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

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


Introduction
In this paper we investigate, via a Hamilton-Jacobi-Bellmann approach, a final state constrained optimal control problem with a Radon measure term in the dynamics.
Several real applications can be described by optimal control problems involving discontinuous trajectories. For instance, in space navigation area, when steering a multi-stage launcher, the separation of the boosters (once they are empty) lead to discontinuities in the mass variable [9]. In resource management, discontinuous trajectories are also used to modelize the problem of sequential batch reactors (see [19]). Many other applications can be found in the Refs. [8,14,15,17]. Consider the controlled system: Here, in problem (1.2), the measure is given by the model and the state equation is controlled by means of a measurable function α. Our main goal is to use the HJB approach in order to characterize the value function v and then to study a numerical method to compute this function.
Since the value function v fulfils a Dynamic Programming Principle (DPP), we can derive, at least formally, the following HJB equation v(X, T ) = ϕ(X).

(1.3)
Clearly, the main difficulty is to give a meaning to the term "Dv · µ" knowing that one can not expect to have a differentiable value function. In order to overcome this problem, following the ideas in [12], we define a new value functionv such that: v(X, τ ) =v(X, W(τ )), where W is the known change of variable coming form the graph completion technique (See Theorem 2.4). The advantage is that now the HJB equation forv has a t-measurable Hamiltonian and not a measure term. More precisely, we can prove thatv is a solution of the following equation: −v s (X, s) + H(s, X, Dv(X, s)) = 0; v(X, 1) = ϕ(X); (1.4) where H(t, x, p) = sup a∈A {−p · F(t, x, a)} and F(t, x, a) is a t-measurable dynamics (see Section 2.2 for the definition of F). Due to the double presence of an only t-measurable Hamiltonian and a lsc final data, the definition of viscosity solution is still not classical.
We recall that, in the case when ϕ is continuous, the definition of viscosity solution for tmeasurable Hamiltonians has been introduced by Ishii in 1985 (see [20]) and then studied for the second order case by Nunziante in [24]- [25](see also the work of Lions-Perthame [21] and Briani-Rampazzo [13]). Moreover, a very general stability result has been proved more recently by Barles in [3]. On the other side, to deal with the case when the Hamiltonian H is continuous with respect to the time variable and the final data ϕ is lsc, the definition of bilateral viscosity solution has been introduced by Barron and Jensen in 1990 ( [6]) and by Frankowska [18].
In this paper, since we deal with target problem, the function ϕ is lsc and the Hamiltonian in (1.4) is only t-measurable. We introduce a new definition of viscosity solution of (1.4), namely the definition of L 1 -bilateral viscosity solution (Definition 3.2 below). This definition allows to characterizev as the unique L 1 -bilateral viscosity solution of equation (1.4) (Theorem 3.4). It gives also a suitable framework to deal with the numerical approximation ofv (and then of v by the change of variable W). More precisely, we prove in Theorem 3.6 a convergence result for monotone, stable and consistent numerical schemes, and give an example of a scheme satisfying these properties. Some numerical tests are presented in Subsection 3.2.
On the other hand, we study the properties of L 1 -bilateral viscosity solution for a general HJB equation. In particular, we derive under classical assumptions on the Hamiltonian (see in Section 4), the consistency of the definition (Theorem 4.7), a general stability result w.r. to the Hamiltonian (Theorem 4.9), a stability result w.r. to the final data (Theorem 4.10), and uniqueness result (Theorem 4.11).
This paper is organized as follows. In Section 2 we set the optimal control problem we are considering. Subsection 2.1 is devoted to the definition of solution for the state equation while Subsection 2.2 to the construction of the reparametrised optimal control problem and the definition ofv. In Section 3 we consider the optimal control problem for the t-measurable HJB equation, we state the definition of L 1 -bilateral viscosity solution, and we prove that the value functionv is the L 1 -bilateral viscosity solution of equation (1.4) in Theorem 3.4. Subsection 3.1 is devoted to the convergence result and to the construction of a good approximating scheme while in Subsection 3.2 we give some numerical test.
Finally in Section 4 we will prove the consistency (Theorem 4.7), stability (Theorem 4.9 and 4.10) and uniqueness (Theorem 4.11) result for L 1 -bilateral viscosity solution.
Notations. For each r > 0, x ∈ R N we will denote by B r (x) the closed ball of radius r centered in x. Given a Radon measure µ we will denote by L 1 µ (R) the space of L 1 -functions with respect to the measure µ.
For In all the sequel, we will use the classical notations: And finally, we will denote by AC([0, 1]; R N ) the set of absolutely continuous functions from [0, 1] to R N .

The optimal control problem with BV trajectories
In this section we state the final state constrained optimal control problem we consider. First, we recall the definition of solution for the state equation introduced by Dal Maso and Rampazzo in [16] and we recall the graph completion construction. Then, we define the value function, we construct the reparametrised optimal control problem and we prove that the two value functions are linked by a change of variable.
(Hg3) There exist K > 0 such that Following [16], we introduce the left continuous primitive B of the Radon measure µ, i.e. B ∈ BV − ([0, T ]; R M ) and his distributional derivativeḂ coincides with µ on [0, T [. In all the sequel, we will denote by T := {t i , i ∈ N} the countable subset of [0, T ) which contains 0 and all the discontinuity points of B and by E c the set of all continuity points of B. Furthermore, let (ψ t ) t∈T := (ψ 1 t , . . . , ψ M t ) be a family of Lipschitz continuous maps from [0, 1] into R M such that (if t = 0 we require only ψ t (1) = B(0 + )). We will denote by ξ the solution of: Definition 2.1. Fix an initial datum and time (X, τ ) and control variable α ∈ A, the function and Y (τ − ) = X. Moreover, if τ ∈ T we have Y (τ + ) = ξ(X, ψ τ ).
In order to prove the uniqueness of this solution we set The graph completion of B corresponding to the family (ψ t ) t∈T is then defined by: (2.10) We are ready now to construct the reparametrisation of system (2.5). Let σ := W(τ ), for each control α ∈ A and initial datum X we denote by Z α X,σ : Z(σ) = X (2.11) where µ a is the absolutely continuous part of the measure µ with respect to the Lebesgue measure, i.e. µ(t) = µ a (t)dt + µ s . Note that the derivatives of φ 0 , φ i are measurable functions, therefore assumptions (Hg1)-(Hg2) ensure the applicability of Caratheodory's Theorem to obtain the existence of a unique solution of (2.11) in AC([σ, 1]; R N ).  [10,11] the case when the g i do not depend on the control and are continuous in t, have been studied. When the g i depends on the control α we refer to [23] for a precise discussion.
However, in this paper, the dependency on the choice of ψ t does not imply any specific difficulty in the sequel.

The control problem
Given a lower semicontinuous function ϕ : R N → R and a final time T , our aim is to calculate the following value function v(X, τ ) : where Y α X,τ is the solution of equation (2.5). It is easy to prove that the following Dynamic Programming Principle holds: (2.14) Therefore we can formally derive a HJB equation: where the Hamiltonian is As we said before, in the introduction of the paper, the problem is to give a meaning to the term Dv · µ knowing that one can not expect to have a differentiable value function. In view of Theorem 2.2, it is then natural to consider the trajectories Z α X,σ solution of the the reparmetrised system (2.11). We define then the corresponding value function as follows: The link between the two problems is given by the following result.
where we respectively denote by v ♯ andv ♯ the lower semicontinuous envelope of v andv w.r. to both variable (X, τ ) and (X, s).
Since, by construction W(τ ) is monotone increasing in [0, T ] and continuous in any τ ∈ T , (2.19) and (2.20) easily follow. (2.20) we stressed the link between the lsc envelopes of v andv because is indeed the functionv ♯ (X, s) that will be characterized as solution of an HJB equation.
Thanks to Theorem 2.4, it is clear that we turn now our attention to the HJB equation for the functionv. The advantage is that we do not have any more measure in the dynamics.
The new value functionv satisfies also a DPP: From this DPP, one could expect to characterizev through the following HJB equation: where the Hamiltonian is Note that, by definition (2.10), the graph completion (φ 0 , φ i ) is a Lipschitz function, therefore we can not expect to have a time continuous Hamiltonian. Moreover, our final condition ϕ is only lower semicontinuous. Thus, we should first give a precise meaning to the definition of the viscosity solution of the equation (2.22).

Optimal control problems with measurable time-dependent dynamics
In this Section we prove that the value functionv is the unique L 1 -bilateral viscosity solution of (2. 22), and that the latter can be solved numerically. For the sake of generality we will prove our results in the following more general framework. The set of controls is A : . Fix a final time T , given x ∈ R N , τ ≥ 0 and a control a ∈ A, we consider the trajectory y a x,τ , solution of the following system: For each initial point and time (x, τ ) ∈ R N × R + we set: We assume the following : (HF3) There exists a K > 0 such that (Hid) The function ϕ : R N → R is lower semi continuous and bounded.
. Therefore, all the results in this section will apply, in particular, to the value functionv defined in (2.17).
In all the sequel, we denote V the lower semicontinuous envelope of ϑ defined by: Our first aim is then to prove that we can characterize the function V in (3.26) as the unique L 1 -bilateral viscosity solution (see the definition below) of the following HJB equation: where the Hamiltonian is Moreover, the final condition is satisfied in the following sense:

Remark 3.3. For the sake of clarity, we will state and prove, in Section 4, the consistency, stability and uniqueness result for the viscosity sense (L1Bvs) defined in Definition 3.2.
Let us now prove the characterization of the value function. Proof. This proof follows the ideas of Barron and Jensen in [7]. First, it is easy to verify that V fulfills the final condition V(x, T ) = ϕ(x) in the sense given by Definition 3.2. Moreover, consider (ϕ n ) n a monotone increasing sequence of continuous functions, from R N to R, pointwise converging to ϕ. For each n ∈ N we set V n (x, τ ) := inf a∈A ϕ n (y a x,τ (T )) . The proof will be divided in two steps.
Step 1. We first remark that by definition we have V n (x, T ) = ϕ n (x), thus the final condition is fulfilled. Moreover, V n is the unique continuous solution of (3.27), with final condition V n (·, T ) = ϕ n (·), in the sense of Definition 4.6. By the consistency result of Theorem 4.7, we get that V n is solution of (3.27) also in the sense of Definition 3.2.
Step 2. By using the same arguments as in [7], we can prove that V n converges pointwise to V. Therefore, the conclusion follows from the stability with respect to the final condition proved in Theorem 4.10. Furthermore, the uniqueness follows by Theorem 4.11.
In the case when the Hamiltonian is continuous (both in time and in space), numerical discretization of Hamilton-Jacobi equations has been studied by many authors. The general framework of Barles-Souganidis [4] ensures that the numerical scheme is convergent (to the viscosity solution) whenever this scheme is consistent, monotone and stable and the HJB equation satisfies a strong comparison principle. The class of schemes satisfying these properties is very large and includes upwind finite differences, Semi-Lagrangian methods, Markov-Chain approximations.
In this section, we extend the result of [4] to the case of equation (3.27), where the Hamiltonian is only t-measurable, and show that the t-measurable viscosity notion is still a good framework to analyze the convergence of numerical approximations. We give also an example of a monotone, stable and consistent scheme of (3.27) based on finite differences approximations. Finally, a numerical example is given in Subsection 3.2.
Let G be a space grid on R N with a uniform mesh size ∆x > 0 (of course a nonuniform grid could also be considered), and let ∆t > 0 be a time step (we assume that T /∆t belongs to N). In the sequel, we will use the following notations: Consider an approximation scheme of the following form: we assume the following:
(C) Consistency. For every point (x 0 , t 0 ), for any b ∈ L 1 (0, T ) and any function φ(x) such that: An example of scheme fulfilling the above assumptions, when the Hamiltonian is given by (3.28), is the following where we classically denoted g + := max(g, 0) and g − := min(g, 0). Proof. Fist remark that the Stability condition (S) easily follows from the boundedness of F and (HF3). Moreover, the monotonicity (M) follows from condition (3.38) by standard arguments.
To prove consistency, we fix (x 0 , t 0 ) and consider a function ψ(x, t) = t 0 b(s) ds + φ(x) for b ∈ L 1 (0, T ) and φ ∈ C 1 (R N ). By using the regularity of ψ and assumption (HF3) on F, we get: The general convergence result is the following.
Proof. The proof will be given in two steps.
Step 1. We first suppose that the final data is continuous (ϕ m ≡ ϕ). We consider a ∆ k = (∆x k , ∆t k ) and denote by v ∆ k the solution of (3.39) corresponding to ∆ k and ϕ m ≡ ϕ. We will prove that, as k → 0, the sequence v ∆ k converges locally uniformly to the unique L 1 -viscosity solution of (3.27).
For the sake of simplicity for each k we will set (x k , t k ) := (x j k , t n k ) where (x j k , t n k ) are the point defined in (3.31) when ∆ is ∆ k .
Let us first observe that by the stability assumption (S) the sequence v ∆ k is bounded, therefore the following weak semi-limits are well defined: Note that both v * and v * trivially satisfy the final condition in (3.27). Therefore, the convergence result will follows once we prove that v * and v * are respectively a L 1 -viscosity supersolution and a L 1 -viscosity subsolution of (3.27). Indeed, if this is true, by the comparison result [20, Theorem 8.1], we have v * ≤ v * . Since the reverse is true by definition, the two weak semi limits coincide and the thesis follows.
Let us now prove that v * is a L 1 -viscosity subsolution of (3.27). (The proof of v * being a L 1 -viscosity supersolution of (3.27) is completely similar and will not be detailed.) Following Definition 4.6 below, for any b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local maximum Note that, without loss of generality, we can assume that ( There exists then a sequences of points (x k , t k ) such that Thanks to (b), we can apply the monotonicity assumption where we also used that v ∆ k is a solution of (3.32). Fix now a δ > 0, by (a) and the regularity of φ we can always find a δ k ≤ δ such that (Dφ(x 0 )). Therefore, also by the Consistency assumption (C) and (3.42) we have: Inequality (3.41) follows then by letting δ → 0 + (which implies δ k → 0 + ).
Step 2. For every m ≥ 1, by Step 1, as k → 0, the sequence (v ∆ k ,m ) k converges to v m the unique L 1 -viscosity solution of (3.44) With the same arguments as in Step 1 of the proof of Theorem 3.4, we conclude the pointwise convergence of v m to V. [22], for a particular numerical scheme.

A numerical test.
In this section, we use the scheme given in (3.37) to solve Hamilton-Jacobi equations coming from a simple control problem with BV trajectories.
Consider the target C := B(0, r), which is the ball centred at the origin and of radius r = 0.25. Consider also a trajectory Y where C1 := 0.5, C2 := 0.2, and δ u (for u = 1, 2) denotes the Dirac measure at time t = u. The control variable c takes its values in a compact set U . Here we will consider two cases: • Case 1: U ≡ {0.5} which amounts saying that we are allowed to move in any direction in the sphere centred at the origin and with radius 0.5.
• Case 2: U = [0, 0.5], which means that we can move in any direction in the Ball centred at the origin and with radius 0.5.
It is not difficult to compute the parametrized function:  14 15 < s < 1. Let us notice that in Case 2, the value functionv corresponding to the parametrized problem is lsc. (3.45) In the two cases, we compute first the value functionv corresponding to the parametrized control  problem, and then we deduce the original value function by using a change of variable. The latter step is very easy to perform numerically, since v turns to be just the restriction ofv on [0, 1 15 ] ∩ [ 7 15 , 8 15 ] ∩ [ 14 15 , 1]. In Figs. 1 & 2, we plot only the 0-level sets. Fix T > 0, and consider the general Hamilton-Jacobi-Bellman equation (4.46) On the Hamiltonian H : R + × R N × R N → R we assume the following: (H0) The function H(t, x, p) is measurable in t and continuous in x and p. Moreover, for each (x, p) ∈ R N × R N we have H(·, x, p) ∈ L 1 (R + ).
(H3) For each (t, x) the function H(t, x, ·) is convex and there exists a constant L > 0 such that On the final data we suppose: (Hid) The function ϕ : R N → R is lower semi continuous and bounded. In order to give an equivalent formulation of the definition of L 1 -bilateral viscosity solution we need to introduce the following sets of functions. Fix (x 0 , t 0 ) and a function φ ∈ C 1 (R N × R + ) we set (Dφ(x 0 , t 0 )), a. e. t ∈ B δ (t 0 ) and some δ > 0} for all x ∈ B δ (x 0 ), p ∈ B δ (Dφ(x 0 , t 0 )), a. e. t ∈ B δ (t 0 ) and some δ > 0} Definition 4.2. L 1 -bilateral viscosity solution (L1Bvs) II Let u : R N × R + → R be a bounded lower semi-continuous function. We say that u is a L 1 -bilateral viscosity solution (L1Bvs) of (4.46) if: Moreover, the final condition is satisfied in the following sense: Remark 4.3. Note that in fact, there are many more formulations. We can take the test function φ(x, t) ∈ C 1 (R N × (0, T )), (i.e. C 1 -depending also on t in Definition 3.2) and φ ∈ C 1 (R N ), (i.e. depending only on the x-variable) in Definition 4.2. We can replace φ ∈ C 1 (R N ) by φ ∈ C 2 (R N ), ..., C ∞ (R N ). Moreover, by classical arguments in the theory of viscosity solutions, we may replace the local minimum by global, or local strict or global strict. Proof. We first remark that for any b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local minimum The equivalence of the two definitions follows then by observing that in Definition 4.2 we can consider test functions φ depending only on the x-variable. (See also Remark 4.3.)

Consistency
We prove now that our definition is consistent with the definitions of viscosity solutions given for a more regular HJB equation. In particular we consider the case of a time-continuous Hamiltonian and/or a continuous initial data. For the sake of completeness let us recall here the definition of viscosity solution in those cases.
Definition 4.5 (bilateral viscosity solution (Bvs), See [6]). Assume that H is continuous w.r. to the time variable. Let u ∈ LSC(R N × (0, T )) be a bounded function. We say that u is a bilateral viscosity solution (Bvs) of (4.46) if: Moreover, the final condition is satisfied in the following sense: ϕ(x) = inf lim inf n→∞ u(x n , t n ) : x n → x , t n ↑ T Definition 4.6 (L 1 -viscosity solution (L1vs), [20,21]). Assume that the final condition ϕ is a continuous function on R N . We say that u ∈ LSC(R N × (0, T )) is a L 1 -viscosity supersolution (L1vsp) of (4.46) if: for We say that u ∈ U SC(R N × (0, T )) is a L 1 -viscosity subsolution (L1vsb) of (4.46) if: for We say that u ∈ C(R N × (0, T )) is a L 1 -viscosity solution (L1vs) if it is both a L 1 -viscosity subsolution and a L 1 -viscosity supersolution and the final condition is satisfied pointwise:

(a) If the final condition ϕ is a continuous function, then
u is a L 1 -bilateral viscosity solution ⇐⇒ u is a L 1 -viscosity solution.

(b) If the Hamiltonian H is continuous also in the t-variable, then
u is a L 1 -bilateral viscosity solution ⇐⇒ u is a bilateral viscosity solution.
Proof of (a).
The key tool to prove this equivalence is a Lemma introduced in the [6] to prove the equivalence between a Bvs and a Cvs. For the sake of completeness we recall here this result.
Lemma 4.8. [6, Theorem 1.1] Let W be a continuous function on [0, ∞) × R n such that W has a zero maximum (minimum) at (τ, ξ). Let ε > 0. Then there is a smooth function ψ, a finite set of numbers α k ≥ 0 summing to one, and a finite collection of points (t k , x k ) such that Since u is a L1vs, in particular is a L 1 -viscosity supersolution therefore inequality (4.49) is satisfied.
To prove (4.50), for each δ > 0 we apply Lemma 4.8 above choosing ε small enough the ensure the existence of an η > 0 such that t)). Therefore, there exists a smooth function ψ and a finite set of points (x k , t k ) such that u − t 0 b − (φ + ψ) has a zero maximum at (x k , t k ) and for each k (4.51) Thus ess inf Since u is a L1vs, in particular is a L 1 -viscosity subsolution therefore in each point (t k , x k ) we have Letting δ going to 0 + (⇒ η → 0 + ) in (4.52) we obtain (4.50) and conclude the proof.
Proof of u is a L1Bvs ⇒ u is a L1vs. We first remark that, by Definition 3.2 if u is a L1Bvs, is in particular a L 1 -viscosity supersolution. Therefore, to prove that u is a L 1 -viscosity subsolution fix b ∈ L 1 (0, T ), φ ∈ C 1 (R N ) and (x 0 , t 0 ) local maximum point of u(x, t) − t 0 b(s)ds − φ(x) our thesis is (4.53) As above, for each δ > 0 we apply Lemma 4.8 choosing ε small enough the ensure the existence of an η > 0 such that t). Therefore, there exists a smooth function ψ and a finite set of points (x k , t k ) such that u − t 0 b − (φ + ψ) has a zero minimum at (x k , t k ) and for each k Since u is a L1Bvs we have (3.30) at each point (t k , x k ), i.e. lim η→0 + ess inf Letting δ going to 0 + (⇒ η → 0 + ) in (4.55) we obtain (4.53) and conclude the proof.
Proof of u is a Bvs ⇒ u is a L1Bvs.
. By standard mollification's arguments we can approximate b in L 1 (0, T ) by a sequence (b ε ) ε>0 ∈ C ∞ ((0, T )) and we have the existence of a sequence of points (x ε , t ε ) such that as ε → 0, (x ε , t ε ) → (x 0 , t 0 ) and fix ε, (x ε , t ε ) is a local minimum point for u(x, t) Note that for each δ > 0 fixed we can find an ε small enough to have ess inf Therefore, letting δ → 0, we respectively obtain (3.30) and (3.29) and this ends the proof.

Stability
We will prove here the stability with respect to the final datum and the one with respect to the Hamiltonian. The latest will be proved under a very weak convergence in time that as been proved for L 1 -viscosity solution by Barles in [3]. (Our proof is indeed an adaptation to L1Bvs of the proof of [3, Theorem 1.1]). Note that in this proof we only need assumptions (H0)-(H1) on the Hamiltonian.
Theorem 4.9. Stability w.r.to H. For each n ∈ N let u n be a L 1 -bilateral viscosity solution of (4.56) We assume that: i) For each n ∈ N the Hamiltonian H n fulfils hypotheses (H0)-(H1) for some modulus m n = m n (K) such that m n (·, r) L 1 (0,T ) → 0 as r → 0 uniformly with respect to n, for any compact subset K.
iii) The final condition ϕ fulfils (Hid), (4.57) Proof. Following Definition 4.2 we have to prove that In order to prove 1, let us fix a (x 0 , t 0 ), Fix now a small δ > 0, we consider a large compact subset K of R N × R N and the functions m, m n given by assumptions i),ii). We construct a new sequence (u δ n ) n defined by Note that for each n, δ the function u δ n is a L1Bvs of Moreover, if we set u δ (x, t) := inf (xn,tn)→(x,t) lim inf n→∞ u δ n (x n , t n ), by the properties of m, m n we have u ≤ u δ ≤ u + O δ (1). Therefore, by classical results, since (x 0 , t 0 ) is a strict local minimum point of u(x, t) − t 0 b(s)ds − φ(x, t), for δ small enough there exists a local minimum point of u δ (x, t) − t 0 b(s)ds − φ(x, t), that we will denote (x δ , t δ ). Note that (x δ , t δ ) → (x 0 , t 0 ) as δ → 0. We set now ψ n (s) := H n (s, x δ , Dφ(x δ , t δ )) − H(s, x δ , Dφ(x δ , t δ )).
By definition of L1Bvs, condition (4.61) and the fact that (x n δ , t n δ ) is a local minimum point of u δ n (x, t) − φ(x, t) − t 0 b(s)ds + t 0 ψ n (s)ds imply that −φ t (x n δ , t n δ ) + G(x n δ , t n δ , Dφ(x n δ , t n δ )) ≤ 0. Therefore letting n → ∞ and δ → 0 by the continuity of G we obtain (4.58) and conclude the proof of 1.
Point 2 can be proved with the same argument by remarking that the functions  Assume that, for each n ∈ N, the function ϕ n ∈ C(R N ) and is bounded, moreover the sequence (ϕ n ) n∈N is monotone increasing and

Uniqueness
We finally prove the uniqueness result. Proof. This proof will follow the idea of G.Barles of using the inf-convolution in the proof of uniqueness for bilateral viscosity solution [2,Theorem 5.14].
Suppose that there exist v and u two L 1 -bilateral viscosity solution of (4.46). Since v is in particular a L 1 -viscosity supersolution the main point is to look for a sequence of L 1 -viscosity subsolutions of (4.46) approximating u. The thesis will then follow by comparison result for L 1viscosity solution.
The construction of the approximating sequence can be summarised in the following Lemma. The proof being an adaptation of the proof given in [2, Lemme 5.5] will be not detailed (see also [5,Lemma 19]). Since (u ε ) * is a L 1 -viscosity subsolution of (4.62) the function (u ε ) * − K ∞ e 1 2 KT M ε is a L 1 -viscosity subsolution of (4.46), therefore, by the comparison result for L 1 -viscosity solutions (see [20,Theorem 8.1] or [25]) we obtain where we used also (4.63). Letting ε → 0 we have u(x, t) ≤ v(x, t) ∀(x, t) ∈ R N × (0, T ), thus, reversing the roles of u and v, the uniqueness follows.