Deterministic Differential Games in Infinite Horizon Involving Continuous and Impulse Controls

We consider a two-player zero-sum deterministic differential game where each player uses both continuous and impulse controls in infinite-time horizon. We assume that the impulses supposed to be of general term and the costs depend on the state of the system. We use the dynamic programming principle and viscosity solutions approach to show existence and uniqueness of a solution for the Hamilton-Jacobi-Bellman-Isaacs (HJBI) partial differential equations (PDEs) of the game. We prove under Isaacs condition that the upper and lower value functions coincide.