Ariela Briani
email: ariela.briani@lmpt.univ-tours.fr

Hasnaa Zidani
email: hasnaa.zidani@ensta.fr

Characterization of the value function of final state constrained control problems with BV trajectories *

Keywords: Optimal control problem, differential systems with measures as data, measurable functions, Hamilton-Jacobi equations. AMS Classification: 49J15, 35F21, 34A37

This paper aims to investigate a control problem governed by differential equations with Radon measure as data and with final state constraints. By using a known reparametrization method (by Dal Maso and Rampazzo [18]), we obtain that the value function can be characterized by means of an auxiliary control problem of absolutely continuous trajectories, involving time-measurable Hamiltonian. We study the characterization of the value function of this auxiliary problem and discuss its numerical approximations.

Introduction

In this paper we investigate, via a Hamilton-Jacobi-Bellman approach, a final state constrained optimal control problem with a Radon measure term in the dynamics.

Several real applications can be described by optimal control problems involving discontinuous trajectories. For instance, in space navigation area, when steering a multi-stage launcher, the separation of the boosters (once they are empty) lead to discontinuities in the mass variable [START_REF] Bokanowski | Hamilton-Jacobi-Bellman approach for the climbing problem[END_REF]. In resource management, discontinuous trajectories are also used to modelize the problem of sequential batch reactors (see [23]). Many other applications can be found in the Refs. [START_REF] Baumeister | On optimal control of a fishery[END_REF]16,[START_REF] Clark | The optimal exploitation of renewable stocks[END_REF]19].

Consider the controlled system:

dY (t) = M i=1 g i (t, Y (t))dµ i + g 0 (t, Y (t), α(t))dt for t ∈ (τ, T ] (1a) Y (τ -) = X. (1b)
where x ∈ R N , the measurable control α : (0, +∞) → R m takes values in a compact set A ⊂ R m , and µ = (µ 1 , . . . , µ M ) is a given Radon measure. Let ϕ : R N → R be a given lower semicontinuous (lsc) function and consider the control problem:

v(X, τ ) := inf{ϕ(Y α X,τ (T )) : α(•) ∈ L ∞ (0, T ; A) and Y α X,τ satisfies (1)}.

(2)

Due to the presence of the measure µ, the definition of solution for the state equation ( 1) is not classical. We will refer to the definition introduced by Dal Maso and Rampazzo in [18] using the technique of graph completion (Definition 2.1 in Section 2 below). Roughly speaking, by a suitable change of variable in both time and the primitive of µ, we can reduce (1) to usual controlled ordinary differential equation with a measurable time-dependent dynamics (see Theorem 2.2 below).

Since the 80's impulsive control problems, i.e. when the measures appear as controls, have been extensively studied by many authors. Let us mention the pioneering works [26], [START_REF] Sarychev | Nonlinear systems with impulsive and generalised functions controls[END_REF] and [10], [11] and [32]. We refer to [30] for the study of the existence results of optimal trajectories, and to [1] for first and second necessary optimality conditions.

In problem (2), the measure is fixed and the state equation is controlled by means of a measurable function α. Our main goal is to use the HJB approach in order to characterize the value function v and then to study a numerical method for the approximation of this function.

It is easy to check that the the value function v fulfills a Dynamic Programming Principle (DPP). Formally, we can derive then, the following HJB equation:

     -v t (X, t) + sup a∈A -Dv(X, t) • g 0 (t, X, a) + M i=1 g i (t, X)µ i = 0;
v(X, T ) = ϕ(X).

(3)

However, the sense of solutions for this equations is not clear. The main difficulty is to give a meaning to the term "Dv • µ" knowing that one can not expect to have a differentiable value function. In order to overcome this difficulty, following the ideas in [14], we define a new value function v such that:

v(X, τ ) = v(X, W(τ )),
where W is the known change of variable coming form the graph completion technique (See Theorem 2.3). The advantage of this cange of variable comes from the fact that the HJB equation for v has a t-measurable Hamiltonian and not a measure term. More precisely, we can prove that v is a solution, in a suitable sense, of the following equation:

-v s (X, s) + H(s, X, Dv(X, s)) = 0; v(X, 1) = ϕ(X);

where H(t, x, p) = sup a∈A {-p • F(t, x, a)} and F (t, x, a) is a t-measurable dynamics (see Section 2.2 for the definition of F ). The main contribution of this paper is to study a precise sense of solution for (4) with t-measurable Hamiltonian and a lsc final function ϕ.

We recall that, in the case when ϕ is continuous the definition of viscosity solution for t-measurable Hamiltonians has been introduced by Ishii in 1985 (see [24]) and extended to second order by Nunziante in [28]- [29](see also the work of and ). Moreover, a very general stability result has been proved more recently by Barles in [4]. On the other hand, to deal with the case when the Hamiltonian H is continuous with respect to the time variable and the final data ϕ is lsc, the definition of bilateral viscosity solution has been introduced by Barron and Jensen in 1990 ([7]) and by Frankowska [20] in 1993.

In this paper, since we are interested in a target problem, the function ϕ is lsc. Moreover, the Hamiltonian in ( 4) is only t-measurable. Similar problems has been studied, via non smooth analysis, by H. 22].

Here we introduce a new definition of viscosity solution of (4), namely the definition of L 1 -bilateral viscosity solution (Definition 3.1 below). This definition allows to characterize v as the unique L 1 -bilateral viscosity solution of equation ( 4) (Theorem 3.2). It gives also a suitable framework to deal with the numerical approximation of v (and then of v by the change of variable W). More precisely, we prove in Theorem 3.3 a convergence result for monotone, stable and consistent numerical schemes, and give an example of a scheme satisfying these properties. Some numerical tests are presented in Subsection 3.2.

On the other hand, we study the properties of L 1 -bilateral viscosity solution for a general HJB equation. In particular, in Section 4, we derive the consistency of the definition (Theorem 4.4), a general stability result w.r. to the Hamiltonian (Theorem 4.6) and uniqueness result (Theorem 4.8).

This paper is organized as follows. In Section 2 we set the optimal control problem we are considering. Subsection 2.1 is devoted to the definition of solution for the state equation and we construct the reparametrized optimal control problem in Subsection 2.2. In Section 3 we state the definition of L 1 -bilateral viscosity solution and we study numerical approximation results for t-measurable HJB equations. Finally, in Section 4 we will prove the consistency (Theorem 4.4), stability (Theorem 4.6 and 4.7) and uniqueness (Theorem 4.8) result for L 1 -bilateral viscosity solution.

Notations. For each r > 0, x ∈ R N we will denote by B r (x) the closed ball of radius r centered in x. Given a Radon measure µ we will denote by L 1 µ (R) the space of integrable functions with respect to the measure µ. In all the sequel, we will use the classical notations:

f (t + ) := lim s→t + f (s) and f (t -) := lim s→t - f (s).
And finally, we will denote by AC([0, 1]; R N ) the set of abso-

lutely continuous functions from [0, 1] to R N .
2 The optimal control problem with BV trajectories

In this section we will first recall the definition of solution for the state equation as introduced by Dal Maso and Rampazzo in [18] and we recall the graph completion construction. Then, we define the reparametrized optimal control problem.

The state equation

Let us fix 0 ≤ τ < T , an initial datum X ∈ R N , a given Radon measure µ = (µ 1 , . . . , µ M ), a control variable α ∈ A, and consider the controlled trajectory Y α X,τ : R + → R N solution of:

     dY (t) = M i=1 g i (t, Y (t))dµ i + g 0 (t, Y (t), α(t))dt for t ∈ (τ, T ] Y (τ -) = X.
(5)

We assume the following:

(Hco) The set of admissible controls is

A := {α : (0, T ) -→ A measurable}, where A is a compact subset of R m , m ≥ 1. (
(•, Y, a) ∈ L 1 (R + ) and g i (•, Y ) ∈ L 1 µi (R + ), (i = 1, . . . , M ). (Hg2) There exists a function k 0 ∈ L ∞ (R + ; R + ) such that |g i (t, Y ) -g i (t, Z)| ≤ k 0 (t)|Y -Z| ∀ Y, Z ∈ R N , a.e. t ∈ R + , i = 1, . . . , M. |g 0 (t, Y, a) -g 0 (t, Z, a)| ≤ k 0 (t)|Y -Z| ∀ Y, Z ∈ R N , ∀a ∈ A, a.e. t ∈ R + .
(Hg3) There exists K > 0 such that

|g i (t, Y )| ≤ K ∀ Y ∈ R N , a.e. t ∈ R + , i = 1, . . . , M. |g 0 (t, Y, a)| ≤ K ∀ Y ∈ R N , ∀a ∈ A, a.e. t ∈ R + .
Following [18], we introduce the left continuous primitive B of the Radon measure µ, i.e. B ∈ BV -([0, T ]; R M ) and its distributional derivative Ḃ coincides with µ on [0, T [. In all the sequel, we will denote by T := {t i , i ∈ N} the countable subset of [0, T ) which contains 0 and all the discontinuity points of B, and by E c the set of all continuity points of B. Furthermore, let (ψ t ) t∈T := (ψ 1 t , . . . , ψ M t ) be a family of Lipschitz continuous maps from

[0, 1] into R M such that t∈T V 1 0 (ψ t ) < ∞, ψ t (0) = B(t -) and ψ t (1) = B(t + ) ∀t ∈ T ; (6)
(if t = 0 we require only ψ t (1) = B(0 + )). We will denote by ξ the solution of:

dξ dσ = M i=1 g i (σ, ξ(σ)) dψ i t dσ for σ ∈ (0, 1] ξ(0) = ξ,
and we set ξ( ξ, ψ t ) := ξ(1) -ξ.

The definition of solution introduced by Dal Maso and Rampazzo in [18, Definition 5.1] is then the following: Definition 2.1. Given an initial datum and time (X, τ ), a control α ∈ A and a family of Lipschitz continuous maps (ψ t ) t∈T fulfilling (6), the function

Y α X,τ ∈ BV -([τ, T ]; R N ) is a solution of (5) if for each Borel subset B of ]τ, T [ we have B dY (t) = B g 0 (t, Y (t), α(t))dt + M i=1 B∩Ec g i (t, Y (t))dµ i + t∈T ∩B ξ(Y (t -), ψ t ) (7) and Y (τ -) = X. Moreover, if τ ∈ T we have Y (τ + ) = ξ(X, ψ τ ).
In order to prove the uniqueness of this solution we set

a i := V 1 0 (ψ ti ), a := +∞ i=1 a i , w(t) := t + V t 0 (B) T + V T 0 (B)
, and we define W : [0, T ] → [0, 1] as follows:

W(t) := 1 1 + a w(t) + ti<t a i . (8)
The graph completion of B corresponding to the family (ψ t ) t∈T is then defined by:

Φ(s) := (φ 0 ; φ 1 , . . . , φ M )(s) (9a) = (t; B(t)) if s = W(t) t ∈ [0, T ] \ T (t i ; ψ ti s-W(ti) W(t + i )-W(ti) ) if s ∈ [W(t i ), W(t + i )] t i ∈ T . (9b)
We are now ready to construct the reparametrization of system (5). Let σ := W(τ ), for each control α ∈ A and initial datum X we denote by Z α X,σ :

[σ, 1] → R N the solution of                dZ ds (s) = M i=1 g i (φ 0 (s), Z(s)) µ a i (φ 0 (s)) dφ 0 ds (s) + dφ i ds (s) + g 0 (φ 0 (s), Z(s), α(φ 0 (s)) dφ 0 ds (s) for s ∈ (σ, 1] Z(σ) = X (
10) where µ a is the absolutely continuous part of the measure µ with respect to the Lebesgue measure, i.e. µ(t) = µ a (t)dt + µ s . Note that the derivatives of φ 0 , φ i are measurable functions, therefore assumptions (Hg1)-(Hg2) ensure the applicability of Caratheodory's Theorem to obtain the existence of a unique solution of (10

) in AC([σ, 1]; R N ).
Theorem 2.2. Assume (Hco) and (Hg1)-(Hg3). Let µ be a Radon measure and (ψ t ) t∈T be a family fulfilling (6).

Then Y α X,τ ∈ BV -([τ, T ]; R N ) is a solution of (5) if and only if there exists a solution Z α X,σ ∈ AC([σ, 1]; R N ) of (10) corresponding to the graph completion Φ defined in (9) such that Z α X,σ (W(t)) = y(t) ∀t ∈ [τ, T ] ( 11
)
where W is given by [START_REF] Baumeister | On optimal control of a fishery[END_REF].

Moreover, for each Radon measure µ and each family (ψ t ) t∈T equation (5) has a unique solution (up to a set of zero Lebesgue measure).

Proof. The equivalence (11) can be obtained by adapting the proof given for M = N = 1 in [14,Theorem 2.8]. On the other hand, the uniqueness of the solution is a consequence of Caratheodory's Theorem applied to equation (10).

Remark 1. We point out that this definition depends on the family (ψ t ) t∈T we choose. It is now a classical result that under commutativity conditions on the vector fields g i (i = 1, . . . , M ) the solution does not depend on this choice as studied in the pioneering works of Bressan and Rampazzo [12,13]. However, in this paper, the dependence on the choice of ψ t does not imply any specific difficulty in the sequel. Of course, from an application point of view, one have to be aware of this dependence and made an accurate choice of the reparametrization.

The control problem

Let us now describe our optimal control problem. Given a lower semicontinuous function ϕ : R N → R and a final time T , our aim is to calculate the following value function v(X, τ

) := inf α∈A ϕ(Y α X,τ (T )) (12)
where Y α X,τ is the solution of equation ( 5). It is easy to prove that the following Dynamic Programming Principle (DPP) holds: for each τ ≤ h ≤ T we have

v(X, τ ) = inf α∈A v(Y α X,τ (h), h).
Therefore we can formally derive a HJB equation:

-v t (X, t) + H(t, X, Dv(X, t)) = 0 for (X, t) ∈ R N × (0, T ), v(X, T ) = ϕ(X) for X ∈ R N
where the Hamiltonian is

H(t, X, P ) = sup a∈A -P • g 0 (t, X, a) + M i=1 g i (t, X)µ i .
As we pointed out in the Introduction, the problem is to give a meaning to the term Dv • µ knowing that one can not expect to have a differentiable value function.

In view of Theorem 2.2, it is then natural to consider the trajectories Z α X,σ solution of the the reparametrized system (10). We define then the corresponding value function as follows:

v(X, σ) = inf α∈A ϕ(Z α X,σ (1)). ( 13
)
The link between the two problems is given by the following result.

Theorem 2.3. Let v and v be respectively defined in ( 12) and ( 13).

For each

X ∈ R N and τ ∈ [0, T ] we have v(X, τ ) = v(X, W(τ )) ( 14
)
where W is given by [START_REF] Baumeister | On optimal control of a fishery[END_REF]. Moreover

v ♯ (X, τ ) = v♯ (X, W(τ )) ∀X ∈ R N , ∀τ ∈ [0, T ] \ T (15) and v ♯ (X, τ ) ≥ v♯ (X, W(τ )) ∀X ∈ R N , ∀τ ∈ T , ( 16
)
where we respectively denote by v ♯ and v♯ the lower semicontinuous envelope of v and v w.r. to both variable (X, τ ) and (X, s).

Proof. By Theorem 2.2 above we have Y α X,τ (T ) = Z α X,W(τ ) (W(T )) = Z α X,σ (1
) then ( 14) follows by the definitions of v and v.

Since, by construction, W(τ ) is monotone increasing in [0, T ] and continuous in any τ ∈ T , ( 15) and ( 16) easily follow.

Remark 2. In ( 15), ( 16) we stressed the link between the lsc envelopes of v and v because is indeed the function v♯ (X, s) that will be characterized as solution of an HJB equation.

Thanks to Theorem 2.3, it is clear that we turn now our attention to the HJB equation for the function v. The advantage is that we do not have any more measure in the dynamics.

The new value function v satisfies also a DPP:

v(X, σ) = inf α∈A v(Z α X,σ (h), h) ∀σ ≤ h ≤ 1, ∀X ∈ R N . (17)
From this DPP, one could expect to characterize v through the following HJB equation:

-v s (X, s) + H(s, X, Dv(X, s)) = 0 for (X, s) ∈ R N × (0, 1), v(X, 1) = ϕ(X) for X ∈ R N (18)
where the Hamiltonian is

H(s, X, P ) = sup a∈A -P • g 0 (φ 0 (s), X, a) dφ 0 ds (s)+ + M i=1 g i (φ 0 (s), X) µ a i (φ 0 (s)) dφ 0 ds (s) + dφ i ds (s) . (19)
Note that, by definition ( 9), the graph completion (φ 0 , φ i ) is a Lipschitz function, therefore we can not expect to have a time continuous Hamiltonian. Moreover, our final condition ϕ is only lower semicontinuous. Thus, we should first give a precise meaning to the definition of the viscosity solution of the equation ( 18). This will be the aim of the following Section. schemes of equation ( 18). We will prove our results in the following more general framework.

Fix a final time T , given x ∈ R N , τ ≥ 0 and a control α ∈ A, we consider the trajectory y α

x,τ , solution of the following system:

ẏ(t) = F (t, y(t), α(t)), for t ∈ (τ, T ) y(τ ) = x. ( 20
)
For each initial point and time (x, τ ) ∈ R N × R + we set:

ϑ(x, τ ) := inf α∈A ϕ(y α x,τ (T )). ( 21
)
We assume the following :

(HF1) F (t, x, a) : R + × R N × A → R N is measurable in t and continuous in x and a. Moreover, for each (x, a) ∈ R N × A we have F (•, x, a) ∈ L 1 (R + ). (HF2) There exists k 0 ∈ L ∞ (R + ; R + ) such that |F(t, x, a) -F(t, z, a)| ≤ k 0 (t)|x -z| ∀x, z ∈ R N , a ∈ A, t ∈ R + .
(HF3) There exists a K > 0 such that

|F(t, x, a)| ≤ K ∀x ∈ R N , a ∈ A, t ∈ R + .
(Hid) The function ϕ : R N → R is lower semi continuous and bounded.

Remark 3. Let us point out that if we assume (Hg1)-(Hg3), then the function

F (s, x, a) := M i=1 g i (φ 0 (s), x) µ a (φ 0 (s)) dφ 0 ds (s) + dφ i ds (s) + g 0 (φ 0 (s), x, a) dφ 0 ds (s)
fulfills (HF1)-(HF3). Therefore, all the results in this section will apply, in particular, to the value function v defined in (13).

In all the sequel, we denote V the lower semicontinuous envelope of ϑ defined by:

V(x, t) := lim inf y→x,s→t ϑ(y, s). ( 22
)
Our first aim is then to prove that we can characterize the function V in ( 22) as the unique L 1 -bilateral viscosity solution (see the definition below) of the following HJB equation:

-V t (x, t) + H(t, x, DV(x, t)) = 0 for (x, t) ∈ R N × (0, T ), V(x, T ) = ϕ(x) x ∈ R N (23)
where the Hamiltonian is

H(t, x, p) = sup a∈A {-p • F(t, x, a)} . ( 24
)
Definition 3.1. L 1 -bilateral viscosity solution (L1Bvs) Let u : R N × (0, T ) → R be a bounded lsc function. We say that u is a L 1 -bilateral viscosity solution (L1Bvs) of ( 23) if: for any b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local minimum point for u(x, t) -

t 0 b(s)ds -φ(x) we have lim δ→0 + ess sup |t-t0|≤δ sup x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≥ 0 and lim δ→0 + ess inf |t-t0|≤δ inf x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≤ 0.
Moreover, the final condition is satisfied in the following sense:

ϕ(x) = inf lim inf n→∞ u(x n , t n ) : x n → x , t n ↑ T .
Remark 4. For the sake of clarity, we will state and prove below (Section 4), the consistency, stability and uniqueness result for the viscosity sense (L1Bvs) defined in Definition 3.1.

Let us now prove the characterization of the value function.

Theorem 3.2. Assume (HF1)-( HF3) and (Hid). The function V, defined in (22), is the unique L 1 -bilateral viscosity solution of (23), when the Hamiltonian is given in (24).

Proof. The proof of V being a L 1 -bilateral viscosity solution of ( 23) can be obtained with the same kind of argument given in [2, Proposition 5.3, Chapter V]. The uniqueness follows by Theorem 4.8.

Numerical approximations of (23).

In the case when the Hamiltonian is continuous (both in time and in space), numerical discretization of Hamilton-Jacobi equations has been studied by many authors. The general framework of ensures that the numerical scheme is convergent (to the viscosity solution) whenever this scheme is consistent, monotone and stable and the HJB equation satisfies a strong comparison principle. The class of schemes satisfying these properties is very large and includes upwind finite differences, Semi-Lagrangian methods, Markov-Chain approximations.

In this section, we extend the result of [5] to the case of equation ( 23), where the Hamiltonian is only t-measurable, and show that the t-measurable viscosity notion is still a good framework to analyze the convergence of numerical approximations. We give also an example of a monotone, stable and consistent scheme of (23) based on finite differences approximations. Finally, a numerical example is given in Subsection 3.2.

Let G be a space grid on R N with a uniform mesh size ∆x > 0 (of course a nonuniform grid could also be considered), and let ∆t > 0 be a time step (we assume that T /∆t belongs to N). In the sequel, we will use the following notations:

∆ := (∆x, ∆t), t n := n∆t, x j is a node in G, N T := T ∆t . ( 25
)
Consider an approximation scheme of the following form:

S ∆ (t n , x j , v n j , v n+1 ) = 0 ∀x j ∈ G, n = 0, • • • , N T -1; v NT j = ϕ(x j ) ∀x j ∈ G.
(26) Thus, if v is a continuous function defined on [0, T ]×R N , the approximation scheme reads

S ∆ (t, x, v(x, t), v(•, t + ∆t)) = 0 in (0, T ) × R N . ( 27
) On S ∆ : (0, T ) × R N × R × L ∞ (R N )
we assume the following:

(M) Monotonicity. For each u ≥ v we have

S ∆ (t, x, r, u) ≤ S ∆ (t, x, r, v) ∀t ∈ (0, T ), x ∈ R N , r ∈ R.
(S) Stability. There exists

K > 0 such that, if v ∆ is solution of (27) then v ∆ L ∞ ≤ K,
K being independent of ∆x, ∆t.

(C) Consistency. For every point (x 0 , t 0 ), for any b ∈ L 1 (0, T ) and any function φ(x) such that: φ ∈ C 1 (R N ), by setting ψ(x, t) := t 0 b(s) ds + φ(x), we have:

ess sup |t-t0|≤∆t sup x∈B∆x(x0), p∈B∆x(Dφ(x0)) {-b(t) + H(t, x, p)} ≥ ≥ S ∆ (t 0 , x 0 , ψ(x 0 , t 0 ), ψ(•, t 0 + ∆t)) + o ∆x (1) ≥ ≥ ess inf |t-t0|≤∆t inf x∈B∆x(x0), p∈B∆x(Dφ(x0))
{-b(t) + H(t, x, p)}. (28)

An example of scheme fulfilling the above assumptions, when the Hamiltonian is given by (24), is the following

S ∆ (t, x, u(x, t), u(•, t + ∆t)) := u(x, t) -u(x, t + ∆t) ∆t + + 1 ∆t t+∆t t sup a∈A (-F) + (s, x, a) • u(x, t + ∆t) -u(x -∆x, t + ∆t) ∆x + +(-F) -(s, x, a) • u(x + ∆x, t + ∆t) -u(x, t + ∆t) ∆x ds, (29)
where we classically denoted g + := max(g, 0) and g -:= min(g, 0).

Proposition 1. Assume that F fulfills assumptions (HF1)-(HF3), and consider the Hamiltonian in (24). Let ∆ = (∆x, ∆t) be mesh sizes satisfying:

∆t ∆x |F(s, x, a)| ≤ 1 for a.e. s ∈ (0, T ), ∀x ∈ R N , a ∈ A. ( 30
)
Then, the scheme S ∆ given in (29) satisfies conditions (M), (S) and (C).

Proof. Fist remark that the Stability condition (S) easily follows from the boundedness of F and (HF3). Moreover, the monotonicity (M) follows from condition (30) by standard arguments.

To prove consistency, we fix (x 0 , t 0 ) and consider a function ψ(x, t) = t 0 b(s) ds+ φ(x) for b ∈ L 1 (0, T ) and φ ∈ C 1 (R N ). By using the regularity of ψ and assumption (HF3) on F , we get:

S ∆ (t 0 , x 0 , ψ(x 0 , t 0 ), ψ(•, t 0 +∆t)) = 1 ∆t t0+∆t t0
{-b(s) + H(s, x 0 , Dφ(x 0 ))} ds+o ∆x (1).

Condition (C) follows.

The general convergence result is the following.

Theorem 3.3. Assume (HF1)-(HF3). Let V be defined as in ( 22) with ϕ fulfilling assumption (Hid). Consider a sequence of continuous and bounded functions ϕ m : R N → R (for m ≥ 1) such that (ϕ m ) m∈N is monotone increasing and

lim m→∞ ϕ m (x) = ϕ(x) ∀x ∈ R N .
Let ∆ = (∆x, ∆t) be a mesh size such that the scheme S ∆ fulfills conditions (M), (S) and (C), and let v ∆,m := (v n j ) n,j be the solution of :

S ∆ (t n , x j , v n j , v n+1 ) = 0 ∀x j ∈ G, n = 0, • • • , N T -1; v NT j = ϕ m (x j ) ∀x j ∈ G. (31
) Then, as ∆t → 0, ∆x → 0 and m → +∞, v ∆,m converges pointwise to the function V.

Proof. The proof will be given in two steps.

Step 1. We first suppose that the final data is continuous (ϕ m ≡ ϕ). We consider a ∆ k = (∆x k , ∆t k ) and denote by v ∆ k the solution of (31) corresponding to ∆ k and ϕ m ≡ ϕ. We will prove that, as k → 0, the sequence v ∆ k converges locally uniformly to the unique L 1 -viscosity solution of (23).

For each k, we set (x k , t k ) := (x j k , t n k ) where (x j k , t n k ) are the points defined in (25) when ∆ is ∆ k . Let us first observe that by the stability assumption (S) the sequence v ∆ k is bounded, therefore the following weak semi-limits are well defined:

v * (x, t) := lim inf k→0 lim x k →x,t k →t v ∆ k (x k , t k ) v * (x, t) := lim sup k→0 lim x k →x,t k →t v ∆ k (x k , t k ).
Note that both v * and v * trivially satisfy the final condition in (23). Therefore, the convergence result will follows once we prove that v * and v * are respectively a L 1 -viscosity supersolution and a L 1 -viscosity subsolution of (23). Indeed, if this is true, by the comparison result [24, Theorem 8.1], we have v * ≤ v * . Since the reverse is true by definition, the two weak semi limits coincide and the thesis follows.

Let us now prove that v * is a L 1 -viscosity subsolution of ( 23). (The proof of v * being a L 1 -viscosity supersolution of ( 23) is completely similar and will not be detailed.) Following Definition 4.3 below, for any b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local maximum point of v * (x, t) -t 0 b(s)dsφ(x) we have to prove that lim

δ→0 + ess inf |t-t0|≤δ inf x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≤ 0. ( 32
)
Note that, without loss of generality, we can assume that (x 0 , t 0 ) is a strict local zero maximum of v * (x, t) -t 0 b(s)dsφ(x). There exists then a sequences of points

(x k , t k ) such that (a) (x k , t k ) → (x 0 , t 0 ) as k → 0. (b) (x k , t k ) is a local maximum point of v ∆ k (x, t) - t 0 b(s)ds -φ(x). (c) ξ k := v ∆ k (x k , t k ) - t k 0 b(s)ds -φ(x k ) → 0 = v * (x 0 , t 0 ) - t0 0 b(s)ds -φ(x 0 ) as k → 0.
Thanks to (b), we can apply the monotonicity assumption (M)

with v = v ∆ k , u = φ(x) + t 0 b(s)ds + ξ k and r = v ∆ k (x k , t k ) = ξ k + φ(x k ) + t k 0 b(s)ds and obtain S ∆ k t k , x k , ξ k + φ(x k ) + t k 0 b(s)ds, ξ k + φ(•) + t k +∆t k 0 b(s)ds ≤ ≤ S ∆ k (t k , x k , v ∆ k (x k , t k ), v ∆ k (•, t k + ∆t k )) = 0, (33)
where we also used that v ∆ k is a solution of ( 26).

Fix now a δ > 0, by (a) and the regularity of φ we can always find a

δ k ≤ δ such that min(∆x k , ∆t k ) ≤ δ k , B δ k (t k , x k ) ⊆ B δ (x 0 , t 0 ), and B δ k (Dφ(x k )) ⊆ B δ (Dφ(x 0 )
). Therefore, also by the consistency assumption (C) and ( 33) we get:

ess inf |t-t0|≤δ inf x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≤ ≤ ess inf |t-t k |≤δ k inf x∈B δ k (x k ), p∈B δ k (Dφ(x k )) {H(t, x, p) -b(t)} ≤ ≤ S ∆ k t k , x k , ξ k + φ(x k ) + t k 0 b(s)ds, ξ k + φ(•) + t k +∆t k 0 b(s)ds +o δ k (1) ≤ o δ k (1).
(34) Inequality (32) follows then by letting δ → 0 + (which implies δ k → 0 + ).

Step 2. For every m ≥ 1, by Step 1, as k → 0, the sequence (v ∆ k ,m ) k converges to v m the unique L 1 -viscosity solution of

-v t (x, t) + H(t, x, Dv) = 0 in R N × (0, T ) v(x, T ) = ϕ m (x) in R N . ( 35
)
The pointwise convergence of v m to V then follows by the stability result Theorem 4.7 below.

Remark 5. In the case of Eikonal equation with t-measurable velocity function, a similar convergence result is proved, in the recent work of A. Monteillet [27], for a particular numerical scheme.

A numerical test.

In this section, we use the scheme given in (29) to solve Hamilton-Jacobi equations coming from a simple control problem with BV trajectories. Consider the target C := B(0, r), which is the ball centered at the origin and of radius r = 0.25. Consider also a trajectory Y (α,c) τ,X , depending on the control variables α : (0, T ) → A := [0, 2π] and c : (0, T ) → U , and governed by the following dynamics

Ẏ (t) = c(t) cos(α) sin(α) + C 1 1 1 δ 1 + C 2 0 1 δ 2 , Y (τ ) = X
where C 1 := 0.5, C 2 := 0.2, and δ u (for u = 1, 2) denotes the Dirac measure at time t = u. The control variable c takes its values in a compact set U . Here we will consider two cases:

• Case 1: U ≡ {0.5} which amounts saying that we are allowed to move in any direction in the sphere centered at the origin and with radius 0.5.

• Case 2: U = [0, 0.5], which means that we can move in any direction in the Ball centered at the origin and with radius 0.5.

In both cases, at time t = 1 and t = 2 the trajectories jump. We consider the value function corresponding to the Rendez-Vous problem:

v(t, X) := inf{ϕ(Y α,c τ,X (T )); α ∈ L ∞ (0, T ; A), c ∈ L ∞ (0, T ; U )}
, where T = 3, and ϕ(x) = 0 when x ∈ C and 1 otherwise.

It is not difficult to compute the reparametrized function: Fig. 1 shows the numerical solution in the Case 1, while Fig. 2 shows the results corresponding to Case 2. These numerical experiments are performed by using the finite differences scheme with 150 2 grid points. Computations are done on the domain [-1.5, 3] 2 . The final cost function is approximated by a function (with n = 10):

Φ(s) =            (15s, 0, 0) 0 ≤ s ≤ 1 15 ; (1, 15 6 s -1 6 , 0)
ϕ n (X) := 1/n min 1, x -0.5) .

In the two cases, we compute first the value function v corresponding to the parametrized control problem, and then we deduce the original value function by using a change of variable. The latter step is very easy to perform numerically, since v turns to be just the restriction of v on [0, 4 Properties of the L 1 -bilateral viscosity solution of HJB equations.

This section is devoted to the main properties of the L 1 -bilateral solutions defined in Definition 3.1. First, we give an equivalent formulation of this definition and we - prove that it is consistent with the definitions of viscosity solutions given for a more regular HJB equation (Subsection 4.1). The stability result is given in Subsection 4.2. Fix T > 0, and consider the general Hamilton-Jacobi-Bellman equation

-2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 (a) 0-level sets of v (b) 0-level sets of v
2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1 0 1 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 (a) 0-level sets of v (b) 0-level sets of v
-u t (x, t) + H(t, x, Du) = 0 in R N × (0, T ) u(x, T ) = ϕ(x) in R N . ( 36
)
On the Hamiltonian H : R + × R N × R N → R we assume the following:

(H0) The function H(t, x, p) is measurable in t and continuous in x and p. Moreover, for each (x,

p) ∈ R N × R N we have H(•, x, p) ∈ L 1 (R + ). (H1) For each compact subset K of R N × R N there exists a modulus m = m(K) : (0, T ) × R + → R + such that t → m(t, r) ∈ L 1 (0, T ) for all r ≥ 0, m(t, r) is increasing in r, m(•, r) → 0 in L 1 (0, T ) as r → 0, and
|H(t, x, p) -H(t, y, q)| ≤ m(t, |x -y| + |p -q|)
for almost every t and for any (x, p), (y, q) ∈ K.

Moreover, in the following, we may need some assumptions stronger than (H1):

(H2) There exists a function

k 0 ∈ L ∞ (R + ; R + ) such that |H(t, x, p)-H(t, y, p)| ≤ k 0 (t)(1+|p|)(|x-y|) for all p ∈ R N , t ∈ R + , x, y ∈ R N .
(H3) For each (t, x) the function H(t, x, •) is convex and there exists a constant L > 0 such that

|H(t, x, p) -H(t, x, q)| ≤ L|p -q| for all p, q ∈ R N , t ∈ R + , x ∈ R N .
On the final data ϕ we suppose (Hid).

Remark 6. It is easy to check that if the dynamics F fulfills assumptions (HF1)-(HF3), then the Hamiltonian defined in (24) satisfies assumptions (H0)-(H3).

In order to give an equivalent formulation of the definition of L 1 -bilateral viscosity solution we need to introduce the following sets of functions. Fix (x 0 , t 0 ) and a function φ ∈ C 1 (R N × R + ), and set:

H -(t 0 , x 0 , Dφ(x 0 , t 0 )) := = G(t, x, p) ∈ C(R + × R N × R N ), convex in p , b(t) ∈ L 1 (R + ) such that G(t, x, p) + b(t) ≤ H(t, x, p) for all x ∈ B δ (x 0 ), p ∈ B δ (Dφ(x 0 , t 0 )), a. e. t ∈ B δ (t 0 ) and some δ > 0} H + (t 0 , x 0 , Dφ(x 0 , t 0 )) := = G(t, x, p) ∈ C(R + × R N × R N ), convex in p , b(t) ∈ L 1 (R + )
such that G(t, x, p) + b(t) ≥ H(t, x, p) for all x ∈ B δ (x 0 ), p ∈ B δ (Dφ(x 0 , t 0 )), a. e. t ∈ B δ (t 0 ) and some δ > 0} .

Definition 4.1. L 1 -bilateral viscosity solution (L1Bvs) II Let u : R N × R + → R be a bounded lower semi-continuous function. We say that u is a L 1 -bilateral viscosity solution (L1Bvs) of (36) if:

1. for any (x 0 , t 0 ), φ ∈ C 1 (R N × R + ), (G, b) ∈ H -(t 0 , x 0 , Dφ(x 0 , t 0 )) such that (x 0 , t 0 ) is a local minimum point for u(x, t) -t 0 b(s)dsφ(x, t) we have

-φ t (x 0 , t 0 ) + G(t 0 , x 0 , Dφ(x 0 , t 0 )) ≤ 0, 2. for any (x 0 , t 0 ), φ ∈ C 1 (R N × R + ), (G, b) ∈ H + (t 0 , x 0 , Dφ(x 0 , t 0 )) such that (x 0 , t 0 ) is a local minimum point for u(x, t) - t 0 b(s)ds -φ(x, t) we have -φ t (x 0 , t 0 ) + G(t 0 , x 0 , Dφ(x 0 , t 0 )) ≥ 0.
3. The final condition is satisfied in the following sense:

ϕ(x) = inf lim inf n→∞ u(x n , t n ) : x n → x , t n ↑ T .
Remark 7. Other formulations can be considered to define the L 1 -bilateral viscosity notion. For instance, one can take the test function φ(x, t)

∈ C 1 (R N × (0, T )) in Definition 3.1 or φ ∈ C 1 (R N ) in Definition 4.1. Of course, one can also replace φ ∈ C 1 (R N ) by φ ∈ C 2 (R N ), ..., C ∞ (R N ).
On the other hand, by classical arguments in the theory of viscosity solutions, we may replace the local minimum by global, or local strict or global strict.

Proposition 2. Assume (H0) and (H1). Then, Definitions 3.1 and 4.1 are equivalent.

Proof. The equivalence follows by remarking that for any b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local minimum point for u(x, t) - G(t 0 , x 0 , Dφ(x 0 )).

Consistency

Let us now prove that Definition 4.1 is consistent with the definitions of viscosity solutions given for more regular HJB equations. In particular those considered for a time-continuous Hamiltonian and/or a continuous initial data. For the sake of completeness let us recall here the definition of viscosity solution in those cases.

Definition 4.2 (bilateral viscosity solution (Bvs), [7]). Assume that H is continuous w.r. to the time variable. Let u ∈ LSC(R N × (0, T )) be a bounded function. We say that u is a bilateral viscosity solution (Bvs) of ( 36) if for any φ ∈ C 1 (R N × (0, T )) and (x 0 , t 0 ) local minimum point of u(x, t)φ(x, t) we have -φ t (x 0 , t 0 ) + H(t 0 , x 0 , Dφ(x 0 , t 0 )) = 0, and if the final condition is satisfied in the following sense:

ϕ(x) = inf lim inf n→∞ u(x n , t n ) : x n → x , t n ↑ T .
Definition 4.3 (L 1 -viscosity solution (L1vs), [24,25]). Assume that the final condition ϕ is a continuous function on R N . We say that u ∈ LSC(R N × (0, T )) is a L 1 -viscosity supersolution (L1vsp) of ( 36) if: for any b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local minimum point of u(x, t) -

{H(t, x, p) -b(t)} ≥ 0. We say that u ∈ U SC(R N × (0, T )) is a L 1 -viscosity subsolution (L1vsb) of (36) if: for any b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local maximum point of u(x, t) - t 0 b(s)ds -φ(x) we have lim δ→0 + ess inf |t-t0|≤δ inf x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≤ 0.
We say that u ∈ C(R N × (0, T )) is a L 1 -viscosity solution (L1vs) if it is both a L 1 -viscosity subsolution and a L 1 -viscosity supersolution and the final condition is satisfied pointwise:

u(x, T ) = ϕ(x) in R N .
The link between these two definitions and our Definition 4.1 is stated in the following Theorem. Proof. The proof of statement (a) is based on some results introduced and developed in ([7, Theorem 1.1]). We recall here this result for the convenience of the reader.

Lemma 4.5. Let W be a continuous function on [0, ∞) × R n such that W has a zero maximum (minimum) at (τ, ξ). Let ε > 0. Then there is a smooth function ψ, a finite set of numbers α k ≥ 0 summing to one, and a finite collection of points

(t k , x k ) such that 1. W -ψ has a zero minimum (maximum) at (t k , x k ); 2. (t k , x k ) ∈ B oε(1) √ ε (s, y) for some (s, y) ∈ B oε(1) (τ, ξ); 3. |D t,x ψ(t k , x k )| = oε(1) √ ε ; 4. k α k D t,x ψ(t k , x k ) = 0.
Step 1. Assume that u is a L1vs and let us show that u is also a L1Bvs. For this let b be in L 1 (0, T ), φ be in C 1 (R N ) and (x 0 , t 0 ) be a local minimum point for u(x, t) -t 0 b(s)dsφ(x), we have to show that lim

δ→0 + ess sup |t-t0|≤δ sup x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≥ 0 ( 37
)
and lim

δ→0 + ess inf |t-t0|≤δ inf x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≤ 0. ( 38
)
Since u is a L1vs, then we have (37). To prove (38), for each δ > 0 we apply Lemma 4.5 above choosing ε small enough the ensure the existence of an η > 0 such that oε(1) √

ε + o ε (1) + η ≤ δ and o ε (1) √ ε + o ε (1) + η ≤ δ (and with W (t, x) = u(x, t) + t 0 b(s)ds -φ(x, t)).
Therefore, there exists a smooth function ψ and a finite set of points (

x k , t k ) such that u - t 0 b -(φ + ψ) has a zero maximum at (x k , t k ) and for each k B η (x k , t k ) ⊂ B δ (t 0 , x 0 ), B η (Dφ(x k ) + Dψ(x k , t k )) ⊂ B δ (Dφ(x 0 )). ( 39
) Thus ess inf |t-t0|≤δ inf x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≤ ≤ ess inf |t-t k |≤η inf x∈Bη(x k ), p∈Bη (Dφ(x k )+Dψ(x k ,t k )) {H(t, x, p) -b(t)} . ( 40
)
Since u is a L1vs, in particular is a L 1 -viscosity subsolution therefore in each point (t k , x k ) we have

lim η→0 + ess inf |t-t k |≤η inf x∈Bη(x k ), p∈Bη (Dφ(x k )+Dψ(x k ,t k )) {H(t, x, p) -b(t)} ≤ 0.
Letting δ going to 0 + (⇒ η → 0 + ) in ( 40) we obtain (38) and conclude the proof.

Step 2. Assume that u is a L1Bvs and let us show that u is also a L1vs. We first remark that, by Definition 3.1 if u is a L1Bvs, is in particular a L 1 -viscosity supersolution. Therefore, to prove that u is a L

1 -viscosity subsolution fix b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local maximum point of u(x, t) - t 0 b(s)ds -φ(x) our thesis is lim δ→0 + ess inf |t-t0|≤δ inf x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p) -b(t)} ≤ 0. ( 41
)
As above, for each δ > 0 we apply Lemma 4.5 choosing ε small enough the ensure the existence of an η > 0 such that oε(1) √ ε + o ε (1)+ η ≤ δ and o ε (1)

√ ε + o ε (1)+ η ≤ δ,
and with W (t, x) = u(x, t) + t 0 b(s)dsφ(x, t). Therefore, there exists a smooth function ψ and a finite set of points (x k , t k ) such that u - Letting δ going to 0 + (⇒ η → 0 + ) in ( 43) we obtain (41) and conclude the proof.

The proof of (b) is straightforward.

Stability

We prove here a stability result with respect to the Hamiltonian. It will be proved assuming a very weak convergence in time as done, for L 1 -viscosity solution, by Barles in [4]. (Our proof is indeed an adaptation to L1Bvs of the proof of [4, Theorem 1.1]). Note that in this proof we only need assumptions (H0)-(H1) on the Hamiltonian.

Theorem 4.6. Stability w.r.to H. For each n ∈ N let u n be a L 1 -bilateral viscosity solution of -u t (x, t) + H n (t, x, Du) = 0 in R N × (0, T ) (44)

We assume that:

i) For each n ∈ N the Hamiltonian H n fulfills hypotheses (H0)-(H1) for some modulus m n = m n (K) such that m n (•, r) L 1 (0,T ) → 0 as r → 0 uniformly with respect to n, for any compact subset K. sequence of L 1 -viscosity subsolutions of (36) approximating u. The thesis will then follow by comparison result for L 1 -viscosity solution.

The construction of the approximating sequence can be summarised in the following Lemma. The proof being an adaptation of the proof given in [3, Lemme 5.5] will be not detailed (see also [6,Lemma 19]).

Lemma 4.9. Under the assumption of Theorem 4.8, if u is L 1 -bilateral viscosity solution of (36), let u ε be defined by Then, the upper semi continuous envelope (u ε ) * is a L 1 -viscosity subsolution of -(u ε ) t + H(t, x, Du ε )-k 0 ∞ e 1 2 KT M ε = 0 for (x, t) ∈ R N × (0, T ), ( 50)

for K big enough and where M = 2 u ∞ and k 0 is given in assumption (H2). Moreover, (u ε ) * (x, T ) ≤ ϕ(x) for x ∈ R N .

(51)

Since (u ε ) * is a L 1 -viscosity subsolution of (50) the function (u ε ) * -( k 0 ∞ e 1 2 KT M ε)t is a L 1 -viscosity subsolution of (36), therefore, by the comparison result for L 1 -viscosity solutions (see [24,Theorem 8.1] or [29]) we obtain

(u ε ) * -k 0 ∞ e 1 2 KT M ε ≤ v(x, t) ∀(x, t) ∈ R N × (0, T )
where we used also (51). Letting ε → 0 we have u(x, t) ≤ v(x, t) ∀(x, t) ∈ R N × (0, T ), thus, reversing the roles of u and v, the uniqueness follows.

  For a function f : [a, b] → R N we will denote by V b a (f ) the classical variation on [a, b] and by BV ([0, T ]; R N ) the set of functions f : [0, T ] → R N with bounded variation on [0, T ]. Moreover, we will denote by BV -([0, T ]; R N ) the set of left continuous functions of BV ([0, T ]; R N ) which are continuous at 0.

Figure 1 :

 1 Figure 1: Case 1: Numerical solutions with 150 2 grid nodes.

Figure 2 :

 2 Figure 2: Case 2: Numerical solutions with 150 2 grid nodes.

  sup x∈B δ (x0), p∈B δ (Dφ(x0)) {H(t, x, p)b(t)} = = inf (G,b)∈H + (t0,x0,Dφ(x0)) G(t 0 , x 0 , Dφ(x 0 )) b)∈H -(t0,x0,Dφ(x0))

  sup x∈B δ (x0), p∈B δ (Dφ(x0))

  Theorem 4.4. (Consistency). Assume (H0)-(H3) and (Hid).

( a )

 a If the final condition ϕ is a continuous function, thenu is a L 1 -bilateral viscosity solution ⇐⇒ u is a L 1 -viscosity solution.(b) If the Hamiltonian H is continuous also in the t-variable, then u is a L 1 -bilateral viscosity solution ⇐⇒ u is a bilateral viscosity solution.

t 0 b

 0 -(φ + ψ) has a zero minimum at (x k , t k ) and for each kB η (x k , t k ) ⊂ B δ (t 0 , x 0 ), B η (Dφ(x k ) + Dψ(x k , t k )) ⊂ B δ (Dφ(x 0 )). x0), p∈B δ (Dφ(x0)) {H(t, x, p)b(t)} ≤ ≤ ess inf |t-t k |≤η inf x∈Bη(x k ), p∈Bη (Dφ(x k )+Dψ(x k ,t k )) {H(t, x, p)b(t)} . (43)Since u is a L1Bvs we have (3.1) at each point (t k , x k ), i.e.lim η→0 + ess inf |t-t k |≤η inf x∈Bη(x k ), p∈Bη (Dφ(x k )+Dψ(x k ,t k )){H(t, x, p)b(t)} ≤ 0.

ii)

  There exists a function H fulfilling hypotheses (H0)-(H1) such that, for any(x, p) ∈ R N × R N , x n , t n ), is a L 1 -bilateral viscosity solution of -u t (x, t) + H(t, x, Du) = 0 in R N × (0, T )(45)

  u ε (x, t) := inf y∈R N u(y, t) + e -Kt |x -y| 2 ε 2 , ε > 0.

  Hg1) The functions g 0 (t, Y, a) : R + ×R N ×A → R N , g i (t, Y ) : R + ×R N → R N , (i = 1, . . . , M ), are measurable functions in t and continuous in Y, a. Moreover, for each Y ∈ R N we have g 0

Optimal control problems with measurable timedependent dynamicsIn this section we characterize the value function v as the unique L 1 -bilateral viscosity solution of (18). Moreover, we investigate convergence results for numerical

Acknowledgments. We wish to thanks the two anonymous referees for the very important remarks that significantly improved the last version of this paper.

* This work is partially supported by the grant DGA-ENSTA No 06 60 037. The second author is also supported by the European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO

Proof. Following Definition 4.1 we have to prove that (1) for any (x 0 , t 0 ), φ ∈ C 1 (R N × R + ), (G, b) ∈ H -(t 0 , x 0 , Dφ(x 0 , t 0 )) such that (x 0 , t 0 ) is a local minimum point for u(x, t) -t 0 b(s)dsφ(x, t) we have -φ t (x 0 , t 0 ) + G(t 0 , x 0 , Dφ(x 0 , t 0 )) ≤ 0, (

(2) for any (x 0 , t 0 ), φ ∈ C 1 (R N × R + ), (G, b) ∈ H + (t 0 , x 0 , Dφ(x 0 , t 0 )) such that (x 0 , t 0 ) is a local minimum point for u(x, t) -

In order to prove statement (1), let us fix a (x

Fix now a small δ > 0, we consider a large compact subset K of R N × R N and the functions m, m n given by assumptions i),ii). We construct a new sequence (u δ n ) n defined by

Note that for each n, δ the function

Moreover, if we set u δ (x, t) := inf (xn,tn)→(x,t) lim inf n→∞ u δ n (x n , t n ), by the properties of m, m n we have u ≤ u δ ≤ u + O δ (1). Therefore, by classical results, since (x 0 , t 0 ) is a strict local minimum point of u(x, t) -t 0 b(s)dsφ(x, t), for δ small enough there exists a local minimum point of u δ (x, t) -t 0 b(s)dsφ(x, t), that we will denote (x δ , t δ ). Note that (x δ , t δ ) → (x 0 , t 0 ) as δ → 0. We set now

Our aim is to use the fact that the function u δ n is a L1Bvs of (48) by testing with the function φ(x, t)

To do this we first observe that the convergence assumption ii) implies

we state now that there exists a n big enough and a δ small enough such that we can find a η > 0 for which

Indeed, since (G, b) ∈ H -(t 0 , x 0 , Dφ(x 0 , t 0 )) there exists a β such that

(where we used also the definition of m, m n ). Thus (49) follows from (x n δ , t n δ ) → (x δ , t δ ) as n → ∞, (x δ , t δ ) → (x 0 , t 0 ) as δ → 0 and the regularity of φ.

By definition of L1Bvs, condition (49) and the fact that (

Therefore letting n → ∞ and δ → 0 by the continuity of G we obtain ( 46) and conclude the proof of (1). Point ( 2) can be proved with the same argument by remarking that the functions

Theorem 4.7. (Stability w.r.to ϕ.) For each n ∈ N, let u n be a L 1 -bilateral viscosity solution of equation ( 36) with final condition

where for each n ∈ N, the function ϕ n ∈ C(R N ) and is bounded. Assume that the sequence (ϕ n ) n∈N is monotone increasing and for every x ∈ R N , we have:

Proof. We first remark that since ϕ n ∈ C(R N ), u n is continuous in R N × [0, T ] (see [24,Corollary 1.10]). Therefore, by uniqueness of L1vs solution, the sequence (u n ) n∈N is monotone increasing. We set w(x, t) := sup n∈N u n (x, t) and we remark that w(x, t) = lim

(see, for instance [2, Lemma 2.18]). Therefore, by Theorem 4.6 above we have that w is a L1Bvs of (36) in R N × (0, T ). Moreover, for each sequence (x n , t n ) → (x, T ), we have

Thus w verify also the final condition w(x, T ) = ϕ(x). By uniquness (Theorem 4.8 below) we can that conclude that w(x, t) = u(x, t) for all (x, t) ∈ R N × (0, T ) and the proof is completed.

Uniqueness

We finally prove the uniqueness result.

Theorem 4.8. Assume (H0)-( H3) and (Hid). Then there exists at most one L 1 -bilateral viscosity solution of (36).

Proof. This proof will follow the idea of G.Barles of using the inf-convolution in the proof of uniqueness for bilateral viscosity solution [3,Theorem 5.14].

Suppose that there exist v and u two L 1 -bilateral viscosity solution of (36). Since v is in particular a L 1 -viscosity supersolution the main point is to look for a


Ariela Briani
email: briani@ensta.fr

Fabio Camilli
email: camilli@dmmm.uniroma1.it.

Hasnaa Zidani
email: zidani@ensta.fr

Convergence Result and Error Estimate

Keywords: 2000 Mathematics Subject Classification. Primary 49L25; Secondary 35J45, 65N15 Monotone systems, viscosity solution, approximation scheme, error estimate

come

introduction

In this paper we study approximation schemes for a system of nonlinear second order partial differential equations (1.1)

F i x, u, Du i , D 2 u i = 0, x ∈ R N , i = 1, . . . , M,
where u = (u 1 , . . . , u M ) denotes the unknown function, and F = (F 1 , . . . , F M ) is a given function.

The theory of viscosity solution, initially developed for the scalar equation has been extended to systems in [START_REF] Engler | Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations[END_REF][START_REF] Ishii | Perron's method for monotone systems of second-order elliptic PDEs[END_REF][START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF][START_REF] Yamada | Viscosity solutions for a system of elliptic inequalities with bilateral obstacle[END_REF]. In this framework the monotonicity of F with respect to the variable u (see (2.2c)) is essential to guarantee the validity of a maximum principle. Note that this property involves not only the single component F i , but all the system at the same time. Given this property and standard regularity assumptions on F , it is possible to prove a strong comparison principle, hence the uniqueness of the viscosity solution to (1.1).

For a large class of monotone systems, the existence of the solution can be obtained either via the Perron's method ( [START_REF] Ishii | Perron's method for monotone systems of second-order elliptic PDEs[END_REF][START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF]) or via control-theoretic representation formulas, ( [START_REF] Engler | Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations[END_REF][START_REF] Krylov | On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients[END_REF]). We refer to [START_REF] Ghosh | Optimal control of switching diffusions with application to flexible manufacturing systems[END_REF][START_REF] González | On unbounded solutions of Bellman's equation associated with optimal switching control problems with state constraints[END_REF] for various applications of systems of PDEs in many areas, in particular we mention [START_REF] Bladt | Pricing derivatives incorporating structural market changes and in time correlation[END_REF] for a Black-Scholes pricing model with jumping volatility.

Here, we consider approximation schemes of the type

(1.2) S i (h, x, u h (x), u h i ) = 0 x ∈ R N , i = 1, . . . , M
where S i are consistent, monotone and uniformly continuous approximation of F i in (1.1) and the coupling among the equations is only in the variable u h (x) (where u h = (u h 1 , . . . , u h M ) represents the solution of the approximate system (1.2), and is expected to be an approximation to the solution u of system (1.1)). Typical approximation of this type, like finite difference methods and semi-Lagrangian schemes, will be discussed in details.

For the case of a scalar fully nonlinear equation, the work of Barles-Souganidis [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] provides a general setting for convergence of approximation schemes. This setting says that any monotone, consistent, continuous scheme converges to the unique viscosity solution of the limit problem, provided that it satisfies a comparison principle. To extend the result of [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] to the case of systems of PDEs, the crucial point is to introduce an appropriate monotonicity condition for (1.2) (the monotonicity of the single scheme S i being not sufficient). Under this appropriate monotonicity, (assumption (C1) in Section 3), and the standard consistency and regularity conditions, by applying the general idea of [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF] based on the use of the half-relaxed limits, we get the convergence result for the monotone systems.

While results on convergence rates for viscosity solutions of 1 st order equation were obtained from the beginning of the viscosity theory, only quite recently Krylov [START_REF] Krylov | On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients[END_REF][START_REF] Krylov | Approximating value functions for controlled degenerate diffusion processes by using piecewise constant policies[END_REF][START_REF] Krylov | The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients[END_REF] and Barles-Jakobsen [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Barles | Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF] success in proving similar rates for second order Hamilton-Jacobi-Bellman equations. Convergence rate of approximation schemes for particular Isaac equations have been obtained in [START_REF] Bonnans | Error bounds for stochastic differential games: the adverse stopping case[END_REF][START_REF] Jakobsen | On error bounds for approximation schemes for non-convex degenerate elliptic equations[END_REF]. We refer also to [START_REF] Caffarelli | A Rate of Convergence for Monotone Finite Difference Approximations to fully Nonlinear, Uniformly Elliptic PDEs[END_REF] for convergence rate in the case of elliptic fully nonlinear equations, and to [START_REF] Fahim | A Probabilistic Numerical Scheme for Fully Nonlinear parabolic PDEs[END_REF] for convergence rate of probabilistic approximation schemes. Our aim, in the second part of the paper, is to extend the previous convergence rates to the case of convex systems. In order to simplify the presentation we choose as a model problem a weakly coupled systems of Hamilton-Jacobi-Bellman equations, but the approach is sufficiently general to hold for other classes of problems, f.e. switching control problems.

To obtain the rate of convergence, we will use the same arguments developed in [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Barles | Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Bonnans | Error estimates for a stochastic impulse control problem[END_REF] and adapt them to the case of monotone system of PDEs. As usual, the upper bound for uu h is easier and it is obtained via a Krylov regularization and shaking coefficient techniques. These techniques allow to define a smooth subsolution to the system. So by using the consistency property, we obtain the upper bound. The proof of the lower bound is more involved and requires an approximation of the weakly coupled system with a switching system with a bigger number of components. By this procedure it is possible to build regular "local" supersolutions of the continuous problems. Then, we derive the lower error estimate by using the consistency and monotonicity conditions. Our result gives an upper bound of h 1/2 and a lower bound of h 1/5 for the finite differences scheme [START_REF] Bonnans | A fast algorithm for two-dimensional HJB equation of stochastic control[END_REF]. For a Semi-Lagrangian scheme these estimates become h 1/4 for the upper bound, and h 1/10 for the lower bound.

The paper is organized as follows. In Section 2 we recall definitions and basic results for the continuous problem. In Section 3 we state the main assumptions for the scheme and we prove the convergence theorem. Section 4 is devoted to the proof of the error estimates, while in section 5.1-5.2 and 5.3 we discuss respectively finite difference schemes and semi-Lagrangian schemes. The Appendix 6 is devoted to the proofs of some technical results.

Notation: We will use the following norms

|f | 0 = ess sup x∈R N |f (x)|, [f ] 1 = |Df | 0 , and |f | 1 = |f | 0 + [f ] 1 .
The space of real symmetric N × N matrices is denoted by S N , and X ≥ Y in S N will mean that X -Y is positive semi-definite.

For a function u : R N → R M , we say that u = (u 1 , . . . , u M ) is upper semicontinuous (u.s.c. for short), respectively lower semicontinuous (l.s.c. for short), if all the components u i , i = 1, . . . , M , are u.s.c., respectively l.s.c..

If u = (u 1 , . . . , u M ), v = (v 1 , . . . , v M ), are two functions defined in a set E we write u ≤ v in E if u i ≤ v i in E, i = 1, . . . , M .

The continuous problem: definitions and assumptions

Consider the system of nonlinear second order equations (2.1)

F i x, u, Du i , D 2 u i = 0, x ∈ R N , i = 1, . . . , M
where u := (u 1 , . . . , u M ) and u i is a real valued function defined in R N and F := (F 1 , . . . , F M ) :

R N × R M × R N × S N → R M is a continuous given function.
Let us first recall the definition of viscosity solution for system (2.1) (see [START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF]).

Definition 2.1. i) An u.s.c. function u : R N → R M is said a viscosity subsolution of (2.1) if whenever φ ∈ C 2 (R N ), i = 1, . . . , M and u iφ attains a local maximum at x ∈ R N , then

F i (x, u(x), Dφ(x), D 2 φ(x)) ≤ 0.
ii) A l.s.c. function v : R N → R M is said a viscosity supersolution of (2.1) if whenever φ ∈ C 2 (R N ), i = 1, . . . , M and v iφ attains a local minimum at x ∈ R N , then

F i (x, v(x), Dφ(x), D 2 φ(x)) ≥ 0.
iii) A continuous function u is said a viscosity solution of (2.1) if it is both viscosity sub-and supersolution of (2.1).

The existence of a solution to (2.1) can be obtained, for a large class of monotone systems, either via Perron's method ( [START_REF] Ishii | Perron's method for monotone systems of second-order elliptic PDEs[END_REF], [START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF]) or via the control-theoretic interpretation of the problem [START_REF] Engler | Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations[END_REF][START_REF] Yamada | Viscosity solutions for a system of elliptic inequalities with bilateral obstacle[END_REF]. To get a comparison principle for system (2.1), we shall assume the following conditions on function F :

F i ∈ C(R N × R M × R N × S N ) i = 1, . . . , M ; (2.2a)
There exists a modulus of continuity ω s.t. if X, Y ∈ S N , β > 1 and

-3β I 0 0 I ≤ X 0 0 Y ≤ -3β I -I -I I (2.2b) then F i (y, r, β(x -y), -Y ) -F i (x, r, β(x -y), X) ≤ ω(β|x -y| 2 + β -1 ); (2.2c)
There exists c 0 > 0 s.t. for r, s ∈ R M satisfying r is i = max j=1,...,M {r js j } ≥ 0, then for all x, y, p ∈ R N , X ∈ S N F i (x, r, p, X) -F i (x, s, p, X) ≥ c 0 (r is i ).

Theorem 2.2 (see [START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF]). Assume

F i : R N × R M × R N × S N → R, i = 1, . . . ,
F i (x, u(x), Du i (x), D 2 i u(x)) := sup α∈A i    L α i (x, u(x), Du i (x), D 2 u i (x)) + M j=1 d ij (x, α)u j (x)   
where the operator L α is defined by:

(2.4)

L α i (x, r, p, X) = - 1 2 T r[a i (x, α)X] -b i (x, α) • p -f i (x, α) + λ i r i .
and with a i (x, α) = σ i (x, α)σ T i (x, α). We assume that the following assumptions hold:

A i are compact metric spaces, i = 1, . . . , M b i : R N × A i -→ R N , σ i : R N × A i → R N ×D , f i : R N × A i → R |f i (•, α)| 1 + |σ i (•, α)| 1 + |b i (•, α)| 1 ≤ L, for any α ∈ A i , i = 1, . . . , M (2.5a) and d ij : R N × A i → R, |d ij (•, α)| 1 ≤ L, for any α ∈ A i , i, j = 1, . . . , M , d ii (x, α) ≥ 0, d ij (x, α) ≤ 0 for i = j, λ i ≥ 0, λ i + M j=1 d ij (x, α) ≥ λ 0 > 0 (x, α) ∈ R N × A i , i = 1, . . . , M.
(2.5b)

It is easy to check that under assumptions (2.5a)-(2.5b), system (2.4) satisfies conditions (2.2a)-(2.2c) (in particular the last condition in (2.5b) implies the monotonicity of the system with c 0 = λ 0 ). Moreover, we have the following result whose proof is given in the appendix. Proposition 2.5. Under assumptions (2.5a)-(2.5b), system (2.3) admits a unique bounded continuous solution u. Moreover, if we have:

λ 0 > max i=1,...,M sup α∈A i |σ i (•, α)| 2 1 + max i=1,...,M sup α∈A i |b i (•, α)| 1 , (2.6)
then u is bounded and Lipschitz continuous in R N .

Finally, let us also mention that system (2.3) comes from infinite horizon optimal control problems of hybrid systems and random evolution processes.

The approximation scheme: definitions and assumptions

For a fixed h > 0, we consider the following approximation scheme:

(3.1) S i (h, x, u h (x), u h i ) = 0 x ∈ R N , i = 1, . . . , M
where the function

u h : R N → R M , u h = (u h 1 , . . . , u h M )
, represents the solution of (3.1). In the sequel, we will state a set of assumptions on the scheme S(h, ., ., .):

(C1) (Monotonicity) There exists c 0 > 0 such that for any h > 0, x ∈ R N , bounded functions u, v such that u ≤ v in R N and r, s ∈ R M such that θ := r i -s i = max j=1,...,M {r j -s j } ≥ 0, then S i (h, x, r, u i + θ) -S i (h, x, s, v i ) ≥ c 0 θ.
(C2) (Regularity) For any h > 0 and any continuous, bounded function φ : R N → R, the functions x → S i (h, x, r, φ) i = 1, . . . , M are bounded and continuous on R N and the functions r → S i (h, x, r, φ) are uniformly continuous for bounded r, uniformly with respect to x ∈ R N .

(C3) (Consistency) Fix i = 1, . . . , M . For all h > 0, x ∈ R N and for any continuous function Φ ∈ C 0 (R N , R M ) with a smooth i-th component Φ i , we have:

|S i (h, x, Φ(x), Φ i ) -F i (x, Φ, DΦ i , D 2 Φ i )| ≤ ω(h)
with ω(h) → 0 for h → 0.

(C4) (Stronger consistency) There exist n, k j > 0, j ∈ J ⊂ {1, . . . , n} and a constant K c > 0 such that: for all h > 0, i = 1, . . . , M , x ∈ R N , and for any continuous

function Φ ∈ C 0 (R N ; R M ) such that its i-th component Φ i is smooth with |D j Φ i (x)| bounded in R N
, for every j ∈ J, we have:

|S i (h, x, Φ(x), Φ i ) -F i (x, Φ, DΦ i , D 2 Φ i )| ≤ K C Q(Φ i ) in R N ,
where Q(φ) := j∈J |D j φ|h k j for every smooth function φ.

Definition 3.1. We say that a function u : R N → R M is a subsolution (respectively, a supersolution) of (3.1) if it satisfies

S i (h, x, u(x), u i ) ≥ 0 (respectively S i (h, x, u(x), u i ) ≤ 0) x ∈ R N , i = 1, . . . , M.
The next result is a comparison principle for the scheme (3.1).

Proposition 3.2. Assume (C1) and (C2). If u and v are bounded sub-and supersolutions of (3.1), respectively, then u ≤ v in R N .

Proof. We assume δ :

= sup i,R N (u i -v i ) > 0 and we derive a contradiction. Let {x n } in R N and {i n } ∈ {1, . . . , M } be such that δ n = u in (x n ) -v in (x n ) = max j {u j (x n ) -v j (x n )} → δ for n → ∞.
Since u and v are respectively sub-and supersolutions, we get:

0 ≥ S in (h, x n , u(x n ), u in ) -S in (h, x n , v(x n ), v in ).
Moreover, we know that u(

x n ) ≤ v(x n )+δ n , 0 = max j (u j (x n )-(v j (x n )+δ n )) and u in ≤ v in +δ in R N .
Then, the monotonicity yields to:

S in (h, x n , u(x n ), u in ) ≥ S in (h, x n , v(x n ) + δ n , v in + δ).
Therefore, by assumption (C2), we have:

0 ≥ S in (h, x n , v(x n ) + δ, v in + δ) -S in (h, x n , v(x n ), v in ) + ω(δ n -δ),
where ω(t) → 0 when t → 0 + . Finally, by using (C1) again, we obtain:

0 ≥ c 0 δ + ω(δ n -δ),
which leads to a contradiction when n → ∞.

In all the sequel, we assume that: (C5) (Existence of discrete solution) For every h > 0, system (3.1) admits a solution u h . We give a convergence result for the scheme based on the classical argument by Barles-Souganidis, [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF].

Proposition 3.3. Assume (C1)-(C3) and (C5). Let u h = (u h 1 , . . . , u h M ) be a locally uniformly bounded family of solutions of (3.1) and let u be the solution of (2.1). Then u h → u for h → 0 locally uniformly in R N .

Proof. Define the relaxed half-limits by:

u i (x) = lim inf h→0,z→x u h i (z), u i (x) = lim sup h→0, z→x u h i (z), i = 1, . . . , M.
Following the arguments introduced in [START_REF] Barles | Convergence of approximation schemes for fully nonlinear second order equations[END_REF], it is sufficient to prove that u is a supersolution and u is a subsolution of (2.1). Then by the comparison principle (Theorem 2.2), it follows that u ≤ u. Since the other inequality is obvious we get u = u = lim h→0 u h and the local uniform convergence in R N . We will only prove that u is a subsolution of (2.1) (the same arguments can be used to prove that u is a supersolution). Let φ be a smooth function, and i ∈ {1, . . . , M }, such that u iφ has a maximum point at x 0 , u i (x 0 ) = φ(x 0 ). By using standard arguments in the viscosity theory, there exists a sequence (h n ) n of positive numbers satisfying:

h n → 0 and x n := x hn → x 0 when n → ∞, u hn
iφ has a maximum point at x n , and

u hn i (x n ) → u i (x 0 ).
Set δ n := (u hn iφ)(x n ), and define the function:

Φ n (x) = (u hn 1 (x), . . . , u hn i-1 (x), φ(x), u hn i+1 (x), . . . , u hn M (x)) in R N .
Up to a subsequence, Φ n (x n ) converges to a vector r ∈ R M , where r i = u i (x 0 ), and r j ≤ u j (x 0 ) for j = i. Then, using the fact that:

u hn i -δ n ≤ φ in R N and δ n = u hn i (x n ) -φ(x n ) = max j (u hn j (x n ) -Φ n j (x n ))
, and taking into account the monotonicity assumption (C1), we get:

0 = S i (h n , x n , u hn (x n ), u hn i ) = S i (h n , x n , u hn (x n ), (u hn i -δ n ) + δ n ) ≥ S i (h n , x n , Φ n (x n ), φ) + c 0 δ n ≥ F i (x n , Φ n (x n ), Dφ(x n ), D 2 φ(x n )) + c 0 δ n -ω(h)
where the last inequality is due to the consistency assumption (C3). Passing to the limit when n → ∞, we get 0 ≥ F i (x 0 , u(x 0 ), Dφ(x 0 ), D 2 φ(x 0 )), which proves that u is a subsolution of (2.1).

The error estimate for the weakly coupled systems

In this section, we consider again system (2.3) considered in Example 2.4, with L α i as in (2.4), i.e.

F i (x, r, p, X) := sup α∈A i    - 1 2 T r[a i (x, α)X] -b i (x, α) • p -f i (x, α) + λ i r i + M j=1 d ij (x, α)r j    .
We will always assume that (2.5) and (2.6) are satisfied and therefore we will denote by u the unique solution given by Proposition 2.5. For this particular system, we will derive an error estimate for |uu h |, where u h is the solution of a scheme satisfying (C1),(C2),(C4) and (C5).

4.1. The upper bound.

Proposition 4.1. There exists C > 0 such that:

(4.1) u i (x) -u h i (x) ≤ Ch γ ∀i = 1, . . . , M
where γ = min j∈J k j j , with J and k j being defined in (C4).

We need a preliminary lemma which will be proved in the Appendix.

Lemma 4.2.

i) There is a unique solution u ε of the system

(4.2) max |e|≤ε F i (x + e, u(x), Du i , D 2 u i ) = 0 x ∈ R N , i = 1, . . . , M
and two constants C 0 , C 1 independent of h, ε such that

(4.3) max i=1,...,M |u ε i | 1 ≤ C 0 and max i=1,...,M |u i -u ε i | 0 ≤ C 1 ε.
where u is a solution of (2.1). ii) Let (ρ ε ) ε be a family of standard mollifiers and define

u ε = (ρ ε * u ε 1 , . . . , ρ ε * u ε M ) := (u 1,ε , . . . , u M,ε ). Then u ε is a classical subsolution of (2.

1).

Proof of Proposition 4.1. Thanks to Lemma 4.2 ii), u ε is classical subsolution of (2.1). Therefore, by (C4) we have, for each i = 1, . . . , M

(4.4) S i (h, x, u ε (x), u i,ε ) ≤ F i (x, u ε , Du i,ε , D 2 u i,ε ) + Q(u i,ε ) ≤ Q(u i,ε ).
Set ε = h γ with γ as in the statement. By [7, Lemma 4.2], we can estimate

Q(u i,ε ) ≤ K|J|C 0 h γ := Ch γ ,
where C 0 as in (4.3). By (C1) and (4.4), we have that

u ε -C c 0 h γ = (u 1,ε -C c 0 h γ , . . . , u M,ε -C c 0 h γ
) is a subsolution of the scheme. Hence, by the comparison principle of the scheme (Proposition 3.2), we get:

(4.5) u i,ε -u h i ≤ C c 0 h γ ∀i = 1, . . . , M.
By estimate (4.3) and (4.5), we conclude

u i -u h i ≤ C c 0 h γ + Ch γ ∀i = 1, . . . , M
and therefore (4.1) is satisfied.

The lower bound.

For the lower bound we need an additional assumption For each control set A i and δ > 0, there is a finite family of controls

{α ij } L i j=1 such that for any α ∈ A i , inf j=1,...,L i |σ i (•, α) -σ i (•, α ij )| 0 + |b i (•, α) -b i (•, α ij )| 0 + |f i (•, α) -f i (•, α ij )| 0 + M l=1 |d il (•, α) -d il (•, α ij )| 0 ≤ δ. (4.6)
Remark 4.3. The previous assumption is satisfied for instance when the sets A i are finite or if the functions σ i , b i , f i , d ij are uniformly continuous in α, uniformly in x. A slight more general assumption is considered in [START_REF] Barles | Error bounds for monotone approximation schemes for parabolic Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Krylov | Approximating value functions for controlled degenerate diffusion processes by using piecewise constant policies[END_REF][START_REF] Krylov | On the rate of convergence of finite-difference approximations for Bellmans equations with variable coefficients[END_REF]. To simplify the notation, we assume in the following w.l.o.g. that all the families {α ij } L i j=1 have the same cardinality L. Proposition 4.4. Assume that (2.5), (2.6) and (4.6) hold. Then there exists C > 0 such that:

(4.7) -Ch γ ≤ u i (x) -u h i (x) ∀i = 1, . . . , M
where γ = min j∈J k j 3j-2 with J, k j being defined in (C4).

For every ℓ > 0, we introduce the following switching system (4.8)

F ε,ℓ ij (x, V ε,ℓ , DV ε,ℓ ij , D 2 V ε,ℓ ij ) = 0 x ∈ R N , i = 1, . . . , M, j = 1, . . . , L where
F ε,ℓ ij (x, R, p, X) = max min |e|≤ε L α ij i (x + e, R ij , p, X) + M l=1 d il (x + e, α ij )R lj , (4.9) M ij (R) , M ij (R) := R ij -min l=1,...,L l =j {R il + ℓ}, (4.10)
where {α ij } L j=1 , i = 1, . . . , M are defined in (4.6). The solution of the system (4.8) is a function from R N to R M ×L . The (i, j) component of the solution of (4.8) is coupled with other L components by means of the switching term M ij (R) and with other (M -1) components by means of the term M l=1 d il (x, α)R lj . Lemma 4.5. Let δ, ε, ℓ > 0 be fixed. Let u be the solution of (2.1) and V ε,ℓ be the solution of (4.8).

i) There exists C > 0 such that:

(4.11) max j=1,...,L |V ε,ℓ ij -u i | 0 ≤ C(ε + ℓ 1 3 + δ) i = 1, . . . , M. ii) Set V ℓ ε,ij := ρ ε * V ε,ℓ ij , i = 1, .
. . , M, j = 1, . . . , L where ρ ε is a standard mollifier. Then (4.12) max j,l=1,...,L

|V ε,ℓ ij -V ℓ ε,il | 0 ≤ C(ε + ℓ) iii) Moreover, for ε ≤ (4 sup ij [V ε,ℓ ij ] 1 ) -1 ℓ and for for any x ∈ R N , if we set j = j(ε, i, x) := arg min l=1,...,J V ℓ ε,il (x), then L α ij i (x, V ℓ ε,ij , DV ℓ ε,ij , D 2 V ℓ ε,ij ) + M l=1 d il (x, α ij )V ℓ ε,lj ≥ 0.
The proof of the previous lemma is postponed to the Appendix.

Proof of Proposition 4.4. The proof is based on the same arguments as in [2, Theorem 3.5].

We fix δ > 0 in such a way that (4.6) is satisfied, we consider the solution V ε,ℓ of (4.8) and its mollification (component by component) V ε . We define a function w : R N → R M by w i := min j=1,...,L V ℓ ε,ij . By Lemma 4.5.(iii), it follows that w is a supersolution of (2.1). We define m = sup where φ(y) = (1 + |y| 2 ) 1/2 . Since w and u h are bounded continuous functions, the supremum in (4. [START_REF] Engler | Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations[END_REF]) is achieved at a point x and at an index i, which is also a maximum point for y -→ u h i (y) -V ℓ ε,ij (y)kφ(y), where j = arg min l=1,...,L V ℓ ε,il (x). By Lemma 4.5.(iii), the definition of φ , (2.5a) and (2.5b), we get sup

α∈A i L α i (x, (V ℓ ε,ij + kφ)(x), D(V ℓ ε,ij + kφ)(x), D 2 (V ℓ ε,ij + kφ)(x)) + M l=1 d il (x, α)(V ℓ ε,lj + kφ)(x) ≥ -Ck. (4.14)
With the consistency assumption (C4), we obtain

(4.15) -Ck ≤ S i (h, x, ({V ℓ ε,lj } M l=1 + kΦ)(x), V ℓ ε,ij + kφ) + Q(V ℓ ε,ij + kφ) where Φ = (φ, . . . , φ). Taking into account the definition of Q(•), V ℓ ε,ij and Φ, we obtain (4.16) -C l∈J ε 1-l h k l + O(k) ≤ S i (h, x, ({V ℓ ε,lj } M l=1 + kΦ)(x), V ℓ ε,ij + kφ).
Moreover by the definition of m k , it follows that

(V ℓ ε,ij + kφ)(y) ≥ u h i (y) -m k ∀y ∈ R N , and
u h i (x) -V ℓ ε,ij (x) -kφ(x) ≥ u h l (x) -w l (x) -kφ(x) ≥ u h l (x) -V ℓ ε,lj (x) -kφ(x) and therefore (u h i (x) -m k ) -(V ℓ ε,ij (x) + kφ(x)) = max l {(u h l (x) -m k ) -(V ℓ ε,lj (x) + kφ(x))}.
This with assumption (C1) yields:

S i (h, x, ({V ℓ ε,lj } M l=1 + kΦ)(x), V ℓ ε,ij + kφ) ≤ S i (h, x, (u h -m k )(x), u h i -m k ) ≤ -c 0 m k + S i (h, x, u h (x), u h i ) = -c 0 m k .
From the previous inequality and (4.16), sending k → 0 we get the estimate (4.17)

c 0 m ≤ C l∈J ε 1-l h k l .
Fix y ∈ R N and l ∈ {1, . . . , M }, then for any j ∈ {1, . . . , L}

u h l (y) -u l (y) ≤ u h l (y) -V ℓ ε,lj (y) + V ℓ ε,lj (y) -u l (y) ≤ u h l (y) -w l (y) + V ℓ ε,lj (y) -u l (y) ≤ m + V ℓ ε,lj (y) -u l (y)
By (4.17), Lemma 4.5(i) and (ii), we get

u h l (y) -u l (y) ≤ C( l∈J ε 1-l h k l + ε + ℓ + ℓ 1/3 + δ),
and the statement of the theorem follows by taking ε = max l∈J h 3k l 3l-2 and ℓ = 4 sup ij [V ε,ℓ ij ] 1 ε and sending δ to 0.

Examples of approximation schemes

5.1. Finite differences, one dimensional problem. Let x be in R, φ in C 4 (R), h in R * + and define δ ± φ(x) = φ(x ± h) -φ(x) ±h , ∆φ(x) = φ(x + h) -2φ(x) + φ(x -h) h 2 .
In particular, by a Taylor expansion, we obtain

|δ ± φ(x) -Dφ(x)| ≤ 1 2 h|D 2 φ|, |∆ h φ(x) -D 2 φ(x)| ≤ 1 12 h 2 |D 4 φ|.
Now, an approximation u h to the solution of the coupled system (2.4) in dimension N = 1, can be obtained by the finite difference scheme in R:

sup α∈A i - 1 2 a i (x, α)∆u h i (x) -b + i (x, α)δ + u h i (x) -b - i (x, α)δ -u h i (x) + λ i u h i (x) -f i (x, α) + M j=1 d ij (x, α)u h j (x)
   = 0, for i = 1, . . . , M , where b + i (x, α) = max(b i (x, α), 0), and b - i (x, α) = min(b i (x, α), 0). This scheme can be rewritten as:

S i (h, x, u h (x), u h i ) = 0 in R, for i = 1, . . . , M
where the operator S i is defined on

R * + × R × R M × C 0 (R) by: S i (h, x, r, w) := sup α∈A i - 1 2 a i (x, α) w(x + h) -2r i + w(x -h) h 2 -b + i (x, α) w(x + h) -r i h (5.1) -b - i (x, α) r i -w(x -h) h + λ i r i -f i (x, α) + M j=1 d ij (x, α)r j    ,
From the Taylor expansion, one can easily check that the monotonicity (C1) holds (with c 0 = λ 0 ), and the consistency hypothesis (C4) is satisfied with Q(φ) = |D 2 φ|h + |D 4 φ|h 2 , i.e. k 2 = 1 and k 4 = 2. Then, by Propositions 4.1 and 4.4, we have

-Ch 1/5 ≤ u -u h ≤ Ch 1/2 .
5.2. The generalized finite differences scheme. We consider the generalized finite differences scheme defined in [START_REF] Bonnans | Consistency of Generalized Finite Difference Schemes for the Stochastic HJB Equation[END_REF]. Let φ be a real valued function. Let h > 0, ξ ∈ Z N and consider the finite difference operator along direction ξ:

∆ ξ φ(x) := φ(x + ξh) + φ(x -ξh) -2φ(x) h 2 .
On the other hand, we consider

(D ± φ(x)) j =      φ(x + he j ) -φ(x) h if [b i (x, α)] j ≥ 0, φ(x) -φ(x -he j ) h if [b i (x, α)] j ≤ 0.
Let S be a finite set of Z \ {0} containing {e 1 , . . . , e N }. We consider the following scheme:

sup α∈A i    - 1 2 ξ∈S γi ξ (x, α)∆ ξ Φ i (x) -b i (x, α)D ± Φ i (x) + λ i Φ i (x) -f i (x, α) (5.2) + M j=1 d ij (x, α)Φ j (x)    = 0,
where the coefficients γi ξ are given by:

a i (x, α) - ξ∈S γi ξ ξξ T = min (γ ξ )∈(R + ) |S| a i (x, α) - γ ξ ξξ T .
For fast computations of the coefficients γ i ξ we refer to [START_REF] Bonnans | A fast algorithm for two-dimensional HJB equation of stochastic control[END_REF]. In the sequel, we make the strong consistency hypothesis (see also [START_REF] Krylov | The rate of convergence of finite-difference approximations for Bellman equations with Lipschitz coefficients[END_REF]):

(5.3) a i (x, α) = ξ∈S γi ξ (x, α)ξξ T , ∀ x ∈ R N , ∀ α ∈ A i .
The scheme defined in (5.2), can be rewritten as S i (h, x, u h (x), u h i ) = 0, where for i = 1, . . . , M , the operator S i is defined in

R * + × R N × R M × C 0 (R N ) by: S i (h, x, r, w) := sup α∈A i    - 1 2 ξ∈S γi ξ (x, α) w(x + hξ) -2r i + w(x -hξ) h 2 - N j=1 max (0, [b i (x, α)] j ) w(x + he j ) -r i h -min (0, [b i (x, α)] j ) r i -w(x -he j ) h +λ i r i -f i (x, α) + M j=1 d ij (x, α)r j    .
With straightforward calculations, one can check that the above scheme satisfies (C1) and (C2). Moreover, under condition (5.3), if we consider a function Φ ∈ C 0 (R N , R M ) with Φ i ∈ C 4 (R N ), then by applying a Taylor expansion, we obtain

|F i (x, Φ(x), DΦ i , D 2 Φ i ) -S i (x, h, Φ(x), Φ i )| ≤ sup α∈A i |b i (., α)| 0 |D 2 Φ i | 0 h + sup α∈A i |σ i (•, α)| 2 0 |D 4 Φ i | 0 h 2 .
Then the scheme satisfies the strong consistency (C4) with k 2 = 1 and k 4 = 2. We conclude that when the stencil S is chosen in such way condition (5.3) is satisfied, for h sufficiently small, the upper bound of the error estimate for the generalized finite difference scheme is of order h 1/2 and the lower bound is of order h 1/5 .

Remark 5.1. Equation (5.2) can be rewritten as a fixed-point equation with a contraction operator. Indeed, let us introduce a fictitious time step τ > 0 and introduce the operator

T : C 0 (R N , R M ) → C 0 (R N , R M ) defined by [T w] i (x) := (1 -τ λ i w i (x)) -τ sup α∈A i    - 1 2 ξ∈S γξ (x, α)∆ ξ w i (x) -b i (x, α)D ± w i (x) -f i (x, α) + M j=1 d ij (x, α)w j (x)    .
Then, we can easily see that (5.2) is equivalent to

(5.4) u(x) = [T u](x), on R N .
Moreover, for τ small enough, the operator T is a monotone contraction, then the fixed-point equation u = T u admits a unique solution, and this solution is limit to any sequence defined by u n+1 = T u n , with u 0 ∈ C(R N , R M ). This iterative process, called value iterations method can be used to compute a numerical solution u h on a grid G.

5.3.

Semi-Lagrangian schemes. Semi-Lagrangian schemes for second order Hamilton-Jacobi equations have been already studied in several papers, we refer to [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Camilli | An approximation scheme for the optimal control of diffusion processes[END_REF][START_REF] Camilli | A finite element like scheme for integro-partial differential Hamilton-Jacobi-Bellman equations[END_REF][START_REF] Kushner | Numerical methods for stochastic control problems in continuous time[END_REF][START_REF] Munos | Consistency of a simple multidimensional scheme for HJB equations[END_REF] for more details. Here, we use the Semi-Lagrangian scheme to approximate the weakly coupled system given in Example 2.4. We recall that the system (2.1) with F i as in (2.3) is the dynamic programming equation of an infinite horizon optimal control problem with dynamics given by the stochastic differential equation (5.5)

dX t = b νt (X t , α t )dt + σ νt (X t , α t )dW t
where X 0 = x, W t is a standard Brownian motion, α t is the control process and ν t is a continuous-time random process with state space {1, . . . , M } for which (5.6)

P{ν t+∆t = j | ν t = i, X t = x, α t = α} = c ij (x, α)∆t + O(∆t) for ∆t → 0. j = i,
for any i, j = 1, . . . , M , i = j. We consider an approximation of (5.5) via a discrete-time control process (X n , ν n ) ∈ R N × {1, . . . , M } which evolves according to the following rule

     X 0 = x, X n+1 = X n + hb νn (X n , α n ) + √ h d m=1 σ νn,m (X n , α n )ξ m n δ νn,ν n+1 , P{ν n+1 = j | ν n = i, X n = x, α n = α} = hc ij (x, α) j = i (5.7)
for n ∈ N, where σ i,m denote the m-th column of σ i and ξ m n , m = 1, . . . , D, are random variables taking values in {-1, 0, 1} such that

P[{ξ i n = ±1}] = 1 2D
and P[{ξ i n = 0} ∩ {ξ j n = 0}] = 0, i = j and δ n,n = 1 and δ n,m = 0 for n = m. The discrete control {α n } is a random variable which is measurable with respect to the σ-algebra generated by X 1 , . . . , X n and such that α n ∈ A νn .

Set d ii = M j=1 c ij , and d ij = -c ij for every j = i, then the generator of the discrete process is

L h i,α (x, Φ(x), Φ i ) = E x,i [Φ ν 1 (X 1 )] -Φ i (x) h = L h i,α (x, Φ(x), Φ i ) - M j=1 d ij (x, α)Φ j (x)
for Φ ∈ C 0 (R N , R M ), where

L h i,α (x, r, φ) = 1 2Dh D m=1 φ(x + hb i (x, α) + √ hσ i,m (x, α)) + φ(x + hb i (x, α) - √ hσ i,m (x, α)) -2r i , (5.8)
Let u h = (u h 1 , . . . , u h M ) be a solution of the system (5.9)

S i (h, x, u h (x), u h ) = 0 in R N , i = 1, . . . , M
where S i (h, x, r, φ) := sup

α∈A i    -(1 -λ i h)L h i,α (x, r, φ) -f i (x, α) -(1 -λ i h) M j=1 d ij (x, a)r j   
and L h i,a is as in (5.8). It is easy to see that the scheme (5.9) satisfies assumption (C1)-(C3). Moreover, for 0 < h < 1, i = 1, . . . , N and for any Φ ∈ C 4 (R N , R M ) satisfying

|Φ i | 0 + • • • + |D 4 Φ i | 0 < ∞, we have (5.10) |F i (x, Φ(x), DΦ i (x), D 2 Φ i (x))-S i (h, x, Φ(x), Φ i )| ≤ C 1 h(|D 2 Φ i | 0 +|D 3 Φ i | 0 +|D 4 Φ i | 0 )
also assumption (C4) is satisfied, giving a rate of convergence of order h 1/4 as in the case of the single equation (see [START_REF] Barles | Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF]).

Appendix

We start by proving the regularity result given in Proposition 2.5.

Proof of Proposition 2.5. The bound on the solution u follows by the comparison principle after checking that C λ 0 and -C L are, respectively, a super and a subsolution of (2.1). To get the bound on the gradient of u consider m := sup i=1,...,M, x,y∈R N {u i (x)u i (y) -L|x -y|}.

If, by choosing a

L > max i=1,...,M sup α∈A i |f i (•, a)| 1 + max i,j=1,...,M sup α∈A i |d i,j (•, a)| 1 max i=1,...,M |u i | 0 λ 0 -max i=1,...,M sup α∈A i |σ i (•, a)| 2 1 -max i=1,...,M sup α∈A i |b i (•, a)| 1
we con conclude that m ≤ 0, the proof is achieved. Assume, for simplicity that the maximum is attained at (x, ŷ) (if it is not one can modify the test function in a standard way). If x = ŷ then m = 0 and we are done. If not, at (x, ŷ) the function L|x -y| is smooth and the proof just follows a classical doubling argument and the same computation as in the proof of Proposition 6.1 below. For more detail see also the proof of [START_REF] Ishii | On the equivalence of two notions of weak solutions, viscosity solutions and distribution solutions[END_REF]Theorem 5].

In order to prove Lemma 4.2, we need the the following continuous dependence estimate.

Proposition 6.1. Let u, v ∈ C 0,1 (R N ) be a solution of (2.3) with L α i as in (2.4) with coefficients {σ i , b i , λ i , f i , d ij } M i=1 and, respectively, {σ i , b i , λ i , f i , d ij } M i=1 satisfying (2.5a), (2.5b) (with the same constant λ 0 ). Then there is a constant C such that

λ 0 max i=1,...,M |u i -v i | 0 ≤ C sup i=1,...,M sup α∈A i {|λ i -λ i |(|u i | 0 ∧ |v i | 0 ) + |σ i (•, a) -σ i (•, a)| 0 + |b i (•, a) -b i (•, a)| 0 + |f i (•, a) -f i (•, a)| 0 + M j=1 |d ij (•, a) -d ij (•, a)| 0 }
Proof. The proof is a modification of [1, Theorem A.1], therefore we only details the difference with that. Define m = sup x∈R N ,i=1,...,M (u iv i ), φ(x, y) := α|x -y| 2 + ε(|x| 2 + |y| 2 ) and ψ i (x, y) = u i (x)v i (y)φ(x, y). Then set m α,ε := sup (x,y)∈R N ×R N ,i=1,...,M ψ i (x, y). A standard computation gives that there exists (x 0 , y 0 ) ∈ R N × R N and i 0 ∈ {1, . . . , M } such that m α,ε = ψ i 0 (x 0 , y 0 ). In the following we drop any dependence on α and ε. Arguing as in [1, Lemma A.2], we get 0 ≤ sup

α∈A i 0 - 1 2 T r[a i 0 (y 0 , a)Y -a i 0 (x 0 , a)X] -b i 0 (y 0 , a)(2α(x 0 -y 0 ) -2εy 0 ) + b i 0 (x 0 , a)(2α(x 0 -y 0 ) + 2εx 0 ) -f i 0 (y 0 , a) + f i 0 (x 0 , a) + M j=1 (d i 0 j (y 0 , a)v j (y 0 ) -d i 0 j (x 0 , a)u j (x 0 )) + λ i 0 v i 0 (y 0 ) -λ i 0 u i 0 (x 0 ) .
for some matrices X, Y ∈ S N . The only additional term to estimate with respect to Lemma [1, Theorem A.1] is the last line of the previous inequality. By m α,ε = u i 0 (x 0 )v i 0 (y 0 )φ(x 0 , y 0 ) ≥ u j (x 0 )v j (y 0 )φ(x 0 , y 0 ), we get

v i 0 (y 0 ) -u i 0 (x 0 ) ≤ -m α,ε u j (x 0 ) -v j (y 0 ) ≤ m α,ε + φ(x 0 , y 0 ). Therefore, recalling that d ii ≥ 0, d ij ≤ 0 we get M j=1 (d i 0 j (y 0 , a)v j (y 0 ) -d i 0 j (x 0 , a)u j (x 0 )) + λ i 0 v i 0 (y 0 ) -λ i 0 u i 0 (x 0 ) ≤ M j=1 |d i 0 j (y 0 , a) -d i 0 j (x 0 , a)||v j (y 0 )| -m α,ε ( M j=1 d i 0 j (x 0 , a) + λ i 0 ) + |λ i 0 -λ i 0 ||v i 0 (y 0 )| ≤ M j=1 |d i 0 j (•, a) -d i 0 j (•, a)| 0 max i |v i | 0 + |λ i 0 -λ i 0 | max i |v i | 0 -λ 0 m α,ε
and we conclude as in [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF]Lemma A.2].

Proof of Lemma 4.2. For existence and uniqueness of the solution to (4.2), we refer to [START_REF] Ishii | Perron's method for monotone systems of second-order elliptic PDEs[END_REF][START_REF] Ishii | Viscosity solutions for monotone systems of second-order elliptic PDEs[END_REF] (see also Theorem 2.2). The regularity of the solution, see (4.3), follows by Proposition 2.5, while the second inequality in (4.3) is consequence of Proposition 6.1.

For the statement ii) in (4.3), we refer to [START_REF] Barles | On the convergence rate of approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF], Lemma 2.7. Note that, although here we are dealing with a system, the weakly coupling term M j=1 d ij (x, a)u j is linear in u j , j = 1, . . . , N therefore the proof of [1, Lemma A3] can be straightforward adapted.

We now give the proof of Lemma 4.5. We need a preliminary result. Lemma 6.2. Let V : R N → R M ×L be the solution of (6.1)

F ij (x, V, DV ij , D 2 V ij ) = 0 x ∈ R N , i = 1, . . . , M, j = 1, . . . , L where
F ij (x, R, p, X) = max sup α∈A ij L α i (x, R ij , p, X) + M l=1 d il (x, α)R lj , M ij (R) ,
where A ij ⊂ A i , i = 1, . . . , M and M ij (R) as in (4.10). Let u : R N → R M be the solution of the system where C depends only on the bounds in the assumptions (2.5a) and (2.5b).

(6.2) sup α∈∪ M j=1 A ij L α i (x, u i , Du i , D 2 u i ) + M l=1 d il (x, α)u l = 0 i = 1, . . . , M. Then (6.3) 0 ≤ V ij -u i ≤ Cℓ
Proof. The proof is an adaptation of Theorem 2.3 in [START_REF] Barles | Error bounds for monotone approximation schemes for Hamilton-Jacobi-Bellman equations[END_REF] to the system (6.1) so we just sketch it. First observe that the function W : R N → R M ×L such that W ij = u i , for any j = 1, . . . , M , where u is the solution of (6.2), is a subsolution of (6.1), hence (6.4) u i ≤ V ij j ∈ {1, . . . , M }. Now consider the system (6.5)

F ε ij (x, V ε , DV ε ij , D 2 V ε ij ) = 0 x ∈ R N , i
L α i (x + e, V ε ij , DV ε ij , D 2 V ε ij ) + M l=1 d il (x + e, α)V ε lj ≤ 0.
Therefore, by change of variable, for any |e| ≤ ε, V ε ij (xe) is a subsolution of the system (6.6) sup

α∈A ij L α i (x, W ij , DW ij , D 2 W ij ) + M l=1
d il (x, α)W lj = 0. Now define V ε : R N → R N ×M by V ε,ij = V ε ij * ρ ε . Then, by Lemma 4.2 ii), V ε is a subsolution to (6.6). Moreover since V ε is a subsolution of (6.5) we have

V ε ij ≤ min l =j
V ε il + ℓ, i = 1, . . . , M, j = 1, . . . , L.

It follows that (6.7) 0 ≤ max

j V ε ij -min l V ε il ≤ ℓ hence |V ε ij -V ε il | 0 ≤ ℓ, i = 1, .
. . , M, l, j = 1, . . . , L. By the definition of V ε , we get

|D n V ε,ij -D n V ε,il | 0 ≤ C ℓ ε n i = 1, .
. . , M, j, l = 1, . . . , L, n ∈ N where C only depends on the mollifier ρ ε . Therefore for any j, l = 1 . . . , L sup

α∈A ij L α i (x, V ε,ij , DV ε,ij , D 2 V ε,ij ) + M r=1 d ir (x, α)V ε,rj -sup α∈A ij L α i (x, V ε,il , DV ε,il , D 2 V ε,il ) + M r=1 d ir (x, α)V ε,rl ≤ C ℓ ε 2
where C now depends also on the bounds L in assumptions (2.5a) and (2.5b). Since V ε is a subsolution of (6.6), we get that ∀i = 1, . . . , M

sup α∈∪ L j=1 A ij L α i (x, V ε,ij , DV ε,ij , D 2 V ε,ij ) + M l=1 d il (x, α)V ε,lj ≤ C ℓ ε 2
It follows that V ε,ij -1 λ 0 C ℓ ε 2 , where λ 0 as in (2.5b), is a subsolution of (6.2). By the comparison principle for (6.2), we get V ε,iju i ≤ C λ 0 ℓ ε 2 , i = 1, . . . , M, j = 1, . . . , L.

Hence we conclude

V ij -u i ≤ V ij -V ε,ij + V ε,ij -u i ≤ Cε + C λ 0 ℓ ε 2 ,
recalling (6.4) and minimizing with respect to ε we get the result.

Proof of Lemma 4.5. The statement i) follows by assumption (4.6) and Proposition 6.1 for the estimate in ε and δ and by Lemma 6.2 for the estimate in ℓ.

We now consider ii). By the definition of V ε , it follows that (6.8)

|V ε,ij -V ε ij | 0 ≤ Cε i = 1, . . . , M, j = 1, . . . , L
where C = max i,j [V ε ij ] 1 . Recalling (6.7) in the proof of Lemma 6.2, by (6.8) we get (4.12). We conclude with the proof of iii). Let x ∈ R N , i ∈ {1, . . . , M } and j be such that j = arg min l=1,...,L {V ε,il }.

Then V ε,ij (x) -M ij V ε (x) = max l =j {V ε,ij -V ε,il -ℓ} ≤ -ℓ

  y∈R N ,i=1,...,M {u h i (y)w i (y)} and (4.13) m k = sup y∈R N ,i=1,...,M {u h i (y)w i (y)kφ(y)}

1 3 i = 1 ,

 131 . . . , M, j = 1, . . . , L

= 1 ,

 1 . . . , M, j = 1, . . . , L whereF ε ij (x, R, p, X) = max sup α∈A ij ,|e|≤ε L α i (x + e, R ij , p, X) + M l=1 d il (x + e, α)R lj , M ij (R)and let V ε the corresponding solution.For any e such |e| ≤ ε, we have supα∈A ij

  M , to be continuous and satisfy (2.2). Let u and v be respectively a bounded subsolution and a bounded supersolution of (2.1). Then u ≤ v in R N .

	Remark 2.3. Assumption (2.2c) is the condition giving the monotonicity of the system (2.1),
	while (2.2a), (2.2b) are standard regularity assumptions in viscosity solution theory.
	Example 2.4. Weakly coupled system: Consider the weakly coupled system of M equations:
	(2.3)

This work is partially supported by European Union under the 7th Framework Programme FP7-PEOPLE-2010-ITN Grant agreement number 264735-SADCO..

since V ε,ij (x) ≤ V ε,il for any l = 1, . . . , L. By (4.12), we have

Hence by (4.8), we get

By this point the proof goes as in [2, Lemma 3.4(b)]. Note that the only difference linked with the fact that here we are dealing with a system can be proved by arguing as in the proof of Lemma 4.2 ii).