Mathematics

We study the asymptotic behavior as $\lambda\rightarrow 0^+$ of $\lambda v_\lambda$, 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 case in which the state of the system is required to stay in the closure of a bounded domain $\Omega\subset{\Bbb R}^n$ with sufficiently smooth boundary (for example $\partial \Omega\in {\cal C}^2$). Under the uniform approximate controllability assumption of one player, we extend the convergence result of the term $\lambda v_\lambda (x)$ to a constant as $\lambda\rightarrow 0^+$ to Differential Games . We also show how to contruct nonanticipative strategies which satisfies some "good" estimates in order to obtain Holder regularity of the value function.