A Zero-Sum Deterministic Impulse Controls Game in Infinite Horizon with a New HJBI-QVI.

In the present paper, we study a two-player, zero-sum, deterministic differential game with both players adopting impulse controls in infinite-time horizon, under rather weak assumptions on the cost functions. We prove by means of the dynamic programming principle that the lower and upper value functions are continuous and viscosity solutions to the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We define a new HJBI-QVI for which, under a proportional property assumption on the maximizing player cost, the value functions are the unique viscosity solution. We then prove that the lower and upper value functions coincide. [ABSTRACT FROM AUTHOR]