Hamilton–Jacobi equations for optimal control on multidimensional junctions with entry costs

We consider an infinite horizon control problem for dynamics constrained to remain on a multidimensional junction with entry costs. We derive the associated system of Hamilton–Jacobi equations (HJ), prove the comparison principle and that the value function of the optimal control problem is the unique viscosity solution of the HJ system. This is done under the usual strong controllability assumption and also under a weaker condition, coined ‘moderate controllability assumption’.


Introduction
In this paper, we consider an infinite horizon control problem for the dynamics of an agent constrained to remain on a multidimensional junction on R 3 , i.e. a union of N ≥ 2 half-planes P i which share a straight line Γ, see Fig. 2. The controlled dynamics are given by a system of ordinary differential equations, where in each P i it is given by a drift f i (·, ·) and to which is associated a running cost i (·, ·). Moreover, the agent pays a cost c i (·) each time it enters the halfplane P i from Γ. The goal of this work is to study the properties of the value function of this control problem and derive the associated Hamilton-Jacobi equation (HJ) under some regularity conditions on the involved dynamics, running and entry cost functions. Although we will not discuss it in this paper, the optimal control problem with exit costs, i.e. instead of paying an entry cost each time the agent enters the half-plane, it pays a cost each time it exits it, can be solved similarly. Oudet [25] considers a similar optimal control problem but without entry or exit costs from the interface to the half-planes.
When the interface Γ is reduced to a point, the junction becomes a simple network with one vertex, i.e. a 1-dimensional junction. Optimal control problems (without entry costs) in which the set of admissible states are networks attracted a lot of interest in recent years. Being among the first papers discussing this topic, Achdou et al. [2], derived an HJ equation associated to an infinite horizon optimal control on networks and proposed a suitable notion of viscosity solutions, where the admissible test-functions whose restriction to each edge are C 1 are applied. Independently and at the same time, Imbert et al. [18] proposed an equivalent notion of viscosity solution for studying an HJ approach to junction problems and traffic flows. Both [2] and [18] contain first results on the comparison principle. In the particular case of eikonal equations on networks, Schieborn and Camilli [26] considered a less general notion of viscosity solution. For that later case, Camilli and Marchi [12] showed the equivalence between the definitions notion of viscosity solution given in [2,18] and [26]. Optimal control on networks with entry costs (and exit costs) has recently been considered by the first author [14].
An important feature of the effect of the entry costs is a possible discontinuity of the value function. Discontinuous solutions of HJ equations have been studied by various authors, see for example Barles [5] for general open domains in R d , Frankowska and Mazzola [15] for state constraint problems, and in particular Graber et al. [16] for a class of HJ equations on networks.
In the case considered in the present work, the effect of entry costs induces a discontinuity of the value function V at the interface Γ, while it is still continuous on each P i \Γ. This allows us to adopt the techniques which apply to the continuous solution case in the works of Barles et al. [7] and Oudet [25], where we split the value function V into the collection {v 1 , . . . , v N } of functions, where each v i is continuous function defined on P i and satisfies We note that the existence of the limit in the above formula comes from the fact that the value functions is Lipschitz continuous on the neighborhood of Γ (see Lemma 3.3), thanks to the 'strong controllability assumption', which is introduced below. The first main result of the present work is to show that (v 1 , . . . , v N , V| Γ ) is a viscosity solution of the following system λv i (x) + H i (x, ∂v i (x)) = 0, if x ∈ P i \Γ, where H i is the Hamiltonian corresponding to the half-plane P i , V| Γ is the restriction of our value function on the interface and H Γ is the Hamiltonian defined on Γ. At x ∈ Γ, the definition of the Hamiltonian has to be particular, in order to consider all the possibilities when x is in the neighborhood of Γ. More specifically, • the term H + i (x, ∂u i (x)) accounts for the situation in which the trajectory does not leave P i , ∂e0 (x)) accounts for situations in which the trajectory remains on Γ.
This feature is quite different from the one induced by the effect of entry costs in a network (i.e. when Γ is reduced to a point) considered in [14], where the value function at the junction point is a constant which is the minimum of the cost when the trajectory stays at the junction point forever and the cost when the trajectory enters immediately the edge that has the lowest possible cost.
The paper is organized as follows. In Sect. 2, we formulate the optimal control problem on a multidimensional junction on R 3 with entry cost. In Sect. 3, we study the control problem under the strong controllability condition, where we derive the system of HJ equations associated with the optimal control problem, propose a comparison principle, which leads to the well-posedness of (1.1)-(1.3), and prove that the value function of the optimal control problem is the unique discontinuous solution of the HJ system. We suggest two different proofs of the comparison principle. The first one is inspired from the work by Lions and Souganidis [21] and uses arguments from the theory of PDEs, and the second one uses a blend of arguments from optimal control theory and PDE techniques suggested in [3,7,8] and [25]. Finally, in Sect. 4, the same program is carried out when the strong controllability is replaced by the weaker one that we coin 'moderate controllability near the interface'. The proof of the comparison principle under the moderate controllability condition is carried on by only using the PDE techniques provided in Lions and Souganidis [21].
The results obtained in the present work extend easily to multidimensional junction on R d , i.e. a union of N ≥ 2 half-hyperplanes P i which share an affine space Γ of dimension d − 2, and to the more general class of ramified sets, i.e. closed and connected subsets of R d obtained as the union of embedded manifolds with dimension strictly less than d, for which the interfaces are non-intersecting manifolds of dimension d − 2, see Fig. 1a for example. We do not know whether these results apply to the ramified sets for which interfaces of dimension d − 2 cross each other (see Fig. 1b). Recent results on optimal control and HJ equations on ramified sets include Bressan and Hong [11], Camilli et al. [13], Nakayasu [24] and Hermosilla and Zidani [17] and the book of Barles and Chasseigne [9].  Let {e i } 0≤i≤N be distinct unit vectors in R 3 such that e i · e 0 = 0 for all i ∈ {1, . . . , N}. The state of the system is given by the junction S which is the union of N closed half-planes P i = Re 0 × R + e i . The half-planes P i are glued at the straight line Γ := Re 0 (see Fig. 2). If x ∈ S\Γ, there exist unique i ∈ {1, . . . , N}, x i > 0 and x 0 ∈ R such that Let x = (x i , x 0 ) ∈ P i and y = (y j , y 0 ) ∈ P j , the geodesic distance d(x, y) between two points x, y ∈ S is , if x ∈ P i and y ∈ P j .

The optimal control problem
We consider an infinite horizon optimal control problem which has different dynamics and running costs for each half-plane. For i = 1, . . . , N, • the set of controls (action set) on P i is denoted by A i , • on P i the dynamics of the system is deterministic with associated dynamic f i , • the agent has to pay the running cost i while (s)he is on P i .
Hereafter, we will use the notation Entry costs. {c 1 , . . . , c N } is a set of entry cost functions, where c i : Γ → R + is Lipschitz continuous and bounded from below by some positive constant C.
[A2] Convexity of dynamics and costs. For x ∈ P i , the following set is non empty, closed and convex. Remark 2.1. In [A0], the assumption that the set A i are disjoint is not restrictive since we can always replace is made to avoid the use of relaxed control (see the definition for relaxed control in [4]). Many of these conditions can be weakened at the cost of keeping the presentation of the results easy to follow.

Controlled dynamics.
Let M be the closed set given by if x ∈ Γ and a ∈ A i .
For x ∈ S, the set of admissible trajectories starting from x is Thanks to the Filippov implicit function lemma (see [23]), it is shown in [ Thus, from now on, we will denote y x by y x,α if (y x , α) ∈ T x . By continuity of the trajectory y x,α , the set T Γ x,α := {t ∈ R + : y x,α (t) ∈ Γ} containing all the times at which the trajectory stays on Γ is closed and therefore, the set T i x,α := {t ∈ R + : y x,α (t) ∈ P i \Γ} is open.
. . , n} if the trajectory y x,α enters P i n times, K i = N if the trajectory y x,α enters P i infinite times and K i = ∅ if the trajectory never enters P i .

Remark 2.2.
From the previous definition, we see that t ik is an entry time in P i \Γ and η ik is an exit time from P i \Γ. Hence We now define a cost functional and a value function corresponding to the optimal problem.

Cost functional and value function.
Definition 2.3. The cost functional associated to the trajectory (y x , α) ∈ T x is defined by where λ > 0 and the running cost : The value function of the infinite horizon optimal control problem is defined by Remark 2.4. By the definition of the value function, we are mainly interested in admissible control laws α for which J(x, α) < +∞. In such a case, even if the set K i may be infinite, it is possible to reorder {t ik , η ik : k ∈ N} such that Indeed, because of the positivity of the entry cost functions, if there exists a cluster point, J(x, α) has to be infinite which leads to a contradiction, since we assumed that J(x, α) < +∞. This means that the state cannot switch half-planes infinitely many times in finite time, otherwise the cost functional becomes obviously infinite.
The following example shows that the value function with entry costs can possibly be discontinuous at the interface Γ.
Example 2.5. Consider a simple junction S with two half-planes P 1 and P 2 . To simplify, we may identify S ≡ R 2 and and entry costs functions c 1 ≡ C 1 , where C 1 is a positive constant and For x ∈ P 2 \Γ, then V(x) = v 2 (x) = 0 with optimal strategy which consists of choosing α ≡ (a 2 = 1, a 0 = 0). For x ∈ P 1 , we can check that The optimal trajectory starting from Fig. 3.
and the value function The optimal trajectory starting from x = (x 1 , x 0 ) ∈ P 1 in case |x 1 | < 1 is plotted in blue in Fig. 3.
To sum up, there are two cases The graph of the value function with entry costs satisfying inf x∈Γ c 2 (x) ≥ 1/λ is plotted in Fig. 4a.

Hamilton-Jacobi system under strong controllability condition near the interface
In this section we derive the Hamilton-Jacobi system (HJ) associated with the above optimal control problem and prove that the value function given by (2.2) is the unique viscosity solution of that (HJ) system, under the following condition: [A3] (Strong controllability) There exists a real number δ > 0 such that for any i = 1, . . . , N and for all x ∈ Γ, Remark 3.1. If x is close to Γ, we can use [A3] to obtain the coercivity of the Hamiltonian which will be needed in Lemma 3.17 below to prove the Lipschitz continuity of the viscosity subsolution of the HJ system.

Lemma 3.2. Under Assumptions [A1] and [A3]
, there exist two positive numbers r 0 and C such that for all Proof. The proof is classical and similar to the one in [14], so we skip it.
Proof. This lemma is a consequence of Lemma 3.2, see [1] and [14] for more details.
For x ∈ Γ, we set and . Let us define a viscosity solution of the switching Hamilton-Jacobi equation on the interface Γ: where H Γ is the Hamiltonian on Γ defined by Definition 3.4. An upper (resp. lower) semi-continuous u Γ : Γ → R is a viscosity subsolution (resp. supersolution) of (3.4) if for any x ∈ Γ, any ϕ ∈ C 1 (Γ) such that u Γ − ϕ has a local maximum (resp. minimum) point at x, then The continuous function u Γ : Γ → R is called viscosity solution of (3.4) if it is both viscosity sub and supersolution of (3.4).
We have the following characterization of the value function V on the interface.  The proof of Theorem 3.5 is made in several steps. The first step is to prove that V| Γ is a viscosity solution of an HJ equation with an extended definition of the Hamiltonian on Γ. For that, we consider the following larger relaxed vector field: for x ∈ Γ, We have Proof. See Appendix.
The second step consists of proving the following lemma.
in the viscosity sense.
From Lemma 3.6, it suffices to prove that According to (3.7) and the dynamic programming principle, for all n ∈ N, (y x,n (t) , α n (t)) e −λt dt + V| Γ (y x,n (t n )) (e −λtn − 1). Dividing both sides by t n , the goal is to take the limit as n tends to ∞. On the one hand, we have On the other hand, since y x,n (t n ) = x + t n (ζ + o(1)e 0 ), we obtain Hence, in view of (3.9) and (3.10), we have Thus (3.11) holds for any (ζ, ξ) ∈ f Γ (x) and therefore (3.8) holds.

Lemma 3.8. Under Assumptions [A] and [A3], for all
(a) Consider any control law α such that (y x , α) ∈ T x . Let α z,x be a control law which connects z to x (which exists thanks to Lemma 3.2) and consider the control laŵ This means that the trajectory goes from z to x with the control law α z,x and then proceeds with the control law α. Therefore, Since α is chosen arbitrarily and i is bounded by M , we obtain Let z tend to x (then τ z,x tends to 0 by Lemma 3.2), we conclude v i (x) ≤ V(x). (b) Consider any control law α z such that (y z , α z ) ∈ T z and use Lemma 3.2 to pick a control law α x,z connecting x to z. Consider the control laŵ for which the trajectory y x,α goes from x to z using the control law α x,z and then proceeds with the control law α z . Therefore,  Since α z is chosen arbitrarily and i is bounded by M , we obtain Let z tend x (then τ x,z tends to 0, by Lemma 3.2), we conclude From Lemmas 3.7 and 3.8, we conclude that V| Γ is a viscosity subsolution of (3.4). The last step of of the proof of Theorem 3.5 is to prove that V| Γ is a viscosity supersolution of (3.4).
in the viscosity sense.
Proof. Let x ∈ Γ and assume that it suffices to prove that V(x) satisfies in the viscosity sense. Let {ε n } be a sequence which tends to 0. For any n, let α n be an ε n -optimal control, i.e. V(x) + ε n > J(x, α n ), and τ n be the first time the trajectory y x,αn leaves Γ, i.e.
We note that τ n is possibly +∞, in which case the trajectory y x,αn stays on Γ for all s ∈ [0, +∞). We consider the two following cases: Case 1: There exists a subsequence of {τ n } (which is still denoted {τ n }) such that τ n → 0 as n → +∞ and at time τ n the trajectory enters P i0 , for some i 0 ∈ {1, . . . , N}. This implies Since is bounded by M , sending n to +∞, yields which leads to a contradiction to (3.12).
where o(τ )/τ → 0 as τ → 0 and the last inequality is obtained by using the boundedness of . Let ϕ ∈ C 1 (Γ) such that V| Γ − ϕ has a minimum on Γ at x, i.e. (3.14) From Lemma 3.6, it suffices to prove that Since lim n→∞ ε n = 0, it is possible to choose a sequence {t n } such that 0 < t n < C and ε n /t n → 0 as n → ∞. Thus from (3.13) and (3.14), we obtain (y x,n (t), α n (t))dt is bounded in Γ × R. Therefore, we can extract a subsequence of this sequence which converges to (ζ,ξ) as n → +∞. Obviously, we have ζ ,ξ ∈ f Γ (x). Hence, sending n to ∞ in (3.16), we obtain and thus (3.15) holds.

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Hamilton-Jacobi equations for optimal control Page 15 of 42 23 a)·e i ≥ 0} and consider the following Hamilton-Jacobi system

Definition 3.11. (Viscosity solution with entry costs)
. . , N} and u Γ ∈ C(Γ, R), is called a viscosity subsolution of (3.17) if for any (ϕ 1 , . . . , ϕ N , ϕ Γ ) ∈ R(S), any i ∈ {1, . . . , N} and any x i ∈ P i , x ∈ Γ such that u i − ϕ i has a local maximum point on P i at x i and u Γ − ϕ Γ has a local maximum point on Γ at x, then . . , N} and u Γ ∈ C(Γ; R), is called a viscosity supersolution of (3.17) if for any (ϕ 1 , . . . , ϕ N , ϕ Γ ) ∈ R(S), any i ∈ {1, . . . , N} and any x i ∈ P i and x ∈ Γ such that u i − ϕ i has a local minimum point on P i at x i and u Γ − ϕ Γ has a local minimum point on Γ at x, then • A functions U := (u 1 , . . . , u N , u Γ ) where u i ∈ C (P i ; R) for all i ∈ {1, . . . , N} and u Γ ∈ C(Γ; R), is called a viscosity solution of (3.17) if it is both a viscosity subsolution and a viscosity supersolution of (3.17).

Relations between the value function and the HJ system
In this section, we wish to prove that Proof. By Theorem 3.5, V| Γ is a viscosity solution of (3.4). Furthermore, if x ∈ P i \Γ , for any i ∈ {1, . . . , N} and (y x , α) ∈ T x , there exists a time τ small enough so that y x,α (t) ∈ P i \Γ for 0 ≤ t ≤ τ . Thus, the proof in this case is classical by using dynamic programming principle (see [4,6]) and we do not detail it. Now assume x ∈ Γ, we shall prove that for all i ∈ {1, . . . , N}, the function v i satisfies (3.18) in the viscosity sense. The proof of this case is a consequence of Lemma 3.13 and Lemma 3.15 below.
in the viscosity sense.
. Hence, it suffices to prove that in the viscosity sense. Let a i ∈ A i be such that f i (x, a i ) · e i > 0. By the Lipschitz continuity of f i (·, a i ), there exist r > 0 such that f i (z, a i ) · e i > 0 for all z ∈ B(x, r) ∩ (P i \Γ). Thus, there exists τ > 0 such that for all z ∈ B(x, r) ∩ (P i \Γ), there exists (y z , α z ) ∈ T z for which whereα is chosen arbitrarily. It follows that y z (t) ∈ P i \Γ for all t ≤ τ . In other words, the trajectory y z cannot approach Γ since the speed pushes it away from Γ, for y z (t) ∈ P i ∩ B (Γ, r). Note that it is not sufficient to choose

This inequality holds for anyα, thus
and by Grönwall's inequality, yielding that y z (s) tends to y x (s) and τ 0 i (y z (s), α z (s))ds tends to τ 0 i (y x (s), α x (s))ds when z tends to x. Hence, from (3.19), by letting z → x, we obtain By letting τ tend to 0, we obtain that The proof is complete.
Before we give a proof of the fact that v i is a viscosity supersolution of (3.18), we prove the following useful lemma.

Lemma 3.14. Let x ∈ Γ and assume that
(3.20) Then, there existτ > 0 and r > 0 such that for any z ∈ (P i \Γ) ∩ B(x, r), any ε sufficiently small and any ε-optimal control law α ε z for z, y z,α ε z (s) ∈ P i \Γ, for all s ∈ [0,τ ]. This lemma means that if (3.20) holds, then any trajectories starting from z ∈ (P i \Γ) ∩ B(x, ε) still remains on P i \Γ for a fixed amount of time. Hence, this lemma takes into account the situation that the trajectory does not leave P i \Γ.
Then, according to (3.21), Hamilton-Jacobi equations for optimal control Page 19 of 42 23 where the notation o ε (1) is used for a quantity which is independent on τ and tends to 0 as ε tends to 0. For a positive integer k, the notation o(τ k ) is used for a quantity that is independent on ε and such that o(τ k )/τ k → 0 as τ → 0. Finally, O(τ k ) stands for a quantity independent on ε such that O(τ k )/τ k remains bounded as τ → 0. From (3.22), we obtain that Since y ε (τ ) ∈ P i for all ε, we have Hence, from (3.23) Let ε n → 0 as n → ∞ and τ m → 0 as m → ∞ such that Therefore, Hence,  (x, a) , i (x, a)) . (3.26) On the other hand, by Lemma 3.14, y εn (s) ∈ P i \Γ for all s ∈ [0, τ m ]. This yields Since y εn (τ m ) · e i > 0, then Let ε n → 0 then let τ m → 0, to obtain f i (x, a) · e i ≥ 0, thus a ∈ A + i (x). Hence, from (3.25) and (3.26), replacing ε by ε n and τ by τ m , let ε n → 0, then let τ m → 0, we finally obtain

A comparison principle and uniqueness
In this section we establish a comparison principle for the Hamilton-Jacobi system (3.17). From the comparison principle, it easily follows that V := (v 1 , . . . , v N , V| Γ ) is the unique viscosity solution of (3.17). We are going to give two proofs of Theorem 3.16. The first one, given below, is inspired by Lions and Souganidis [21,22] by using arguments from the theory of PDE. The second one (displayed in the appendix) is inspired by the works of Achdou et al. [3] and Barles et al. [7,8] by using arguments from the theory of optimal control and PDE techniques. Both proofs make use of the following important properties of viscosity subsolutions displayed in the next lemma. Proof. The proof of Lemma 3.17 is based on the fact that if U = (u 1 , . . . , u N , u Γ ) is a viscosity subsolution of (3.17), then for any i ∈ {1, . . . , N}, u i is a viscosity subsolution of (3.27) Therefore, the proof is complete by applying the result in [25,Section 3.2.3] (which is based on the proof of Ishii [19]).
A first proof of Theorem 3. 16. First of all, we claim that there exists a positive constantM such that (φ 1 , .
Thus, using Cauchy-Schwarz inequality, there existsM > 0 such that The claim for φ j is proved and we can prove similarly the one for φ Γ . Next, for 0 < μ < 1, μ close to 1, setting u μ is a viscosity subsolution of (3.17). Moreover, since u μ j and u μ Γ tend to −∞ as |x| and |z| tend to +∞ respectively, the functions u μ j − w j and u μ Γ − w Γ have maximum values M μ j and M μ Γ which are reached at some pointsx j andx Γ respectively. We argue by contradiction, through considering the two following cases: is a viscosity subsolution of (3.17), by Lemma 3.17, there exists a positive number L such that u μ i is Lipschitz continuous with Lipschitz constant L in P i ∩ B(x i , r). We consider the function Ψ i,ε : P i × P i → R which is defined by where ε > 0, δ(ε) := (L + 1) ε and x = ( as ε → 0. positive when ε is small enough. Furthermore, since w i is bounded and u μ i is bounded from above, we have |x ε −x i | 2 is bounded and x ε − y ε → 0 as ε → 0. Hence, after extraction of a subsequence, x ε , y ε →x ∈ P i as ε → 0, for somex ∈ P i . Thus, from (3.30) we obtain The claim is proved. From now on in this proof we only consider the case whenx i ∈ Γ, since otherwise, the proof follows by applying the classical theory (see [4,6]). We claim that x ε / ∈ Γ for ε small enough. Indeed, assume by contradiction that x ε ∈ Γ, i.e. x i ε = 0, we have (a) If y ε / ∈ Γ, then let z ε = (y i ε , x 0 ε ), we have Since u μ i is Lipschitz continuous in B (x i , r) ∩ P i , we see that for ε small enough Therefore, if y i ε,γ = 0, then L ≥ L + 1 − |x ε + z ε − 2x i |, which leads to a contradiction, since x ε , z ε tend tox i as ε → 0.
This implies that L ≥ L + 1/2 − |x ε + z ε − 2x i |, which yields a contradiction since x ε , z ε tend tox i as ε → 0. The second claim is proved. We consider the following three possible cases Case A.1 There exists a subsequence of {y ε } (still denoted by {y ε }) such that y ε ∈ Γ and w i (y ε ) ≥ w Γ (y ε ). Since x ε , y ε →x i as ε → 0 and u μ i is continuous, for ε small enough, we have Recall that the second inequality of (3.31) holds since M μ i > 0 and the last inequality of (3.31) holds since u μ i and u μ Γ satisfy in the viscosity sense. From (3.31), On the one hand, since H + i (x, p) ≤ H i (x, p) for all x ∈ Γ and p ∈ Re i × Re 0 , we have On the other hand, we have a viscosity inequality for u μ i at x ε ∈ P i \Γ: Subtracting (3.33) from (3.34), we obtain In view of [A1] and [A2], there exists C i > 0 such that for any x, y ∈ P i and p, q ∈ R Letting ε to ∞ and applying (3.29), we obtain that max Pi {u μ i − w i } = M μ i ≤ 0, which leads to a contradiction. Case A. 3 There exists a subsequence of {y ε } such that y ε ∈ P i \Γ. Since the inequalities (3.33) and (3.34) still hold, a contradiction is obtained by using the similar arguments as in the previous case. Case B Assume that We consider the function By classical arguments, Φ ε attains its maximum K ε at (ζ ε , ξ ε ) ∈ Γ×Γ and ⎧ ⎪ ⎨ We consider the following two cases: Case B.1 There exists a subsequence {ξ ε } (still denoted by {ξ ε }) such that We also have a viscosity inequality for u μ Γ at ζ ε By applying the classical arguments, one has (ζ ε − ξ ε ) 2 /ε → 0 and ζ ε , ξ ε →x Γ as ε → 0. Subtracting the two above inequalities and sending ε to 0, we obtain that u μ Γ (x Γ ) − w Γ (x Γ ) = M μ Γ ≤ 0, which contradicts (3.35).

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Hamilton-Jacobi equations for optimal control Page 25 of 42 23 Recall that the second inequality holds since M μ Γ > 0 and the last one comes from the fact that u μ Γ satisfies in the viscosity sense. This implies that Letting ε → 0 and applying (3.36), we get M μ Now we apply similar arguments as in the proof of Case A by considering the function Ψ k,ε : P k × P k → R which is defined in (3.28) with the index i replacing the index k. Remark that we may assumex k =x Γ sincex Γ is a maximum point of u μ k − w k by (3.39 there exists a subsequence of {y ε } (still denoted by {y ε }) such that w k (y ε ) ≥ w Γ (y ε ). Letting ε → 0, one gets w k (x Γ ) ≥ w Γ (x Γ ). In the other hands, letting ε → 0 in (3.37), one also gets w Γ (x Γ ) ≥ w k (x Γ ) + c k (x Γ ) > w k (x Γ ) which leads to a contradiction. Finally, the two last cases are proved by using the same arguments as in the proofs of Case A.2 and Case A.3.

Hamilton-Jacobi system under a moderate controllability condition near the interface
In this section we derive the Hamilton-Jacobi system (HJ) associated with the above optimal control problem and prove that the value function given by (2.2) is the unique viscosity solution of that (HJ) system, under the condition [Ã3] below, which is weaker that the strong controllability condition [A3] used above.