Ergodic problem for the Hamilton-Jacobi-Bellman equation. II

Mariko Arisawa

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We study the asymptotic behavior of $λ v_{λ}$ as $λ \to 0^{+}$ , where $v_{λ}$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) $λ v_{λ} + H (x, D v_{λ}) = 0,$ with $H (x, p) : = min_{b \in B} max_{a \in A} {- f (x, a, b) \cdot p - l (x, a, b)} .$ We discuss the cases in which the state of the system is required to stay in an $n$ -dimensional torus, called periodic boundary conditions, or in the closure of a bounded connected domain $Ω \subset ℝ^{n}$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case...

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Geometric restrictions for the existence of viscosity solutions

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