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On ergodic problem for Hamilton-Jacobi-Isaacs equations

Published online by Cambridge University Press:  15 September 2005

Piernicola Bettiol
Piernicola Bettiol*
Affiliation:
SISSA/ISAS  via Beirut, 2-4 - 34013 Trieste, Italy; bettiol@ma.sissa.it
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Abstract

We study the asymptotic behavior of $\lambda v_\lambda$ as $\lambda\rightarrow 0^+$ , where $v_\lambda$ is the viscosity solution of the following Hamilton-Jacobi-Isaacs equation (infinite horizon case) \[ \lambda v_\lambda + H(x,Dv_\lambda)=0,\] 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 $\Omega\subset{\xR}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat boththe case of the Neumann boundary conditions(reflection on the boundary) andthe case of state constraints boundary conditions.Under the uniform approximate controllability assumption of one player, we extendthe uniform convergence result of the value function to a constant as $\lambda\rightarrow 0^+$ to differential games.As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous.

Keywords

Hamilton-Jacobi-Isaacs equationsviscosity solutionsasymptotic behaviordifferential gamesboundary conditionsergodicity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 4 , October 2005 , pp. 522 - 541
Copyright
© EDP Sciences, SMAI, 2005

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