltonians, a method due to Ishii and Lions [START_REF] Ishii | Viscosity solutions of fully nonlinear second-order elliptic partial differential equations[END_REF] allows to take profit of the uniform ellipticity of the equation to control the sublinear terms coming from the Hamiltonian. Whereas, up to our knowledge, the strategy to prove (1.8) for superlinear Hamiltonians relies on a weak Bernstein method needing an exponential-type change of function like e w ǫ = φ ǫ -min T N φ ǫ + 1 in order to take advantage of the superlinear property of the Hamiltonian. In the case of systems, one has to perform this exponential change in all equations, producing quadratic terms of the form |σ(x)Dw ǫ | 2 . These latter terms are dramatic in equations with sublinear Hamiltonians since they are not anymore under the control of the ellipticity using the usual proof.

The new idea to overcome this difficulty is to establish first uniform oscillation bounds osc(φ ǫ ) := sup

T N φ ǫ -inf T N φ ǫ ≤ K where K is independent of ǫ, (1.10)
for the solution of (1.7). The point is that the proof of the oscillation bound does not need any exponential change of variable and so will work for asymmetric systems. This gives some uniform bounds for the new function w ǫ and we are able to "localize" the proof (see (2.26) for details) allowing to control the bad quadratic term. It is worth mentioning that our proof of the oscillation bound works in very general settings and is a new result interesting by itself. For instance, (1.10) holds for the solutions of (1.7) as soon as

H(x, p) |p| → +∞ as |p| → +∞ uniformly with respect to x. (1.11)
Taking advantage of this oscillation bound, we are in fact able to produce a kind of unified proof of the sublinear and superlinear cases of [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF]. More precisely, we obtain the gradient bound (1.8) for (1.7) when

H = H + H,
where H is a sublinear Hamiltonian, i.e, having a sublinear growth,

|H(x, p)| ≤ C(1 + |p|),
and H is a superlinear one. An important feature, which will be crucial when dealing with asymmetric systems, is that we allow H or/and H to be zero which is natural for H but not so for H. As far as the precise definition of superlinear Hamiltonian is concerned, we propose two definitions, one when (1.7) is uniformly elliptic, see (2.8), which generalizes slightly the one of [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] and a stronger one, see (2.40), which allows to deal with degenerate equations (1.7). Both of them include of course typical superlinear Hamiltonians like (1.9) and H may be zero in some cases. We refer the reader to Section 2.4 for comments and examples.

Using this new gradient bound for scalar equations, we are in position to extend it to systems

ǫφ ǫ 1 -trace(A 1 (x)D 2 φ ǫ 1 ) + H 1 (x, Dφ ǫ 1 ) + φ ǫ 1 -φ ǫ 2 = 0, ǫφ ǫ 2 -trace(A 2 (x)D 2 φ ǫ 2 ) + H 2 (x, Dφ ǫ 2 ) + φ ǫ 2 -φ ǫ 1 = 0, x ∈ T N .
(1.12)

An immediate consequence is that we can solve the ergodic problem (1.4) extending the classical by now proofs of [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF][START_REF] Arisawa | On ergodic stochastic control[END_REF] to the case of our systems. We then prove the convergence (1.3).

As in [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF], the proof of the convergence is based on the strong maximum principle but let us mention that we establish a new version of the strong maximum principle for systems which may contain some degenerate equations, see Theorem 3.4 for details. In particular, the result holds for (1.5); See Section 3.5 for more examples and discussions.

Let us turn to an overview of related results in the litterature. The ideas of the proof of gradient bounds in viscosity theory using the uniform ellipticity of the equation are due to Ishii and Lions [START_REF] Ishii | Viscosity solutions of fully nonlinear second-order elliptic partial differential equations[END_REF], see also [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Barles | C 0,α -regularity and estimates for solutions of elliptic and parabolic equations by the Ishii & Lions method[END_REF] and the references therein. Gradient bounds for superlinear-type Hamiltonians can be found in Lions [START_REF] Lions | Generalized solutions of Hamilton-Jacobi equations[END_REF] and Barles [START_REF] Barles | A weak Bernstein method for fully nonlinear elliptic equations[END_REF], see also Lions and Souganidis [START_REF] Lions | Homogenization of degenerate second-order PDE in periodic and almost periodic environments and applications[END_REF]. These ideas were used in Barles and Souganidis [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] as explained above. Our approach is mainly based on this latter work. For superlinear Hamiltonians satisfying H(x, p) ≥ a -1 |p| m -a, a, m > 1, some Hölder or gradient estimates were obtained in Capuzzo Dolcetta et al. [START_REF] Capuzzo Dolcetta | Hölder estimates for degenerate elliptic equations with coercive Hamiltonians[END_REF], Barles [4], Cardaliaguet [START_REF] Cardaliaguet | A note on the regularity of solutions of Hamilton-Jacobi equations with superlinear growth in the gradient variable[END_REF]. Recently, oscillations and Hölder bounds for nonlinear degenerate parabolic equations were proved in Cardaliaguet and Sylvestre [START_REF] Cardaliaguet | Hölder continuity to Hamilton-Jacobi equations with superquadratic growth in the gradient and unbounded right-hand side[END_REF] but the bounds depends on the L ∞ norm of the solution.

The large time behavior of such kind of nonlinear equations or systems in the periodic setting were extensively studied. For Hamilton-Jacobi equations (the totally degenerate case when A ≡ 0), we refer the reader to [START_REF] Namah | Remarks on the long time behaviour of the solutions of Hamilton-Jacobi equations[END_REF][START_REF] Fathi | Sur la convergence du semi-groupe de Lax-Oleinik[END_REF][START_REF] Barles | On the large time behavior of solutions of Hamilton-Jacobi equations[END_REF][START_REF] Davini | A generalized dynamical approach to the large time behavior of solutions of Hamilton-Jacobi equations[END_REF][START_REF] Barles | A new PDE approach to the large time asymptotics of solutions of Hamilton-Jacobi equations[END_REF] and the references therein. In this framework, the gradient bounds are not a difficult step but the proof of the convergence is more delicate since one does not have any strong maximum principle. Such kind of results were extended to systems of Hamilton-Jacobi equations in [START_REF] Camilli | Large time behavior of weakly coupled systems of firstorder Hamilton-Jacobi equations[END_REF][START_REF] Mitake | Remarks on the large time behavior of viscosity solutions of quasi-monotone weakly coupled systems of Hamilton-Jacobi equations[END_REF][START_REF] Nguyen | Some results on the large time behavior of weakly coupled systems of first-order hamiltonjacobi equations[END_REF][START_REF] Mitake | A dynamical approach to the large-time behavior of solutions to weakly coupled systems of hamilton-jacobi equations[END_REF]. For second order nonlinear equations, the asymptotics results of [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] were recently generalized in [START_REF] Ley | Large time behavior for nonlinear degenerate parabolic equations[END_REF] to some superlinear degenerate equations which are totally degenerate on some subset Σ of T N and uniformly parabolic outside Σ using the gradient bound of Theorem 2.6 and some strong maximum principle type ideas. Similar results for uniformly convex degenerate equations were established in Cagnetti et al. [START_REF] Cagnetti | A new method for large time behavior of convex Hamilton-Jacobi equations I: degenerate equations and weakly coupled systems[END_REF] using a different approach based on a nonlinear adjoint method [START_REF] Evans | Adjoint and compensated compactness methods for Hamilton-Jacobi PDE[END_REF]. Their results also apply to systems with uniformly convex quadratic Hamiltonians with quite general degeneracy assumptions since the proof is not based on strong maximum principle-type arguments. However, such a method does not seem to be applicable for fully nonlinear equations and the system in [START_REF] Cagnetti | A new method for large time behavior of convex Hamilton-Jacobi equations I: degenerate equations and weakly coupled systems[END_REF] is not asymmetric.

The paper is organized as follows. The bounds (1.10)-(1.8) are established in Section 2 for the scalar equation (1.7) with H = H + H, firstly when the equation is uniformly elliptic and secondly in some degenerate cases. Some examples of applications are collected in Section 2.4. Section 3 is devoted to systems. The gradient bounds for asymmetric systems are obtained in Section 3.1 and a new strong maximum principle is obtained in Section 3.2. Then the ergodic problem is solved and the main application of large time behavior of asymmetric is investigated. The section ends with some examples of applications and extensions. Several results are collected in the appendix. In particular, since the equations under consideration do not satisfy the classical assumptions in viscosity solutions (due to the possibly superlinear growth of the Hamiltonian for instance), we recall several versions of the comparison principle which apply in our case. Finally a control theoritical interpretation is given.

Gradient bounds for nonlinear parabolic equations

For ǫ > 0, we consider the approximated equation

ǫφ ǫ -trace(A(x)D 2 φ ǫ ) + H(x, Dφ ǫ ) = 0, x ∈ T N . (2.1)
We will always assume

A = σσ T , σ ∈ W 1,∞ (T N ; R N ×N ), H ∈ C(T N × R N ). (2.2)
2.1. A general result for the oscillation. We first show that the oscillation of the solution of (2.1) is uniformly bounded under a very general hypothesis. This result is interesting by itself.

   There exists L > 1 such that for all x, y ∈ T N ,

if |p| = L, then H(x, p) ≥ |p| H(y, p |p| )+|H(•, 0)| ∞ +N 3/2 |σ x | 2 ∞ .
(

φ ǫ (x) -φ ǫ (x ǫ ) ≤ L|x -x ǫ | for all x ∈ T N ,
where L is the constant (independent of ǫ) which appears in (2.3).

An immediate consequence is

osc(φ ǫ ) := max φ ǫ -min φ ǫ ≤ √ N L.
Proof of Lemma 2.1. For simplicity, we skip the ǫ superscript for φ ǫ writing φ instead. The constant L which appears below is the one of (2.3). Consider

M = max

x,y∈T N {φ(x) -Lφ(y) + (L -1) min φ -L|x -y|}.

We are done if M ≤ 0. Otherwise, the above positive maximum is achieved at (x, y) with x = y. Notice that the continuity of φ is crucial at this step. The theory of second order viscosity solutions (see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF] and Lemma 2.4) yields, for every ̺ > 0, the existence of (p, X) ∈ J 2,+ φ(x)

and (p/L, Y /L) ∈ J 2,-φ(y), p = L x-y |x-y| , such that -trace(A(x)X -A(y)Y ) ≥ -LN 3/2 |σ x | 2 ∞ + O(̺) and    ǫφ(x) -trace(A(x)X) + H(x, p) ≤ 0, ǫφ(y) -trace(A(y) Y L ) + H(y, p L )} ≥ 0. It follows ǫ(φ(x) -Lφ(y)) -trace(A(x)X -A(y)Y ) + H(x, p) -LH(y, p L ) ≤ 0.
We have

ǫ(φ(x) -Lφ(y)) > -(L -1)ǫ min φ ≥ -L|H(•, 0)| ∞ since ǫ min φ ≤ |H(•, 0)| ∞
by the maximum principle (see (4.4)). Combining all the above information, we get

H(x, p) -L H(y, p L ) + |H(•, 0)| ∞ + N 3/2 |σ x | 2 ∞ < 0.
Applying (2.3) yields a contradiction.

2.2. Gradient bounds for uniformly elliptic equations. In this section, we suppose that (2.1) is uniformly elliptic, i.e., there exists ν > 0 such that A(x) ≥ νI, x ∈ T N . (2.4) In this setting, we consider Hamiltonians under the special form H = H + H where H is a sublinear Hamiltonian and H is of superlinear type in the sense defined below. This form will be useful later to deal with asymmetric systems. We rewrite (2.1) as

ǫφ ǫ -trace(A(x)D 2 φ ǫ ) + H(x, Dφ ǫ ) + H(x, Dφ ǫ ) = 0. (2.5)
We say that the H is a sublinear Hamiltonian if

|H(x, p)| ≤ C(1 + |p|), (x, p) ∈ T N × R N . (2.6)
We consider the following superlinear-type assumptions for H. The first one is needed to obtain an oscillation bound for the solution and the second one is slightly stronger to get the gradient bound.

There exists C > 0, L, µ > 1 such that for all x, y ∈ T

N , |p| ≥ L, H(x, p) -µH(y, p µ ) ≥ -C|p|. (2.7)       
There exists C > 0, L > 1 such that for all x, y ∈ T N , |p| ≥ L, and µ ≥ 1+L|x -y|, H(x, p) -µH(y, p µ ) ≥ -C|p|.

(2.8)

From the inequality in (2.6), we see that H has a sublinear growth in the classical sense. But, let us point out that the superlinear H may be zero in (2.7) and (2.8). This fact will allow to treat some cases of asymmetric systems. We chose to keep the terminology superlinear since (2.8) is a consequence of the superlinear-type assumption

H ∈ W 1,∞ loc (T N × R N )
, and there exists L > 1 such that for a.e.

x ∈ T N , |p| ≥ L, L H p p -H -|H x | ≥ 0 (2.9)
introduced in [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF]. Moreover, (2.8) is satisfied for the typical superlinear Hamiltonian H(x, p) = a(x)|p| 1+α + ℓ(x), α > 0, a > 0 we have in mind. We refer the reader to Section 2.4 for more discussions and examples showing that our assumptions are quite general.

We state the main result of this section.

Theorem 2.2. Assume (2.2), (2.4), (2.6) and (2.8). For all ǫ > 0, there exists a unique continuous viscosity solution φ ǫ ∈ C(T N ) of (2.5) and a constant K > 0 independent of ǫ such that

|Dφ ǫ | ∞ ≤ K.
The proof relies on two important lemmas. The first one establishes a uniform bound for the oscillation and the second one improves this bound into a gradient bound. We first state and prove the lemmas and then give the proof of the theorem. Lemma 2.3. Under the hypotheses of Theorem 2.2, where (2.8) could be replaced by the weaker condition (2.7), there exists a constant K > 0 (independent of ǫ) such that, if φ ǫ is a continuous solution of (2.5), then

osc(φ ǫ ) := max φ ǫ -min φ ǫ ≤ K.
Proof of Lemma 2.3. For simplicity, we skip the ǫ superscript for φ ǫ writing φ instead. 1. Construction a concave test function. Consider the function

Ψ(s) = A 1 A 2 (1 -e -A 2 s ) for 0 ≤ s ≤ √ N = diameter(T N ), (2.10)
where A 1 , A 2 > 0 will be chosen later. It is straightforward to see that Ψ is a C ∞ concave increasing function satisfying Ψ(0) = 0 and, for all s ∈ [0,

√ N ],

Ψ ′′ + A 2 Ψ ′ = 0, A 1 e -A 2 √ N = Ψ ′ ( √ N ) ≤ Ψ ′ (s) ≤ Ψ ′ (0) = A 1 . (2.11) 2. Viscosity inequalities. Consider M µ = max x,y∈T N {φ(x) + (µ -1) min φ -µφ(y) -Ψ(|x -y|)}, (2.12)
with µ given in (2.7). If M µ ≤ 0 then the lemma holds with K = A 1 /A 2 . From now on, we argue by contradiction assuming that the maximum is positive and achieved at (x, y) with x = y.

The theory of second order viscosity solutions yields, for every ̺ > 0, the existence of (p, X)

∈ J 2,+ φ(x), (p/µ, Y /µ) ∈ J 2,-φ(y) such that X 0 0 -Y ≤ A + ̺A 2 , (2.13) with q = x -y |x -y| , p = Ψ ′ (|x -y|)q, B = 1 |x -y| (I -q ⊗ q), (2.14) A = Ψ ′ (|x -y|) B -B -B B + Ψ ′′ (|x -y|) q ⊗ q -q ⊗ q -q ⊗ q q ⊗ q (2.15)
and the following viscosity inequalities hold for the solution φ of (2.5),

     ǫφ(x) -trace(A(x)X) + H(x, p) + H(x, p) ≤ 0, ǫφ(y) -trace(A(y) Y µ ) + H(y, p µ ) + H(y, p µ ) ≥ 0. It follows ǫφ(x) -ǫµφ(y) -trace(A(x)X -A(y)Y ) (2.16) +H(x, p) -µH(y, p µ ) + H(x, p) -µH(y, p µ ) ≤ 0.
3. Trace estimates. We have the following estimates which will be useful in the sequel, see for instance [START_REF] Ishii | Viscosity solutions of fully nonlinear second-order elliptic partial differential equations[END_REF][START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF][START_REF] Barles | C 0,α -regularity and estimates for solutions of elliptic and parabolic equations by the Ishii & Lions method[END_REF]. A proof is given in the Appendix.

Lemma 2.4. Under assumption (2.2),

-trace(A(x)X -A(y)Y ) ≥ -N |σ x | 2 ∞ |x -y|Ψ ′ (|x -y|) + O(̺). If, in addition, (2.4) holds, then -trace(A(x)X -A(y)Y ) ≥ -4νΨ ′′ (|x -y|) -CΨ ′ (|x -y|) + O(̺), (2.17) where C = C(N, ν, |σ| ∞ , |σ x | ∞ ) is given by (4.3).
4. End of the proof. At x such that φ(x) = max φ, we get

ǫ min φ + H(x, 0) + H(x, 0) ≤ ǫ max φ + H(x, 0) + H(x, 0) ≤ 0. It follows ǫ min φ ≤ |(H + H)(•, 0)| ∞ =: H 0 . (2.18)
So, using that the maximum is positive in (2.12), we obtain

ǫ(φ(x) -µφ(y)) > -(µ -1)ǫ min φ ≥ -(µ -1)H 0 .
We have

|H(x, p) -µH(y, p µ )| ≤ 2C(1 + |p|) = 2C(1 + Ψ ′ (|x -y|)), from (2.6) and
-tr(A(x)X -A(y)Y ) ≥ -4νΨ ′′ (|x -y|) -CΨ ′ (|x -y|) + O(̺)
from Lemma 2.4 (2.17). Choosing

A 1 = Le A 2 √ N ,
where L is the constant of (2.7), we obtain |p| = Ψ ′ (|x -y|) ≥ L from (2.11), (2.14) and

H(x, p) -µH(y, p µ ) ≥ -CΨ ′ (|x -y|)
from (2.7). Using these estimates in (2.16) and sending ̺ to 0, we have (2.11), we obtain a contradiction with (2.19) with the choice

-4νΨ ′′ -( C + 2C + C)Ψ ′ -(µ -1)H 0 -2C ≤ 0. (2.19) Recalling that Ψ ′′ + A 2 Ψ ′ = 0 by
A 2 = 1 4ν C + 2C + C + (µ -1)H 0 + 2C + 1 L .
It ends the proof.

Lemma 2.5. Under the hypotheses of Theorem 2.2. Let φ ǫ be a continuous viscosity solution of (2.5) and define

exp(w ǫ ) = φ ǫ -min T N φ ǫ + 1.
Then, there exists a constant

K (independent of ǫ) such that |Dw ǫ | ∞ ≤ K.
Proof of Lemma 2.5. For simplicity, we skip the ǫ superscript.

1. New equation for w. The function w solves ǫ e -w(x) (min φ -1) + ǫ -tr(AD 2 w) + G(x, w, Dw) + G(x, w, Dw) = 0, (2.20) where G(x, w, p) = e -w H(x, e w p) -|σ(x) T p| 2 , G(x, w, p) = e -w H(x, e w p).

(2.21)

Definition of the test function. We define Ψ(s) =

A 1 A 2 (1 -e -A 2 s
) as in (2.10) for 0 ≤ s ≤ √ N = diameter(T N ) and set for further purpose

A 2 = 1 4ν C + 4C + C + H 0 + 2|σ| ∞ |σ x | ∞ osc(φ) and A 1 = (L + osc(φ))e A 2 √ N , (2.22)
where ν, C, C, L are the constants appearing in the assumptions (2.4), (2.6), (2.8), H 0 is defined in (2.18) and osc(φ) is bounded independently of ǫ by Lemma 2.3.

Since Ψ(0) = 0 and

Ψ( √ N ) = (L + osc(φ)) e A 2 √ N -1 A 2 ≥ L + osc(φ) > osc(φ), there exists r ∈ [0, √ N ] such that Ψ(r) = osc(φ). (2.23) We then consider max x,y∈T N {w(x) -w(y) -Ψ(|x -y|)}. (2.24)
If this maximum is nonpositive, then for all x, y ∈ T N , we have

w(x) -w(y) ≤ Ψ(|x -y|) ≤ A 1 |x -y|,
where the latter inequality follows from the concavity of Ψ. This yields the desired result.

From now on, we argue by contradiction, assuming that the maximum in (2.24) is positive. This implies that it is achieved at (x, y) ∈ T N ×T N with x = y since w is continuous. Noticing that

w(x) -w(y) ≤ e w(x)-w(y) -1 ≤ e w(x) -1 ≤ φ(x) -min φ ≤ osc(φ), (2.25) we get 0 < w(x) -w(y) -Ψ(|x -y|) ≤ osc(φ) -Ψ(|x -y|).
Using that Ψ is increasing and (2.23), we infer

|x -y| < r. (2.26)
This latter inequality is a kind of localization of points of maxima in (2.24). It will allow us to control the quadratic term coming from (2.21) in term of the oscillation, see (2.36).

Viscosity inequalities for (2.20). Writing the viscosity inequalities as in

Step 2 of the proof of Lemma 2.3, we obtain

ǫe -w(x) (min φ -1) + ǫ -trace(A(x)X) + G(x, w(x), p) + G(x, w(x), p) ≤ 0, ǫe -w(y) (min φ -1) + ǫ -trace(A(y)Y ) + G(y, w(y), p) + G(y, w(y), p) ≥ 0.
Therefore, ǫ(e -w(x) -e -w(y) )(min φ -1) -trace(A(x)X -A(y)Y ) (2.27) +G(x, w(x), p) -G(y, w(y), p) + G(x, w(x), p) -G(y, w(y), p) ≤ 0.

The end of the proof consists in reaching a contradiction in the above inequality. 4. Estimates of the terms in (2.27). From (2.18) and the fact that w(x) > w(y) ≥ 0, we have ǫ(e -w(x) -e -w(y) )(min φ -1) > -H 0 .

(2.28) From Lemma 2.4 (2.17), we have

-trace(A(x)X -A(y)Y ) ≥ -4νΨ ′′ (|x -y|) -CΨ ′ (|x -y|) + O(̺). (2.29)
Using (2.2), (2.6) and recalling that |p| = Ψ ′ (|x -y|), we get

|G(x, w(x), p) -G(y, w(y), p)| (2.30) ≤ |e -w(x) H(x, e w(x) p)| + |e -w(y) H(y, e w(y) p)| + ||σ(x) T p| 2 -|σ(y) T p| 2 | ≤ 2C(1 + Ψ ′ (|x -y|)) + 2|σ| ∞ |σ x | ∞ |x -y|Ψ ′ 2 (|x -y|).
We now estimate G(x, w(x), p) -G(y, w(y), p) using (2.8). Set P := e w(x) p and µ := e w(x)-w(y) , we have

G(x, w(x), p) -G(y, w(y), p) = e -w(x) H(x, P ) -µH(y, P µ ) . (2.31)
From the choice of A 1 in (2.22) and the concavity of Ψ, we get

|P | ≥ |p| = Ψ ′ (|x -y|) ≥ Ψ ′ (r) = A 1 e -A 2 r = (L + osc(φ))e A 2 ( √ N -r) ≥ L. (2.32)
Since the maximum in (2.24) is positive, it follows

µ ≥ 1 + w(x) -w(y) > 1 + Ψ(|x -y|) ≥ 1 + Ψ ′ (|x -y|)|x -y| ≥ 1 + L|x -y|. (2.33) From (2.25) and since |p| ≤ Ψ ′ (0) = A 1 , we notice L ≤ |p| ≤ A 1 = A 1 (σ, C, C, L, osc(φ)) and 1 + L|x -y| ≤ µ ≤ osc(φ) + 1, (2.34)
which will be useful in the proof of Theorem 2.2. It follows that we can apply (2.8) to (2.31) to get

G(x, w(x), p) -G(y, w(y), p) ≥ -Ce -w(x) |P | = -CΨ ′ (|x -y|). (2.
-4νΨ ′′ (|x -y|) -( C + 2C + C)Ψ ′ (|x -y|) -2|σ| ∞ |σ x | ∞ |x -y|Ψ ′ 2 (|x -y|) -H 0 -2C < 0. Since |x -y| ≤ r by (2.26), using that for all s ∈ [0, r], Ψ ′′ (s) + A 2 Ψ ′ (s) = 0 and |x -y|Ψ ′ 2 (|x -y|) ≤ Ψ(|x -y|)Ψ ′ (|x -y|) ≤ Ψ(r)Ψ ′ (|x -y|) = osc(φ)Ψ ′ (|x -y|), (2.36)
we can rewrite the above estimate as

4νA 2 -( C + 2C + C + 2|σ| ∞ |σ x | ∞ osc(φ)) Ψ ′ (|x -y|) -H 0 -2C < 0.
It is then straightforward to see that (2.22) leads to a contradiction in the above inequality. Finally, we obtain the result with K = A 1 .

We turn to the proof of Theorem 2.2. To use the previous lemmas, we need first to build a continuous viscosity solution of (2.5). It is straightforward to build a discontinuous viscosity solution for (2.5) using Perron's method for viscosity solutions of second order equations, see [START_REF] Ishii | Perron's method for Hamilton-Jacobi equations[END_REF][START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]. The continuity of this solution usually follows from a strong comparison principle for (2.5), i.e., a comparison principle between USC viscosity subsolutions and LSC viscosity supersolutions. But, classical structure assumptions on the first order nonlinearity H + H like Lipschitz continuity in x-variable, do not hold here. It follows that a strong comparison result may not hold. We are only able to compare a discontinuous sub-or supersolution with a Lipschitz continuous solution, see Theorem 4.1 in the Appendix. That is why, we need another approach inspired from [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] to build a continuous solution of (2.5). It is based on a truncation of the nonlinearity. Another natural approach would be to use the classical regularity theory for uniformly elliptic equations but it would not apply for degenerate equations we will consider in the next section.

Proof of Theorem 2.2. We fix ǫ > 0 and skip the ǫ-dependence in the proof for simplicity. 1. Truncated Hamiltonian. For all n > 0, we define the continuous Hamiltonian

H n := H n + H n (2.37) with H n (x, p) (resp. H n (x, p)) = H(x, p) (resp. H(x, p)) if |p| ≤ n, H(x, n p |p| ) (resp. H(x, n p |p| )) if |p| ≥ n. 2.
Construction of a continuous viscosity solution for the truncated equation. We have a strong comparison principle between discontinuous solutions for (2.5) where H, H are replaced with H n , H n respectively, see Theorem 4.2. By Perron's method, see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF][START_REF] Ishii | Perron's method for Hamilton-Jacobi equations[END_REF], there exists a continuous viscosity solution φ n . 3. Uniform Lipschitz bound for solutions of the truncated equation. We first notice that H n satisfies (2.6) with the same constant C as H. Moreover, if we choose n bigger than L which appears in (2.7), then, by Lemma 2.3, we obtain a bound for the oscillation of φ n which is independent of n, ǫ. Moreover, if n is chosen bigger than the right hand side of (2.34), then (2.8) hold for H n with the same constants as for H for all p satisfying (2.34). As noticed in the proof of Lemma 2.5, it is enough to obtain a gradient bound K for w n defined by exp(w n ) = φ n -min

T N φ n + 1.
The crucial point is that this gradient bound K does not depend on n since the constants in (2.6), (2.7), (2.8) are the same for all H n , H n . The uniform bound for the oscillation of φ n yields a L ∞ bound for w n which is independent of n, ǫ. It follows

|Dφ n | ∞ ≤ Kexp(2 √ N K). (2.38)
4. Convergence of φ n . In addition to (2.38), from (4.4), we have a L ∞ bound for φ n which is independent of n. Then, by Ascoli-Arzela's theorem, we obtain that, up to a subsequence,

φ n → φ in C(T N ) and φ = φ ǫ is Lipschitz continuous with |Dφ| ∞ ≤ Ke 2 √ N K , which is independent of ǫ. Noticing that H n → H, H n → H.
By the stability result for viscosity solution, we conclude that φ is a Lipschitz continuous viscosity solution of (2.5). 5. Uniqueness. The uniqueness of φ in the class of continuous viscosity solutions relies on a comparison principle between the Lipschitz solution φ with any continuous solution φ of (2.5), see Theorem 4.1 in the Appendix.

2.3.

Gradient bounds for degenerate elliptic equations. We now consider degenerate elliptic equations, i.e., (2.4) does not necessarily hold. In this case, we suppose that the sublinear Hamiltonian H ≡ 0 in (2.5) and we reinforce the assumptions (2.7)-(2.8) in order that the superlinear Hamiltonian H controls all the terms in the equation.

The assumptions are

        
There exists L, µ > 1 such that for all x, y ∈ T N , |p| ≥ L,

H(x, p) -µH(y, p µ ) ≥ (µ -1) |H(•, 0)| ∞ + N |σ x | 2 ∞ |p| , (2.39)
         There exists L > 1 such that for all x, y ∈ T N , |p| ≥ L and µ ≥ 1 + L|x -y|, H(x, p) -µH(y, p µ ) ≥ (µ -1) |H(•, 0)| ∞ + N |σ x | 2 ∞ |p| + 2|σ x | ∞ |σ| ∞ |p| .
(2.40)

As for the uniformly elliptic case, the first assumption is needed to get the oscillation bound and the stronger one to obtain the gradient bound. Some discussion about these assumptions and examples are given in Section 2.4.

Theorem 2.6. Assume (2.2), (2.40) and suppose that H ≡ 0. For all ǫ > 0, there exists a unique continuous viscosity solution φ ǫ ∈ C(T N ) of (2.5) and a constant K > 0 independent of ǫ such that

|Dφ ǫ | ∞ ≤ K.
The proof of the above theorem is similar to the one of Theorem 2.2, so we skip it. It relies on the auxiliary Lemmas 2.3 and 2.5 where (2.7)-(2.8) are replaced by (2.39)-(2.40). The proofs of the auxiliary lemmas follow the same lines, the changes occur in the estimates of the terms in (2. [START_REF] Evans | Adjoint and compensated compactness methods for Hamilton-Jacobi PDE[END_REF]) and (2.27), so we only rewrite Step 4 of the proof of Lemma 2.5 where the main changes occur.

Proof of Lemma 2.5 for Theorem 2.6 under assumptions (2.40). We estimate the different terms appearing in (2.27). We set P := e w(x) p and µ := e w(x)-w(y) and we recall that, if the maximum is positive in (2.24) and with a suitable choice of A 1 , A 2 , r in (2.10), then (2.32) and (2.33)). From (2.18), we get

|P | = |e w(x) p| ≥ L > 1 and µ > 1 + |p||x -y| ≥ 1 + L|x -y| (see
ǫ(e -w(x) -e -w(y) )(min φ -1) > e -w(x) (1 -µ)ǫ(min φ -1) (2.41) ≥ -e -w(x) (µ -1)|H(•, 0)| ∞ .
From Lemma 2.4 (2.17), we have

-trace(A(x)X -A(y)Y ) ≥ -e -w(x) (µ -1)N |σ x | 2 ∞ |P | + O(̺). Since H ≡ 0, we have G(x, w(x), p) -G(y, w(y), p) = -|σ(x) T p| 2 + |σ(y) T p| 2 ≥ -2|σ| ∞ |σ x | ∞ |x -y||p| 2 ≥ -2e -w(x) (µ -1)|σ| ∞ |σ x | ∞ |P |.
As far as the superlinear Hamiltonians are concerned, we have G(x, w(x), p) -G(y, w(y), p) = e -w(x) H(x, P ) -µH(y, P µ ) .

Plugging 1) .

H(x, p) -µH(y, p µ ) ≥ -C(1 + |p|) -Cµ(1 + | p µ |) ≥ -3C(1 + |p|) ≥ -6C|p|. (ii) When 0 ≤ α ≤ 1, then (2.6) holds. For α > 1, H(x, p) -µH(y, p µ ) ≥ (a -µ 1-α A)|p| α -2B|p| ≥ -2B|p| provided µ ≥ (A/a) 1/(α-
Notice that no regularity assumption (except continuity) is needed to obtain the oscillation bound. To obtain a gradient bound, we need to reinforce (2.7)-(2.39) into (2.8)-(2.40). These new assumptions contain a kind of regularity assumption with respect to (x, p). Actually, (2.8) is very close to (2.9) in [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] and (2.40) is very close to

       H ∈ W 1,∞ loc (T N × R N ) and there exists L such that if |p| ≥ L, then, for a.e. (x, p) ∈ T N × R N , L (H) p p -H -|H(•, 0)| ∞ -N |σ x | 2 ∞ |p| -2|σ| ∞ |σ x | ∞ |p| -|(H) x | ≥ 0, (2.42)
which is an extension of (2.9) for x-dependent degenerate diffusion matrices. In (2.9)-(2.42), the Hamiltonian is supposed to be locally Lipschitz with respect to (x, p).

= |B(x)p| k + b(x), p +ℓ(x) with B ∈ C(T N ; R N ×N ), b ∈ N ; R N ), ℓ ∈ C(T N ) satisfies (2.8) if |b x | ∞ , |B x | ∞ ≤ C and B(x)B(x) T > 0.
Example 2.13. Let define Ĥ(p) = Ĥ(|p|) radial by Ĥ(0) = 0 and, for all t ∈ [n, n + 1], Ĥ(t) = (n + 1)t -n(n + 1)/2. We notice that Ĥ ∈ W 1,∞ loc (R N ) and Ĥp (p)p -Ĥ = 0 a.e. It follows that H(x, p) = Ĥ(p) + ℓ(x) satisfies (2.8) if ℓ is continuous but does not satisfy (2.40) (even if ℓ is Lipschitz continuous). The Hamiltonian H(x, p) = H(x, p)+ Ĥ(p)+ ℓ(x) where H satisfies (2.6) fulfills the assumptions of Theorem 2.2 if the diffusion matrix satisfies (2.2)-(2.4) but such a case cannot be handled with the results of [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF].

Remark 2.14. We can extend the results with easy adaptations to fully nonlinear equations like

ǫφ ǫ + sup θ∈Θ {-trace(A θ (x)D 2 φ ǫ ) + H θ (x, Dφ ǫ ) + H θ (x, Dφ ǫ )} = 0, x ∈ T N , (2.43)
where Θ is a compact metric space and there exists a constant C > 0 such that

A θ = σ θ σ T θ satisfies |σ θ (x)| ≤ C, |σ θ (x) -σ θ (y)| ≤ C|x -y|, x, y ∈ T N , θ ∈ Θ,
and both H θ , H θ are continuous satisfying, for all R > 0, there exists a modulus of continuity m R such that

|H θ (x, 0)| ≤ C, |H θ (x, p) -H θ (y, p)| ≤ m R (|x -y|), x, y ∈ T N , |p| ≤ R, θ ∈ Θ. ( 2

Asymmetric Systems

We consider the weakly coupled system

eφ ǫ 1 -trace(A 1 (x)D 2 φ 1 ) + H 1 (x, Dφ 1 ) + φ 1 -φ 2 = 0, x ∈ T N , eφ ǫ 2 -trace(A 2 (x)D 2 φ 2 ) + H 2 (x, Dφ 2 ) + φ 2 -φ 1 = 0, (3.1)
where the A i 's and H i 's satisfy the steady assumption (2.2).

3.1.

Gradient bounds for systems. The aim is to obtain uniform gradient bounds (i.e., independent of ǫ) for (3.1) when H i = H i + H i . We define

H := sup x∈T N ,1≤j≤2 |H j (x, 0)|. (3.2)
and each equation satisfies a different set of assumptions. (3.4) tells that the second one may be degenerate with superlinear Hamiltonian. So the system is asymmetric. This case is the one in interest in this work but let us mention that the case when both equations satisfy either (3.3) or (3.4) is also possible with easier arguments in the proof of the theorem which follows. See Section 3.5 for examples and extensions.

A 1 , H 1 ,
ǫ = (φ ǫ 1 , φ ǫ 2 ) ∈ C(T N ) 2 of (3.1)
and K > 0 depending only on the H i 's such that |Dφ ǫ i | ∞ ≤ K for all ǫ > 0, i = 1, 2. Similarly to the case of scalar equations, the proof consists in two main steps: first, we prove an uniform bound for the oscillation and we then improve it to a uniform gradient bound. The key lemmas are Lemma 3.2. Under the hypotheses of Theorem 3.1. Let φ ǫ = (φ ǫ 1 , φ ǫ 2 ) ∈ C(T N ) 2 be a solution of (3.1). There exists K > 0 depending only on the H i 's such that osc(φ ǫ i ) ≤ K for all ǫ > 0, i = 1, 2. We skip the proof of Lemma 3.2 since it is similar to the case of scalar equations. Lemma 3.3. Under the hypotheses of Theorem 3.1, let φ ǫ be a continuous viscosity solution of (3.1) and define w ǫ by exp(w ǫ i ) = φ ǫ i -min

T N φ ǫ i + 1 for all ǫ, i.
Then, there exists a constant K independent of ǫ such that |Dw ǫ i | ∞ ≤ K for all ǫ, i. Proof of Theorem 3.1. It is sufficient to see that the proof of Theorem 2.2 can be extended to systems like (3.1). For i = 1, 2, we define H in like in (2.37). By Perron's method for systems, see [START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF] and Theorem 4.2, there exists a continuous viscosity solution φ n of (3.1) where the H i are replaced with the H in 's. It is now possible to apply Lemma 3.3 to φ n to obtain a gradient bound which is independent of ǫ, n. We conclude as in the proof of Theorem 2.2.

Proof of Lemma 3.3. The proof is almost the same as the one for scalar equations. The main change is the estimate (3.7). So we only write some important steps. 1. Set m i = min T N φ i , the new equation satisfied by w i is

-trace(A i (x)D 2 w i ) + G i (x, w i , Dw i ) + G i (x, w i , Dw i ) + b i (w i ) + 1 -e w i+1 -w i = 0,
where

G i (x, w, p) = e -w H i (x, e w p) -|σ i (x) T p| 2 , G i (x, w, p) = e -w H i (x, e w p), b i (w) = e -w [m i -m i+1 + ǫ(m i -1)] + ǫ,
where we identify m 3 = m 1 and w 3 = w 1 .

Consider max

T N ×T N , i=1,2 {w i (x) -w i (y) -Ψ(|x -y|)}, (3.5)
where Ψ is of the form in (2.10) with A 1 , A 2 to be chosen later. We are done if the maximum is nonpositive. 2. Otherwise, the maximum is positive and hence is achieved at (x, y) with x = y for some i ∈ {1, 2}. So we can write the viscosity inequalities for the ith equation at this point. For every ̺ > 0, there exist (p, X) ∈ J

G i + G i + B + C ≤ 0, (3.6)
where

G i = -tr(A i (x)X -A i (y)Y ) + G i (x, w i (x), p) -G i (y, w i (y), p), G i = G i (x, w i (x), p) -G i (y, w i (y), p), B = b i (w i (x)) -b i (w i (y)),
C = e w i+1 (y)-w i (y) -e w i+1 (x)-w i (x) , where we identify w 3 = w 1 .

Since the maximum in (3.5) is positive, we have C ≥ 0. Next, we claim that Since we have different assumptions for each equation, we distinguish two cases: "uniformly elliptic" (Step 3) and "degenerate elliptic" (step 4) to get a contradiction in (3.6). 3. When i = 1, since the first equation is "uniformly elliptic", we use the arguments of Step 4 in the proof of Lemma 2.5. We obtain

m i -m i+1 + ǫm i ≤ 3H, ( 3
G 1 + G 1 + B + C (3.8) ≥ -4νΨ ′′ -( C + 2C + C + 2|σ 1 | ∞ |(σ 1 ) x | ∞ |x -y|Ψ ′ )Ψ ′ -2C -3H.
With a suitable choice of A 1 , A 2 in the definition of Ψ, there exists r ≥ |x -y| such that |x -y|Ψ ′ ≤ Ψ(r) = max j osc(φ j ), which is bounded by Lemma 3.2. It is then possible to make the right-hand side of (3.8) positive. 4. When i = 2, since the second equation is "degenerate elliptic" so we control the 0th order terms using the superlinear Hamiltonian. We repeat readily the arguments of the proof of Lemma 2.5 in Section 2.3 (case of degenerate elliptic equations) estimating G 2 using now (3.4). We omit the details.

Finally Steps 3 and 4 lead to a contradiction in (3.6). 5. It remains to prove (3.7). Let x 1 such that φ 1 (x 1 ) = min T N φ 1 . Using the first equation of (3.1), we get min

T N φ 2 -min T N φ 1 ≤ φ 2 (x 1 ) -min T N φ 1 ≤ ǫφ 1 (x 1 ) + H 1 (x 1 , 0) ≤ 2H.
Taking into account (4.4), we get the desired inequality.

3.2.

Strong maximum principle for systems. The following extension of the strong maximum principle to parabolic systems is a crucial ingredient in the proof of the large time behavior.

Theorem 3.4. Assume that, for i = 1, 2, A i (x) ≥ ν i (x)I with ν i ∈ C(T N ), ν i ≥ 0 and for all x ∈ T N , i=1,2 ν i (x) > 0. (3.9)
If u is a continuous viscosity subsolution of

           ∂u 1 ∂t -trace(A 1 (x)D 2 u 1 ) -C|Du 1 | + u 1 -u 2 = 0, x ∈ T N ×(0, +∞), ∂u 2 ∂t -trace(A 2 (x)D 2 u 2 ) -C|Du 2 | + u 2 -u 1 = 0, u i (x, 0) = u 0i (x), x ∈ T N , (3.10)
which attains a maximum at (x, t) ∈ T N × (0, +∞), then u is constant. (ii) This result contains, as a particular case, stationary systems.

Proof of Theorem 3.4.

Suppose that

M := sup (x,t)∈T N ×[0,+∞), j=1,2 u j (x, t) = u i (x, t), t > 0. (3.11)
We do a formal calculation. A rigorous one can be made using classical viscosity techniques. At the point (x, t), we have ∂u i ∂t = 0, Du i = 0 and -D 2 u i ≥ 0. From the ith equation, we obtain

u i (x, t) -u i+1 (x, t) ≤ 0, where we identify u 3 = u 1 . Therefore u 1 (x, t) = u 2 (x, t). It follows that (x, t) is the common maximum point of u 1 , u 2 in T N × [0, +∞).
2. By (3.9), for all x ∈ T N , there exists j ∈ {1, 2} such that ν j (x) > 0. By continuity, there exists r x > 0 such that B(x, r x ) ⊂ Ω j := {x ∈ T N : ν j (x) > 0}. It follows

T N = x∈T N B(x, r x )
and, by compactness, there exists a finite covering

T N = n p=1
B p with, for all p, B p ⊂ Ω j for some j ∈ {1, 2}. (3.12) It follows that there exist p, j such that x ∈ B p ⊂ Ω j . By continuity of ν j and compactness, we have inf y∈Bp,|ξ|=1

A j (y)ξ, ξ ≥ inf y∈Bp ν j (y) =: ν > 0.
Hence the jth equation is uniformly parabolic in B p . Moreover, the maximum in (3.11) is also a maximum on

B p × [0, +∞). We set u M k = u k -M ≤ 0 for all k. So we have ∂u M j ∂t -trace(A j D 2 u M j ) -C|Du M j | + u M j ≤ ∂u M j ∂t -trace(A j D 2 u M j ) -C|Du M j | + u M j -u M j+1 identifying u 3 = u 1 , = ∂u j ∂t -trace(A j D 2 u j ) -C|Du j | + u j -u j+1 ≤ 0 in B p .
By the strong maximum principle for viscosity solutions of single parabolic equations (see, e.g., [START_REF] Lio | Remarks on the strong maximum principle for viscosity solutions to fully nonlinear parabolic equations[END_REF]), we obtain u M j ≡ 0 in B p × [0, +∞). Coming back to Step 1, we infer that the maximum in (3.11) is achieved for i = 1, 2 at every (y, t) ∈ B p × [0, +∞). 3. By (3.12), there exist p ′ and x ′ such that x ′ ∈ B p ∩ B p ′ . It follows that the u j 's achieve their maximum over B p ′ × [0, +∞) at (x ′ , t). Repeating Step 2, we conclude that the u j 's are constant in B p ′ × [0, +∞). From (3.12), we conclude that u j ≡ M for all j and (x, t) ∈ T N × [0, +∞). The following assumption is used to "linearize" the system in order to apply the strong maximum principle-Theorem 3.4.

H i ∈ W 1,∞ loc (T N × R N ). (3.13)
Theorem 3.6 (Ergodic problem). Suppose that the assumptions of Theorem 3.1 hold for (3.1). Then, there exists a solution (c, v) ∈ R 2 × W 1,∞ (T N ) 2 of (1.4). The ergodic constant c is unique and c = (c 1 , c 1 ). If, in addition, the H i 's satisfy (3.13), then v is unique up to an additive constant vector.

For m = 1, we find the classical results for scalar equations (see [START_REF] Lions | Homogenization of Hamilton-Jacobi equations[END_REF][START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF]).

Proof of Theorem 3.6. Let φ ǫ be the continuous solution of (3.1) given by Theorem 3.1. We first claim that there exists a constant C ′ independent of ǫ such that

φ ǫ i (x) -φ ǫ i+1 (x) ≤ C ′ , x ∈ T N , i = 1, 2 and we identify φ ǫ 3 = φ ǫ 1 . (3.14)
Indeed, choosing x i as the maximum point of φ ǫ i , using the ith equation and (4.4), we have

| max φ ǫ 1 -max φ ǫ 2 | ≤ C.
From Theorem 3.1, we obtain

φ ǫ i (x) -φ ǫ i+1 (x) = [max φ i -max φ i+1 ] + [φ ǫ i (x) -φ ǫ i (x i )] + [φ ǫ i+1 (x i+1 ) -φ ǫ i+1 (x)] ≤ C + osc(φ 1 ) + osc(φ 2 ).
The lower bound is established in the same way by introducing x i as the minimum point of φ i .

Fix x * ∈ T N and set

ρ ǫ i := φ ǫ i (x * ) -φ ǫ i+1 (x * ) and v ǫ i (x) := φ ǫ i (x) -φ ǫ i (x *
) for all i = 1, 2. From Theorem 3.1, (4.4) and (3.14), using Ascoli-Arzela's theorem, there exists a sequence ǫ k → 0 so that

v ǫ k i → v i , ǫ k φ ǫ k i → -c i , ρ ǫ k i → ρ i , uniformly on T N as k → ∞ for some v i ∈ W 1,∞ (T N ), c i , ρ i ∈ R. Note that c i is a constant independent of x * while v i , ρ i depend on x * .
Now, we rewrite (3.1) as

ǫφ ǫ 1 -trace(A 1 (x)D 2 v ǫ 1 ) + H 1 (x, Dv ǫ 1 ) + v ǫ 1 -v ǫ 2 + ρ ǫ 1 = 0, ǫφ ǫ 2 -trace(A 2 (x)D 2 v ǫ 2 ) + H 2 (x, Dv ǫ 2 ) + v ǫ 2 -v ǫ 1 + ρ ǫ 2 = 0.
(3.15)

Passing to the limit along the subsequence ǫ k , by stability, we obtain that

v = (v 1 , v 2 ) ∈ W 1,∞ (T N ) 2 is solution of -trace(A 1 (x)D 2 v 1 ) + H 1 (x, Dv 1 ) + v 1 -v 2 + ρ 1 = c 1 , -trace(A 2 (x)D 2 v 2 ) + H 2 (x, Dv 2 ) + v 2 -v 1 + ρ 2 = c 2 .
Multiplying (3.15) by ǫ and passing to the limit along the subsequence ǫ k , we get

c 1 = c 2 . Note that ρ ǫ 1 + ρ ǫ 2 = 0, so ρ 1 = -ρ 2 . We conclude by setting ṽ1 (x) = v 1 (x) + ρ 1 , ṽ2 = v 2 (x).
We finally mention that we can easily see the uniqueness of the ergodic constant by the comparison principle for (1.2) (Theorem 4.3). We claim that the uniqueness up to constant of solutions comes from the strong maximum principle (Theorem 3.4). Indeed, let v, ṽ be Lipschitz continuous solutions of (1.4) (the constant c is the same by the above). Since |Dv| ∞ , |Dṽ| ∞ ≤ K for some K and H i ∈ W 1,∞ (B(0, K)), classical arguments in viscosity solutions imply that v -ṽ is a viscosity subsolution of the stationary version of (3.10) in T N . Therefore v = ṽ + C where C ∈ R m is a constant.

3.4.

Large time behavior result. We first give a general result and then apply it to asymmetric systems. Theorem 3.7. (Large time behavior) Suppose that (3.9), (3.13) hold. Suppose there exists a viscosity solution

u ∈ C(T N × [0, +∞)) 2 of (1.2), a solution ((c 1 , c 1 ), v) ∈ R 2 × W 1,∞ (T N ) 2 of the ergodic problem (1.4) and K > 0 such that |Du i (•, t)| ∞ , ≤ K, t ≥ 0, i = 1, 2.
Then, there exists a constant ℓ ∈ R such that u(x, t) + (c 1 , c 1 )t + (ℓ, ℓ) → v(x) unif ormly as t → +∞.

Proof of Theorem 3.7. Several parts of this proof are inspired by [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF]. 1. Since |Du i (•, t)| ∞ ≤ K, we deduce from Theorem 4.3 that u is the unique continuous viscosity solution of (1.2).

Defining c = (c 1 , c 1 ), we see that v ± (x, t) = v(x) -ct ± (C, C) is a super and subsolution, respectively, of (1.2) for C large enough. Thanks to the comparison principle, we have

v i (x) -C ≤ u i (x, t) + ct ≤ v i (x) + C, x ∈ T N , i = 1, 2.
It follows that u i (x, t) + ct is bounded; by the change of function u → u + ct, we may assume without loss of generality that u is bounded and the ergodic constant c is 0.

Set m(t) = max x∈T N , i=1,2 {u i (x, t) -v i (x)}. The comparison principle claims that m is nonincreasing and hence, m(t) → ℓ as t → ∞. 2. A by-product of Step 1 is that {u(•, t), t > 0} is relatively compact in W 1,∞ (T N ) 2 . So we can extract a sequence, t j → +∞ such that u(•, t j ) → u ∈ W 1,∞ (T N ) 2 . Applying Theorem 4.3 for (1.2), we obtain

|u i (x, t + t p ) -u i (x, t + t q )| ≤ max y∈T N , j=1,2
|u j (y, t p ) -u j (y, t q )|, x ∈ T N , t ≥ 0, p, q ∈ N, which proves that (u(•, • + t p )) p is a Cauchy sequence in C(T N × [0, +∞)) 2 . We call u ∞ its limit. Notice, on the one hand, that |Du ∞i (•, t)| ∞ ≤ K for all i, t and, on the other hand, by stability, u ∞ is solution of (1.2) with initial data u.

3. Using the uniform convergence of u i (., t + t j ), we pass to the limit with respect to j in

m(t + t j ) = max i,x (u i (x, t + t j ) -v i (x)) to obtain ℓ = max i,x (u ∞i (x, t) -v i (x)) for any t > 0. 4. Since u ∞ is solution of (1.
2) and v is solution of (1.4) and, up to increase K, both are K-Lipschitz continuous in x, thanks to the Lipschitz continuity of H i with respect to p, see (3.13), we obtain that u ∞ -v is subsolution of (3.10). The strong maximum principle (Theorem 3.4), then implies

ℓ = u ∞i (x, t) -v i (x) (x, t) ∈ T N × [0, +∞), i = 1, 2.
Noticing that (ℓ, ℓ) + v(x) does not depend on the choice of subsequences, we obtain u i (x, t)ℓ -v i (x) → 0 uniformly in x as t → ∞, for i = 1, 2.

We apply the previous result for the particular systems we studied in Section 3.

Corollary 3.8. (Large time behavior) Assume (3.9), (3.13) and suppose that the assumptions of Theorem 3.1 are in force. Then, for any initial condition u 0 ∈ W 1,∞ (T N ) 2 , (i) The system (1.2) has a unique viscosity solution u ∈ C(T N × [0, +∞)) 2 and there exists K > 0 such that |Du i (•, t)| ∞ ≤ K for all t ≥ 0, i = 1, 2. (ii) There exist a unique ergodic constant c = (c 1 , c 1 ) ∈ R and v ∈ W 1,∞ (T N ) 2 solution of (1.4) such that u(x, t) + ct → v(x) uniformly as t → +∞.

Remark 3.9. Remember that the classical assumptions on the Hamiltonians in (1.2) do not hold in order to have comparison and uniqueness of viscosity solutions. It is why, we prove first the existence of a (Lipschitz) continuous in space viscosity solution of (1.2). To this purpose, we need the initial condition to be Lipschitz continuous though this regularity assumption is not necessary to obtain the long time behavior.

Proof of Corollary 3.8.

1. We introduce the truncated evolutive system (1.2) in T N × [0, +∞) with the H in defined by (4.5). From Theorem 4.3 and Perron's method, there exists a unique continuous viscosity solution u n in T N × [0, +∞).

In the following, M is a constant which may vary line to line but is independent of n. 2. Applying Theorem 3.6 with the H in 's, we obtain the existence of (

c n , v n ) ∈ R 2 ×W 1,∞ (T N ) 2 .
Since the H in 's satisfy (3.3)-(3.4) with constants independent of n for large n, and c n → c when n → +∞, we obtain from Theorem 3.

1 that |v n | ∞ , |Dv n | ∞ ≤ M.
3. Noticing that v n (x) -c n t satisfies (1.2) with the H in 's, by Theorem 4.3, we obtain

v ni (x) -M ≤ u ni (x, t) + c ni t ≤ v ni (x) + M, (x, t) ∈ T N × [0, +∞), i = 1, 2. Therefore |u n (•, t) + c n t| ∞ , osc(u n (•, t)) ≤ M. 4. We claim that |Du n (•, t)| ∞ ≤ M.
To prove this fact, we repeat the proof of Lemma 3.3 for the evolutive system (1.2) with the H in 's. Let T > 0 and consider max

(x,y)∈(T N ) 2 ,t∈[0,T ],1≤i≤m {w ni (x, t) -w ni (y, t) -ψ(|x -y|)},
where e w ni (x,t) = u ni (x, t) -min T N u ni (•, t) + 1, and ψ is given by (2.10). If the maximum is nonpositive we are done. Otherwise it is positive and achieved at some (x, y, t, i) with x = y and we can write the viscosity inequalities for the ith equation, see [START_REF] Crandall | User's guide to viscosity solutions of second order partial differential equations[END_REF]Theorem 8.3]: For every ̺ > 0, there exist (a, p, X) ∈ P

Consider            ∂u 1 ∂t -d 1 (x)∆u 1 + a 1 (x)|Du 1 | 1+α 1 + u 1 -u 2 = f 1 (x), x ∈ T N × (0, +∞), ∂u 2 ∂t -d 2 (x)∆u 2 + a 2 (x)|Du 2 | 1+α 2 + u 2 -u 1 = f 2 (x), x ∈ T N × (0, +∞), u 1 (x, 0) = u 01 (x), u 2 (x, 0) = u 02 (x), x ∈ T N ,
where d i ≥ 0, a i , f i , u 0i are Lipschitz continuous, α i > 0 and a i > 0. Suppose moreover that for any x ∈ T N , we have either d 1 (x) > 0 or d 2 (x) > 0 (this implies (3.9)). So, the hypotheses of Corollary 3.8 hold.

When there is at least one sublinear Hamiltonian, we need that the diffusion matrix of the corresponding equation is uniformly elliptic. But we permit other diffusion matrices to be degenerate everywhere.

Example 3.11. (Asymmetric systems with degenerate equations) In the following system, the first equation is of sublinear type while the second is of superlinear type,

           ∂u 1 ∂t -trace(A 1 D 2 u 1 ) + sup θ∈Θ {-b θ1 (x), Du 1 -f θ1 (x)} + u 1 -u 2 = f 1 (x), ∂u 2 ∂t -trace(A 2 D 2 u 2 ) + a 2 (x)|Du 2 | 1+α + u 2 -u 1 = f 2 (x), x ∈ T N × (0, +∞), u 1 (x, 0) = u 01 (x), u 2 (x, 0) = u 02 (x), x ∈ T N , (3.16)
where the functions b θ1 , f θ1 , a 2 , f 2 , u 01 , u 02 satisfy

|f (x)| ≤ C, |f (x) -f (y)| ≤ C|x -y|, x, y ∈ T N , θ ∈ Θ (3.17)
and α > 0 and a 2 (x) > 0. If we suppose moreover that A 1 > 0 in T N (notice that we only assume A 2 ≥ 0), then the hypotheses of Corollary 3.8 hold.

We end with some possible extensions to fully nonlinear systems with m ≥ 2 equations of the form (1.1) with H θi = H θi + H θi . We assume that the system is monotone and irreducible, i.e., the coupling matrix D = (d ij ) 1≤i,j≤m satisfies In this case, it is possible to find Λ ∈ R m with positive components Λ i > 0 such that D T Λ = 0 ( [START_REF] Camilli | Large time behavior of weakly coupled systems of firstorder Hamilton-Jacobi equations[END_REF]) and to define λ D := max 1≤i≤m 1 Λ i j =i Λ j . Then it is possible to generalize the previous results: We assume that A θi satisfies (2.43) and H θi , H θi satisfy (2.44).

d ii ≥ 0, d ij ≤ 0 for i = j and
Xζ, ζ -Y ξ, ξ ≤ Ψ ′ ζ -ξ, B(ζ -ξ) + Ψ ′′ ζ -ξ, (q ⊗ q)(ζ -ξ) + O(̺).
We estimate trace(A(x)X) and trace(A(y)Y ) using two orthonormal bases (e 1 , • • • , e N ) and (ẽ 1 , • • • , ẽN ) in the following way:

T := trace(A(x)X -A(y)Y ) = N i=1 Xσ(x)e i , σ(x)e i -Y σ(y)ẽ i , σ(y)ẽ i ≤ N i=1 Ψ ′ ζ i , Bζ i + Ψ ′′ ζ i , (q ⊗ q)ζ i + O(̺) ≤ Ψ ′′ ζ 1 , (q ⊗ q)ζ 1 + N i=1 Ψ ′ ζ i , Bζ i + O(̺), (4.1)
where we set ζ i = σ(x)e i -σ(y)ẽ i and noticing that Ψ ′′ ζ i , (q ⊗ q)ζ i = Ψ ′′ ζ i , q 2 ≤ 0 since Ψ is concave.

We now build suitable bases in two following cases. In the case where σ is degenerate, we choose any orthonormal basis such that e i = ẽi . It follows

T ≤ N i=1 Ψ ′ (σ(x) -σ(y))e i , B(σ(x) -σ(y))e i + O(̺) ≤ Ψ ′ N |σ(x) -σ(y)| 2 |B| + O(̺) ≤ Ψ ′ N |σ x | 2 ∞ |x -y| + O(̺) from (2.2) and since |B| ≤ 1/|x -y|.
When (2.4) holds, i.e., A(x) ≥ νI for every x, we can set

e 1 = σ(x) -1 q |σ(x) -1 q| , ẽ1 = - σ(y) -1 q |σ(y) -1 q|
, where q is given by (2.14).

If e 1 and ẽ1 are collinear, then we complete the basis with orthogonal unit vectors e i = ẽi ∈ e ⊥ 1 , 2 ≤ i ≤ N. Otherwise, in the plane span{e 1 , ẽ1 }, we consider a rotation R of angle π 2 and define

e 2 = Re 1 , ẽ2 = -Rẽ 1 .
Finally, noticing that span{e 1 , e 2 } ⊥ = span{ẽ 1 , ẽ2 } ⊥ , we can complete the orthonormal basis with unit vectors e i = ẽi ∈ span{e 1 , e 2 } ⊥ , 3 ≤ i ≤ N.

From (2.4), we have

ν ≤ 1 |σ(x) -1 q| 2 ≤ |σ| 2 ∞ ≤ C 2 . (4.2) It follows ζ 1 , (q ⊗ q)ζ 1 = 1 |σ(x) -1 q| + 1 |σ(y) -1 q| 2 ≥ 4ν.
From (2.14), we deduce Bq = 0. Therefore

ζ 1 , Bζ 1 = 0. For 3 ≤ i ≤ N, using (2.2), ζ i , Bζ i = (σ(x) -σ(y))e i , B(σ(x) -σ(y))e i ≤ |σ x | 2 ∞ |x -y| ≤ √ N C 2 .
since |B| ≤ 1/|x -y| and |x -y| ≤ √ N . We have

|ζ 2 | = |(σ(x) -σ(y))Re 1 + σ(y)R(e 1 + ẽ1 )| ≤ C|x -y| + C|e 1 + ẽ1 |.
It remains to estimate

|e 1 + ẽ1 | ≤ 1 |σ θ (x) -1 q| |σ(x) -1 q -σ(y) -1 q| + |σy) -1 q| 1 |σ(x) -1 q| - 1 |σ(y) -1 q| ≤ 2|σ| ∞ |σ x | 2 ∞ ν |x -y| = 2C 3 ν |x -y|, from (4.2) and |(σ -1 ) x | ∞ ≤ |σ x | 2 ∞ /ν. From (4.1), we finally obtain the conclusion T ≤ 4νΨ ′′ + CΨ ′ + O(̺) where C = C(N, ν, |σ| ∞ , |σ x | ∞ ) := C 2 √ N (N -2 + (1 + 2C 3 ν ) 2 ) with C := max{|σ| ∞ , |σ x | ∞ }. (4.3) 4.2.
Comparison principles for stationary systems. The following results are stated in [START_REF] Barles | Space-time periodic solutions and long-time behavior of solutions to quasi-linear parabolic equations[END_REF] in the case of a scalar equation without control. We state them for systems. Theorem 4.1. Let ψ ǫ ∈ U SC(T N ) 2 and φ ǫ ∈ LSC(T N ) 2 be respectively a viscosity subsolution and supersolution of (1.12). Assume that either ψ ǫ or φ

ǫ ∈ W 1,∞ (T N ) 2 . Then ψ i ≤ φ i in T N for i = 1, 2.
A useful consequence of Theorem 4.1 is that, for any viscosity solution φ ǫ of (1.12), After subtracting write the viscosity inequalities at (x, y) of (1.12), we get ǫ(ψ i (x) -φ i (y)) -trace(A i (x)X -A i (y)Y ) + H i (x, p) -H i (y, p)} +[ψ i (x) -φ i (y)] -[ψ i+1 (x) -φ i+1 (y)] ≤ 0.

|ǫφ ǫ i | ∞ ≤ sup
Since ψ j (x) -φ j (y) ≤ ψ i (x) -φ i (y), we get [ψ i (x) -φ i (y)] -[ψ i+1 (x) -φ i+1 (y)] ≥ 0. Moreover -trace(A i (x)X -A i (y)Y ) ≥ o α (1). Therefore ǫ(ψ i (x) -φ i (y)) ≤ o α (1), this yields M 0 ≤ 0 as desired.

In the proofs of Theorems 2.2 and 3.1, we need a discontinuous comparison result for truncated system when H i in (1.12) is replaced by Due to the truncation in the gradient variable, we do not need to assume that the sub-or supersolution is Lipschitz continuous. The proof follows the same lines as above. We obtain Theorem 4.2. Let ψ ∈ U SC(T N ) 2 and φ ∈ LSC(T N ) 2 be respectively a viscosity subsolution and supersolution of (1.12) with H in defined by (4.5). Then ψ i ≤ φ i in T N for all i. The proofs are easy adaptations of the proofs of Theorems 4.1 and 4.2 in the case of degenerate parabolic systems.

H in := H i (x,

4.4.

A Control-theoretic interpretation for (3.16). We can give an interpretation of weakly coupled systems as dynamic programming equations of hybrid systems with pathwise stochastic trajectories with random switching, see [START_REF] Fleming | Risk-sensitive production planning of a stochastic manufacturing system[END_REF].

Let (Ω, F, (F t ) t≥0 , P) be a filtered probability space, W t be a F t -adapted standard N -Brownian motion such that W 0 = 0 a.s. Consider the controlled random evolution process (X t , ν t ) with dynamics dX t = b θtνt (X t )dt + √ 2 σ νt (X t )dW t , t > 0, (X 0 , ν 0 ) = (x, i) ∈ T N × {1, 2}, where the control law θ t : [0, ∞) → Θ is a measurable function.

For every θ t there exists a F t -adapted solution (X t , ν t ), where X t : [0, ∞) → T N is piecewise C 1 and ν t is a continuous-time Markov chain with state space {1, 2} and probability transitions given by By choosing Θ = R N , setting A i = σ i σ T i , i = 1, 2 and assuming that b θ1 , f θ1 , u 01 , u 02 satisfy (3.17), b θ2 = 2θ and f θ2 = |θ| 2 -f 2 with f 2 continuous, we obtain (3.16) with a 2 = 1 and α = 1. If moreover A 1 > 0, then the first equation is uniformly parabolic of sublinear type and the second one is possibly degenerate superlinear with a quadratic Hamiltonian. The assumptions of Theorems 3.1, 3.6 and 3.8 hold.

Roughly speaking, the Lipschitz regularization of u 1 is provided by the nondegenerate diffusion in mode 1 whatever the bounded drift does. While the Lipschitz regularity of u 2 comes from the controlable unbounded drift in mode 2 even if the diffusion degenerates. Assumption (3.9) is obviously satisfied since ν 1 (x) := min |ξ|=1 A 1 (x)ξ, ξ > 0 for all x ∈ T N . So the strong maximum principle holds and we have the convergence when t → +∞. The point is that the nondegeneracy of A 1 implies also the convergence for u 2 . This seems to be due to the combined effects of the nondegenerate Brownian motion in mode 1 together with the number of switchings which tends to +∞ as t → +∞ (since the transition probabilities (4.7) are positive and independent of t). It follows that the process visits any open subset of T N in each mode as t → +∞ yielding the convergence to an equilibrium state for (u 1 , u 2 ).

Theorem 3 . 1 .

 31 Assume (3.3)-(3.4). There exists a unique continuous viscosity solution φ

. 7 )

 7 see Step 5 for its proof. It follows easily that B ≥ -3H.

Remark 3. 5 .

 5 (i) A new feature of the above result is that one only need a partial nondegeneracy condition for the diffusion matrices A i in the following sense. At each point of T N , there exists at least one equation such that A i (x) is nondegenerate. It can be interpreted using optimal control as follows. When considering the stochastic control problem associated with the equation in(3.10), it means that the controlled process visits any open set of T N almost surely for any open time interval (see Section 4.4 for the control interpretation).

3. 3 .

 3 Ergodic problem. The uniform gradient bound established for (3.1) allows us to solve the ergodic problem.

d

  ij = 0 for all i, for all subset I {1, • • • , m}, there exists i ∈ I and j ∈ I such that d ij = 0.

•

  Theorem 3.1 holds if, for each 1 ≤ i ≤ m, the ith-equation satisfies, uniformly with respect to θ, either (3.3) or (3.4) with |H(•, 0)| replaced by (2λ D + 1)H where H is defined in (3.2) with a supremum over all 1 ≤ j ≤ m, θ ∈ Θ. Notice that, when m = 1 (resp. m = 2), λ D = 0 (resp. λ D = 1) and we recover exactly Theorem 2.6 (resp. Theorem 3.1). • Theorem 3.6 holds under the assumptions above and (3.13) uniformly with respect to θ. • Corollary 3.8 holds under the assumptions above and (3.9) with A θi ≥ ν i I.

4 . Appendix 4 . 1 .

 441 Proof of Lemma 2.4. From (2.13), for every ζ, ξ ∈ R N , we have

j=1, 2

 2 |H j (•, 0)| ∞ =: H, i = 1, 2. (4.4)Proof of Theorem 4.1. We only give a sketch of proof since it is classical. We suppose without loss of generality that supi |Dψ i | ∞ =: L < ∞. Set M 0 = max x∈T N , j=1,2 {ψ j (x) -φ j (x)}, M α = max x,y∈T N , j=1,2 {ψ j (x) -φ j (y) -α 2 |x -y| 2 } = ψ i (x) -φ j (y) -α 2 |x -y| 2 .We have some classical estimates α 2 |x -y| 2 → α→∞ 0, |p| ≤ L where p = 2α 2 (x -y), lim sup α→∞ M α = M 0 .

4. 3 .Theorem 4 . 3 .. 6 ) 4 . 4 .

 343644 Comparison for the evolutive problem. Let u ∈ U SC(T N × [0, +∞)) 2 and v ∈ LSC(T N × [0, +∞)) 2 be respectively a viscosity subsolution and supersolution of (1.2). Assume that either u(•, t) or v(•, t) ∈ W 1,∞ (T N ) 2 uniformly in t ∈ [0, +∞). Then u i (x, t) -v i (x, t) ≤ sup y∈T N ,j=1,2 (u j (y, 0) -v j (y, 0)) + , (x, t) ∈ T N × [0, +∞), i = 1, 2. (4Theorem Let u ∈ U SC(T N × [0, +∞)) 2 and v ∈ LSC(T N × [0, +∞))2 be respectively a viscosity subsolution and supersolution of (1.2) with H in defined by (4.5). Then (4.6) holds.

P{ν t+∆t = 2 |

 2 ν t = 1} = ∆t + o(∆t), P{ν t+∆t = 1 | ν t = 2} = ∆t + o(∆t). (4.7)We introduce the value functions of the optimal control problemsu i (x, t) = inf θt∈L ∞ ([0,t],Θ) E x,i { t 0 f θsνs (X s )ds + u 0νt (X t )}, i = 1, 2,where E x,i denote the expectation of a trajectory starting at x in the mode i.The function u = (u 1 , u 2 ) satisfies the system σ 1(x)σ 1 (x) T D 2 u 1 ) + sup θ∈Θ {-b θ1 (x), Du 1 -f θ1 (x)} + u 1 -u 2 = 0 ∂u 2 ∂t -trace(σ 2 (x)σ 2 (x) T D 2 u 2 ) + sup θ∈Θ {-b θ2 (x), Du 2 -f θ2 (x)} + u 2 -u 1 = 0u 1 (x, 0) = u 01 (x), u 2 (x, 0) = u 02 (x).

  the previous estimates in(2.27) and applying (2.40), we reach a contradiction with the strict inequality in (2.41). It ends the proof.2.4. Comments on the assumptions, examples and extensions. then (2.3) holds. For instance, if H(x, p) = a(x)sin(h(|p|))|p| 1+α + ℓ(x), where α > 0, a > 0, ℓ, h are continuous and lim sup r→+∞ sin(h(r)) > 0, then (2.3) holds. If there exists α ≥ 0, a > 0, A, B, L ≥ 0 such that, for all x ∈ T N , |p| ≥ L, A|p|

	Example 2.7. (Hamiltonian satisfying (2.3)) If lim sup |p|→+∞ respect to x Lemma 2.8.	H(x,p) |p|	= +∞ uniformly with
	(i) A sublinear Hamiltonian H satisfying (2.6) satisfies (2.7).
	(ii)

α + B|p| ≥ H(x, p) ≥ a|p| α -B|p|, then (2.7) holds Proof of Lemma 2.8. (i) Using (2.6), we have, for µ = 2 and |p| > 1,

  When the Hamiltonians are locally Lipschitz, we can prove that (2.8)-(2.40) and (2.9)-(2.42) are equivalent but our assumptions allow to deal with some non Lipschitz continuous Hamiltonians as shown in the following example. Such a H satisfies (2.6). Notice that the Lipschitz continuity is actually not needed in the proof of the gradient bound.

	We turn to some examples of sub-and superlinear Hamiltonians.
	Example 2.10. (Typical sublinear Hamiltonians from optimal control problem)
	H(x, p) = sup
	Example 2.11. (Typical superlinear Hamiltonian)
	H(x, p) = a(x)|p| 1+α + b(x)|p| + c(x),
	with a > 0, b, c ∈ W 1,∞ (T N ). Then H satisfies (2.8) and (2.40) as soon as (2.2) holds. It follows that the gradient bound of Theorem 2.2 holds for (2.5) even if σ is degenerate (i.e.,
	(2.4) does not hold).
	Example 2.9. (A non Lipschitz continuous Hamiltonian satisfying (2.8)) Let H(x, p) =
	|p| 2 + h(x, p) with h continuous bounded. For all x, y ∈ T N , p ∈ R N and µ > 1, we have H(x, p) -µH(y, p µ ) = (1 -1 µ )|p| 2 + h(x, p) -µh(y, p µ ) ≥ -(1 + µ)|h| ∞ .
	From (2.34), we see that (2.8) has to hold only for bounded µ ≥ 1 + L|x -y| and |p| ≥ L > 1. So (2.8) holds.

θ∈Θ {-b θ (x), p -ℓ θ (x)},

where b θ , ℓ θ ∈ W 1,∞ (T N ) uniformly with respect to θ.

Example 2.12. H(x, p)

  2,+ w ni (x, t) and (b, p, Y ) ∈ P 2,-w ni (y, t) such that(2.16) holds since a -b = 0. We achieve a contradiction as in the proof of Lemma 3.3. 5. From Ascoli-Arzela's theorem and the stability result for systems, by letting n → +∞, we obtain a continuous viscosity u solution of (1.2) in T N × [0, +∞), with |Du(•, t)| ∞ ≤ M. This solution is unique thanks to Theorem 4.3. It ends the proof of (i). The proof of (ii) is an immediate consequence of Theorems 3.6 and 3.7. 3.5. Examples and extensions. We give some examples such that our convergence result, Corollary 3.8, holds. Example 3.10. (Systems with possibly degenerate equations and superlinear Hamiltonians)

  p) if |p| ≤ n, H i (x, n p |p| ) if |p| ≥ n.

Acknowledgement. We thank Guy Barles, Hiroyoshi Mitake and Hung Tran for fruitful discussions. This work was partially supported by the ANR (Agence Nationale de la Recherche) through HJnet project ANR-12-BS01-0008-01 and WKBHJ project ANR-12-BS01-0020.


Jean-Luc Dekeyser
email: jean-luc.dekeyser@inria.fr

A Shadi Aljendi

Adopting New Learning Strategies for Computer Architecture in Higher Education Case Study: Building the S3 Microprocessor in 24 Hours

Keywords: computer architecture teaching, higher education, FPGA, PBL, revised Bloom's taxonomy I

Teaching computer architecture is a difficult task, especially to inexperienced students who prefer developing their java programing skills rather than understanding how machine instructions are executed effectively on a silicon black box. Because manipulating electronic components in a classroom is highly complex and complicated, teaching activities used to be aligned to simulation and exercises on paper. During the last 2 years we have developed a set of practical experimentations on FPGA boards. From scratch the students, in twelve 2 hour lessons, build their own processors and execute small programs on them with their own assembler and assembly language. A step by step approach permits the students to understand by design the different concepts of processors that are introduced during the lectures. All the labs are based on intuitive realization of elementary bricks and then combining them like a game of Legos. The results after two years are more than positive and some students have chosen to pursue a Masters of Computer Science with a particular interest in architecture design. The S3 processor can be considered as PBL (Project Based Learning) that covers all revised Bloom's taxonomy levels of knowledge acquisition.

INTRODUCTION

Since the 1990s, a lot of computer architecture curricula in higher education are dependent on simulations [START_REF] Grunbacher | Teaching Computer Architecture[END_REF][START_REF] Shine | Teaching Computer Architecture Using Simulation Tools[END_REF]. Simulations are often used to tackle the complexity of hardware systems and the difficulty in covering the high variety of components [START_REF] Wolffe | Teaching computer organization/architecture with limited resources using simulators[END_REF]. Cifredo-Chacón et al. summarized many attempts of teaching computer architecture practically in [START_REF] Cifredo-Chacón | Computer architecture and FPGAs: A learning-by-doing methodology for digital-native students[END_REF]. Many of these attempts achieved limited success because of, among other things, the inability to fulfill requirements and prerequisites.

In [START_REF] Machanick | Experience of applying Bloom's Taxonomy in three courses[END_REF], Bloom's taxonomy was applied on three different courses, one of which is computer architecture. The paper was limited to the original Bloom's taxonomy, not the revised one [START_REF] Forehand | Bloom's taxonomy[END_REF]. The conclusion of the paper stated that the use of Bloom's taxonomy in teaching these courses was at least as good as using traditional approaches.

A taxonomy derived from Bloom's original, revised Bloom and problem solving taxonomies was used in [START_REF] Kastelan | An approach to the evaluation of embedded engineering study programs[END_REF]. This is a 3levels approach, covering exercises, problems and projects.

Our approach of teaching computer architecture using the S3 microprocessor allows a comprehensive coverage of learning outcomes required in highly standardized core computer science curricula such as those mentioned in the ACM/IEEE 2013 curricula recommendations [START_REF]Computer Science Curricula 2013 Curriculum Guidelines for Undergraduate Degree Programs in Computer Science[END_REF]. This includes: Digital Logic and Digital Systems, Machine Level Representation of Data, Assembly Level Machine Organization, Memory System Organization and Architecture, Interfacing and Communication, Functional Organization and Performance Enhancements, the last two being electives in [START_REF]Computer Science Curricula 2013 Curriculum Guidelines for Undergraduate Degree Programs in Computer Science[END_REF]. It also covers all the six classes of the learning thinking model of the revised bloom's taxonomy. Furthermore, it is a Project Based Learning (PBL) approach that permits the learners to develop a real life product over a long period of learning.

The remainder of this paper is as follows: in the next section we will cover all aspects of our computer architecture teaching using the S3 processor; this includes educational as well as technical aspects. In section III, the effectiveness of our approach is discussed before the conclusion and the future directions in section IV.

II. THE S3 PROCESSOR

A. Educational aspects

In the PBL paradigm, learning is organized around long term complex tasks, with an essential autonomous role of the learner within small groups, which result in a realistic product, without direction from the facilitators [START_REF] Solomon | Project-based learning: A primer[END_REF]. PBL is used in [START_REF] Martínez-Monés | Multiple case studies to enhance project-based learning in a computer architecture course[END_REF] to enhance computer architecture learning. However, in this attempt, no real processor was built by the learners.

The authors have high level expertise in teaching computer architecture in different academic environments; from very tightly resource constrained environments to very wealthy and developed ones. One of the main features of the S3 processor course is that it fits easily in all kinds of environments regardless of their level of resource availability. In tightly resource constrained environments, free online available simulations can be used, and in wealthier environments, learners can build their processors on real world FPGA boards. FPGAs allow for interactive learning [START_REF] Cifredo-Chacón | Computer architecture and FPGAs: A learning-by-doing methodology for digital-native students[END_REF].

Not only is the learner able to follow the functionality of the processor on different levels of abstraction, but also they can develop and build the functionality that they desire in the processor, passing through all levels of the revised Bloom's taxonomy, and being able to make the "cognitive leap" from abstract to concrete [START_REF] Djordjevic | Flexible web-based educational system for teaching computer architecture and organization[END_REF]. PBL approach is used in building the S3 processor; Activities are organized so that learners start from the very beginning to build blocks that will be later used in the processor. Almost in every session, there is a new block or aspect that is used in the processor which is built or learned by the students. Learners are essentially autonomous during the sessions and also they have further suggested work to do as exercises if they want. They also work in small groups of 2 learners, and these groups also communicate and discuss difficult issues. At the end of the semester, a real processor with a real assembly language that can be used in order to program realistic algorithms is achieved.

B. Learning ecosystem

The course is a required module for second year computer science students. It does not have digital systems or computer architecture prerequisites. Even basic logical algebra knowledge is addressed in the theoretical part of the course and practiced in the first few practical lessons.

No previous VHDL knowledge is required. Also, building the processor itself does not need profound knowledge in VHDL. Only some basic aspects are needed during the course and it is explained during the sessions. Most of the work is done using schematics which are relatively easy to understand by learners.

Students have access to FPGA boards during practical sessions and access to computer simulation software throughout the day. The simulation tool is freeware, when high speed internet connection is available, learners can also easily download and install the simulation tool on their personal computers. Even when high speed internet is not available, when tested on an individual level in less developed regions and countries, learners could find solutions to install the simulation software and build their S3 processors.

Consequently, at the end of the semester, not only did learners build a complete high level modern processor but also learnt critical computer science and computer architecture aspects such as Registers, Buses, ALUs, Control Units, Learners had to understand the French language to follow the course. However, by February 2015 an English version was added to the website.

At the University of Science and Technology of Lille we have 5 groups of 30 students every year.

C. Course structure

The third semester offers 12 weeks of teaching (S3 processor name comes from 3rd Semester). Each week students follow:

 a lecture to introduce concepts, 1.5 hour  an exercise session per group of 30 students to apply the concepts and to prepare their usage on the board, 1.5 hour per group  the practical lab on PC Linux with their nexys boards, 2 hours per group (24 lab. hours)

This program is a step by step discovery of architecture design. The first web page of the course explains how to install the different tools (ISE webpack from Xilinx) to their own machines. A university pre install was done for all the PC classrooms.

The course is composed of the following units:

D. Technical aspects

Many attempts were and are undertaken to create microprocessors in the higher education context. The DidaComp in [START_REF] Cifredo-Chacón | Computer architecture and FPGAs: A learning-by-doing methodology for digital-native students[END_REF], which manipulates 4 bits data with only 4 instructions and with a very limited amount of memory, is an example. In [START_REF] Poduel | Design and Implementation of Synthesizable 32-bit Four Stage Pipelined RISC Processor in FPGA Using Verilog/VHDL[END_REF] a 32-bit Harvard RISC processor is proposed. The processor is synthesizable on FPGA using Verilog and VHDL. The processor has a constant restricted 32 instruction set. In [START_REF] Lee | Pipelined cpu design with fpga in teaching computer architecture[END_REF], a basic model is provided to the learners. The model already contains the CPU, Memory, Address Decoder, Timer, and General Purpose Input/Output. Also, the system programming necessitates the use of the Eclipse IDE which is a relatively complicated environment.

The S3 process is a Harvard RISC processor with a very simple yet very easy to evolve architecture and microarchitecture. It provides a 16-bit processor, that executes one instruction per clock cycle with 2 stage pipeline (Load/decode, execute). The pipeline used delayed branch to avoid pipeline break, keeping a simple FSM design. Only a minimal instruction set is suggested for the S3 process, which allows the user/learner to add and implement the instructions needed for the running program.

The processor contains, as depicted in figure 1, 5 general purpose registers; R1 to R5. The user interface of the processor is realized using 3 registers; Rsw, which stores the 8-bit input data entered on the switches of the board, and Rled and R7seg, connected to the 8 LEDs and to the 7-segment display of the board which are used as the output means of the processor. The CO register and the RI register are used during the loading phase in order to point to the next instruction and store the executed instruction respectively. Ram and Rdm are registers used with the data memory. Finally, the Rsrc1, Rsrc2 and Rdest are the ALU registers.

E. Instruction set

All instructions are coded on 16 bits. We propose 4 fields of [START_REF] Cifredo-Chacón | Computer architecture and FPGAs: A learning-by-doing methodology for digital-native students[END_REF] This file allows for 27 instructions. Learners, and users, can add other instructions and functionality depending on their applications and needs. Adding new instructions (up to few thousands) is a very simple task; once the functionality of the instruction is determined, it must be implemented in the ALU and added to the FSM and to the SED file. By now, it is obvious that the learner has acquired advanced processor architecture using modern technical and learning strategies. From the technical point of view, the learner has acquired sufficient knowledge about microprocessor functionality and architecture, as well as the use of FPGAs and IPs. This is a very important feature of the S3 processor; while still very simple, it covers without any added complication advanced topics such as hardware reusability and the use of IP cores. On the other hand, from the learning point of view, learners used all revised Bloom's taxonomy levels; remembering, understanding, applying, analyzing, evaluating and creating.

III. OUTCOMES

A. Learning outcomes

At the end of the semester, learners succeeded in building a very powerful microprocessor from scratch, including all needed hardware and software architecture aspects and principles. They could build the processor's register set, an ALU that is capable of including as much functionality as desired.

It is evident that learners have acquired solid knowledge in all computer architecture aspects, from basic and simple, as logic ports and input/output, up to very detailed ones such as microprocessor's microarchitecture.

They also learn how to implement these aspects on FPGAs using multi-level schematics and some VHDL programming. Advanced topics in digital design were covered and used by the learners, such as IP cores and hardware reusability.

Learners at the end of the semester are completely aware of all the details of the processor they have built. They can use it as a real processor and they can run and simulate the programs they write for it. Figure 4 is an S3 assembly program that allows the user to enter two 8-bit words on the switches, display them on the 7-segment display, calculate the sum and then display this sum on the 7-segment display. The figure also shows the hexadecimal machine language code for the program. Figure 5 shows the simulation of the execution of the program in Figure 4 using the Isim simulation tool.

Table I shows the outcomes of the course and how these outcomes were met. Learners were assessed using quizzes, exams, bonus points and particularly by following them during the whole semester. More than 75% of them completed at least ten lessons. After 10 weeks we could verify for these students functional processors that they could use to input/output and process data. 10-20% of the learners acquired very good knowledge in architecture and were motivated to work on the course at their homes. In each cell of this table, we list the outcome in bold. This table is constructed in the same manner as the "information literacy competency" taxonomy table in [START_REF] Vitolo | Taxonomy of information literacy competencies[END_REF]. However, the table in [START_REF] Vitolo | Taxonomy of information literacy competencies[END_REF] is built using the original Bloom's taxonomy. While we could easily construct the table for the original taxonomy, we would prefer here to present a table that we constructed and called the revised Bloom's taxonomy in computer architecture education.

B. Website statistics

From September 2014 more than 25000 pages were accessed, of which 85% from France because the English version was only released recently and 75% from Lille by our students during the semester. Since January, starting a new access count, we identified 31 different countries, with more than 300 different users (teaching in Lille finished in December). The course is conceived as a PBL course, which requires working on long term tasks. Learners need to recall many aspects during the semester Leaners will recall general algorithmic aspects and use them in programming the processor Learners write assembly language programs for the processor, which needs basic algorithmic knowledge, that learners have acquired in a previous semester Understand

Learners will explain what the components of a hardware system do Because learners build the system in a multilevel Lego game manner, they master the role of each component in the system

Learners will describe the role of software in a system Understanding the software architecture of the processor (machine language, micro-code, assembling programs) allows learners to assimilate the role of software in the processor Apply Learners will identify When would the hardware suit their needs

The main objective of our course is to build a functional processor, so that learners understand well when and where the complete system, as well as each of its components, meet concrete needs Leaners will choose which software fits a certain situation Learners acquire knowledge regarding the difference between hardware and software solutions, and when it is better to build functionality in hardware vs. when they should write a program to acquire it (hardware Inc is one cycle time less than software Add) Analyze

Learners will discover how a piece of hardware works Learners build their processor using basic and simple digital circuits / components. Thus they fully understand the functionality of all pieces of hardware and of the overall system at the end

Learners will explicate how a piece of software works

As in the case of hardware, the assembler is built using simple and basic tools. Accordingly, learners understand the functionality of the processor's software architecture Evaluate Learners will select ways to improve hardware design Learners discover different ways to solve one hardware problem and learn basics about slecting one of them

Learners will explain how to produce quality software

Learners are briefly presented to software architecture optimization concepts such as pipelining, delayed branching as well as control flow instructions Create

Learners will be able to improve the design Some learners continue their work in answering the challenging questions at the end of each lesson. These questions propose improvements to the component they have built in lab.

Learners will build functionality and instructions to the processor and write programs

Being flexible, Learners can easily add instructions to the processor when they realize situations that require hardware functionality rather than software

With international access the English version's access should increase. In universities, Nexys3 is a well-known board throughout the world and a lot of students and teachers are looking for exercises and tutorials on google.

We are also aware of self-learning individuals, living in different countries that used the course to acquire knowledge about FPGAs, and computer architecture. They expressed their satisfaction with the course.

IV. CONCLUSION AND FUTURE DIRECTIONS

In this paper we presented a PBL approach to teach computer architecture in computer science higher education. The approach allows the learners to acquire knowledge in the considered field passing through all six levels of the revised Bloom's taxonomy.

The approach is based on building a processor from scratch. Practically, no previous digital design or computer architecture knowledge is needed. While based on simple aspects, building the S3 processor allows the learners to acquire basic knowledge in processor architecture and modern digital design using FPGAs. This knowledge is acquired thanks to the PBL approach. Learners build simple logic blocks and combine them on a semester long period in order to get more complex blocks.

Outcomes attainment was assessed using continuous follow of learners as well as quizzes, exams and bonus points which were given to motivated learners that could go far in the construction of the processor. Learners considered this approach more as a game, which improved considerably their motivation.

The Lego project approach of the course was very much appreciated by our new generation learners and we could evaluate this by their high motivation compared to approaches adapted in previous years. The simplicity of the course; no need to learn advanced VHDL or enter in the complicated details of the components, allowed the learners to acquire lasting knowledge about architecture, that we can easily identify two years later when they start master studies.

The authors are planning to transform the course into a mooc (Massively Open Online Course). While previous attempts to allow web-based teaching of computer architecture exist, up to our knowledge, a comprehensive computer architecture course such as the one considered in this paper has never been proposed.

A subsequent course for postgraduate students is available. It is based on the HoMade processor developed for research activity. This processor is developed by using different IPs in VHDL to extend the instruction set for mono and multiprocessor usages on Nexys board. Each student used his own board during the semester, thanks to the French ministry of education financial support.

  Memories, Instruction formats, microcommands, I/O interfaces, Von Neumann architecture, Pipeline notion, etc. The course is available online under the CC BY-NC 4.0 Creative Commons license at: https://sites.google.com/site/l2s3ae/ In the classroom course, the learners worked in pairs, on desktop computers. Each computer have Xilinx ISE design suite. Each pair used a Digilent [12] nexys2 or nexys3 FPGA board. Upgrade to Nexys4 or Basys3 board is trivial and should be done soon on the website.

1 .

 1 ISE discovery for Nexys: a. Learner builds first simple design b. I / O concepts of the Nexys2 or Nexys3 c. Use of Xilinx and Digilent tools d. Learner passes through their first complete development chain e. Use a simple schematic diagram f. Create the bitstream file using ISE g. Test the electronic behavior on the board h. Link between schematic and pins of the FPGA 2. Logical gates: a. Build of first schematic using logic gates b. Use of the ISE libraries c. Simulate the behavior of the circuit using the Isim interactive interface (the simulation tool inside ISE) 3. 4-bits adder (This is the first part of the S3 ALU): a. Concept of reusability b. Build more complex circuits c. Advanced simulation methods 4. 16-bits display on Nexys: a. The concept of sequential circuits b. Basic concept of an FSM (Finite State Machine) and time in circuit design c. Seven segments display 5. FSM design and elementary concepts of VHDL (This lesson is the most VHDL oriented lesson): a. Introduction to FSM behavior with clocked design b. Real life application: Simple traffic light experimentation simulation and board validation using 6 LEDs and a few buttons to change the FSM behavior c. Basics of FSM VHDL coding d. Concept of a control Unit 6. Registers and buses: a. Introduction to the registers b. Concept of a bus c. Beginning of the S3 design with 15 registers connected to a bus (Fig. 1) d. Register transfer (achieved by a trivial FSM) 7. Instruction memory and load phase in Control Unit: a. Concept of memory (ROM) b. Concept of IP (Intellectual Property) c. Manipulating the program execution using a program counter register d. Understand and build the different instruction execution phases 8. Move instruction and first program: a. Format of instruction b. The first instruction MOV c. The concept of execution pipeline d. Instruction decoding using the FSM 9. ALU (supports a few thousands of different processing instructions if needed) : a. Concept of no-operand ALU b. Adding functionality to the ALU and control unit 10. Conditional instruction and immediate operand: a. Conditional MOV instruction (according to the value of the last ALU operation) b. Building branch instructions (conditional MOV to PC) c. Immediate addressing 11. Assembly language and S3 program development: a. Basic and simple assembly language (possible thanks to the simplicity of the instruction format) b. Learners create their own language c. Use of substitute string tool (such as SED [13]) to generate the configuration file of the ROM component 12. Data memory, stack and Call/Return implementation: a. Data memory with READ and WRITE instructions b. Autonomous stack By now, the S3 processor is almost ready to develop interesting programs running on the board with their own development for each pair of students. The final processor architecture is depicted in figure 2.

Fig. 1 .

 1 Fig. 1. The register structure of the S3 processor

Fig. 3 .

 3 Fig. 3. Example of code transformation Figure 3 shows an example of code transformation using the previous SED file. The user/learner writes the assembly language program (in the Instruction column) and passes it to the SED instruction along with the previous SED file in order to binary/hexa code that should be stored in the instruction memory of the S3 processor.

Fig. 4 .Fig. 5 .

 45 Fig. 4. S3 assembly language program (Entering 2 words of 8 bits from the switches to the ALU input register and transfer the result of ADDing them to the 7 segment output register) with the assembled machine language code

TABLE I .

 I COMPUTER ARCHITECTURE EDUCATION TAXONOMY Hardware

Software Remember Learners will recognize and recall different architecture aspects and concepts

ACKNOWLEDGMENTS

The authors would like to acknowledge the efforts of Ms. Diana Samson for reviewing the educational aspects of the paper.