On ergodic problem for Hamilton-Jacobi-Isaacs equations

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 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