Boualem Djehiche
email: boualem@math.kth.se.

Said Hamadène ‡ Marie-Amelie Morlais
email: amelie.morlais@univ-lemans.fr

Viscosity Solutions of Systems of Variational Inequalities with Interconnected Bilateral Obstacles

Keywords: AMS Classification subjects: 49N70, 49L25, 60H30, 90C39, 93E20 Switching games, variational inequalities, oblique reflection, penalization, backward stochastic differential equation, Bellman-Isaacs equation, Perron's method

We study a general class of nonlinear second-order variational inequalities with interconnected bilateral obstacles, related to a multiple modes switching game. Under rather weak assumptions, using systems of penalized unilateral backward SDEs, we construct a continuous viscosity solution of polynomial growth.

Moreover, we establish a comparison result which in turn yields uniqueness of the solution.

Introduction

In this paper we study systems of variational inequalities with interconnected lower and upper obstacles. This type of inequalities arises as the Bellman-Isaacs equation in a multiple modes switching game between two players.

Besides their classical fields of applications, multiple modes switching games are attracting a lot of interest in the management of power plants (see [START_REF] Bernhart | Modélisation et méthodes d'évaluation de contrats gaziers: Approches par contrôle stochastique[END_REF] ([3]) and Perninge (2011) ( [START_REF] Perninge | A Stochastic control approach to include transfer limits in power system operation[END_REF]), where they are successfully used to design optimal stopping and starting strategies for power flow control through activation of regulating bids on a regulated power market.

The objective of this work is to establish existence and uniqueness of a continuous viscosity solution of the following system of variational inequalities with oblique reflection:

       min v ij -L ij [ v] (t, x), max v ij -U ij [ v] (t, x), -∂ t v i,j (t, x) -Lv ij (t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x v ij (t, x)) = 0, v ij (T, x) = h ij (x), (1.1)
for every pair (i, j) in the finite set of modes Γ 1 × Γ 2 , where, for any (t, x)

∈ [0, T ] × R k , Lϕ(t, x) := b(t, x)D x ϕ(t, x) + 1 2 Tr[σσ ⊤ (t, x)D 2 xx ϕ(t, x)],
and to any solution v = (v ij ) (i,j)∈Γ 1 ×Γ 2 we associate the lower obstacle operator

L ij [ v](t, x) := max k∈Γ 1 , k =i v kj -g ik (t, x),
and the upper obstacle operator

U ij [ v](t, x) := min l∈Γ 2 , l =j v il + ḡjl (t, x),
where, g ik (resp. ḡjl ) stands for the switching cost incurred when the first (resp. second) player decides to switch from mode i to mode k (resp. from mode j to mode l). Finally, the function f ij stands for the instantaneous payoff when the first player is in mode i and the second one in mode j.

The system (1.1) and related switching games have been studied by several authors. The most recent work discussing this topic includes the papers by [START_REF] Hu | Switching Games of Backward Stochastic Differential Equations[END_REF] ( [START_REF] Hu | Switching Games of Backward Stochastic Differential Equations[END_REF]) and Tang and Hou (2007) ( [START_REF] Tang | Switching games of stochastic differential systems[END_REF]) (see also the references therein) which deal with switching games related to (1.1), when the switching costs do not depend on the state variable. To the best of our knowledge, Ishii and Koike (1991) ( [START_REF] Ishii | Viscosity solutions of a system of Nonlinear second order PDE's arising in switching games[END_REF]) are the latest most general existence and uniqueness results for the system (1.1), for state-dependent switching costs. They derive existence and uniqueness of viscosity solutions of the elliptic version of (1.1) in a bounded domain of R k whose boundary is of class C 2 , when the so-called Fichera functions are strictly negative (see [START_REF] Ishii | Viscosity solutions of a system of Nonlinear second order PDE's arising in switching games[END_REF], Proposition 4.3).

The main result of the present paper, which is given in Theorem 5.4, establishes existence and uniqueness of a continuous viscosity solution of the system (1.1), when the state space is the whole R k and under rather weak assumptions on the involved coefficients. Our approach is probabilistic and makes use of penalization schemes that allow us to connect the related penalized PDEs with systems of reflected backward SDEs with unilateral interconnected obstacles which, for instance, have been studied in [START_REF] Djehiche | A finite horizon optimal multiple switching problem[END_REF], [START_REF] El Asri | The finite horizon optimal multi-modes switching problem: the viscosity solution approach[END_REF], [START_REF] Hamadène | Switching problem and related system of reflected backward stochastic differential equations Stoch[END_REF] or [START_REF] Hu | Multi-dimensional BSDE with oblique reflection and optimal switching[END_REF]. With the help of these sequences of solutions of reflected BSDEs and their connection with PDEs, via Feynman-Kac's formula, we are able to construct in Propositions 5.1 and 5.2 both a viscosity subsolution and a supersolution for the system (1.1). Relying next both on the comparison result established in Theorem 4.2 and adapting the Perron's method we construct a solution for (1.1) which is therefore unique. Finally, using again the uniqueness result, we identify the limit of the penalized decreasing scheme as the solution of the system (1.1).

We made this detour instead of trying to solve a related system of reflected BSDEs with interconnected bilateral obstacles, as one would expect, simply because they satisfy neither the so-called Mokobodski condition nor the condition of strict separation of the two obstacles, which would guarantee existence and uniqueness of the solutions of the system of BSDEs, since these obstacles depend on the solution. The structure of these bilateral obstacles suggests a rather new type of conditions to guarantee existence and uniqueness result for the related system of reflected BSDEs. This problem is beyond the scope of the present paper and therefore left for future research.

Our plan for this paper is as follows. In Section 2, we provide all the notations used in the paper, state the whole list of required assumptions and define viscosity sub-and supersolutions along with equivalent characterizations.

In Section 3 we construct two approximation schemes (an increasing and a decreasing on), consisting of sequences of penalized reflected BSDEs associated with standard switching problems. The counterpart of the decreasing scheme (resp. the increasing one) in terms of PDEs stands for the penalized scheme of system (1.1) (resp. system (1.2) given below). Section 4 is devoted to the proof of a comparison result related to the sub-and supersolutions of (1.1). In Section 5, the decreasing limit is identified as a viscosity subsolution of (1.1) while a super-solution is exhibited. Finally, we use Perron's method to construct a viscosity solution of (1.1) and, thanks to the uniqueness result, its connection with the limit of the decreasing scheme is obtained. As a by product, we also obtain a similar characterization of the limit of the increasing scheme as the unique solution in viscosity sense of the following system of variational inequalities: For every (i,

j) ∈ Γ 1 × Γ 2 ,        max v ij -U ij [ v] (t, x), min v ij (t, x) -L ij [ v] (t, x), -∂ t v ij (t, x) -Lv ij (t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x v ij (t, x)) = 0, v ij (T, x) = h ij (x).
(1.2)

We do not know whether the solutions of (1.1) and (1.2) coincide or not. We note that this is a very important issue since this will enable to characterize this common solution as the value of the zero-sum switching game.

Since it is beyond the scope of the paper, this question is left for further research.

Preliminaries and notation

Let T (resp. k, d) be a fixed positive constant (resp. two integers) and Γ 1 (resp. Γ 2 ) denote the set of switching modes for player 1 (resp. 2). For later use, we shall denote by Λ the cardinal of the product set Γ 1 × Γ 2 and for

(i, j) ∈ Γ 1 × Γ 2 , (Γ 1 ) -i := Γ 1 -{i} and (Γ 2 ) -j := Γ 2 -{j}. Next, for y = (y kl ) (k,l)∈Γ 1 ×Γ 2 ∈ R Λ , (i, j) ∈ Γ 1 × Γ 2 ,
and y ∈ R, we denote by [ y -(ij) , y] the matrix which is obtained from y by replacing the element y ij with y.

Next, let us introduce the following functions. For any (i,

j) ∈ Γ 1 × Γ 2 , b : (t, x) ∈ [0, T ] × R k → b(t, x) ∈ R k ; σ : (t, x) ∈ [0, T ] × R k → σ(t, x) ∈ R k×d ; f ij : (t, x, y, z) ∈ [0, T ] × R k+Λ+d → f ij (t, x, y, z) ∈ R ; g ik : (t, x) ∈ [0, T ] × R k → g ik (t, x) ∈ R (k ∈ (Γ 1 ) -i ); ḡjl : (t, x) ∈ [0, T ] × R k → ḡjl (t, x) ∈ R (l ∈ (Γ 2 ) -j ); h ij : x ∈ R k → h ij (x) ∈ R. A function Φ : (t, x) ∈ [0, T ] × R k → Φ(t, x) ∈ R
|Φ(t, x)| ≤ C(1 + |x| γ ), (t, x) ∈ [0, T ] × R k .
Hereafter, this class of functions is denoted by Π

g . Let C 1,2 ([0, T ] × R k ) (or simply C 1,2
) denote the set of realvalued functions defined on [0, T ] × R k , which are once (resp. twice) differentiable w.r.t. t (resp. x) and with continuous derivatives.

In this paper, we investigate existence and uniqueness of viscosity solutions v(t, x) := (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 of the following system of variational inequalities with upper and lower interconnected obstacles: For any (i, j)

∈ Γ 1 ×Γ 2 ,        min v ij -L ij [ v] (t, x), max v ij -U ij [ v] (t, x), -∂ t v ij (t, x) -Lv ij (t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x v ij (t, x)) = 0 v ij (T, x) = h ij (x) (2.1)
where, for any (t,

x) ∈ [0, T ] × R k , Lϕ(t, x) := b(t, x)D x ϕ(t, x) + 1 2 Tr[σσ ⊤ (t, x)D 2 xx ϕ(t, x)],
and (i,

j) ∈ Γ 1 × Γ 2 , L ij [ v](t, x) := max k∈(Γ 1 ) -i (v kj (t, x) -g ik (t, x)) and U ij [ v](t, x) = min l∈(Γ 2 ) -j (v il (t, x) + ḡjl (t, x)).
The functions f ij stand for the instantaneous payoff when the first player is in mode i and the second one in mode j, and g ik (resp. ḡjl ) stands for the switching cost incurred when the first (resp. second) player decides to switch from mode i to mode k (resp. from mode j to mode l).

The lower obstacle L ij [ v] and an upper obstacle U ij [ v] are called interconnected because each of them depends on the underlying solution

v := (v kl ) (k,l)∈Γ 1 ×Γ 2 .
In a way, the system (2.1) is the Bellman-Isaacs system of equations associated with the zero-sum switching game with utility functions (f ij ) (i,j)∈Γ 1 ×Γ 2 , terminal payoffs (h ij ) (i,j)∈Γ 1 ×Γ 2 and switching costs for the maximizer (resp. minimizer) given by (g ij

) (i,j)∈Γ 1 ×Γ 2 ) (resp. (ḡ ij ) (i,j)∈Γ 1 ×Γ 2 )).
The following assumptions are in force throughout the rest of the paper.

(H0) The functions b and σ associated with the second order operator L are jointly continuous in (t, x), of linear growth in (t, x) and Lipschitz continuous w.r.t. x, meaning that there exists a non-negative constant C such that for any (t, x, x ′ ) ∈ [0, T ] × R k+k we have:

|b(t, x)| + |σ(t, x)| ≤ C(1 + |x|) and |σ(t, x) -σ(t, x ′ )| + |b(t, x) -b(t, x ′ )| ≤ C|x -x ′ |. (H1) Each function f ij (i) is continuous in (t,
x) uniformly w.r.t. the other variables ( y, z) and for any (t, x) and the mapping

(t, x) → f ij (t, x, 0, 0) is of polynomial growth.
(ii) satisfies the standard hypothesis of Lipschitz continuity with respect to the variables (

y := (y ij ) (i,j)∈Γ1×Γ2 , z), i.e. ∀ (t, x) ∈ [0, T ] × R k , ∀ ( y 1 , y 2 ) ∈ R Λ × R Λ , (z 1 , z 2 ) ∈ R d × R d , |f ij (t, x, y 1 , z 1 ) -f ij (t, x, y 2 , z 2 )| ≤ C (| y 1 -y 2 | + |z 1 -z 2 |) ,
where, | y| stands for the standard Euclidean norm of y in R Λ .

(H2) Monotonicity: Let y = (y kl ) (k,l)∈Γ 1 ×Γ 2 , then for any (i, j) ∈ Γ 1 × Γ 2 and any (k, l) = (i, j) the mapping y k,l → f ij (s, y, z) is non-decreasing.

(H3) The functions h ij , which are the terminal conditions in the system (2.1), are continuous with respect to x, belong to class Π g and satisfy

∀ (i, j) ∈ Γ 1 × Γ 2 and x ∈ R k , max k∈(Γ 1 ) -i h kj (x) -g ik (T, x) ≤ h ij (x) ≤ min l∈(Γ 2 ) -j h il (x) + ḡjl (T, x) .
(H4) The no free loop property: The switching costs g ik and ḡjl are non-negative, jointly continuous in (t, x), belong to Π g and satisfy the following condition:

For any loop in Γ 1 ×Γ 2 , i.e., any sequence of pairs (i 1 , j 1 ), . . . , (i N , j N ) of Γ 1 ×Γ 2 such that (i N , j N ) = (i 1 , j 1 ), card{(i 1 , j 1 ), . . . , (i N , j N )} = N -1 and ∀ q = 1, . . . , N -1, either i q+1 = i q or j q+1 = j q , we have

∀(t, x) ∈ [0, T ] × R k , q=1,N -1 ϕ iq iq+1 (t, x) = 0, (2.2)
where, ∀ q = 1, . . . , N -1, ϕ iq iq+1 (t, x) = -g iq iq+1 (t, x)1

1 iq =iq+1 + ḡjqiq+1 (t, x)1 1 jq =jq+1 (resp. ϕ iq iq+1 (t, x) = g iq ,iq+1 (t, x)1 1 iq =iq+1 -ḡjq,iq+1 (t, x)1 1 jq =jq+1 ). 4
This assumption implies in particular that

∀ (i 1 , . . . , i N ) ∈ (Γ 1 ) N such that i N = i 1 and card{i 1 , . . . , i N } = N -1, N -1 p=1 g i k ,i k+1 > 0 (2.3)
and ∀ (j 1 , . . . , j N ) ∈ (Γ 2 ) N such that j N = j 1 and card{j 1 , . . . , j N } = N -1,

N -1 p=1 ḡj k ,j k+1 > 0. (2.4)
By convention we set ḡj,j = g i,i = 0.

Conditions (2.3) and (2.4) are classical in the literature of switching problems and usually referred to as the no free loop property.

Finally, let us mention that if we set

g ij (t, x) = |i -j|g(t, x) and ḡij (t, x) = |i -j|ḡ(t, x), (i, j) ∈ Γ 1 × Γ 2 ,
where both g and ḡ are functions such that, for any (t, x) ∈ [0, T ] × R k , ḡ(t,x) g(t,x) is not a rational number, then assumption (2.2) holds.

We now define the notions of viscosity super (or sub)-solution of the system (2.1). This is done in terms of the notions of subjet and superjet which we recall here.

Definition 1. (Subjet and superjet)

(i) For a lower semicontinuous (lsc) (resp. upper semicontinuous (usc)) function u :

[0, T ] × R k → R, we denote by J -u(t, x) (resp. J + u(t, x)) the parabolic subjet (resp. superjet) of u at (t, x) ∈ [0, T ] × R k , as the set of triples (p, q, M ) ∈ R × R k × S k satisfying u(t ′ , x ′ ) ≥ (resp. ≤) u(t, x) + p(t ′ -t) + q ⊤ (x ′ -x) + 1 2 (x ′ -x) ⊤ M (x ′ -x) + o |t ′ -t| + |x ′ -x| 2
where S k is the set of symmetric real matrices of dimension k.

(ii) For a lsc (rep. usc) function u : [0, T ] × R k → R, we denote by Ju(t, x) (resp. J+ u(t, x)) the parabolic limiting subjet (resp. superjet) of u at (t, x) ∈ [0, T ] × R k , as the set of triples (p, q, M ) ∈ R × R k × S k such that:

(p, q, M ) = lim n (p n , q n , M n ), (t, x) = lim n (t n , x n ) with (p n , q n , M n ) ∈ J -u(t n , x n ) (resp. J + u(t n , x n )) and u(t, x) = lim n u(t n , x n ).
Finally, given a locally bounded R-valued deterministic function u, we denote by u * (resp. u * ) its lower (resp. upper) semicontinuous envelope defined by:

∀ (t, x) ∈ [0, T ] × R k , u * (t, x) = lim (t ′ ,x ′ )→(t,x); t ′ <T u(t ′ , x ′ ) and u * (t, x) = lim (t ′ ,x ′ )→(t,x); t ′ <T u(t ′ , x ′ ). ( 2

.5)

We now give the definition of a viscosity solution for the system (2.1).

Definition 2. Viscosity solution to (2.1)

(i) A function v = (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 : [0, T ] × R k → R Λ such that for any (i, j) ∈ Γ 1 × Γ 2 , v ij is lsc (resp.
usc), is called a viscosity supersolution (resp. a viscosity subsolution) to (2.1) if for any (i, j) ∈ Γ 1 × Γ 2 , for any

(t, x) ∈ [0, T ) × R k and any (p, q, M ) ∈ J-v ij (t, x) (resp. J+ v ij (t, x)) we have:        min v ij (t, x) -L ij [ v](t, x), max -p -b(t, x).q -1 2 T r[(σσ ⊤ )(t, x)M ] -f ij (t, x, v(t, x), σ ⊤ (t, x)q); v ij (t, x) -U ij [ v](t, x) ≥ 0, v i (T, x) ≥ h ij (x), (2.6)
resp.

       min v ij (t, x) -L ij [ v](t, x), max -p -b(t, x).q -1 2 T r[(σσ ⊤ )(t, x)M ] -f ij (t, x, v(t, x), σ ⊤ (t, x)q); v ij (t, x) -U ij [ v](t, x) ≤ 0, v ij (T, x) ≤ h ij (x) .
(2.7)

(ii) A locally bounded function v = (v kl ) (k,l)∈Γ 1 ×Γ 2 : [0, T ] × R k → R Λ is
(v ij, * )(t, x) of v * ) by the subjet J -(v ij * )(t, x) (resp. by the superjet J + (v ij, * )(t, x)
). This results from the continuity of the functions b, σ, (f ij , h ij , g ij , ḡij ) (i,j)∈Γ 1 ×Γ 2 and the monotonicity property (H2) of

(f ij ) (i,j)∈Γ 1 ×Γ 2 .
3 Systems of reflected BSDEs and approximation schemes of the solutions Let (Ω, F , P) be a fixed probability space on which is defined a standard d-dimensional Brownian motion B = (B t ) 0≤t≤T whose natural filtration is (F 0 t := σ{B s , s ≤ t}) 0≤t≤T . Let F = (F t ) 0≤t≤T be the completed filtration of (F 0 t ) 0≤t≤T with the P-null sets of F , hence (F t ) 0≤t≤T satisfies the usual conditions, i.e., it is right continuous and complete. Furthermore, let

• P be the σ-algebra on [0, T ] × Ω of F-progressively measurable sets;

• H 2,ℓ (ℓ ≥ 1) be the set of P-measurable and R ℓ -valued processes w = (w t ) t≤T such that E[

T 0 |w s | 2 ds] < ∞; • S 2,ℓ (ℓ ≥ 1) be the subset of H 2,ℓ of continuous processes such that E[sup t≤T |w t | 2 ] < ∞. Finally let A +,2 be the subset of S 2,1 of non-decreasing processes K = (K t ) t≤T such that K 0 = 0 and E[K 2 T ] < ∞. Next, for n, m ≥ 0, let (Y ij,n,m , Z ij,n,m ) (i,j
)∈Γ 1 ×Γ 2 be the solution of the following system of BSDEs.

       (Y ij,n,m , Z ij,n,m ) ∈ S 2,1 × H 2,d ; dY ij,n,m s = -f ij,n,m (s, X t,x s , (Y kl,n,m s ) (k,l)∈Γ 1 ×Γ 2 , Z ij,n,m s )ds + Z ij,n,m s dB s , s ≤ T, Y ij,nm T = h ij (X t,x T ), (3.1)
where,

f ij,n,m (s, X t,x s , (y ij ) (i,j)∈Γ 1 ×Γ 2 , z) := f ij (s, X t,x s , (y kl ) (k,l)∈Γ 1 ×Γ 2 , z)+ n(y ij -max k∈(Γ 1 ) -i {y kj -g ik (s, X t,x s )}) --m(y ij -min l∈(Γ 2 ) -j {y il + ḡjl (s, X t,x s )}) + .
( [START_REF] Pardoux | Adapted solution of a backward stochastic differential equation[END_REF]), the solution (Y ij,n,m , Z ij,n,m ) (i,j)∈Γ 1 ×Γ 2 exists and is unique. Since the generators and the terminal values depend on (t, x), the processes Y ij,n,m and Z ij,n,m also depend on (t, x) but, to avoid overload notation, we do not mention this dependence in the sequel. Furthermore, the following monotonicity properties holds for the double sequence (Y ij,n,m ) n,m .

Proposition 3.1. For any (i, j) ∈ Γ 1 × Γ 2 and n, m ≥ 0 we have

P -a.s., Y ij,n,m ≤ Y ij,n+1,m and Y ij,n,m ≥ Y ij,n,m+1 , (i, j) ∈ Γ 1 × Γ 2 . (3.2)
Moreover, for any (i, j) ∈ Γ 1 × Γ 2 and n, m ≥ 0, there exists a deterministic continuous function v ij,n,m in Π g such that, for any t ≤ T , BSDEs (we recall in Appendix (A1)). To prove this, it is enough to show that for any t, (y ij

Y ij,n,m s = v ij,n,m (s, X t,x s ), s ∈ [t, T ]. (3.3) Finally, for any (i, j) ∈ Γ 1 × Γ 2 and n, m ≥ 0, v ij,n,m (t, x) ≤ v ij,n+1,m (t, x) and v ij,n,m (t, x) ≥ v ij,n,m+1 (t, x), (t, x) ∈ [0, T ] × R k . ( 3
) (i,j)∈Γ 1 ×Γ 2 , (ȳ ij ) (i,j)∈Γ 1 ×Γ 2 ∈ R Λ and (z ij ) (i,j)∈Γ 1 ×Γ 2 , (z ij ) (i,j)∈Γ 1 ×Γ 2 ∈ (R d ) Λ we have: -4 (i,j)∈Γ 1 ×Γ 2 y - ij f ij,n,m (s, X t,x s , (y + kl + ȳkl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) -f ij,n+1,m (s, X t,x s , (ȳ kl ) (k,l)∈Γ 1 ×Γ 2 , zij ) ≤ 2 (i,j)∈Γ 1 ×Γ 2 1 [yij<0] z ij -zij 2 + C (i,j)∈Γ 1 ×Γ 2 (y - ij ) 2 ,
where, C is a constant, y - ij = max(-y ij , 0) and y + ij = max(y ij , 0). This inequality follows easily from the fact that, for any (i,

j) ∈ Γ 1 × Γ 2 , (i) f ij,n,m (s, X t,x s , (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) ≤ f ij,n+1,m (s, X t,x s , (ȳ kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ), (ii) For any (u kl ) (k,l)∈Γ 1 ×Γ 2 ∈ R Λ such that u ij = 0 and u kl ≥ 0, for (k, l) = (i, j), f ij,n,m (s, X t,x s , (y kl + u kl ) (i,j)∈Γ 1 ×Γ 2 , z ij ) ≥ f ij,n,m (s, X t,x s , (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ).
(iii) f ij depends only on z ij and not on the other components z kl , (k, l) = (i, j).

Consequently, we have

P -a.s., Y ij,n,m ≤ Y ij,n+1,m , (i, j) ∈ Γ 1 × Γ 2 .
In the same way one can show that Y ij,n,m+1 ≤ Y ij,n,m . Finally, the inequalities of (3.4) are obtained by taking

s = t in (3.2) in view of the representation (3.3) of Y ij,n,m by v ij,n,m and X t,x .
We now suggest two approximation schemes obtained from the sequence (Y ij,n,m , (i, j) ∈ Γ 1 × Γ 2 ) n,m of the solution of the system (3.1). The first scheme is a sequence of decreasing reflected BSDEs with interconnected lower obstacles and the second one is an increasing sequence of reflected BSDEs with interconnected upper obstacles.

Let us first introduce the decreasing approximation scheme by considering the following system of reflected BSDEs with interconnected obstacles:

∀(i, j) ∈ Γ 1 × Γ 2 ,              Ȳ ij,m ∈ S 2,1 , Zij,m ∈ H 2,d and Kij,m ∈ A 2,+ ; Ȳ ij,m s = h ij (X t,x T ) + T s f ij,m (r, X t,x r , ( Ȳ kl,m r ) (k,l)∈Γ 1 ×Γ 2 , Zij,m r )dr + Kij,m T -Kij,m s - T s Zij,m r dB r , Ȳ ij,m -g ik (s, X t,x s )})d Kij,m s = 0, (3.5)
where, for any (i,

j) ∈ Γ 1 × Γ 2 , m ≥ 0 and (s, y, z ij ), f ij,m (s, X t,x s , y, z ij ) := f ij (s, X t,x s , (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) -m y ij -min l∈(Γ 2 ) -j (y il + ḡjl (s, X t,x s )) + .
Thanks to Assumptions (H1)-(H3) and (2.3), by Theorem 3.5 in Hamadène and Zhang (2010) ( [START_REF] Hamadène | Switching problem and related system of reflected backward stochastic differential equations Stoch[END_REF]), the solution of (3.5) exists and is unique. In fact, this holds again under weaker assumptions (see Hamadène and Morlais (2011)( [START_REF] Hamadène | Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Multi-Modes Switching Problem[END_REF], Theorem 1). For sake of completeness, a statement of this recent result is given in Appendix (A2).

Moreover, we have the following properties.

Proposition 3.2. For any (i, j) ∈ Γ 1 × Γ 2 and m ≥ 0, we have

(i) E[ sup t≤s≤T |Y ij,n,m s -Ȳ ij,m s | 2 ] → 0, as n → ∞, (3.6)
(ii)

P -a.s., Ȳ ij,m ≥ Ȳ ij,m+1 .
(iii) There exists a unique Λ-uplet of deterministic continuous functions

(v kl,m ) (k,l)∈Γ 1 ×Γ 2 in Π g such that, for every t ≤ T , Ȳ ij,m s = vij,m (s, X t,x s ), s ∈ [t, T ]. (3.7) Moreover, for any (i, j) ∈ Γ 1 × Γ 2 and (t, x) ∈ [0, T ] × R k , vij,m (t, x) ≥ vij,m+1 (t, x).
Finally, (v ij,m ) (i,j)∈Γ 1 ×Γ 2 is the unique viscosity solution in the class Π g of the following system of variational inequalities with inter-connected obstacles.

∀ (i, j) ∈ Γ 1 × Γ 2 ,              min{v ij,m (t, x) -max k∈(Γ 1 ) -i (v kj,m (t, x) -g ik (t, x)); -∂ t vij,m (t, x) -b(t, x)D x vij,m (t, x) -1 2 Tr(σσ ⊤ (t, x)D 2 xx vij,m (t, x)) -f ij,m (t, x, (v kl,m (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x vij,m (t, x))} = 0, vij,m (T, x) = h ij (x) (3.8)
where,

f ij,m (s, x, (y kl ) (k,l)∈Γ 1 ×Γ 2 , z) = f ij (s, x, (y kl ) (k,l)∈Γ 1 ×Γ 2 , z) -m y ij -min l∈(Γ 2 ) -j (y il + ḡjl (s, x)) + .
Proof . Let us prove (i). For this, it is enough to consider the case m = 0, and we will do so, since for any

(i, j) ∈ Γ 1 × Γ 2 , the function (s, x, (y kl ) (k,l)∈Γ 1 ×Γ 2 ) -→ -m y ij -min l∈(Γ 2 ) -j (y il + ḡjl (s, x))) +
has the same properties as f ij displayed in (H1)-(H2).

To begin with, let us show that for any i, j and n ≥ 0,

P -a.s., Y ij,n,0 ≤ Ȳ ij,0 . (3.9)
First andt w.l.o.g. we may assume that f ij is non-decreasing w.r.t. (y kl ) (k,l)∈Γ 1 ×Γ 2 , since thanks to Assumption (H2), it is enough to multiply the solutions by e ̟t , where ̟ is appropriately chosen in order to fall in this latter case, since f ij is Lipschitz in y ij . Now, for fixed n, let us define recursively the sequence ( Ỹ k,ij,n ) k≥0 as follows:

For k = 0 and any (i, j) ∈ Γ 1 × Γ 2 , we set Ỹ 0,ij,n := Ȳ ij,0 and, for any k ≥ 1, let us define ( Ỹ k,ij,n , Zk,ij,n ) ∈ S 2,1 × H 2,d as the solution of the following system of BSDEs:

∀(i, j) ∈ Γ 1 × Γ 2 ,        -d Ỹ k,ij,n s = f ij (s, X t,x s , ( Ỹ k-1,pq,n s ) (p,q)∈Γ 1 ×Γ 2 , Zk,ij,n s )ds+ n{ Ỹ k,ij,n s -max l∈(Γ 1 ) -i ( Ỹ k-1,lj,n s -g ik (s, X t,x s ))} -ds -Zk,ij,n s dB s , Ỹ k,ij,n T = h ij (X t,x T ) (3.10)
The solution of (3.10) exists since it is a multi-dimensional standard BSDE with a Lipschitz coefficient, noting that ( Ỹ k-1,pq,n s ) (p,q)∈Γ 1 ×Γ 2 is already given. Since, n is fixed and the coefficient

ϕ ij,n (s, ω, (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) := f ij (s, X t,x s , (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) + n{y ij -max l∈(Γ 1 ) -i (y lj -g il (s, X t,x s )} - is Lipschitz w.r.t. ((y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ), the sequence ( Ỹ k,ij,n ) k converges in S 2,1 to Y ij,n,0
, as k → ∞, for any i, j and n.

Using an induction argument w.r.t. k, we prove that, for any i, j and n,

P -a.s., Ỹ k,ij,n ≤ Ȳ ij,0 , k ≥ 0.
Indeed, for k = 0 the inequalities hold true through the definition of Ỹ 0,ij,n . Assume now that these inequalities are valid for some k -1, i.e., for any i, j and n,

P -a.s., Ỹ k-1,ij,n ≤ Ȳ ij,0 . (3.11)
Thus, taking into account that Ȳ ij,0 satisfies (3.5) with m = 0 and the fact that f ij is non-decreasing w.r.t.

(y kl ) (k,l)∈Γ 1 ×Γ 2 then for any i, j, n, it holds

f ij (s, X t,x s , ( Ỹ k-1,pq,n s ) (p,q)∈Γ 1 ×Γ 2 , z ij ) + n{ Ȳ ij,0 -max l∈(Γ 1 ) -i ( Ỹ k-1,lj,n s -g il (s, X t,x s ))} - ≤ f ij (s, X t,x s , ( Ȳ pq,0 s ) (p,q)∈Γ 1 ×Γ 2 , z ij ).
Using the standard comparison result of solutions of one dimensional BSDEs we obtain that, for any n, i, j, Ỹ k,ij,n ≤ Ȳ ij,0 . a.s. Thus the property (3.11) is valid for any k. Taking the limit as k tends to ∞, we obtain (3.9).

We can now apply Peng's monotonic limit (see Peng (1999) ( [START_REF] Peng | Monotonic limit theorem of BSDE and nonlinear decomposition theorem of Doob Meyers type[END_REF])) to the increasing sequence (Y ij,n,0 ) n≥0 . This yields the existence of processes Ŷ ij , Ẑij and Kij , (i, j) ∈ Γ 1 × Γ 2 , such that:

(a) Ŷ ij is RCLL and uniformly P-square integrable. Moreover, for any stopping time τ , lim

n→∞ ր Y ij,n,0 τ = Ŷ ij τ . (b) Kij is RCLL non-decreasing, Kij 0 = 0 and for any stopping time τ , lim n→∞ K ij,n,0 τ = Kij τ , P -a.s. (c) Ẑij ∈ H 2,d and for any p ∈ [1, 2), lim n→∞ E[ T 0 |Z ij,n,0 s -Ẑij s | p ds] = 0. (d) For any (i, j) ∈ Γ 1 × Γ 2 , the triple ( Ŷ ij , Ẑij , Kij ) satisfies: ∀s ≤ T , Ŷ ij s = h ij (X t,x T ) + T s f ij (r, X t,x r , ( Ŷ kl r ) (k,l)∈Γ 1 ×Γ 2 , Ẑij r )dr + Kij T -Kij s - T s Ẑij r dB r Ŷ ij s ≥ max k∈(Γ 1 ) -i { Ŷ kj s -g ik (s, X t,x s )}.
(3.12)

The remaining of the proof is the same as in Hamadène and Zhang (2010) (( [START_REF] Hamadène | Switching problem and related system of reflected backward stochastic differential equations Stoch[END_REF]), Theorem 3.2) and it mainly consists in proving both the continuity of Ŷ ij and Kij and the Skorohod condition (these properties are deduced using the no-free loop property (H4)). Finally we obtain that the triple ( Ŷ ij , Ẑij , Kij ) satisfies (3.5) with m = 0.

Hence, by uniqueness of the solutions of (3.5), Ŷ ij = Ȳ ij,0 a.s, which completes the proof of (i).

(ii) is an immediate consequence of (i) and Proposition 3.1.

We now establish (iii). The existence of the deterministic continuous functions (v ij,m ) (i,j)∈Γ 1 ×Γ 2 such that for

any s ∈ [t, T ], Y ij,m s = vij,m (s, X s ), (i, j) ∈ Γ 1 × Γ 2
, m ≥ 0, and which satisfy the system (3.8), is obtained in [START_REF] Hamadène | Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Multi-Modes Switching Problem[END_REF] (see Appendix A2, Theorem 6.2). Finally as Y ij,m ≥ Y ij,m+1 a.s., we deduce that vij,m ≥ vij,m+1 , taking into account (ii), which completes the proof.

We now introduce the increasing approximation scheme by considering the following system of reflected BSDEs with interconnected obstacles: for any (i,

j) ∈ Γ 1 × Γ 2 ,                  Y ij,n ∈ S 2 , Z ij,n ∈ H 2,d and K ij,n ∈ A 2,+ ; Y ij,n s = h ij (X t,x T ) + T s f ij,n (r, X t,x r , (Y kl,n r ) (k,l)∈Γ 1 ×Γ 2 , Z ij,n r )dr -(K ij,n T -K ij,n s ) - T s Z ij,n r dB r , Y ij,n s ≤ min l∈(Γ 2 ) -j {Y il,n s + ḡjl (s, X t,x s )}, s ≤ T, T 0 (Y ij,n s -min l∈(Γ 2 ) -j {Y il,n s + ḡjl (s, X t,x s )})dK ij,n s = 0, (3.13)
where, for any (i, j) ∈ Γ 1 × Γ 2 , n ≥ 0 and (s, y, z ij ),

f ij,n (s, X t,x s , y, z ij ) = f ij (s, X t,x s , (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) + n y ij -max k∈(Γ 1 ) -i {y kj -g ik (s, X t,x s )} -.
Thanks to assumptions (H1)-( H3) and (H4)-(2.4), by Theorems 3.2 and 3.5 in Hamadène and Zhang (2010) ( [START_REF] Hamadène | Switching problem and related system of reflected backward stochastic differential equations Stoch[END_REF]) the solution of (3.5) exists and is unique.

Below is the analogous of Proposition 3.2. We do not give its proof since it is deduced from this latter proposition in considering the equation satisfied by (-

Y ij,n , -Z ij,n , K ij,n ). Proposition 3.3. For any (i, j) ∈ Γ 1 × Γ 2 we have (i) E[ sup t≤s≤T |Y ij,n,m s -Y ij,n s | 2 ] → 0 as m → ∞. (3.14)
(ii) For any n ≥ 0,

P -a.s., Y ij,n ≤ Y ij,n+1 .
(iii) For every n ≥ 0, there exists a unique Λ-uplet of deterministic continuous functions

(v kl,n ) (k,l)∈Γ 1 ×Γ 2 in Π g such that, for every t ≤ T , Y ij,n s = v ij,n (s, X t,x s ), t ≤ s ≤ T. (3.15)
Moreover, for any n ≥ 0 and (i,

j) ∈ Γ 1 × Γ 2 , v ij,n (t, x) ≤ v ij,n+1 (t, x), (t, x) ∈ [0, T ] × R k .
Finally, (v ij,n ) (i,j)∈Γ 1 ×Γ 2 is the unique viscosity solution in the class Π g of the following system of variational inequalities with inter-connected obstacles:

∀ (i, j) ∈ Γ 1 × Γ 2 ,              max{v ij,n (t, x) -min l∈(Γ 2 ) -j (v il,n (t, x) + ḡik (t, x)); -∂ t v ij,n (t, x) -b(t, x)D x v ij,n (t, x) -1 2 Tr(σσ ⊤ (t, x)D 2 xx v ij,n (t, x)) -f ij,n (t, x, (v kl,n (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x v ij,n (t, x))} = 0, v ij,n (T, x) = h ij (x) (3.16)
where,

f ij,n (s, x, (y kl ) (k,l)∈Γ 1 ×Γ 2 , z) = f ij (s, x, (y kl ) (k,l)∈Γ 1 ×Γ 2 , z) + n{y ij -max k∈(Γ 1 ) -i (y kj -g ik (s, x))} -. For (t, x) ∈ [0, T ] × R k ∈ [0, T ] × R k and (i, j) ∈ Γ 1 × Γ 2 , let us define vij (t, x) := lim m→∞ vij,m (t, x) and v ij (t, x) := lim n→∞ v ij,n (t, x).
Then, as a by-product of Propositions 3.2 and 3.3 we have Corollary 1. For any (i, j) ∈ Γ 1 × Γ 2 , the function vij (resp. v ij ) is usc (resp. lsc). Moreover, vij and v ij belong to Π g i.e., there exist deterministic constants C ≥ 0 and γ > 0 such that, for any

(t, x) ∈ [0, T ] × R k , |v ij (t, x)| + |v ij (t, x)| ≤ C(1 + |x| γ ), (t, x) ∈ [0, T ] × R k .
Finally, for any

(t, x) ∈ [0, T ] × R k , v ij (t, x) ≤ vij (t, x). Proof . For any (i, j) ∈ Γ 1 × Γ 2 , vij (resp. v ij
) is obtained as a decreasing (resp. increasing) limit of continuous functions. Therefore, it is usc (resp. lsc). Next, for any (i, j) and n, m,

v ij,n,m (t, x) ≤ v ij,n,0 (t, x), (t, x) ∈ [0, T ] × R k ,
as the sequence (v ij,n,m ) m≥0 is decreasing. Thus, taking the limit as m → ∞ we obtain

v ij,n ≤ v ij,n,0 .
Now, using (3.3) and (3.6) it follows that, for any t ≤ T and s ∈ [t, T ], Y ij,n,0 s = v ij,n,0 (s, X t,x s ) and the processes Y ij,n,0 converges in S 2 , as n → ∞, to Ȳ ij,0 solution of (3.5) with m = 0. Furthermore, by (3.7), there exists a deterministic continuous function vij,0 of class Π g such that for any t ≤ T and s ∈ [t, T ], Ȳ ij,0 s = vij,0 (s, X t,x s ). Then, taking s = t and the limit as n → ∞ to obtain

v ij (t, x) := lim n→∞ v ij,n (t, x) ≤ lim n→∞ v ij,n,0 (t, x) = vij,0 (t, x), ∀(t, x) ∈ [0, T ] × R k .
But, vij,0 and v ij,n belong to Π g and v ij,n ≤ v ij,n+1 . Thus, v ij belongs to Π g , for any (i, j) ∈ Γ 1 × Γ 2 . In the same way one can show that vij belongs to Π g , for any (i, j) ∈ Γ 1 × Γ 2 . The last inequality follows from (3.4) and the definitions of vij and v ij .

A comparison result

In this section we investigate some qualitative properties of viscosity solutions of the system (2.1). In particular, we show in Corollary 2 below that if the system (2.1) admits a viscosity solution in the class Π g , then it is unique and continuous.

We first show that if (u ij ) (i,j)∈Γ 1 ×Γ 2 (resp. (w ij ) (i,j)∈Γ 1 ×Γ 2 ) is an usc subsolution (resp. lsc supersolution) of (2.1) which belongs to Π g , then for any (i, j) ∈ Γ 1 × Γ 2 , u ij ≤ w ij . To begin with, we give an intermediary result which is required in the second step of the proof of the comparison result. Lemma 4.1. Let u := (u ij ) (i,j)∈Γ 1 ×Γ 2 (resp. w := (w ij ) (i,j)∈Γ 1 ×Γ 2 ) be an usc subsolution (resp. lsc supersolution) of (2.1). Let (t, x) ∈ [0, T ] × R k and let Γ(t, x) be the following set.

Γ(t, x) := {(i, j) ∈ Γ 1 × Γ 2 , u ij (t, x) -w i,j (t, x) = max (k,l)∈Γ 1 ×Γ 2 {u kl (t, x) -w kl (t, x)}.
Then, there exists (i 0 , j 0 ) ∈ Γ(t, x) such that u i0j0 (t, x) > L i0j0 [ u](t, x) and w i0j0 (t, x) < U i0j0 [ w](t, x).

(4.1)

Proof: Let (t, x) ∈ [0, T ] × R k be fixed. Since the set Γ 1 × Γ 2 is finite then Γ(t, x) is not empty. Next, let us show the claim by contradiction. So for any (i, j) ∈ Γ(t, x) either

u ij (t, x) ≤ L ij [ u](t, x) or w ij (t, x) ≥ U ij [ w](t, x)
holds. Let us first assume that:

u ij (t, x) ≤ L ij [ u](t, x). (4.2)
Then, there exists some k ∈ (Γ 1 ) -i , such that

u ij (t, x) ≤ L ij [ u](t, x) = u kj (t, x) -g ik (t, x).
But, since w is a supersolution to (2.1) we also deduce

w ij (t, x) ≥ w kj (t, x) -g ik (t, x),
which implies that

u ij (t, x) -u kj (t, x) ≤ -g ik (t, x) ≤ w ij (t, x) -w kj (t, x)
and then

u ij (t, x) -w ij (t, x) ≤ u kj (t, x) -w kj (t, x).
Since, by assumption (i, j) ∈ Γ(t, x), the two previous inequalities are instead equalities, (k, j) belongs to Γ(t, x), k = i and finally it holds that

u ij (t, x) -u kj (t, x) = -g ik (t, x) = w ij (t, x) -w kj (t, x). (4.3) Next, if (4.2) does not hold, then necessarily u ij (t, x) > L ij [ u](t, x) and w ij (t, x) ≥ U ij [ w](t, x). As (u ij ) i,j is a subsolution of (2.1) we obtain u ij (t, x) ≤ U ij [ u](t, x) ≤ u il (t, x) + ḡjl (t, x), l ∈ Γ 2 -j .
On the other hand, for some index l ∈ Γ 2 -j it holds

w ij (t, x) -w il (t, x) ≥ ḡjl (t, x) ≥ u ij (t, x) -u il (t, x).
Once more as (i, j) ∈ Γ(t, x) then the previous inequalities yield that (i, l) ∈ Γ(t, x), l = j and

u ij (t, x) -u il (t, x) = ḡjl (t, x) = w ij (t, x) -w il (t, x). (4.4)
Repeating now this reasoning as many times as necessary, and since Γ 1 × Γ 2 is finite, there exits a loop (i 1 , j 1 ), . . . , (i N -1 , j N -1 ), (i N , j N ) = (i 1 , j 1 ) such that q=1,N -1 ϕ iq ,iq+1 (t, x) = 0, which contradicts Assumption (H4), whence the claim is proved.

We now give the main result of this subsection.

Theorem 4.2. Assume that u = (u ij ) (i,j)∈Γ 1 ×Γ 2 (resp. w = (w ij ) (i,j)∈Γ 1 ×Γ 2
) is an usc (resp. lsc) subsolution (resp. supersolution) of the system (2.1) such that, for any (i, j) ∈ Γ 1 × Γ 2 , both u ij and w ij belong to Π g , i.e., there exist two constants γ and C such that

∀(i, j) ∈ Γ 1 × Γ 2 , ∀(t, x) ∈ [0, T ] × R k , |u ij (t, x)| + |w ij (t, x)| ≤ C(1 + |x| γ ). (4.5)
Then, it holds that for any (i,

j) ∈ Γ 1 × Γ 2 , u ij (t, x) ≤ w ij (t, x), (t, x) ∈ [0, T ] × R k .
Proof. Let us proceed by contradiction and let (u ij ) (i,j)∈Γ 1 ×Γ 2 (resp. w = (w ij ) (i,j)∈Γ 1 ×Γ 2 ) be usc (resp. lsc) and a subsolution (resp. a supersolution) of the system (2.1) such that there exists ε 0 > 0 and (t

0 , x 0 ) ∈ [0, T ]×R k such that max i,j (u ij -w ij )(t 0 , x 0 ) ≥ ǫ 0 . (4.6)
Next, w.l.o.g. we may assume that for any (i, j)

∈ Γ 1 × Γ 2 , lim |x|→∞ (u ij -w ij )(t, x) = -∞. (4.7)
Indeed, if this is not the case, one may replace w ij with w ij,ϑ,µ defined by

w ij,ϑ,µ (t, x) = w ij (t, x) + ϑe -µt |x| 2γ+2 , (t, x) ∈ [0, T ] × R k ,
which is still an usc supersolution of (2.1) for ϑ > 0 and µ ≥ µ 0 which satisfies (4.7) (a proof of the supersolution property for good choices of ϑ and µ can be found in e.g. Pham (2009) (( [START_REF] Pham | Continuous-time stochastic control and optimization with financial applications[END_REF]), pp.76). Therefore, it suffices to show that u ij (t, x) ≤ w ij,ϑ,µ (t, x), (t, x) ∈ [0, T ] × R k , since, by taking the limit as ϑ → 0, we deduce that

u ij (t, x) ≤ w ij (t, x), (t, x) ∈ [0, T ] × R k .
Thus, assume that (4.6) and (4.7) are satisfied. Then, there exists R > 0 such that

max (t,x)∈[0,T ]×R k max i,j {(u ij -w ij )(t, x)} = max (t,x)∈[0,T ]×B(0,R) max i,j {(u ij -w ij )(t, x)} = max ij (u ij -w ij )(t * , x * ) ≥ ǫ 0 > 0, (4.8)
where, (t * , x * ) ∈ [0, T ) × B(0, R), where, B(0, R) denotes the ball in R k with center the origin and radius R,

since by definition u ij (T, x) ≤ w ij (T, x), for all (i, j) ∈ Γ 1 × Γ 2 .
The remaining of the proof is obtained in two steps: the first step which is the main one establishes the comparison result under the additional condition (4.9) and the second step provides the proof in the general case.

Step 1. Let us make the following assumption on the functions (f ij ) (i,j)∈Γ 1 ×Γ 2 . For all

λ > c(f i,j )(Λ -1), (i, j) ∈ Γ 1 × Γ 2 , (t, x, y, z) ∈ [0, T ] × R k × R Λ+d , and (u, v) ∈ R 2 s.t. u ≥ v, f ij (t, x, [ y -(ij) , u], z) -f ij (t, x, [ y -(ij) , v], z) ≤ -λ(u -v), (4.9)
where, c(f ij ) is the Lipschitz constant of f ij w.r.t. (y kl ) (k,l)∈Γ 1 ×Γ 2 . Next, let (i 0 , j 0 ) be an element of Γ(t * , x * ) that satisfies (4.1). For n ≥ 1, let Φ i0,j0 n be the function defined as follows.

Φ i0,j0 n (t, x, y) := (u i0j0 (t, x) -w i0j0 (t, y)) -φ n (t, x, y), (t, x, y) ∈ [0, T ] × R k+k ,
where,

φ n (t, x, y) := n|x -y| 2γ+2 + |x -x * | 2γ+2 + (t -t * ) 2 . Since Φ i0,j0 n is usc in (t, x, y), there exists (t n , x n , y n ) ∈ [0, T ] × B(0, R) 2 such that Φ i0,j0 n (t n , x n , y n ) = max (t,x,y)∈[0,T ]×B(0,R) 2 Φ i0,j0 n (t, x, y). Moreover, Φ i0,j0 n (t * , x * , x * ) = u i0,j0 (t * , x * ) -w i0,j0 (t * , x * ) ≤ u i0,j0 (t * , x * ) -w i0,j0 (t * , x * ) + φ n (t n , x n , y n ) ≤ u i0,j0 (t n , x n ) -w i0,j0 (t n , y n ). (4.10)
and then

-{f i0j0 (t n , x n , (u ij (t n , x n )) (i,j)∈Γ 1 ×Γ 2 , σ(t n , x n ) ⊤ .q n u ) -f i0j0 (t n , y n , (w ij (t n , y n )) (i,j)∈Γ 1 ×Γ 2 , σ(t n , y n ) ⊤ .q n w )} ≤ ̺ n ,
with lim n→∞ ̺ n ≤ 0, using the fact that all the terms in the first line of (4.14) are converging sequences.

Linearizing f i0j0 (see Appendix A3), which is Lipschitz w.r.t. (y ij ) (i,j)∈Γ 1 ×Γ 2 , and using Assumption (4.9), we obtain

λ(u i0j0 (t n , x n ) -w i0j0 (t n , y n )) - (i,j)∈Γ 1 ×Γ 2 ,(i,j) =(i0j0) Θ i,j n (u ij (t n , x n ) -w ij (t n , y n )) ≤ ̺ n ,
where, Θ i,j n is the increment rate of f i0j0 w.r.t. y ij which is uniformly bounded (w.r.t. n) and non-negative thanks to the monotonicity assumption (H2). Therefore,

λ(u i0j0 (t n , x n ) -w i0j0 (t n , y n )) ≤ (i,j)∈Γ 1 ×Γ 2 ,(i,j) =(i0,j0) Θ i,j n (u ij (t n , x n ) -w ij (t n , y n )) + ̺ n ≤ c(f i0j0 ) × (i,j)∈Γ 1 ×Γ 2 ,(i,j) =(i0,j0) ((u ij (t n , x n ) -w ij (t n , y n )) + + ̺ n .
Taking the limit as n → ∞ we obtain

λ(u i0j0 (t * , x * ) -w i0j0 (t * , y * )) ≤ lim n→∞ c(f i0j0 ) (i,j)∈Γ 1 ×Γ 2 ,(i,j) =(i0,j0) (u ij (t n , x n ) -w ij (t n , y n )) + ≤ c(f i0j0 ) (i,j)∈Γ 1 ×Γ 2 ,(i,j) =(i0,j0) lim n→∞ (u ij (t n , x n ) -w ij (t n , y n )) + ≤ c(f i0j0 ) (i,j)∈Γ 1 ×Γ 2 ,(i,j) =(i0,j0) (u ij (t * , x * ) -w ij (t * , x * )) + , since u ij (resp. w ij ) is usc (resp. lsc).
As (i 0 , j 0 ) belongs to Γ(t * , x * ), we obtain

i0j0 (t * , x * ) -w i0j0 (t * , y * )) ≤ c(f i0j0 ) (Λ -1)(u i0j0 (t * , x * ) -w i0j0 (t * , x * )) .
But this is contradictory with (4.8) and (4.9). Thus, for any (i, j)

∈ Γ 1 × Γ 2 , u ij ≤ w ij . ✷
Step 2. The general case.

For

arbitrary λ ∈ R, if (u ij ) (i,j)∈Γ 1 ×Γ 2 ∈Γ 1 ×Γ 2 (resp. (w ij ) (i,j)∈Γ 1 ×Γ 2
) is a subsolution (resp. supersolution) of (2.1) then ũij (t, x) = e λt u ij (t, x) and wij (t, x) = e λt w ij (t, x) is a subsolution (resp. supersolution) of the following system of variational inequalities with oblique reflection. For every (i,

j) ∈ Γ 1 × Γ 2 ,            min{ṽ ij (t, x) -max l∈(Γ 1 ) -i {ṽ lj (t, x) -e λt g il (t, x)}; max{ṽ ij (t, x) -min k∈(Γ 2 ) -j {e λt ḡjk (t, x) + ṽij (t, x)}; -∂ t ṽij (t, x) -Lṽ ij (t, x) + λṽ ij (t, x) -e λt f ij (t, x, (e -λt ṽij (t, x)) (i,j)∈Γ 1 ×Γ 2 , e -λt σ ⊤ (t, x).D x ṽij (t, x))} = 0, ṽij (T, x) = e λT h ij (x). (4.15)
But, by choosing λ large enough the functions

F ij (t, x, (u kl ) (k,l)∈Γ 1 ×Γ 2 , z) = -λu ij + e λt f ij (t, x, (e -λt u kl ) (k,l)∈Γ 1 ×Γ 2 , e -λt z), (i, j) ∈ Γ 1 × Γ 2 ,
satisfy Condition (4.9). Hence, thanks to the result stated in Step 1, we have ũij ≤ ṽij , (i, j) ∈ Γ 1 × Γ 2 . Thus,

u ij ≤ v ij , for any (i, j) ∈ Γ 1 × Γ 2 , which is the desired result.
As a consequence of this comparison result, we obtain the following one related to uniqueness of the solution of (2.1).

Corollary 2. If the system (2.1) admits a viscosity solution in the class Π g , then it is unique and continuous.

Proof. Indeed, assume that (v ij ) (i,j)∈Γ 1 ×Γ 2 is a solution of (2.1) that belongs to Π g . Then, thanks to the previous comparison result, for any (i,

j) ∈ Γ 1 × Γ 2 we have v ij, * ≤ v ij * . Thus, v ij, * = v ij * and then v ij = v ij * = v ij, * , which means that v ij is continuous. Next, if (v ij ) (i,j)∈Γ 1 ×Γ 2 is
another solution of (2.1) in the class Π g , then it is also continuous and by the Comparison Theorem 4.2 we have v ij ≤ vij and v ij ≥ vij . Hence,

v ij = vij , (i, j) ∈ Γ 1 × Γ 2 , i.e.
, uniqueness of the solution of (2.1).

5 Viscosity solution of the system (2.1)

In this section we prove that the family (v ij ) i,j constructed in Section 3 provides the unique continuous solution in viscosity sense of the system (2.1). For sake of clarity, the proof is divided into several steps.

Proposition 5.1. The family (v ij ) (i,j)∈Γ 1 ×Γ 2 is a viscosity subsolution of the system (2.1).

Proof. First recall that for each (i,

j) ∈ Γ 1 × Γ 2 , vij is usc, since vij = lim m ց v ij,m
, where v ij,m is a continuous function solution of the system (3.8). Thus, for any (i,

j) ∈ Γ 1 × Γ 2 , it holds that vij, * = vij . Next, at T we have vij (T, x) = lim m ց v ij,m (T, x) = h ij (x), x ∈ R k .
We shall now prove that, for any

(t, x) in [0, T ) × R k , any ((i, j) ∈ Γ 1 × Γ 2 and (p, q, M ) in J+ vij (t, x), min[ vij -Lij (t, x), max{ vij -Ū ij (t, x), -p -b(t, x)q -1 2 Tr(σσ T (t, x)M ) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x).q)}] ≤ 0, (5.1)
with Lij and Ū ij defined as follows:

Lij (t, x) = max k∈(Γ 1 ) -i vkj (t, x) -g ik (t, x) and Ū ij (t, x) = min l∈(Γ 2 ) -j vil (t, x) + ḡjl (t, x) .
Now, let (i, j) ∈ Γ 1 × Γ 2 be fixed. Then it is equivalent to show that, either vij -Lij (t, x) ≤ 0, (

or max{ vij -Ū ij (t, x), -p -b(t, x)q - 1 2 Tr(σσ T (t, x)M ) -f ij (t, x, v(t, x), σ ⊤ (t, x).q)} ≤ 0. 5.2)
If (5.2) is satisfied then the subsolution property (5.1) holds. Therefore, from now on, we suppose that there

exists ǫ 0 > 0 such that vij (t, x) ≥ Lij (t, x) + ǫ 0 , (5.4)
and show (5.3). Thanks to the decreasing convergence of (v ij,m ) m≥0 to vij , (i, j) ∈ Γ 1 × Γ 2 , there exists m 0 such that for any m ≥ m 0 , we have

v ij,m (t, x) ≥ L ij,m (t, x) + ǫ 0 2 . (5.5)
Next, by continuity of v ij,m and L ij,m , there exists a neighborhood O m of (t, x) such that

v ij,m (t ′ , x ′ ) ≥ L ij,m (t ′ , x ′ ) + ǫ 0 4 , (t ′ , x ′ ) ∈ O m . (5.6)
Now, by Lemma 6.1 in [START_REF] Crandall | User's guide for viscosity solutions[END_REF] there exists a subsequence ((t k , x k )) k≥0 such that

(t k , x k ) → k→∞ (t, x) and vij (t, x) = lim k→∞ v ij,k (t k , x k ). (5.7)
Moreover, there exists a sequence (p

k , q k , M k ) ∈ J+ v ij,k (t k , x k ) such that lim k→∞ (p k , q k , M k ) = (p, q, M ). (5.8)
But, the subsequence ((t k , x k )) k≥0 can be chosen in such a way that for any k ≥ 0, (t k , x k ) ∈ O k . Applying now the viscosity subsolution property of v ij,k (which satisfies (3.8)) at (t k , x k ) and taking into account of (5.6) we obtain

-p k -b(t k , x k ) ⊤ .q k - 1 2 Tr(σσ ⊤ (t k , x k )M k ) -f ij,k (t k , x k , (v pq,k (t k , x k )) (p,q)∈Γ 1 ×Γ 2 , σ(t k , x k ) ⊤ q k ) ≤ 0, (5.9)
where, once more,

f ij,k (s, x, (y pq ) (p,q)∈Γ 1 ×Γ 2 , z) = f ij (s, x, (y pq ) (p,q)∈Γ 1 ×Γ 2 , z) -k y ij -min l∈(Γ 2 ) -j (y il + ḡjl (s, x)) + .
Next, thanks to the boundedness of the sequence ((t k , x k )) k≥0 , the uniform polynomial growth of vpq,k k ≥ 0, (by Proposition 3.2 and Corollary 1), the assumptions (H0)-(H2) on b, σ and f ij , and the convergence of ((p k , q k , M k )) k to (p, q, M ), we deduce from (5.9) that

ǫ k := vij,k (t k , x k ) -min l∈(Γ 2 ) -j vil,k (t k , x k ) + ḡjl (t k , x k ) ) + → 0, k → ∞.
But, for any fixed (t, x) and k 0 , the sequence (v il,k (t, x)) k≥k0 is decreasing and then for any

k ≥ k 0 ≥ m 0 , vij,k (t k , x k ) ≤ min l =j vil,k (t k , x k ) + ḡjl (t k , x k ) + ǫ k ≤ min l =j vil,k0 (t k , x k ) + ḡjl (t k , x k ) + ǫ k .
Taking now the limit as k → +∞, in view of the continuity of vil,k0 , we get

lim k vij,k (t k , x k ) = vij (t, x) ≤ min j =l (v il,k0 (t, x) + ḡjl (t, x)).
Finally, passing to the limit as k 0 goes to +∞ to obtain vij (t, x) ≤ min

j =l (v il (t, x) + ḡjl (t, x)) = Ū ij (t, x). (5.10)
Let us now consider a subsequence of (k), which we denote by (k l ), such that for any (p, q) ∈ Γ 1 ×Γ 2 , the sequence (v pq,k (t k l , x k l )) l is convergent. This subsequence exists since the functions vij,k l are uniformly of polynomial growth (by Proposition 3.2 and Corollary 1). Then, taking the limit w.r.t. l in equation (5.9), we obtain

-p -qb(t, x) -1 2 Tr(σσ ⊤ (t, x)M ) ≤ lim l→∞ f ij,k l (t k l , x k , (v pq,k l (t k l , x k l )) (p,q)∈Γ 1 ×Γ 2 , σ(t k l , x k l ) ⊤ q k l ) ≤ lim l→∞ f ij (t k l , x k , (v pq,k l (t k l , x k l )) (p,q)∈Γ 1 ×Γ 2 , σ(t k l , x k l ) ⊤ q k l ) = f ij (t, x, (lim l→∞ vpq,k l (t k l , x k l )) (p,q)∈Γ 1 ×Γ 2 , σ(t, x) ⊤ q), (5.11)
since f ij is continuous in (t, x, y, z). Now for any (p, q) ∈ Γ 1 × Γ 2 , since vpq,n is continuous and vpq,n ≥ vpq,n+1 ,

∀n ≥ 0, it holds that vpq, * (t, x) = vpq (t, x) = lim t ′ →t,x ′ →x,n→∞ vpq,n (t ′ , x ′ ), (t, x) ∈ [0, T ] × R k .
Therefore, for any (p, q) ∈ Γ 1 × Γ 2 ((p, q) = (i, j)), we have

vpq (t, x) ≥ lim l→∞ vpq,k l (t k l , x k l ) and vij (t, x) = lim l→∞ vij,k l (t k l , x k l ).
(5.12)

As f ij is non-decreasing w.r.t. y kl , (k, l) ∈ Γ 1 × Γ 2 , (k, l) = (i, j), we deduce from (5.11) and (5.12) that

-p -qb(t, x) -1 2 Tr(σσ ⊤ (t, x)M ) ≤ f ij (t, x, (v pq (t, x)) (p,q)∈Γ 1 ×Γ 2 , σ(t, x) ⊤ q). (5.13)
Finally, under the condition (5.4), the relations (5.13), (5.10) imply that (5.3) is satisfied. Thus vij is a viscosity subsolution for

       min[ vij -L ij (t, x), max{ vij -U ij (t, x), -p -b(t, x) ⊤ q -1 2 Tr(σσ ⊤ (t, x)M ) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x).q)}] = 0, vij (T, x) = h ij (x).
Since (i, j) is arbitrary, (v ij ) i,j∈Γ 1 ×Γ 2 is a viscosity subsolution for (2.1). This finishes the proof. Proposition 5.2. Let m 0 be fixed in N. Then, the family (v ij,m0 ) (i,j)∈Γ 1 ×Γ 2 is a viscosity supersolution of the system (2.1).

Proof. First and thanks to Proposition 3.2, we know that the triples (( Ȳ ij,m0 , Zij,m0 , Kij,m0 )) (i,j)∈Γ 1 ×Γ 2 introduced in (3.5), solve the following system of reflected BSDEs: For every (i, j)

∈ Γ 1 × Γ 2 ,              Ȳ ij,m0 ∈ S 2,1 , Zij,m0 ∈ H 2,d and Kij,m0 ∈ A 2,+ ; Ȳ ij,m0 s = h ij (X t,x T ) + T s f ij,m0 (r, X t,x r , ( Ȳ kl,m0 r
) (k,l)∈Γ 1 ×Γ 2 , Zij,m0 r )dr + Kij,m0 T -Kij,m0 s - T s Zij,m0 r dB r Ȳ ij,m0 s ≥ max k∈(Γ 1 ) -i { Ȳ kj,m0 s -g ik (s, X t,x s )}, s ≤ T ; T 0 ( Ȳ ij,m0 s -max k∈(Γ 1 ) -i { Ȳ kj,m0 s -g ik (s, X t,x s )})d Kij,m0 s = 0 (5.14)
where, for any (i, j) ∈ Γ 1 × Γ 2 and (s, y, z ij ),

f ij,m0 (s, X t,x s , y, z ij ) = f i,j (s, X t,x s , (y kl ) (k,l)∈Γ 1 ×Γ 2 , z ij ) -m 0 y ij -min l∈(Γ 2 ) -j (y il + ḡjl (s, X t,x s )) + .
Furthermore, it holds true that

∀(i, j) ∈ Γ 1 × Γ 2 , ∀(t, x) ∈ [0, T ] × R k , ∀s ∈ [t, T ], Ȳ ij,m0 s = vij,m0 (s, X t,x s ).
On the other hand, we note that Ȳ

) (i,j)∈Γ 1 ×Γ 2 , Zij;m0 r )dr+ max k∈(Γ 1 ) -i { Ȳ kj;m0 σ -g ik (σ, X t,x σ )}1 1 [σ<τ ] + { Ȳ ij;m0 τ ∧ min l∈(Γ 2 ) -j { Ȳ il;m0 τ -ḡjl (τ, X t,x τ )}}1 1 [τ ≤σ<T ] +h ij (X t,x T )1 1 [τ =σ=T ] |F s ]. (5.15)
Thus, by Theorem 3.7 in Hamadène-Hassani (2005) ( [START_REF] Hamadène | BSDEs with two reflecting barriers: the general result[END_REF]), it follows that vij,m0 is the unique viscosity solution for the following PDE with two obstacles.

           min[ϑ(t, x) -max k∈(Γ 1 ) -i {v kj,m0 (t, x) -g ik (t, x)}, max{ϑ(t, x) -vij,m0 (t, x) ∨ min l∈(Γ 2 ) -j (v il,m0 (t, x) -ḡjl (t, x)), -∂ t ϑ(t, x) -b(t, x) ⊤ D x ϑ(t, x) -1 2 Tr(σσ ⊤ (t, x)D 2 xx ϑ(t, x))- f ij (t, x, (v kl,m0 (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x).D x ϑ(t, x))}] = 0 ; ϑ(T, x) = h ij (x). Next, let (t, x) ∈ [0, T ] × R k and (p, q, M ) ∈ J-[v ij,m0 ](t, x).
As vij,m0 is a solution in a viscosity sense of the previous PDE with two obstacles then it holds that vij,m0 (t, x) ≥ max k∈(Γ 1 ) -i {v kj,m0 (t, x)g ik (t, x)}

(5.16) and max{v ij,m0 (t, x) -vij,m0 (t, x) ∨ min l∈(Γ 2 ) -j (v il,m0 (t, x) -ḡjl (t, x)) ;

-pb(t, x) ⊤ q -1 2 Tr(σσ ⊤ (t, x)M )f ij (t, x, (v kl,m0 (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x).q)} ≥ 0.

(5.17 have from (5.17), max{v ij,m0 (t, x) -min l∈(Γ 2 ) -j (v il,m0 (t, x) -ḡjl (t, x)) ;

-pb(t, x) ⊤ q -1 2 Tr(σσ ⊤ (t, x)M )f ij (t, x, (v kl,m0 (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x).q)} ≥ 0. Combining this inequality with (5.16) and since v ij,m0 (T, x) = h ij (x) it follows that v ij,m0 is a viscosity supersolution of the system

          
min[ϑ(t, x) -max k∈(Γ 1 ) -i (v kj,m0 (t, x)g ik (t, x)) ; max{ϑ(t, x) -min l∈(Γ 2 ) -j (v il,m0 (t, x) -ḡjl (t, x));

-∂ t ϑ(t, x) -b(t, x) ⊤ D x ϑ(t, x) -1 2 Tr(σσ ⊤ (t, x)D 2 xx ϑ(t, x))- f ij (t, x, (v kl,m0 (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x).D x ϑ(t, x))}] = 0, ϑ(T, x) = h ij (x).
Since (i, j) is arbitrary in Γ 1 × Γ 2 , the system of continuous functions (v ij,m0 ) (i,j)∈Γ 1 ×Γ 2 is a supersolution of (2.1).

Consider now the set U m0 defined as follows.

U = { u := (u ij ) (i,j)∈Γ 1 ×Γ 2 s.t. u is a subsolution of (2.1) and ∀ (i, j) ∈ Γ 1 × Γ 2 , vi,j ≤ u i,j ≤ vij,m0 }. Obviously, U m0 is not empty since it contains (v ij ) (i,j)∈Γ 1 ×Γ 2 . Next for (t, x) ∈ [0, T ] × R k and (i, j) ∈ Γ 1 × Γ 2 , let us set: m0 v ij (t, x) = sup{u ij (t, x), (u kl ) (k,l)∈Γ 1 ×Γ 2 ∈ U m0 }.
We now state the main result of this section.

Theorem 5.3. The family ( m0 v ij ) (i,j)∈Γ 1 ×Γ 2 does not depend on m 0 and is the unique continuous viscosity solution in the class Π g of the system (2.1).

Proof. We first note that for any (i, j) ∈ Γ 1 ×Γ 2 , vij ≤ m0 v i,j ≤ vij,m0 . Since vij and vij,m0 are of polynomial growth, then ( m0 v ij ) (i,j)∈Γ 1 ×Γ 2 belongs to Π g .

The remaining of the proof is divided into two steps and mainly consists in adapting the Perron's method (see Crandall-Ishii-Lions, [START_REF] Crandall | User's guide for viscosity solutions[END_REF] Theorem 4.1, pp 23) to construct a viscosity solution to our general system of PDEs. To ease notation, we denote ( m0 v ij ) (i,j)∈Γ 1 ×Γ 2 by (v ij ) (i,j)∈Γ 1 ×Γ 2 as no confusion is possible.

Step 1. We first show that (v ij ) (i,j)∈Γ 1 ×Γ 2 is a subsolution of (2.1). Indeed, it is clear that for any (t, x) ∈

[0, T ] × R k , vij (t, x) ≤ v ij (t, x) ≤ vij,m0 (t, x
). This implies that vij ≤ v ij, * ≤ vij,m0 since, as pointed out previously, vij is usc and v ij,m0 is continuous. Therefore, for any

x ∈ R k , we have v ij, * (T, x) = h ij (x), since vij (T, x) = vij,m0 (T, x) = h ij (x).
Next, fix (i, j) ∈ Γ 1 × Γ 2 and let (ṽ ij ) (i,j)∈Γ 1 ×Γ 2 be an arbitrary element of U m0 . Then, for any (t, x) ∈ [0, T ) × R k and any (p, q, M ) ∈ J+ ṽi,j, * (t, x) we have

min[ ṽij, * -L ij ((ṽ kl, * ) k,l ) (t, x), max{ ṽij, * -U ij ((ṽ kl, * ) k,l (t, x) , -p -b(t, x) ⊤ q -1 2 Tr(σσ ⊤ (t, x)M ) -f ij (t,
x, (ṽ kl, * (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)q)}] ≤ 0. By definition we have ṽkl ≤ v kl and then ṽkl, * ≤ v kl, * for any (k, l) ∈ Γ 1 × Γ 2 . Since the operators w = (w kl ) k,l → ṽij * -L i,j ((w kl ) k,l ), w = (w kl ) k,l → ṽij * -U ij ((w kl ) k,l are decreasing, in view of the monotonicity property (H2) of the generator f ij , we have

   min[ ṽij, * -L ij ((v kl, * ) k,l ) (t, x) ; max{ ṽij, * -U ij ((v kl, * ) k,l ) (t, x), -p -b(t, x) ⊤ q -1 2 Tr(σσ ⊤ (t, x)M ) -f ij (t, x, [(v kl, * (t, x)) (k,l)∈Γ 1 ×Γ 2 (k,l) =(i,j)
, ṽij, * (t, x)], σ ⊤ (t, x)q)}] ≤ 0 Now, in view of Corollary 2, ( m0 v ij ) (i,j)∈Γ 1 ×Γ 2 is the unique continuous viscosity solution in the class Π g of (2.1).

Thus, using once more uniqueness, we deduce that ( m0 v ij ) (i,j)∈Γ 1 ×Γ 2 does not depend on m 0 .

As above, let us denote by (v i,j ) (i,j)∈Γ 1 ×Γ 2 the family ( m0 v i,j ) (i,j)∈Γ 1 ×Γ 2 . Here is the second main result of the paper.

Theorem 5.4. For any (i, j) ∈ Γ 1 × Γ 2 , vij = v ij , i.e., (v ij ) (i,j)∈Γ 1 ×Γ 2 is continuous and is the unique viscosity solution in the class Π g of the system (2.1).

Proof. For any (i, j) ∈ Γ 1 × Γ 2 and m 0 ∈ N we have vij ≤ v ij ≤ vij,m0 .

Taking the limit as m 0 → ∞ we obtain vij = v ij , for all (i, j) ∈ Γ 1 × Γ 2 . Finally, Theorem 5.3 completes the proof.

As a by-product of this result, we have the following theorem for the family (v ij ) (i,j)∈Γ 1 ×Γ 2 .

Theorem 5.5. The functions (v ij ) (i,j)∈Γ 1 ×Γ 2 are continuous, of polynomial growth and unique viscosity solution in the class Π g of the following system of variational inequalities. For every

(i, j) ∈ Γ 1 × Γ 2 ,        max v ij (t, x) -min l∈(Γ 2 ) -j (v il + ḡjl )(t, x), min v ij (t, x) -max k∈(Γ 1 ) -i (v kj -g ik )(t, x), -∂ t v ij (t, x) -Lv ij (t, x) -f ij (t, x, (v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , σ ⊤ (t, x)D x v ij (t, x)) = 0 v ij (T, x) = h ij (x).
(5.23)

Proof. It is enough to consider (-v ij ) (i,j)∈Γ 1 ×Γ 2 which, in view of Theorem 5.4, is continuous, of polynomial growth and the unique viscosity solution of the following system. For all (i, j)

∈ Γ 1 × Γ 2 ,        min v ij (t, x) -max l∈(Γ 2 ) -j (v il -ḡjl )(t, x), max v ij (t, x) -min k∈(Γ 1 ) -i (v kj + g ik )(t, x), -∂ t v ij (t, x) -Lv ij (t, x) + f ij (t, x, (-v kl (t, x)) (k,l)∈Γ 1 ×Γ 2 , -σ ⊤ (t, x)D x v ij (t, x)) = 0 v ij (T, x) = -h ij (x).
(5.24)

Using now the result by Barles ([2], pp. 18), we obtain that (v ij ) (i,j)∈Γ 1 ×Γ 2 are continuous, of polynomial growth and unique viscosity solution in the class Π g of system (5.23), which is the desired result.

Remark 2. We do not know whether or not we have

v ij = vij , (i, j) ∈ Γ 1 × Γ 2 .
(i) The function (t, x) → f i (t, x, y 1 , . . . , y m , z) is continuous uniformly w.r.t. ( -→ y , z) := (y 1 , . . . , y m , z).

(ii) The function f i is uniformly Lipschitz continuous with respect to ( -→ y , z) := (y 1 , . . . , y m , z), i.e., for some

C ≥ 0, |f i (t, x, y 1 , . . . , y m , z) -f i (t, x, ȳ1 , . . . , ȳm , z)| ≤ C(|y 1 -ȳ1 | + • • • + |y m -ȳm | + |z -z|).
(iii) The mapping (t, x) → f i (t, x, 0, . . . , 0) is B([0, T ] × R k )-measurable and of polynomial growth i.e. it belongs to Π g ;

(iv) Monotonicity. ∀i ∈ J , for any k ∈ J -i , the mapping y k ∈ R → f i (t, x, y 1 , . . . , y k-1 , y k , y k+1 , . . . , y m ) is non-decreasing whenever the other components (t, x, y 1 , . . . , y k-1 , y k+1 , . . . , y m ) are fixed.

[Ag]. (i) The function g ij is jointly continuous in (t, x), non-negative, i.e., g ij (t, x) ≥ 0, ∀(t, x) ∈ [0, T ] × R k and belongs to Π g .

(ii) The no free loop property. for any (t, x) ∈ [0, T ] × R k and for any sequence of indexes i 1 , . . . , i k such that

i 1 = i k and card{i 1 , . . . , i k } = k -1 we have g i1i2 (t, x) + g i2i3 (t, x) + • • • + g i k-1 i k (t, x) + g i k i1 (t, x) > 0, ∀(t, x) ∈ [0, T ] × R k .
As a convention we assume hereafter that g ii (t, x) = 0 for any (t, x)

∈ [0, T ] × R k and (i, j) ∈ Γ 1 × Γ 2 .
[Ah1]. h i is continuous, belongs to Π g and satisfies:

∀x ∈ R, h i (x) ≥ max j∈J -i (h j (x) -g ij ((T, x)).
[Ah2]. The function h i is continuous, belongs to Π g and satisfies

∀x ∈ R, h i (x) ≥ min j∈J -i (h j (x) + g ij ((T, x)).
Then, we have Theorem 6.2. Assume that [Ah], [Ag] and [Ah1] are fulfilled. Then, there exist m triples of processes ((Y i;t,x , Z i;t,x , K i;t,x )) i∈J that satisfy: ∀ i ∈ J , Moreover there exist m deterministic functions (v i (t, x)) i∈J continuous and belonging to Π g such that:

           Y i , K i ∈ S 2 , Z i ∈ H
∀s ∈ [t, T ], Y i;t,x s = v i (s, X t,x s ).

Finally (v i (t, x)) i∈J is the unique solution, in the sub-class of Π g of continuous functions, of the following system of variational inequalities with inter-connected obstacles:

∀ i ∈ J         
min v i (t, x) -max j∈J -i (-g ij (t, x) + v j (t, x)), -∂ t v i (t, x) -Lv i (t, x)f i (t, x, v 1 (t, x), . . . , v m (t, x), σ ⊤ (t, x)D x v i (t, x)) = 0 ; v i (T, x) = h i (x). (6.2) Remark 3. In equations (6.1) and (6.2), if instead we have required an upper barrier reflection, then one would have obtained a similar result which can be stated as follows.

Assume that [Af ], [Ag] and [Ah2] are fulfilled. Then there exist m triples of processes (( Ỹ i;t,x , Zi;t,x , Ki;t,x )) i∈J that satisfy, for all i ∈ J , Moreover, there exist m deterministic functions (ṽ i (t, x)) i∈J continuous and belong to Π g such that:

           Ỹ i , Ki ∈ S 2 , Zi ∈ H
Ỹ i;t,x s = ṽi (s, X t,x s ), s ∈ [t, T ].
Finally, (ṽ i (t, x)) i∈J is the unique solution, in the subclass of Π g of continuous functions, of the following system of variational inequalities with interconnected obstacles. For all i ∈ J

        
max ṽi (t, x) -min j∈J -i (g ij (t, x) + ṽj (t, x)), -∂ t ṽi (t, x) -Lṽ i (t, x)f i (t, x, ṽ1 (t, x), . . . , ṽm (t, x), σ ⊤ (t, x)D x ṽi (t, x)) = 0 ; ṽi (T, x) = h i (x).

(6.4)

The proof of this result is obtained straightforward from Theorem 6.2, in considering the equations satisfied by ((-Ỹ i , -Zi , Ki )) i∈J .

A3. Linearization procedure of Lipschitz functions. Let f be a function from R 2 to R which with (x 1 , x 2 ) associates f (x 1 , x 2 ) which is Lipschitz in its arguments. Then, we can write where, C(f ) is the Lipschitz constant of f . Moreover, if f is non-decreasing with respect to x 1 (resp. x 1 ) when

x 2 (resp. x 1 ) is fixed, then a i ≥ 0, i = 1, 2.

Linearizing f consists of writing f as f (x 1 , x 2 )f (y 1 , y 2 ) = a 1 (x 1 , x 2 , y 1 ).(x 1y 1 ) + a 2 (x 2 , y 1 , y 2 ).(x 2y 2 ).

A4. Representation of a penalization scheme of two barriers reflected BSDE.

For n ≥ 0 let (Y n , Z n , K +,n ) be the solution of the following one barrier reflected BSDE. where, the processes L and U belong to S 2,1 , (g(s)) s≤T ∈ H 2,1 , ξ is square integrable and F T -measurable.

       (Y n , Z n , K +,n ) ∈ S
Moreover, we require that L ≤ U and L T ≤ ξ. Under these conditions, the solution (Y n , Z n , K +,n ) exists and is unique (see e.g. [START_REF] El Karoui | Reflected solutions of backward SDEs and related obstacle problems for PDEs[END_REF]). Next, for n ≥ 0 and t ≤ T , set K where τ and σ are F -stopping times.

. 4 )

 4 Proof. The second claim is just the representation of solutions of standard BSDEs by deterministic functions in the Markovian framework (see e.g. El Karoui et al. (1997) ([START_REF] El Karoui | Backward SDEs in Finance[END_REF]) for more details). As for the first one, it is based on the result by[START_REF] Hu | On comparison theorem for multi-dimensional BSDEs[END_REF] ([22]) related to the comparison of solutions of multi-dimensional

  ) But, for any constants a, b ∈ R we have a -(a ∨ b) ≤ ab and thus a -(a ∨ b) ≥ 0 ⇒ ab ≥ 0. Therefore, we

f (x 1 , x 2 )(x 2 -y 2 )= a 1

 12221 f (y 1 , y 2 ) = f (x 1 , x 2 )f (y 1 , x 2 ) + f (y 1 , x 2 )f (y 1 , y 2 ) = 1 1 x1-y1 =0 f (x1,x2)-f (y1,x2) x1-y1 (x 1y 1 ) + 1 1 x2-y2 =0 f (y1,x2)-f (y1,y2) x2-y2 (x 1 , x 2 , y 1 ).(x 1y 1 ) + a 2 (x 2 , y 1 , y 2 ).(x 2y 2 ) (6.5)where, a 1 and a 2 are measurable functions and bounded, i.e.,|a 1 (x 1 , x 2 , y 1 )| ∨ |a 2 (x 2 , y 1 , y 2 )| ≤ C(f ), i = 1, 2,

  ∈ Γ 1 × Γ 2 , v kl * = v kl * , then (v kl ) (k,l)∈Γ 1 ×Γ 2 isa continuous viscosity solution of (2.1).

	called a viscosity solution of (2.1) if the
	associated lower (resp. upper) semicontinuous envelope (v kl * ) (k,l)∈Γ 1 ×Γ 2 (resp. (v kl * ) (k,l)∈Γ 1 ×Γ 2 ) defined in (2.5)
	is a viscosity supersolution (resp. subsolution) of (2.1).
	If, in addition, for any (k, l) Remark 1. Under Assumptions (H0)-(H4), the above definition of a viscosity solution for (2.1) can be relaxed
	replacing the limiting subjet J-(v ij * )(t, x) of the supersolution v

* (resp. the limiting superjet J+

  ij,m0 is the value function of a zero-sum Dynkin game (see Appendix A4),

	i.e., it satisfies, for all s ≤ T
	Ȳ ij;m0 s	= ess sup σ≥s ess inf τ ≥s E[	σ∧τ s	f ij (r, X t,x r , ( Ȳ ij;m0 r

  2,d , K i non-decreasing and K i

		0 = 0;
	Y i s = h i (X t,x T ) +	T s f i (r, X t,x r , Y 1 r , . . . , Y m r , Z i r )dr + K i T -K i s -	(6.1)
	T 0 (Y i s -max j∈J -i {Y j s -g ij (s, X t,x s )})dK i s = 0.

T s Z i r dB r , ∀ s ≤ T Y i s ≥ max j∈J -i {Y j sg ij (s, X t,x s )}, ∀s ≤ T

  2,d , Ki non-decreasing and Ki

	0 = 0;
	Ỹ i s = h i (X t,x T ) + Ỹ i s ≤ min j∈J -i { Ỹ j T s f i (r, X t,x r , Ỹ 1 r , . . . , Ỹ m r , Zi s + g ij (s, X t,x s )}, t ≤ s ≤ T,	T s	Zi r dB r ,	t ≤ s ≤ T,	(6.3)
	T 0 ( Ỹ i s -min j∈J -i { Ỹ j s + g ij (s, X t,x s )})d Ki s = 0.

r )dr -( Ki T -Ki s ) -

  2,1 × H 2,d × A +,2

	Y n t = ξ +	T t g(s)ds -n	T t (Y n s -U s ) + ds + K +,n T	-K n,+ t	-	T t Z n s dB s , t ≤ T ;	(6.6)
	Y n t ≥ L s	= 0

t , 0 ≤ t ≤ T and T 0 (Y n s -L s )dK +,n

  U s ) + ds. Then, K -,n ∈ A +,2 and T 0 (Y n s -Y n s ∨ U s )dK -,n s = 0. Therefore, the equation (6.6) can be expressed as a BSDE with two reflecting barriers in the following manner. For all t ≤ T , Thus, a result by Cvitanic and Karatzas[START_REF] Cvitanic | Backwards stochastic differential equations and Dynkin games[END_REF] or Hamadène and Lepeltier[START_REF] Hamadène | Zero-sum stochastic differential games and backward equations[END_REF] allows to represent Y n as a value function of a Dynkin game, i.e., it holds true that for any t ≤ T , = ess sup σ≥t ess inf τ ≥t E[σ∧τ t g(s)ds + L σ 1 1 [σ<τ ] + (Y n τ ∨ U τ )1 1 [τ ≤σ<T ] + ξ1 1 [τ =σ=T ] |F t ] = ess inf τ ≥t ess sup σ≥t E[ σ∧τ t g(s)ds + L σ 1 1 [σ<τ ] + (Y n τ ∨ U τ )1 1 [τ ≤σ<T ] + ξ1 1 [τ =σ=T ] |F t ].

	n,-t s - = n t 0 (Y n    Y n t = ξ + T t g(s)ds + (K +,n T -K +,n t ) -(K -,n T -K -,n t	) -	T t Z n s dB s ;
		L t ≤ Y n t ≤ Y n t ∨ U t ,				(6.7)
	  	T 0 (Y n s -L s )dK +,n s	=	T 0 (Y n s -Y n s ∨ U s )dK -,n s	= 0.
	Y n t

. Financial support from GEANPYL and Svensk Export Kredit (SEK) is gratefully acknowledged.

The definition of φ n together with the growth condition of u ij and w ij implies that (x ny n ) n≥1 converges to 0.

Next, for any subsequence ((t n l , x n l , y n l )) l which converges to ( t, x, x) we deduce from (4.10) that u i0j0 (t * , x * )w i0j0 (t * , x * ) ≤ u i0,j0 ( t, x)w i0j0 ( t, x), since u i0j0 is usc and w i0j0 is lsc. As the maximum of u i0j0w i0j0 on [0, T ] × B(0, R) is reached in (t * , x * ), then this last inequality is actually an equality. Using the definition of φ n and (4.10), we deduce that the sequence ((t n , x n , y n )) n converges to (t * , x * , x * ) which also implies n|x ny n | 2γ+2 → 0 and (u i0j0 (t n , x n ), w i0j0 (t n , y n )) → (u i0j0 (t * , x * ), w i0j0 (t * , x * )), as n → ∞. This latter convergence holds, since we first obtain from (4.10) and in taking into account that u i0j0 and w i0j0 are lsc and usc respectively,

It follows that the sequence (u i0j0 (t n , x n )) n converges to u i0j0 (t * , x * ) and then (w i0j0 (t n , y n )) n converges also to

Next, recalling that u i0j0 (resp. w i0j0 ) is usc (resp. lsc) and satisfies (4.1), then, for n large enough and at least for a subsequence which we still index by n, we obtain

and

.12)

Applying now Crandall-Ishii-Lions's Lemma (see e.g. [START_REF] Crandall | User's guide for viscosity solutions[END_REF] or [START_REF] Fleming | Controlled Markov processes and viscosity solutions[END_REF], pp.216) with Φ i0j0 n and φ n at the point (t n , x n , y n ) (for n large enough in such a way that this latter triple will belong to [0, T ]× B(0, R) 2 ), there exist (p n u , q n u , M n u ) ∈ J2,+ (u i0j0 )(t n , x n ) and (p n w , q n w , M n w ) ∈ J2,-(w i0j0 )(t n , y n ) such that

where,

) is a subsolution (resp. supersolution) of (2.1) and using once more (4.11) and (4.12) we get

and

Taking the difference between these two inequalities yields

where

) is a subsolution of the following equation.

, w], σ ⊤ (t, x)q)}] = 0.

Next, relying on the lower semi continuity of the function

, w], σ ⊤ (t, x)q)]} and using Lemma 4.2, in Crandall et al. (1992) ( [START_REF] Crandall | User's guide for viscosity solutions[END_REF], pp.23), related to suprema of subsolutions, combined with the above result, it holds that v ij is a subsolution of the following equation:

(5.18)

Step 2. In this step we use the so called Perron's method to show that (v ij ) (i,j)∈Γ 1 ×Γ 2 is a viscosity supersolution of (2.1).

Indeed, we first note for any (i,

since vij,m0 is continuous and v ij is lsc. Therefore, for any

Next, assume that (v i,j ) (i,j)∈Γ 1 ×Γ 2 is not a supersolution for (2.1).

Then, taking into account of (5. [START_REF] Hamadène | Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Multi-Modes Switching Problem[END_REF]) and Remark 1, there exists at least one pair (i, j) such that v i,j does not satisfy the viscosity supersolution property: this means that for some point (t 0 , x 0 ) ∈ [0, T ) × R k there exists a triple (p, q, M ) in

(5.20)

We now follow the same idea as in Crandall et al. (1992) ( [START_REF] Crandall | User's guide for viscosity solutions[END_REF], pp.24). For any positive δ, γ and r, set u δ,γ and B r as follows:

Using (5.20) and continuity of all the data, choosing δ, γ small enough we obtain

Next, let us define the function Υ as follows.

where,

First we note that from (5.21), Υ(t 0 , x 0 ) < 0, since u δ,γ (t 0 , x 0 ) = v ij * (t 0 , x 0 ) + δ. On the other hand by the continuity of u δ,γ , Assumptions (H1)-(H2) on f ij and finally the lower semi-continuity of v kl * , (k, l) ∈ Γ 1 × Γ 2 , we can easily check that the function Υ is usc. Thus, for any ε > 0, there exists η > 0 such that for any (t, x) ∈ B η we have Υ(t 0 , x 0 ) ≥ Υ(t, x)ǫ. Since Υ(t 0 , x 0 ) < 0, then choosing ε small enough we deduce that Υ(t, x) ≤ 0 for any (t, x) ∈ B η with an appropriate η. It follows that the function u δ,γ is a viscosity subsolution on B η of the

Since, for any (k, l) ∈ Γ 1 × Γ 2 , v kl * ≤ v kl, * and since f ij satisfies the monotonicity condition (H2), u δ,γ is also a viscosity subsolution on B η of the system min ̺(t, x) -max k =i v kj, *g ik (t, x); max{̺(t, x) -min l =j v il, * + ḡjl (t, x),

In view of the definition of u δ,γ and taking δ = r 2 8 γ, it is easily seen that

as soon as r 2 < |xx 0 | ≤ r and r small enough. We now take r ≤ η and consider the function ũij :

Then taking into account of (5.22) and using Lemma 4.2 in Crandall et al. (1992) ( [START_REF] Crandall | User's guide for viscosity solutions[END_REF]), it follows that ũij is a subsolution of (5.18). Next, as ũij ≥ v ij and using once more the monotonicity assumption (H2) on f kl , (k, l) ∈ Γ 1 × Γ 2 , we get that [(v kl ) (k,l) =(i,j) , ũij ] is also a subsolution of (2.1) which belongs to Π g . Thus, thanks to the Comparison Theorem 4.2, [(v kl ) (k,l) =(i,j) , ũij ] belongs also to U m0 . Finally, in view of the definition of v ij * , there exists a sequence (t n , x n , v ij (t n , x n )) n≥1 that converges to (t 0 , x 0 , v ij * (t 0 , x 0 )). This implies that lim

This means that there are points (t n , x n ) such that ũij (t n , x n ) > v ij (t n , x n ). But this contradicts the definition of v ij , since [(v kl ) (k,l) =(i,j) , ũij ] belongs also to U m0 . Therefore, (v ij ) (i,j)∈Γ 1 ×Γ 2 is also a supersolution for (2.1). (ii) the functions f (t, ω, y, z) and f (t, ω, y, z) defined on [0, T ] × Ω × R k+k×d are R k -valued, Lipschitz in (y, z)

Appendix

uniformly in (t, ω) and the process (f (t, ω, 0, 0)) t≤T (resp. ( f (t, ω, 0, 0)) t≤T ) belongs to H 2,k ;

(iii) for any i = 1, . . . , k, the ith component f i (resp. fi ) of f (resp. f ) depends only on the ith row of the matrix z.

If there exists a constant C ≥ 0, such that for any y, ȳ ∈ R k , z, z ∈ R k×d -4 where y + i = max(y i , 0) and y - i = max(-y i , 0). Then for any i = 1, . . . , k, P-a.s., Y i ≤ Ȳ i .

A2. Systems of reflected BSDEs with one inter-connected barrier and their related systems of variational inequalities (see e.g. [START_REF] Hamadène | Switching problem and related system of reflected backward stochastic differential equations Stoch[END_REF] or [START_REF] Hamadène | Viscosity Solutions of Systems of PDEs with Interconnected Obstacles and Multi-Modes Switching Problem[END_REF]).

Let J := {1, . . . , m} and let us consider the following functions: for i, j ∈ J , f i : (t, x, y 1 , . . . , y m , z) ∈ [0, T ] × R k+m+d → f i (t, x, y 1 , . . . , y m , z) ∈ R ;

We now make the following assumptions.

[Af ]. For i ∈ J , f i satisfies


Online Feedforward/Feedback Structure Adaptation for Heterogeneous CACC Strings

Carlos Flores, Vicente Milanés and Fawzi Nashashibi.

Abstract-Cooperative Adaptive Cruise Control has been extensively studied in recent years, however most of efforts have been oriented to homogeneous strings control. When a heterogeneous string is formed, the differences between vehicles dynamics introduce disturbances in the closed loop system affecting the string stability. This paper presents an online CACC feedforward adaptation with a fractional-order feedback controller for stable heterogeneous strings of vehicles. A particle filter algorithm is employed to perform an online parameter identification over the preceding vehicle longitudinal model, allowing further adaptation of the control structure. Simulations demonstrate the advantages over conventional homogeneous structures as well as system's capability to both enhance stability and guarantee string stability regardless the vehicles distribution.

I. INTRODUCTION

In the last years, Cooperative Adaptive Cruise Control (CACC) has become a promising solution for improving traffic flow [START_REF] Shladover | Cooperative adaptive cruise control: Definitions and operating concepts[END_REF] with the introduction of longitudinal automation, ranging sensors and vehicle-to-vehicle (V2V) communication links. Several projects have been deployed to investigate the benefits of CACC and platooning when applied over conventional vehicles [START_REF] Rajamani | An experimental comparative study of autonomous and co-operative vehicle-follower control systems[END_REF], [START_REF] Ploeg | connect & drive " c&d c-acc for reducing congestion dynamics[END_REF], heavy duty trucks 1 and mixed traffic 2 . These works have proved the advantages of such techniques on highway driving without requiring major modification of the infrastructure, showing significant improvements in terms of traffic capacity [START_REF] Van Arem | The impact of cooperative adaptive cruise control on traffic-flow characteristics[END_REF].

From a control point of view, CACC approaches require each vehicle to track efficiently, smoothly and with robustness its preceding one at a reference distance defined by a spacing policy. Among the different spacing policies, constant time gap (CTG) [START_REF] Swaroop | A review of constant time headway policy for automatic vehicle following[END_REF] strategy is the most employed since it provides the most intuitive way to set the distance between vehicles as it varies proportionally to the ego-speed. The system gap-regulation performance and its stability strongly depend on the desired time gap-i.e. keeping a car-following formation with shorter time gaps requires more performing reference tracking capabilities-.

String stability plays a key role when it comes to design CACC systems. This concept was firstly introduced by [START_REF] Swaroop | String stability of interconnected systems[END_REF] and widely studied in the work of [START_REF] Ploeg | Controller synthesis for string stability of vehicle platoons[END_REF], [START_REF] Vugts | String-stable cacc design and experimental validation[END_REF] among others. It Carlos Flores and Fawzi Nashashibi are with the Robotics and Intelligent Transportation Systems (RITS) Team, INRIA Paris, 2 Rue Simone Iff, 75012 France {carlos.flores-pino, fawzi.nashashibi}@inria.fr; Vicente Milanés is with the Research Department, Renault SAS, 1 Avenue de Golf, 78280 Guyancourt, France {vicente.milanes@renault.com}.

1 https://www.eutruckplatooning.com/default.aspx 2 http://www.gcdc.net/en/ refers to the string capability to attenuate disturbance propagations in upstream direction. This can be mathematically interpreted as:

Γ(s) = Z i (s) Z i-1 (s) ; i ≥ 2 (1)
where the term Z i (s) might refer to different vehicle states. A sufficient condition to guarantee the system string stability is to ensure that the infinite norm of the string stability function is less or equal than unity for all vehicles-i.e. Γ(s) ∞ ≤ 1; ∀ i ≥ 2. In the case where all the string members have identical dynamics (homogeneous strings), the string stability can be studied defining Z i (s) as the vehicle position, speed, acceleration, spacing error or control effort. However, when the CACC is integrated by vehicles with different dynamics (heterogeneous strings), spacing errors and control effort signals evolve differently for each vehicle, because they are closely dependent to the ego and preceding vehicles' dynamics. For this reason, this work employs the string stability criterion defined in GCDC 2011 [START_REF] Ploeg | Introduction to the special issue on the 2011 grand cooperative driving challenge[END_REF].

Regarding to heterogeneous strings, firsts analysis have been presented on [START_REF] Naus | String-stable cacc design and experimental validation: A frequency-domain approach[END_REF]. Some first approaches include a graph-based control for multi-lane convoy in [START_REF] Navarro | Distributed graph-based control of convoys of heterogeneous vehicles using curvilinear road coordinates[END_REF]. In [START_REF] Xiao | Practical string stability of platoon of adaptive cruise control vehicles[END_REF], the effect of different time lags and system delays in an ACC string is studied. A CACC control strategy is proposed in [START_REF] Shaw | String stability analysis for heterogeneous vehicle strings[END_REF], [START_REF]Controller design for string stable heterogeneous vehicle strings[END_REF] that relies on leader-predecessor topology, showing improvement in the performance and ensuring a bounded spacing error regardless the string configuration-i.e. fast-slow, slow-fast, among others-. Although good results have been obtained, communication with the leader vehicle may not always be available due to congestion or if the string length exceeds the secure communication range. A feedforward/feedback (FF/FB) control strategy (which has been widely employed in the CACC literature [START_REF] Milanés | Cooperative adaptive cruise control in real traffic situations[END_REF], [START_REF] Ploeg | Design and experimental evaluation of cooperative adaptive cruise control[END_REF]) is proposed to control a heterogeneous string in [START_REF] Wang | String stable heterogeneous vehicle platoon using cooperative adaptive cruise control[END_REF], where the effect of having vehicles with different behavior is analysed. The ideal FF/FB controllers in function of the formation dynamics, actuators lag, and communications delays is also revised. The influence of the difference between string vehicles' dynamics is analized in [START_REF] Wang | String stable heterogeneous vehicle platoon using cooperative adaptive cruise control[END_REF], considering an offline set FF controller that filters the preceding vehicle reference signal. However, CACC formations should handle real traffic split/merge maneuvers, which means that the system may face a change of the preceding vehicle, requiring to adapt its structure in real time.

Having this in mind, the objectives of this paper are:

• Mitigating the disturbances introduced by the difference between vehicles' dynamics in the string

• Developing a string stable control system for heterogeneous strings • Taking into account communication links only available between the ego and preceding vehicle • Adapting the control response online in function of the preceding vehicle dynamics with robustness against modelling uncertainties This work proposes a FF/FB control strategy which adapts the FF filter in function of the ego and preceding vehicles' dynamics, guaranteeing string stability. The online adaptation is done applying an identification algorithm based on a Particle Filter (PF) over the received V2V data to determine the preceding vehicle longitudinal behavior. A fractional-order feedback controller is designed to be robust against system disturbances and to guarantee the desired tracking stability within a given loop bandwidth without requiring high control efforts. Simulation results demonstrate the advantages of applying the proposed online-adapted structure over static structures.

II. EMPLOYED FF/FB STRUCTURE

The following configurations of CACC controllers can be found in the state-of-the-art: leader-predecessor following [START_REF] Shaw | String stability analysis for heterogeneous vehicle strings[END_REF], [START_REF]Controller design for string stable heterogeneous vehicle strings[END_REF], two-vehicle look-ahead [START_REF] Ploeg | Controller synthesis for string stability of vehicle platoons[END_REF], bidirectional control [START_REF] Zegers | Consensus-based bi-directional cacc for vehicular platooning[END_REF] and one-vehicle look-ahead topology [START_REF] Ploeg | Design and experimental evaluation of cooperative adaptive cruise control[END_REF]. This work considers only preceding vehicle information.

Fig. 1 shows the proposed structure, where Gpf i (s), C i (s), H(s), F i (s) represent the vehicle model (X i (s)/U i (s)), controller, spacing policy and FF filter respectively. The communication temporal delay is represented by D(s) = e -θs ; θ > 0. The closed-loop transfer function of the structure in Fig. 1 is given by the expression:

X i (s) X i-1 (s) = D(s)F i (s) Gpi(s) Gpi-1(s) + Gpf i (s)C i (s) 1 + Gpf i (s)C i (s)H i (s) ; i ≥ 2
(2) which ∞-norm has to be maintained below unity to ensure the string stability. The expression Gp i (s) refers to the low level longitudinal speed behavior (V real (s)/U i (s)) and can be also defined as Gp i (s) = s • Gpf i (s). Let define P i (s) as the term Gpi(s) Gpi-1(s) . For homogeneous strings this term has no effect since P i (s) = 1, but when Gp i = Gp i-1 (s) the FF loop is affected, compromising the system stability. For this reason, an inverse model control is proposed to be employed so the effect of P i (s) in the closed loop is reduced. This type of control structures has been extensively studied and employed in the literature [START_REF] Xie | Robust two degree-of-freedom control for mimo system with both model and signal uncertainties[END_REF]. The FF controller is set to be of the form F i (s) = ( Pi (s)H i (s)) -1 where Pi (s) is an estimation of P i (s). The term H(s) is employed to guarantee string stability for every time gap h > 0, in case the FF link presents no communication delays [START_REF] Naus | String-stable cacc design and experimental validation: A frequency-domain approach[END_REF], limiting its bandwidth to the interest frequency region as well.

Ideally, perfect tracking of the given reference is achieved when F i (s) ≡ (P i (s)H i (s)) -1 . In practice, it is difficult to determine exactly the term P i (s) and attain a perfect model Ẽff,i (jω) 2 ≤ Ẽfb,i (jω) 2 [START_REF] Devasia | Should model-based inverse inputs be used as feedforward under plant uncertainty?[END_REF], [START_REF] Faanes | Feedforward control under the presence of uncertainty[END_REF]. Assuming no communication delays, the FF/FB Ẽff,i (jω) and FB errors Ẽff,i (jω) functions are given by the expressions:

Ẽ(fb,i) (jω) = X i-1 (jω) (1 + Gpf i (jω)C i (jω)H(jω)) ; Ẽ(ff,i) (jω) = ( Pi (jω) -P i (jω)) • X i-1 (jω) (1 + Gpf i (jω)C i (jω)H(jω)) Pi (jω) ; (3)
where it can be observed that the condition is fulfilled if:

| Pi (jω) -P i (jω)| ≤ | Pi (jω)|; (4)
To satisfy this condition, the objective is to reduce as much as possible the model uncertainty |∆ Pi (jω)| = | Pi (jω) -P i (jω)| in the interest frequency region-i.e. the closed loop bandwidth ω ∈ (0, 1/h]-to yield the best reference tracking as possible, being h the employed time gap. For the estimation of Pi (s) through the FF controller F i (s), the ego-vehicle model is assumed to be known; whereas the preceding car model is not due to changes when preceding vehicle leaves or a cut-in occurs. To approximate Gi-1 (s) → G i-1 (s), the V2V data is employed to estimate G i-1 (s) by using a parameter identification method.

III. ONLINE PRECEDING VEHICLE IDENTIFICATION

As explained above, an identification algorithm is applied over the V2V data to adapt online the FF filter in function of the ego and preceding vehicle dynamics. Generally, vehicles' longitudinal command is based on a hierarchical control strategy composed by a high-level structure that generates a reference speed and a low-level cruise controller in charge of commanding the actuators to track the desired speed. The longitudinal low-level control can be described by a second order model of the form:

Gp(s) = V real (s) U (s) = ω 2 n • e -θs s 2 + 2ξω n s + ω 2 n (5)
where θ, ω n and ξ represent the transport delay, plant natural frequency and damping factor. Given that the identification algorithm requires to compute the distance between the model estimations and the observations at each sample time, a discrete version of the model in Eq. 5 is used for the estimation. For this purpose, the Tustin discretization method is applied:

s = 2 T s 1 -z -1 1 + z -1 ; z ≥ 0 (6)
where T s is the sample time of the identification algorithm and z the discrete operator. Model parameters are determined in real time employing a Particle Filter (PF) algorithm. It provides a fast and robust estimation of the parameters, even if the measurements are corrupted with sensor noise or if the system varies through the time.

A. Online Parameters Modelling Algorithm

An applied version of the Sequential-Importance-Resampling (SIR) PF proposed in [START_REF] Gordon | Novel approach to nonlinear/non-gaussian bayesian state estimation[END_REF] is employed in this work. Since the experimental platform models present a negligible transport delay (θ ≈ 0), only the natural frequency ω n and damping factor ξ parameters are considered to describe all the possible longitudinal low level controllers.

Model parameters are assumed to exist as ξ ∈ [ξ min , ξ max ] and ω n ∈ [ω nmin , ω nmax ]. This assumption is carried out because PF algorithms require distribution in bounded parameters spaces. Each particle is defined as a two component vector:

θ j = log(ω j n ), ξ j ; j ∈ [1, N p ],
having N p as the number of particles. The first component is set as the natural frequency logarithm to ensure a more uniform distribution and resampling over the Bode frequency axis.

Initially, a set of particles Θ k0 is generated with a random distribution over the 2D parameter space. Afterwards, the PF is executed online with a sample period T s recommended to be less than π/ω max to agree with Nyquist sampling principle, but high enough to allow the algorithm online execution without data loss. At each time step k, an inputobservation pair u k , y k (reference and measured vehicle longitudinal speed respectively) is received through V2V and queued to the 2D vector Zk0:k . The maximum size of Zk0:k -i.e. N s -is selected as the highest amount of u k , y k pairs that can be employed to determine the particles score, allowing real time execution.

At sample time k, the PF computes every particle estimations within the time window between k 0 to k as ŷk0:k , employing a discrete-time state space model obtained combining Eq. 5 and 6. Each particle score is evaluated as the Root Mean Square Error (RMSE) between each particle estimations ỹj k0:k and the observations y k0:k along the time window.

After computing and normalizing each particle score, a resampling process [START_REF] Carvalho | Particle learning and smoothing[END_REF] based on such scores is carried out to discard the particles with low fitting and increase the particles population in the areas where high scores have been obtained. For the first executions, as there is no previous knowledge or data to deliver a right parameter estimation with low variance, the preceding vehicle model is assumed to be similar than the ego-model-i.e. P i (s) = 1-. As the vector Zk0:k is filled with input-output data, the PF algorithm converges to a stable solution presenting low variance and the estimated parameters ( θk ) are updated as the best scored particle's parameters (E(θ k )). The algorithm can be summarized as follows:

p(θ j k+1 = θ j k ) = w j k θ j k+1 = θ j k+1 + ν j k ; ν j k ∼ N (0, dev θ )
The configuration of the PF parameters (particles number N p , samples number N s , Gaussian noise standard deviation dev θ and highest tolerated estimation variance σ max ) are chosen to increase the convergence speed of θk → θ real (having θ real as the parameters of Gp i-1 ) and reduce the expectation variance σ θk as much as possible.

The PF outputs the expectation θk = log(ω n ), ξ at each sample time and such values are employed to configure Gp i-1 (s) in the FF filter F i (s). Even though the application of the PF filter yields a good estimation of Gp i-1 (s), measurement noise and uncertainties introduce a variance over the algorithm output with respect to the real model Gp i-1 (s). As explained in Sec. II, the FF/FB structure demands a low |∆ Gpi-1 (s)| to guarantee an enhanced reference tracking. Therefore, it is required to determine the PF accuracy on the parameters identification task. For this, an offline convergence study is performed modelling different type of vehicles (fast/slow and underdamped/overdamped) following a speed profile with several speed changes. Observation noise is considered as N (0, 0.15). The implemented PF parameters configuration for the study are set as follows: The study is depicted in Fig. 2, where it can be appreciated the parameters estimation of 9 different vehicle dynamics. Based on the obtained performance, the preceding vehicle model uncertainty is going to be defined as follows:

ξ = ξ + ∆ ξ ; |∆ ξ | < 0.11; ω n = ωn • (1 + ∆ ωn ); |∆ ωn | < 0.12; (7)
and consequently, the system performance and stability must be guaranteed for such uncertainties over the term G i-1 (s).

In this work, the FB controller design is going to be oriented not only to improve the system stability but also to ensure the robustness against perturbations due to the presented uncertainties.

IV. FEEDBACK CONTROLLER DESIGN Fig. 3 presents the control structure, which FF filter F i (s) is adapted with the preceding vehicle model with its uncertainties (see Eq. 7). In the literature, FF structures have been widely employed with lead controllers because while FF link improves reference tracking, the lead controller enhances the system damping and FB loop stability [START_REF] Flores | Using fractional calculus for cooperative car-following control[END_REF]. A fractionalorder generalization of a lead compensator is employed to gain a more flexible and stable frequency response and avoid high frequency noise amplification. The mathematical form of the fractional-order lead controller is given by:

C(s) = K p • (1 + s α /ω c ) (1 + s α /ω p ) ; (8)
where K p , α, ω c and ω p stand for the controller DC gain, differentiation order, zero and pole frequencies. The differentiation order is bounded as α ∈ (0, 2), modifying the magnitude and phase contribution of the controller as +α20 dB/dec and +απ/2 rad/dec respectively. Controller design objectives are: 1) Ensuring W Ti T i ∞ ≤ 1; ω > 0 for a time gap of h = 0.6 to agree with the conclusions obtained in [26] 2) Minimizing W Si S i ∞ to improve the robustness to perturbations and FB loop stability 3) Reducing the control effort to avoid actuator saturation W Ui C i (jω)S i (jω) ∞ within the closed loop bandwidth The transfer functions T i , S i represent the complementary sensitivity (string stability) and the FB loop sensi-Fig. 3.

Control structure with emphasis in the FF controller and its uncertainty introduced to the FB loop tivity to perturbations respectively. The template functions W Ti , W Si , W Ui are selected as follows:

W Ti = 1; W Si = s w s + s/M s ; W Ui = 1; (9)
where w s is selected as the plant natural frequency and M s = 2. An optimization algorithm (f solve from Matlab) is applied to determine the controller parameters configuration that minimizes the infinite norms stated in the objectives above. For implementation on discrete embedded systems, a Tustin approximation (Eq. 6) is required with a continued fraction expansion (refer to [START_REF] Podlubny | Using continued fraction expansion to discretize fractional order derivatives[END_REF] for more details).

A. Robustness Validation

Three different vehicle dynamics and controllers designed based on their models (Tab. II), are defined for validating the proposed algorithm. Firstly, a sensitivity study is carried out to validate the robustness of the proposed online-adapted FF/FB algorithm. The system performance is evaluated for the worst possible modelling error in Eq. 7 that would result in the highest | Pi (s)|.

The system sensitivity and stability is evaluated in Tab. III analysing S i (s) ∞ ≡ E i (s)/X i-1 (s) ∞ . Three control structures are considered. A FB-only (ACC) where the FF link is replaced with the ego-speed as positive feedback to ensure no stationary state error. A FF/FB approach with the FF filter set as F i (s) = 1/H(s) and the FF/FB structure with the proposed adapted FF filter F i (s) = ( Pi (s)H(s)) -1 .

Every possible ego-preceding combination of the three vehicle types presented on Tab. II is considered to compare the performance between FB-only ( S f b i ∞ ), the conventional FF/FB ( S f f,conv i ∞ ) and the proposed FF/FB approach ( S f f,adap i ∞ ). The analysis considers no temporal delay on the communication link and a time gap set of 0.6s. For the FB-only approach, one can see that the system stability is independent from the preceding vehicle dynamics as would be expected from an ACC structure. In the case where F (s) = 1/H(s), it is clear that when P i (s) ≈ 1 the performance improves. But when the vehicle dynamics differ, the system stability decreases as the magnitude of P i (s) becomes higher, introducing undesired behavior on the closed loop. When the ego-model has faster dynamics the performance results better, while the worst behavior is obtained when its dynamics are slower than the preceding ones.

For the proposed structure, the system robustness/stability is improved for all scenarios. The enhancement with respect to the other solutions becomes more noticeable as the magnitude of P i (s) is higher. Generally, better results are obtained with the proposed solution if the ego-vehicle presents a faster model than the preceding one.

Finally, the string stability is analysed in Fig. 4 to demonstrate the fulfilment of the condition in Eq. 1. As in the previous study, the performances of FB-only, FF/FB with the conventional FF filter and the proposed approach are compared for all the possible ego-preceding combinations from Tab. II. As the communication delay directly affects the system string stability, a θ = 0.1s is set.

For the FB-only and the FF/FB structures with conventional FF filter, the string stability present an overshoot over the unity at higher frequencies if the ego-model has a low ω n . This is due to the incapacity of slower vehicles to track correctly high frequency speed changes. If the ego-model accounts with a bigger ω n , an overshoot is observed at lower frequencies given that faster vehicles overreact to the preceding speed changes.

When the proposed adapted-FF filter is employed, although in some frequency regions the magnitude of T i (s) may be higher than for the other static structures, its peak value is maintained below unity in all frequencies for all the possible scenarios, satisfying the string stability condition.

V. SIMULATION RESULTS

Two scenarios are considered to validate the FF/FB structure. The first where all the vehicles have a previously Having the presented string configuration, speed change perturbations are introduced in the leader vehicle. In the upper plots of Fig. 5, the vehicles' speeds are showed for both scenarios evolving after a leader acceleration and deceleration at t = 55s and t = 80s respectively. The proposed system attenuates uniformly the leader speed changes upstream. Whereas for the conventional FF filter, the second and fifth vehicles amplify their precedings' oscillations. Such behavior is observed when the ego-vehicle has slower dynamics or is less damped than its preceding one, which agrees with the conclusions obtained in [START_REF] Shaw | String stability analysis for heterogeneous vehicle strings[END_REF] and the studies presented in Sec. IV-A.

In the lower plots of Fig. 5 the spacing errors of the controlled vehicles are depicted when using the conventional and the proposed adapted FF filter. The most remarkable differences are that not only the spacing errors are attenuated upstream but also the maximum observed error is reduced from 0.42 to 0.13 meters when applying the online-adapted FF filter.

A. Simulation with Online Preceding Vehicle Modelling

A four vehicle scenario is set with the following string configuration: Type 2, Type 1, Type 3, Type 2; from leader to last vehicle. For the first test, the conventional FF filter is employed for all string members. In the second test, all the vehicles have only previous knowledge of their own models and their preceding vehicles' models are obtained applying the PF over the received V2V data adapt online FF filter. The same communication delay of θ = 0.1s is considered with a reference time gap of 0.6s between vehicles. Fig. 6 shows the vehicles' speed evolution. When the conventional FF filter is applied, the vehicles' speeds do not evolve uniformly as desired. It can be appreciated that slower vehicles present a higher speed overshoot than the faster ones, which introduces considerable spacing errors (see Fig. 7) that can reach up to 1.6 meters for the case of the third vehicle. When the proposed adapted FF filter is applied, in the first speed step the vehicles' speeds evolve as in the conventional scenario due to the lack of knowledge of their preceding vehicle model. For the following speed changes, one can observe in Fig. 8 that the modelling parameters converge to their solutions as more V2V data is received to be modelled. This permits a more stable speed evolution as each vehicle is able to adapt the FF filter with the identified preceding vehicle parameters, compensating the dynamics difference. The spacing error peak value is also reduced from 1.8 up to 0.4 meters.

In CACC homogeneous strings, when the leader accelerates or decelerates the rest of the string members should face a positive or negative spacing error respectively due to the temporal delay in the V2V link. Instead, when the CACC string is heterogeneous, faster vehicles overreact to the speed change with respect to their slower preceding vehicle, which produces a negative distance error (see 2nd and 4th vehicle in upper plot of Fig. 7). If the ego-vehicle accounts with slower dynamics, the spacing error becomes positive as would be expected towards an increment in the leader speed but higher than desired due to the ego-vehicle limitation to track efficiently its faster preceding vehicle speed changes. This also agrees with the results obtained in Sec. IV-A.

To deal with this problem, the adaptive FF filter is configured to process the preceding vehicle's reference considering both vehicles' models. This can be appreciated on the lower plot of Fig. 7, where for the first speed change in time t=10s the spacing errors evolve differently; but as each vehicle adapts its structure, not only the errors' peak values are reduced but also their evolution results more likely as an homogeneous string scenario (positive or negative speed changes produce spacing errors of equal sign).

VI. CONCLUSIONS

An online adapted FF/FB control structure to deal with the CACC heterogeneous string challenge has been proposed, analysed and validated through simulation. Results show that by adapting the FF filter online considering the ego and preceding vehicle models, the disturbances introduced by the vehicles dynamics difference can be attenuated and string stability can be ensured for constant time gap CACC. A PF has been proposed to identify the preceding vehicle model parameters online, using the V2V data. Analysis show that although the PF outputs the parameters within a estimation range, the proposed FB fractional-order lead controller is able to provide an enhanced reference tracking task, as well as ensure the string stability even for the worst uncertainty over the preceding vehicle model. Simulation results demonstrate that the proposed online adapted FF/FB structure outperforms structures that does not take into consideration the difference between vehicles' dynamics, not only in terms of string stability but also on the spacing error regulation task. Future work includes considering uncertainties over the ego-vehicle model, emergency situations and tests over real platforms.

Fig. 1 .

 1 Fig. 1. FF/FB structure employed for CACC systems

Fig. 2 .

 2 Fig. 2. Accuracy study of the PF determining model parameters. In gray the algorithm outputs (250 executions per model), in red the standard deviation illustration for both axis and the ground truth in blue dots

3 Fig. 4 .

 34 Fig. 4. String stability comparison between FB-only (green), FF/FB with the conventional FF filter (blue) and the proposed FF/FB with adaptive FF filter (red)

Fig. 5 .

 5 Fig. 5. Performance comparison between the FF/FB structure with the conventional and the online-adapted FF filter. First to last vehicle (dark to light blue)

Fig. 6 .Fig. 7 .

 67 Fig.6. Vehicles' speeds along the string from first to last (dark to light blue), with the string members adapting their FF filter online and with the conventional FF filter (top and bottom figures respectively)

  54

Fig. 8 .

 8 Fig. 8. Model parameters identification by the PF filter of the first, second and third follower vehicles. In red and green the estimation of ωn and ξ respectively

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ACKNOWLEDGMENTS

This work has been founded by French Research Agency through the VALET project (ANR-15-CE22-0013). The authors would like to thank PhD. Raoul de Charette for valuable discussions during the development of this work.