On ergodic problem for Hamilton-Jacobi-Isaacs equations

Piernicola Bettiol

ESAIM: Control, Optimisation and Calculus of Variations (2005)

  • Volume: 11, Issue: 4, page 522-541
  • ISSN: 1292-8119

Abstract

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We study the asymptotic behavior of λ v λ as λ → 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 ∈ B max a ∈ A { - f ( x , a , b ) · 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 ⊂ ℝ n with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary) and the case of state constraints boundary conditions. Under the uniform approximate controllability assumption of one player, we extend the uniform convergence result of the value function to a constant as λ → 0 + to differential games. As far as state constraints boundary conditions are concerned, we give an example where the value function is Hölder continuous.

How to cite

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Bettiol, Piernicola. "On ergodic problem for Hamilton-Jacobi-Isaacs equations." ESAIM: Control, Optimisation and Calculus of Variations 11.4 (2005): 522-541. <http://eudml.org/doc/245460>.

@article{Bettiol2005,
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)\[ \hspace*\{-71.13188pt\}\lambda v\_\lambda + H(x,Dv\_\lambda )=0, \]with\[ \hspace*\{-71.13188pt\}H(x,p):=\min \_\{b\in B\}\max \_\{a \in A\} \lbrace -f(x,a,b)\cdot p -l(x,a,b)\rbrace . \]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 $\Omega \subset \{\mathbb \{R\}\}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary) and the case of state constraints boundary conditions. Under the uniform approximate controllability assumption of one player, we extend the 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.},
author = {Bettiol, Piernicola},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Hamilton-Jacobi-isaacs equations; viscosity solutions; asymptotic behavior; differential games; boundary conditions; ergodicity; Abelian-Tauberian theorem; Skorokhod problem},
language = {eng},
number = {4},
pages = {522-541},
publisher = {EDP-Sciences},
title = {On ergodic problem for Hamilton-Jacobi-Isaacs equations},
url = {http://eudml.org/doc/245460},
volume = {11},
year = {2005},
}

TY - JOUR
AU - Bettiol, Piernicola
TI - On ergodic problem for Hamilton-Jacobi-Isaacs equations
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2005
PB - EDP-Sciences
VL - 11
IS - 4
SP - 522
EP - 541
AB - 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)\[ \hspace*{-71.13188pt}\lambda v_\lambda + H(x,Dv_\lambda )=0, \]with\[ \hspace*{-71.13188pt}H(x,p):=\min _{b\in B}\max _{a \in A} \lbrace -f(x,a,b)\cdot p -l(x,a,b)\rbrace . \]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 $\Omega \subset {\mathbb {R}}^n$ with sufficiently smooth boundary. As far as the latter is concerned, we treat both the case of the Neumann boundary conditions (reflection on the boundary) and the case of state constraints boundary conditions. Under the uniform approximate controllability assumption of one player, we extend the 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.
LA - eng
KW - Hamilton-Jacobi-isaacs equations; viscosity solutions; asymptotic behavior; differential games; boundary conditions; ergodicity; Abelian-Tauberian theorem; Skorokhod problem
UR - http://eudml.org/doc/245460
ER -

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