Differential Games and Zubov's Method

In this paper we provide generalizations of Zubov’s equation to differential games without Isaacs’ condition. We show that both generalizations of Zubov’s equation (which we call min-max and max-min Zubov equation, respectively) possess unique viscosity solutions which characterize the respective controllability domains. As a consequence, we show that under the usual Isaacs condition the respective controllability domains as well as the local controllability assumptions coincide.