Approximation Schemes for Monotone Systems of Nonlinear Second Order Partial Differential Equations: Convergence Result and Error Estimate

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Approximation Schemes for Monotone Systems of Nonlinear Second Order Partial Differential Equations:

Convergence Result and Error Estimate

Ariela Briani, Fabio Camilli, Hasnaa Zidani

To cite this version:

Ariela Briani, Fabio Camilli, Hasnaa Zidani. Approximation Schemes for Monotone Systems of Nonlin- ear Second Order Partial Differential Equations: Convergence Result and Error Estimate. Differential Equations and Applications, Element, 2012, 4, pp.297-317. �10.7153/dea-04-18�. �inria-00627520�

NONLINEAR SECOND ORDER PARTIAL DIFFERENTIAL EQUATIONS:

CONVERGENCE RESULT AND ERROR ESTIMATE

ARIELA BRIANI, FABIO CAMILLI AND HASNAA ZIDANI

Abstract. We consider approximation schemes for monotone systems of fully nonlinear second order partial differential equations. We first prove a general convergence result for monotone, consistent and regular schemes. This result is a generalization to the well known framework of Barles-Souganidis, in the case of scalar nonlinear equation. Our second main result provides the convergence rate of approximation schemes for weakly coupled systems of Hamilton-Jacobi-Bellman equations. Examples including finite difference schemes and Semi-Lagrangian schemes are discussed.

1. 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, D²u_i

= 0, x∈R^N, i= 1, . . . , M,

where u = (u₁, . . . , u_M) denotes the unknown function, and F = (F₁, . . . , F_M) is a given function.

The theory of viscosity solution, initially developed for the scalar equation has been ex- tended to systems in [13, 17, 19, 26]. 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 ([17, 19]) or via control-theoretic representation formulas, ([13, 22]).

We refer to [15, 16] for various applications of systems of PDEs in many areas, in particular we mention [5] 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

whereS_i are consistent, monotone and uniformly continuous approximation ofF_iin (1.1) and the coupling among the equations is only in the variableu^h(x) (whereu^h = (u^h₁, . . . , u^h_M) rep- resents 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.

2000Mathematics Subject Classification. Primary 49L25; Secondary 35J45, 65N15.

Key words and phrases. Monotone systems, viscosity solution, approximation scheme, error estimate.

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

1

For the case of a scalar fully nonlinear equation, the work of Barles-Souganidis [4] pro- vides 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 [4] 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 schemeS_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 [4] 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^storder equation were obtained from the beginning of the viscosity theory, only quite recently Krylov [22, 21, 23] and Barles- Jakobsen [1, 2] 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 [6, 20]. We refer also to [10] for convergence rate in the case of elliptic fully nonlinear equations, and to [14] 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 [1, 2, 7] and adapt them to the case of monotone system of PDEs. As usual, the upper bound foru−u^h is easier and it is obtained via aKrylov regularization andshaking coefficienttechniques. 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 [9]. For a Semi-Lagrangian scheme these estimates become h^1/4 for the upper bound, andh^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∈RN|f(x)|, [f]₁=|Df|0, and |f|1=|f|0+ [f]₁.

The space of real symmetricN×N matrices is denoted by S^N, andX≥Y inS^N will mean that X−Y is positive semi-definite.

For a function u : R^N → R^M, we say that u = (u₁, . . . , 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.. Ifu = (u₁, . . . , u_M), v= (v₁, . . . , v_M), are two functions defined in a setE we write u≤v inE ifu_i ≤v_i inE,i= 1, . . . , M.

2. The continuous problem: definitions and assumptions Consider the system of nonlinear second order equations

(2.1) F_i x, u, Du_i, D²u_i

= 0, x∈R^N, i= 1, . . . , M

whereu:= (u₁, . . . , u_M) andu_iis a real valued function defined inR^N andF := (F₁, . . . , 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 [19]).

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∈C²(R^N), i= 1, . . . , M and u_i−φattains a local maximum at x∈R^N, then

F_i(x, u(x), Dφ(x), D²φ(x))≤0.

ii) A l.s.c. function v : R^N → R^M is said a viscosity supersolution of (2.1) if whenever φ∈C²(R^N), i= 1, . . . , M and v_i−φattains a local minimum atx∈R^N, then

F_i(x, v(x), Dφ(x), D²φ(x))≥0.

iii) A continuous functionu 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 ([17], [19]) or via the control-theoretic interpretation of the prob- lem [13, 26]. To get a comparison principle for system (2.1), we shall assume the following conditions on functionF:

F_i ∈C(R^N×R^M ×R^N ×S^N) i= 1, . . . , M; (2.2a)

There exists a modulus of continuityω s.t. ifX,Y ∈S^N,β >1 and

−3β

I 0 0 I

≤

X 0 0 Y

≤ −3β

I −I

−I I (2.2b)

thenF_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 c₀>0 s.t. forr,s∈R^M satisfying r_i−s_i = max

j=1,...,M{r_j−s_j} ≥0, then for allx,y,p∈R^N,X ∈S^N F_i(x, r, p, X)−F_i(x, s, p, X)≥c₀(ri−s_i).

Theorem 2.2 (see [19]). Assume F_i : R^N ×R^M ×R^N ×S^N → R, i = 1, . . . , M, to be continuous and satisfy (2.2). Letu andv 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 ofM equations:

(2.3)

F_i(x, u(x), Du_i(x), D_i²u(x)) := sup

α∈Aⁱ







L^αi(x, u(x), Du_i(x), D²u_i(x)) +

M

X

j=1

d_ij(x, α)u_j(x)







where the operator L^α is defined by:

(2.4) L^αi(x, r, p, X) =−1

2T r[ai(x, α)X]−b_i(x, α)·p−f_i(x, α) +λ_ir_i.

and witha_i(x, α) =σ_i(x, α)σ^T_i (x, α). We assume that the following assumptions hold:

Aⁱ are compact metric spaces, i= 1, . . . , M

b_i:R^N × Aⁱ −→R^N, σ_i :R^N× Aⁱ→R^N×D, f_i :R^N × Aⁱ →R

|f_i(·, α)|1+|σ_i(·, α)|1+|b_i(·, α)|1 ≤L,for anyα∈ Aⁱ, i= 1, . . . , M (2.5a)

and

d_ij :R^N × Aⁱ →R,

|d_ij(·, α)|1 ≤L, for anyα∈ Aⁱ, i, j= 1, . . . , M, d_ii(x, α)≥0, d_ij(x, α)≤0 fori6=j,λ_i ≥0, λ_i+

M

X

j=1

d_ij(x, α)≥λ₀>0 (x, α)∈R^N × Aⁱ, 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 withc₀=λ₀). 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:

λ₀ > max

i=1,...,M sup

α∈Aⁱ|σ_i(·, α)|²1+ max

i=1,...,M sup

α∈Aⁱ|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.

3. 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^h₁, . . . , 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 c₀ > 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 =

j=1,...,Mmax {r_j−s_j} ≥0, then

S_i(h, x, r, ui+θ)−S_i(h, x, s, vi)≥c₀θ.

(C2) (Regularity) For any h >0 and any continuous, bounded function φ:R^N → R, the functions

x7→S_i(h, x, r, φ) i= 1, . . . , M are bounded and continuous on R^N and the functions

r7→S_i(h, x, r, φ)

are uniformly continuous for bounded r, uniformly with respect tox∈R^N.

(C3) (Consistency) Fix i = 1, . . . , M. For all h > 0, x ∈ R^N and for any continuous function Φ∈C⁰(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, D²Φ_i)| ≤ω(h) withω(h)→0 forh→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∈C⁰(R^N;R^M) such that its i-th component Φ_i is smooth with |D^jΦ_i(x)| bounded in R^N, for everyj∈J, we have:

|S_i(h, x,Φ(x),Φ_i)−F_i(x,Φ, DΦ_i, D²Φ_i)| ≤K_CQ(Φ_i) inR^N, whereQ(φ) :=P

j∈J|D^jφ|h^kj for every smooth functionφ.

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), ui)≥0 (respectivelyS_i(h, x, u(x), ui)≤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,RN(u_i−v_i)>0 and we derive a contradiction. Let {x_n} inR^N and{i_n} ∈ {1, . . . , M}be such thatδ_n=u_in(xn)−v_in(xn) = maxj{u_j(xn)−v_j(xn)} →δ for n→ ∞. Since u andv are respectively sub- and supersolutions, we get:

0≥S_in(h, xn, u(xn), uin)−S_in(h, xn, v(xn), vin).

Moreover, we know thatu(x_n)≤v(x_n)+δ_n, 0 = max_j(u_j(x_n)−(v_j(x_n)+δ_n)) andu_in ≤v_in+δ inR^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, xn, v(xn) +δ, v_in+δ)−S_in(h, xn, v(xn), vin) +ω(δn−δ), where ω(t)→0 whent→0⁺. Finally, by using (C1) again, we obtain:

0≥c₀δ+ω(δ_n−δ),

which leads to a contradiction when n→ ∞.

In all the sequel, we assume that:

(C5) (Existence of discrete solution) For everyh >0, system (3.1) admits a solutionu^h. We give a convergence result for the scheme based on the classical argument by Barles- Souganidis, [4].

Proposition 3.3. Assume (C1)–(C3) and (C5). Let u^h = (u^h₁, . . . , u^h_M) be a locally uni- formly 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 inR^N.

Proof. Define the relaxed half-limits by:

u_i(x) = lim inf

h→0,z→xu^h_i(z), u_i(x) = lim sup

h→0, z→x

u^h_i(z), i= 1, . . . , M.

Following the arguments introduced in [4], 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→0u^h and the local uniform convergence inR^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, andi∈ {1, . . . , M}, such that u_i −φ has a maximum point at x₀, u_i(x₀) = φ(x₀). By using standard arguments in the viscosity theory, there exists a sequence (hn)n of positive numbers satisfying:

h_n→0 and x_n:=x_hn→x₀ whenn→ ∞,

u^h_iⁿ−φhas a maximum point at x_n, andu^h_iⁿ(xn)→u_i(x0).

Set δ_n:= (u^h_iⁿ−φ)(x_n), and define the function:

Φⁿ(x) = (u^h₁ⁿ(x), . . . , u^h_i−1ⁿ (x), φ(x), u^h_i+1ⁿ (x), . . . , u^h_Mⁿ(x)) inR^N.

Up to a subsequence, Φⁿ(xn) converges to a vectorr∈R^M, wherer_i=u_i(x0), andr_j ≤u_j(x0) for j 6= i. Then, using the fact that: u^h_iⁿ −δ_n ≤ φ in R^N and δ_n = u^h_iⁿ(x_n)−φ(x_n) = max_j(u^h_jⁿ(x_n)−Φⁿ_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^h_iⁿ) =S_i(h_n, x_n, u^hn(x_n),(u^h_iⁿ−δ_n) +δ_n)≥ S_i(h_n, x_n,Φⁿ(x_n), φ) +c₀δ_n≥F_i(x_n,Φⁿ(x_n), Dφ(x_n), D²φ(x_n)) +c₀δ_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(x0, u(x0), Dφ(x0), D²φ(x0)), which proves that u is a subsolution of (2.1).

4. The error estimate for the weakly coupled systems

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

F_i(x, r, p, X) := sup

α∈Aⁱ







−1

2T r[a_i(x, α)X]−b_i(x, α)·p−f_i(x, α) +λ_ir_i+

M

X

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 byuthe unique solution given by Proposition 2.5. For this particular system, we will derive an error estimate for |u−u^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∈Jn

kj

j

o

, withJ 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 solutionu^ε of the system

(4.2) max

|e|≤εF_i(x+e, u(x), Du_i, D²u_i) = 0 x∈R^N, i= 1, . . . , M and two constants C₀, C₁ independent of h, ε such that

(4.3) max

i=1,...,M|u^ε_i|1 ≤C₀ and max

i=1,...,M|u_i−u^ε_i|0 ≤C₁ε.

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^ε₁, . . . , ρ_ε∗u^ε_M) :=

(u1,ε, . . . , u_M,ε). Then u_ε is a classical subsolution of (2.1).

Proof of Proposition 4.1. Thanks to Lemma 4.2ii),u_εis classical subsolution of (2.1). There- fore, by(C4)we have, for each i= 1, . . . , M

(4.4) S_i(h, x, uε(x), ui,ε)≤F_i(x, uε, Du_i,ε, D²u_i,ε) +Q(ui,ε)≤Q(ui,ε).

Setε=h^γ withγ as in the statement. By [7, Lemma 4.2], we can estimate Q(u_i,ε)≤K|J|C₀h^γ:=Ch^γ,

whereC₀as in (4.3). By(C1)and (4.4), we have thatu_ε−_c^C₀h^γ = (u_1,ε−_c^C₀h^γ, . . . , u_M,ε−_c^C₀h^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

c₀h^γ ∀i= 1, . . . , M.

By estimate (4.3) and (4.5), we conclude u_i−u^h_i ≤ C

c₀h^γ+Ch^γ ∀i= 1, . . . , M

and therefore (4.1) is satisfied.

4.2. The lower bound. For the lower bound we need an additional assumption For each control set Aⁱ and δ >0, there is a finite family of controls{α_ij}^Lj=1ⁱ

such that for anyα∈ Aⁱ, inf

j=1,...,Li

|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

X

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 Aⁱ 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 [3, 21, 22]. To simplify the notation, we assume in the following w.l.o.g. that all the families {α_ij}^Lj=1ⁱ 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∈Jn _k

j

3j−2

o

withJ, k_j being defined in (C4).

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

(4.8) F_ij^ε,ℓ(x, V^ε,ℓ, DV_ij^ε,ℓ, D²V_ij^ε,ℓ) = 0 x∈R^N, i= 1, . . . , M, j = 1, . . . , L where

F_ij^ε,ℓ(x, R, p, X) = maxn

|e|≤εmin

L^αi^ij(x+e, R_ij, p, X) +

M

X

l=1

d_il(x+e, α_ij)R_lj , (4.9)

M^ij(R)o , Mij(R) :=R_ij − min

l=1,...,L l6=j

{R_il+ℓ}, (4.10)

where {α_ij}^Lj=1, i = 1, . . . , M are defined in (4.6). The solution of the system (4.8) is a function fromR^N toR^M×L. The (i, j) component of the solution of (4.8) is coupled with other L components by means of the switching termMij(R) and with other (M −1) components by means of the termPM

l=1d_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_ij^ε,ℓ−u_i|0≤C(ε+ℓ¹³ +δ) i= 1, . . . , M.

ii) SetV_ε,ij^ℓ :=ρ_ε∗V_ij^ε,ℓ,i= 1, . . . , M, j= 1, . . . , Lwhere ρ_ε is a standard mollifier. Then

(4.12) max

j,l=1,...,L|V_ij^ε,ℓ−V_ε,il^ℓ |0 ≤C(ε+ℓ)

iii) Moreover, forε≤(4 sup_ij[V_ij^ε,ℓ]₁)⁻¹ℓ and for for anyx∈R^N, if we setj=j(ε, i, x) :=

arg min_l=1,...,JV_ε,il^ℓ (x), then

L^α_i^ij(x, V_ε,ij^ℓ , DV_ε,ij^ℓ , D²V_ε,ij^ℓ ) +

M

X

l=1

d_il(x, α_ij)V_ε,lj^ℓ ≥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 := minj=1,...,LV_ε,ij^ℓ . By Lemma 4.5.(iii), it follows that w is a supersolution of (2.1). We define

m= sup

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)}

whereφ(y) = (1 +|y|²)^1/2. Sincewandu^h are bounded continuous functions, the supremum in (4.13) is achieved at a point x and at an index i, which is also a maximum point for y 7−→ u^h_i(y)−V_ε,ij^ℓ (y)−kφ(y), where j = arg min_l=1,...,LV_ε,il^ℓ (x). By Lemma 4.5.(iii), the definition ofφ, (2.5a) and (2.5b), we get

sup

α∈AⁱL^αi(x,(V_ε,ij^ℓ +kφ)(x), D(V_ε,ij^ℓ +kφ)(x), D²(V_ε,ij^ℓ +kφ)(x)) +

M

X

l=1

d_il(x, α)(V_ε,lj^ℓ +kφ)(x)≥ −Ck.

(4.14)

With the consistency assumption(C4), we obtain

(4.15) −Ck≤S_i(h, x,({V_ε,lj^ℓ }^Ml=1+kΦ)(x), V_ε,ij^ℓ +kφ) +Q(V_ε,ij^ℓ +kφ)

where Φ = (φ, . . . , φ). Taking into account the definition ofQ(·), V_ε,ij^ℓ and Φ, we obtain

(4.16) −CX

l∈J

ε^1−lh^kl+O(k)≤S_i(h, x,({V_ε,lj^ℓ }^Ml=1+kΦ)(x), V_ε,ij^ℓ +kφ).

Moreover by the definition ofm_k, it follows that

(V_ε,ij^ℓ +kφ)(y)≥u^h_i(y)−m_k ∀y∈R^N, and

u^h_i(x)−V_ε,ij^ℓ (x)−kφ(x)≥u^h_l(x)−w_l(x)−kφ(x)≥u^h_l(x)−V_ε,lj^ℓ (x)−kφ(x) and therefore

(u^h_i(x)−m_k)−(V_ε,ij^ℓ (x) +kφ(x)) = max

l {(u^h_l(x)−m_k)−(V_ε,lj^ℓ (x) +kφ(x))}. This with assumption (C1)yields:

S_i(h, x,({V_ε,lj^ℓ }^Ml=1+kΦ)(x), V_ε,ij^ℓ +kφ)≤S_i(h, x,(u^h−m_k)(x), u^h_i −m_k)

≤ −c₀m_k+S_i(h, x, u^h(x), u^h_i) =−c₀m_k. From the previous inequality and (4.16), sendingk→0 we get the estimate

(4.17) c₀m≤CX

l∈J

ε^1−lh^kl.

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_ε,lj^ℓ (y) +V_ε,lj^ℓ (y)−u_l(y)≤u^h_l(y)−w_l(y) +V_ε,lj^ℓ (y)−u_l(y)

≤m+V_ε,lj^ℓ (y)−u_l(y)

By (4.17), Lemma 4.5(i) and (ii), we get u^h_l(y)−u_l(y)≤C(X

l∈J

ε^1−lh^kl+ε+ℓ+ℓ^1/3+δ),

and the statement of the theorem follows by taking ε= max_l∈Jh

3kl

3l−2 and ℓ= 4 sup_ij[V_ij^ε,ℓ]₁ε

and sendingδ to 0.

5. Examples of approximation schemes

5.1. Finite differences, one dimensional problem. Let x be in R,φinC⁴(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)

h² .

In particular, by a Taylor expansion, we obtain

|δ_±φ(x)−Dφ(x)| ≤ 1

2h|D²φ|, |∆_hφ(x)−D²φ(x)| ≤ 1

12h²|D⁴φ|.

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∈Aⁱ

−1

2a_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_iu^h_i(x)−f_i(x, α) +

M

X

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, fori= 1, . . . , M where the operatorS_i is defined onR^∗₊×R×R^M ×C₀(R) by:

S_i(h, x, r, w) := sup

α∈Aⁱ

−1

2a_i(x, α)w(x+h)−2r_i+w(x−h)

h² −b⁺_i (x, α)w(x+h)−r_i (5.1) h

−b⁻_i (x, α)r_i−w(x−h)

h +λ_ir_i−f_i(x, α) +

M

X

j=1

d_ij(x, α)r_j





 ,

From the Taylor expansion, one can easily check that the monotonicity (C1) holds (with c₀=λ₀), and the consistency hypothesis(C4)is satisfied withQ(φ) =|D²φ|h+|D⁴φ|h², i.e.

k₂ = 1 andk₄ = 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 differ- ences scheme defined in [8]. Letφbe a real valued function. Leth >0,ξ∈Z^N and consider the finite difference operator along directionξ:

∆_ξφ(x) := φ(x+ξh) +φ(x−ξh)−2φ(x)

h² .

On the other hand, we consider (D^±φ(x))_j =







φ(x+he_j)−φ(x)

h if [bi(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{e₁, . . . , e_N}. We consider the following scheme:

sup

α∈Aⁱ







−1 2

X

ξ∈S

¯

γ_ξⁱ(x, α)∆_ξΦ_i(x)−b_i(x, α)D^±Φ_i(x) +λ_iΦ_i(x)−f_i(x, α) (5.2)

+

M

X

j=1

d_ij(x, α)Φ_j(x)







= 0, where the coefficients ¯γ_ξⁱ are given by:

ka_i(x, α)−X

ξ∈S

¯

γ_ξⁱξξ^Tk= min

(γξ)∈(R+)^|S|ka_i(x, α)−X

γ_ξξξ^Tk.

For fast computations of the coefficientsγ_ξⁱ we refer to [9]. In the sequel, we make the strong consistency hypothesis (see also [23]):

(5.3) a_i(x, α) =X

ξ∈S

¯

γ_ξⁱ(x, α)ξξ^T, ∀x∈R^N,∀α∈ Aⁱ.

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 operatorS_i is defined in R^∗₊×R^N×R^M×C₀(R^N) by:

S_i(h, x, r, w) := sup

α∈Aⁱ







−1 2

X

ξ∈S

¯

γⁱ_ξ(x, α)w(x+hξ)−2r_i+w(x−hξ) h²

−

N

X

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

+λir_i−f_i(x, α) +

M

X

j=1

d_ij(x, α)rj





 .

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 ∈ C⁰(R^N,R^M) with Φ_i ∈C⁴(R^N), then by applying a Taylor expansion, we obtain

|F_i(x,Φ(x), DΦi, D²Φi)−S_i(x, h,Φ(x),Φi)| ≤ sup

α∈Aⁱ|b_i(., α)|⁰|D²Φi|⁰h+ sup

α∈Aⁱ|σ_i(·, α)|²0|D⁴Φi|⁰h².

Then the scheme satisfies the strong consistency (C4)withk₂ = 1 and k₄ = 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 orderh^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 :C⁰(R^N,R^M)→C⁰(R^N,R^M) defined by

[Tw]_i(x) := (1−τ λ_iw_i(x))−τsup

α∈Aⁱ







−1 2

X

ξ∈S

¯

γ_ξ(x, α)∆_ξw_i(x)−b_i(x, α)D^±w_i(x)

−f_i(x, α) +

M

X

j=1

d_ij(x, α)w_j(x)





 .

Then, we can easily see that (5.2) is equivalent to

(5.4) u(x) = [Tu](x), on R^N.

Moreover, forτ small enough, the operatorT is a monotone contraction, then the fixed-point equationu=Tu admits a unique solution, and this solution is limit to any sequence defined byuⁿ⁺¹ =Tuⁿ, withu⁰ ∈C(R^N,R^M). This iterative process, calledvalue iterations method can be used to compute a numerical solutionu^h on a gridG.

5.3. Semi-Lagrangian schemes. Semi-Lagrangian schemes for second order Hamilton-Jacobi equations have been already studied in several papers, we refer to [1, 11, 12, 24, 25] 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 X₀ = 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. j6=i, for any i, j = 1, . . . , M, i 6= 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







X₀=x,

X_n+1 =X_n+

hb_νn(X_n, α_n) +√ hPd

m=1σ_νn,m(X_n, α_n)ξ_n^m

δ_νn,νn+1, P{ν_n+1 =j|ν_n=i, X_n=x, α_n=α}=hc_ij(x, α) j6=i

(5.7)

for n ∈ N, where σ_i,m denote the m-th column of σ_i and ξ_n^m, m = 1, . . . , D, are random variables taking values in {−1,0,1}such that

P[{ξⁱ_n=±1}] = 1

2D and P[{ξⁱ_n6= 0} ∩ {ξ_n^j 6= 0}] = 0, i6=j

andδ_n,n= 1 andδ_n,m = 0 forn6=m. The discrete control{α_n}is a random variable which is measurable with respect to the σ-algebra generated by X₁, . . . , X_n and such that α_n ∈ A^νn.

Set d_ii =PM

j=1c_ij, andd_ij =−c_ij for every j6=i, then the generator of the discrete process is

L^hi,α(x,Φ(x),Φ_i) = E_x,i[Φ_ν1(X₁)]−Φ_i(x)

h =L^h_i,α(x,Φ(x),Φ_i)−

M

X

j=1

d_ij(x, α)Φ_j(x) for Φ∈C⁰(R^N,R^M), where

L^h_i,α(x, r, φ) = 1 2Dh

D

X

m=1

h

φ(x+hb_i(x, α) +√

hσ_i,m(x, α)) +φ(x+hb_i(x, α)

−√

hσ_i,m(x, α))−2r_ii , (5.8)

Let u^h = (u^h₁, . . . , 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∈Aⁱ







−(1−λ_ih)L^h_i,α(x, r, φ)−f_i(x, α)−(1−λ_ih)

M

X

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 ∈ C⁴(R^N,R^M) satisfying

|Φ_i|0+· · ·+|D⁴Φ_i|0<∞, we have

(5.10) |F_i(x,Φ(x), DΦi(x), D²Φi(x))−S_i(h, x,Φ(x),Φi)| ≤C₁h(|D²Φi|⁰+|D³Φi|⁰+|D⁴Φi|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 [2]).

6. 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 _λ^C

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 >

i=1,...,Mmax sup

α∈Aⁱ|f_i(·, a)|1+ max

i,j=1,...,M sup

α∈Aⁱ|d_i,j(·, a)|1 max

i=1,...,M|u_i|0

λ₀− max

i=1,...,M sup

α∈Aⁱ|σ_i(·, a)|²1− max

i=1,...,M sup

α∈Aⁱ|b_i(·, a)|1

we con conclude that m≤0, the proof is achieved. Assume, for simplicity that the maximum is attained at (ˆx,y) (if it is not one can modify the test function in a standard way). Ifˆ ˆ

x= ˆy thenm = 0 and we are done. If not, at (ˆx,y) the functionˆ 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 [18, 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^αi as in (2.4) with co- efficients {σ_i, b_i, λ_i, f_i, d_ij}^Mi=1 and, respectively, {σ_i, b_i, λ_i, f_i, d_ij}^Mi=1 satisfying (2.5a), (2.5b) (with the same constant λ₀). Then there is a constantC such that

λ₀ max

i=1,...,M|u_i−v_i|0 ≤C sup

i=1,...,M

sup

α∈Aⁱ{|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

X

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∈RN,i=1,...,M(u_i −v_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)∈RN×R^N,i=1,...,Mψ_i(x, y). A standard computation gives that there exists (x₀, y₀) ∈R^N ×R^N and i₀ ∈ {1, . . . , M} such thatm_α,ε=ψ_i0(x₀, y₀). In the following we drop any dependence on α andε. Arguing as in [1, Lemma A.2], we get

0≤ sup

α∈Aⁱ⁰

−1

2T r[a_i0(y₀, a)Y −a_i0(x₀, a)X]−b_i0(y₀, a)(2α(x₀−y₀)−2εy₀) +b_i0(x₀, a)(2α(x₀−y₀) + 2εx₀)−f_i0(y₀, a) +f_i0(x₀, a)

+

M

X

j=1

(d_i0j(y₀, a)v_j(y₀)−d_i0j(x₀, a)u_j(x₀)) +λ_i0v_i0(y₀)−λ_i0u_i0(x₀) .

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_i0(x₀)−v_i0(y₀)−φ(x₀, y₀)≥u_j(x₀)−v_j(y₀)−φ(x₀, y₀), we get

v_i0(y0)−u_i0(x0)≤ −m_α,ε u_j(x₀)−v_j(y₀)≤m_α,ε+φ(x₀, y₀).

Therefore, recalling that d_ii≥0,d_ij ≤0 we get

M

X

j=1

(d_i0j(y₀, a)v_j(y₀)−d_i0j(x₀, a)u_j(x₀)) +λ_i0v_i0(y₀)−λ_i0u_i0(x₀)

≤

M

X

j=1

|d_i0j(y0, a)−d_i0j(x0, a)||v_j(y0)| −m_α,ε(

M

X

j=1

d_i0j(x0, a) +λ_i0) +|λ_i0 −λ_i0||v_i0(y0)|

≤

M

X

j=1

|d_i0j(·, a)−d_i0j(·, a)|0max

i |v_i|0+|λ_i0−λ_i0|max

i |v_i|0−λ₀m_α,ε

and we conclude as in [1, Lemma A.2].