Vanishing viscosity limits for space-time periodic Hamiltonian-Jacobi-Bellman equations

Extending previuos results, we study the vanishing viscosity limit of solutions of space-time periodic Hamiltonian-Jacobi-Belllman equations, assuming that the"Aubry set"is the union of a finite number of hyperbolic periodic orbits of the Hamiltonian flow.

For our main result we will assume that M is the d-dimensional torus T d and the Hamiltonanin satisfies the following growth condition: There is K > 0 such that for all x, p, t with |p| ≥ K (1) (H p · p − H + inf (x,t) H(x, 0, t))K − |H x | ≥ 0.
A natural example that satisfies the hypotheses is given by with V ∈ C k (T d+1 ), since in this case the flow is complete and hypothesis (1) reads For c ∈ R consider the Hamilton-Jacobi equation (2) u t + H(x, Du, t) = c.
It is know ( [CIS]) that there is only one value c = c(L), the so called critical value, such that (2) has a time periodic viscosity solution.
Consider also the Hamilton-Jacobi-Bellman equation (3) φ t + ε∆φ(x, t) + H(x, Dφ(x, t), t) = c(ε) As for equation (2), there is only one constant c(ε) such that (3) admits solutions. However, this time the solution is unique up to an additive constant ( [BS]), and we denote it φ ε . We study the behaviour of φ ε as ε goes to zero. We prove that the family (φ ε ) ε>0 is equi-Lipschitz, so that we can extract subsequences which converge uniformly (see Lemma 10). By the stability theorem for viscosity solutions ( [CEL], [Ba], [BCD]), limits as ε → 0 of such subsequences have to be viscosity solutions of equation (2). We shall assume that the "Aubry set" (section 1) of the Lagrangian system, is the union of a finite number of hyperbolic periodic orbits of the Euler-Lagrange flow. Extending previuos results ( [AIPS], [Be]), we describe the limits in terms of the orbits in the Aubry set that minimize, the normalized integral along the orbit, of the Laplacian of the corresponding "Peierls barrier" (section 1). In particular we prove that the limit is unique if there is only one orbit in the Aubry set that minimizes that normalized integral. For a ≤ b, x, y ∈ M let C(x, a, y, b) be the set of absolutely continuous curves γ :

Preliminaries and statement of main result
For t ∈ R let [t] be the corresponding point in S 1 and t be the integer part of t. Define the action potential Φ : We have −∞ < Φ ≤ h < ∞. The Lagrangian is called regular if the lim inf in (4) is a lim and in that case, for each s, t ∈ R the convergence of the sequence The critical value is the unique number c such that (2) has viscosity solutions u : M × S 1 → R. In fact ( [CIS]), for any p the functions z → h(p, z) and z → −h(z, p) are respectively backward and forward viscosity solutions of (2). Set c = c(L) and let S − (S + ) be the set of backward (forward) viscosity solutions of (2). A subsolution of (2) always means a viscosity subsolution.
If u ∈ S − (S + ), for any ( For such a pair (u − , u + ), we define I(u − , u + ) as the set where u − and u + coincide.
) ∈ M and u be a viscosity subsolution of (2), then for any t ≤ τ we have : (x, [s]) ∈ I(u − , u + )}. We may define the Aubry set either as the set [B] A or as its pre-image under the Legendre transformation [F] The projection of either Aubry set in M × S 1 is Consider the natural projections Pr : T * M × S 1 → T * M , π * : T * M → M . For u a subsolution of (2), we define calibrates u in R} Lemma 3. If u : M × S 1 → R is a subsolution of (2), there is a conjugated pair (u − , u + ) such that u + ≤ u ≤ u − .
It follows from Proposition 4 that for a conjugated pair (u − , u + ) given by Proposition 5 we have A * = I * (H, u − ) = I * (H, u + ).
In this article we assume the Aubry set A * is the union of the hyperbolic periodic orbits Γ * i (t), i ∈ [1, m] of the Hamiltonian flow with periods N i , i ∈ [1, m]. As shown in [CIS] viscosity solutions are completely determined by taking one value in each projected orbitγ Conversely, if this necessary condition is satisfied, then there is a unique φ ∈ S + having these prescribed values. In fact it is given by Because the orbit Γ i is hyperbolic, we have that h i (z) = h(z,x i ) is C 2 in a neighbourhood of the projected orbit. A proof of this fact is similar to that of Proposition 7 below. Define Our main result is THEOREM. Let M = T d and assume H satisfies (1). Suppose φ εn converges to φ 0 for a sequence ε n → 0, then In particular, if there is just one orbit γ I such that λ I =λ, then the solutions φ ε of (3), normalized by φ ε (x I , 0) = 0, converge uniformly to −h I as ε → 0.
Example 1. For k ∈ N let V : R → R be a 1/k-periodic function with nondegenerate maxima 0 ≤ x 1 , . . . , x N < 1 k , max V = 0. Consider the mechanical Hamiltonian and Lagrangian whose projected Aubry set consists of the hyperbolic fixed points Let h a be the Peierls barrier for H a , then φ(

Consider now the time periodic Hamiltonian
with corresponding Lagrangian The projected Aubry set consists of the hyperbolic k-periodic orbitsγ

Critical subsolutions
An important tool for the proof of THEOREM is the existence of strict C k critical subsolutions in our setting, that extends the result of Bernard [B] for the autonomous case Theorem 6. Let H : T * M × R → R be C k satifying the standard hypotheses. Assume the Aubry set A * is the union of a finite number of hyperbolic periodic orbits Γ * i of the the Hamiltonian flow, then there is a C k subsolution u of (2) such that Being a minimizer, Γ * i has no pair of conjugated points and writing θ i = Pr •Γ * i , we have from [CI] i is locally the graph of a C k−1 transformation. Proposition 7. The functions u ± ∈ S ± given by Proposition 5 are C k in a neighborhood of A.
We have the following Lemma for u − , and its analogue for u + We can assume by taking a subsequence that Z * n (T n ) converges to (y, w, [τ ]) and Υ n converges uniformly on compact sets to a limit trajectory Υ : To get Theorem 6 the main tool is the following Lemma proved by D. Masart Lemma 9. [Ma] There is a C 2 non-negative function W : M × S 1 → R, positive outside A(L) and zero inside A(L) such that c(L − W ) = c(L) and A(L − W ) = A(L).
It follows that for any function V : M × S 1 → R, positive outside A(L) and such that 0 ≤ V ≤ W we have c(L − V ) = c(L) and A(L − W ) = A(L). Function V can be chosen so flat on A(L) that the linearized Hamiltonian flow along the orbits Γ * i is the same for H and H + V . As a consequence, the orbits Γ * i remain hyperbolic as orbits of H + V . Applying Proposition 7 to H + V we obtain a solution of the Hamilton-Jacobi equation . This function is a subsolution of (2) which is strict outside A(L) and can be regularized to a C k subsolution of (2). Smoothing in this kind of context has been done in [F], Theorem 9.2.
Proof. We first observe that Indeed, if φ ε has a minimum at (x,t), H(x, 0, t).
Assuming we have this bound we prove that ψ ε is uniformly bounded. Since Dφ ε = −ψ ε Dw ε , we get that Dφ ε is uniformly bounded.
The solution to the viscous equation (3) can be characterized by a variational formula. We need to introduce a probability space (Ω, B, P) endowed with a brownian motion W (t) : Ω → T d on the flat d-torus. We denote by E the expectation defined by the probability measure P.
The solution to equation (3) satisfies Lax's formula where v is an admissible progressively measurable control process, τ is a bounded stopping time and X ε is the solution to the stochastic differential equation See [Fl] Lemma IV 3.1.
Lemma 11. The periodic solutions φ ε of (3) are uniformly semiconvex in the spatial variable.
Proof. We have the following description of the optimal v, see for example [Fl] Theorem IV 11.1.: Introduce the time dependent vector field U ε (x, t) = H p (x, Dφ ε (x, t), t) and consider the solution X ε (s) of the stochastic differential equation then an optimal control in (17) is given by the formula v(s) = U ε (X ε (s), s). Let x ∈ T d , t ∈ [−2, −1] and take that optimal control, then Let |y| < 1 be an increment, the controls U ε (X ε (s), s) ± y t are admissible and then

This is finite by Lemma 10. Define now
An application of Taylor's Theorem gives We will need the following Lemma 12. Suppose the sequence φ εn of solutions of (3) converges uniformly to φ 0 . Assume that φ 0 is differentiable in an open neighborhood V of a periodic orbit. Then Dφ εn converges to Dφ 0 uniformly in every compact subset of V .
This is an easy consequence of Lemma 11 and next theorem, which is a slight extension of Theorem 25.7 in [R] and follows the same proof Theorem 13. Let A ⊂ R n be open convex, B ⊂ R m be open, and f n : A × B → R be a sequence of differentiable functions, convex in the first variable, converging uniformly to a differentiable function f . Then D 1 f n converges pointwise to D 1 f , and in fact uniformly on compact subsets.

Reduction to a regular Lagrangian
In this section we show how to deduce THEOREM from the case when the Lagrangian is regular.
Let N be the least common multiple of the periods of the orbits Γ 1 , . . . , Γ m of the Aubry set. Define , and the Lagrangian L N = L • P N . The corresponding Hamiltonian is given by For a curve γ : m]. According to sections 3, 5 of [B1], the Aubry set of L N is the union of the hyperbolic 1-periodic orbits Γ [t]) and L N is regular. For ε ≥ 0, a function u : T d × R → R is a viscosity solution of (3) if and only if w(x, t) = 1 N v(x, N t) is a viscosity solution of (21) w t + N ε∆w + H N (x, Dw, t) = c(ε).
for k ∈ N, there are in fact only N i accumulation values and then From Lemma 14, in a neighborhood of (γ Proof of THEOREM for L assuming it holds for L N . Let φ ε be a periodic solution of (3) and suppose φ εn converges to φ 0 for a sequence ε n → 0. The solutions From (22), THEOREM for L N and Lemma 14,

Regular Lagrangians
In this section we assume that the Lagrangian is regular. Let f : T d+1 → R be a strict C k subsolution of (2) given by Theorem 6 and consider the Lagrangian . Thus L and L have the same Euler Lagrange flow and projected Aubry set and the Peierls barrier of L is h(z, w) − f (w) + f (z). Moreover, and u is a viscosity solution of (2) if and only if u − f is a viscosity solution of Lemma 15. Assume the Lagrangian also satisfies (23). A φ ∈ S + has a local maximum atγ i if and only if From the continuity of φ and h, if condition (25) holds, there is a neighbourhood Let γ n : [0, n] → T d be a curve joining x i to x j such that Let t n ∈ [0, n] be the first exit time ofγ n (t) out of V , andγ n (t n ) be the first point of intersection with ∂U j . As n goes to infinity, t n and n − t n tend to infinity. This follows from the fact thatγ n (0) has to tend toγ i (0), andγ n (n) has to tend toγ j (0). To justify this, consider v a limit point ofγ n (0), and γ : R → T d the solution to the Euler-Lagrange equation such that γ(0) = x i ,γ(t) = v. From the fact that and the regularity of L, taking limit n → ∞ it follows so that the curve obtained by gluing γ i | [−1,0] with γ| [0,1] minimizes the action between its endpoints. In particular, it has to be differentiable, thus v =γ(0) =γ i (0). Let (y, w, τ ) be a cluster point of (γ n (t n ),γ n (t n ), t n − t n ). From the fact and the uniform convergence of F a,b when b − a → ∞, we obtain This contradiction shows that (26) can not happen.
Corollary 16. Assume the Lagrangian also satisfies (23). Let φ ∈ S + and B = {i :γ i is a local maximum of φ}. Then Thus k = i. We continue until we arrive to l ∈ B with We now take assumption (23) out. Recall that f : T d+1 → R is a strict C k subsolution of (2).
Proof. Let i ∈ B and apply Lemma 15 to the Lagrangian L to get a neighborhood Applying Corollary 17 we have Proof. We will prove that lim inf for an arbitrary r > 0. Take I with λ I =λ and let Φ be a C 3 function that coincides with −h I = −h(·,x I ) in a neighbourhood V ofγ I . Defining U (x, t) = H p (x, DΦ(x, [t]), t), we have thatγ I is a attractive periodic orbit of the vector field (U (x, t), 1). Let X ε be the solution to To easy notation we writeX ε (t) = (X ε (t), t) Let δ > 0 be sufficiently small to have δ Φ C 3 ≤ r and B δ (γ I ) := {(x, [t]) : d(x, γ I (t)) ≤ δ} ⊂ V , and define the stopping time (28) τ (ω) = min{s > 0 : d(X ε (s, ω), γ I (s)) ≥ δ}.
Reasoning as in the proof of Lemma 18, we get Ni 0 ∆h i (γ i (s))ds − 2N i ||h i || C 2 (Vi) E(τ ∧ κ) and we can now pass to the limit κ → +∞ to get By Lemma 12, (u εn ) converges uniformly to H p (x, Dφ 0 (x, t), t) in the neighborhood V i ; the estimate of Freidlin and Wentzell for E(τ ) also applies ( [FW], Chapter 5.3): m = lim inf n→∞ ε n log E(τ ) > 0, and so, letting n grow we obtain lim sup n→∞ c(ε n ) − c(0) ε n ≤ −λ i + r, which, by our choice or r, is possible only if λ i =λ.
Lemma 19 and Corollary 17 imply THEOREM for a regular Lagrangian.