A Zero-Sum Differential Game in a Finite Duration with Switching Strategies

Abstract

A zero-sum differential game of finite horizon with both players using switching controls is studied. Positive switching costs are associated with each player. Under some suitable conditions, it is proved that the Elliot–Kalton upper and lower value functions of the game are the unique viscosity solution of the same Isaacs’ equation, which turns out to be a system of evolutionary quasi-variational inequalities with bilateral obstacles. The existence of the Elliot–Kalton value of the game then follows. Some limiting cases are also discussed.