State Constrained Two Player Differential Games with Decoupled Dynamics

We consider a two player zero-sum differential game with state constraints, in which the dynamics is decoupled: each player has to stay in a closed (nonempty) set. We prove that, under suitable assumptions, the lower and the upper values are locally Lipschitz continuous and we establish that they are solutions, in the viscosity sense, of the Hamilton–Jacobi–Isaacs equation, which involves an appropriate Hamiltonian, called inner Hamiltonian. We finally provide a comparison theorem. It follows that the differential game under consideration admits a value (which coincides with the lower and the upper values). A key step in our analysis is a new nonanticipative Filippov-type theorem, which is valid for general closed sets.