Convergence of the solutions of the discounted equation: the discrete case

Albert Fathi (1)

1 UMPA-ENSL - Unité de Mathématiques Pures et Appliquées

We derive a discrete version of the results of Davini et al. (Convergence of the solutions of the discounted Hamilton–Jacobi equation. Invent Math, 2016). If M is a compact metric space, c:M×M→ℝ a continuous cost function and λ∈(0,1), the unique solution to the discrete λ-discounted equation is the only function uλ:M→ℝ such that∀x∈M,uλ(x)=miny∈Mλuλ(y)+c(y,x).We prove that there exists a unique constant α∈ℝ such that the family of uλ+α/(1−λ) is bounded as λ→1 and that for this α, the family uniformly converges to a function u0:M→ℝ which then verifies∀x∈X,u0(x)=miny∈Xu0(y)+c(y,x)+α.The proofs make use of Discrete Weak KAM theory. We also characterize u0 in terms of Peierls barrier and projected Mather measures.

Domaines: Mathématiques [math]

Albert Fathi. Convergence of the solutions of the discounted equation: the discrete case. Mathematische Zeitschrift, 2016, ⟨10.1007/s00209-016-1685-y⟩. ⟨ensl-01411575⟩