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Pré-Publication, Document De Travail Année : 2016

CONVERGENCE OF THE SOLUTIONS OF THE DISCOUNTED EQUATION: THE DISCRETE CASE

Résumé

We derive a discrete version of the results of our previous work. 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.
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Dates et versions

hal-04502601 , version 1 (13-03-2024)

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  • HAL Id : hal-04502601 , version 1

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Andrea Davini, Albert Fathi, Renato Iturriaga, Maxime Zavidovique. CONVERGENCE OF THE SOLUTIONS OF THE DISCOUNTED EQUATION: THE DISCRETE CASE. 2024. ⟨hal-04502601⟩
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