Generalized differential games (Q6078091)

In this paper, a generalized differential game is a two-player differential game, with a constraint on the controls of the players. More precisely, if \(\eta(\tau)\) and \(\xi(\tau)\) respectively denote the controls of the players at time \(\tau\), the authors consider a constraint of type \(g(\eta(\tau),\xi(\tau))\leq 0\) for all \(\tau\), where \(g\) is a continuous function. They show that, under standard assumptions, both the upper and lower value function are solution in viscosity sense of some Hamilton-Jacobi-Isaacs'-type equation. It follows that, under some also adapted Isaacs' assumption, the game has a value. Finally it is shown that the generalized differential game is the limit of a sequence of standard differential games with penalization.