Viscosity solutions of the Isaacs equation οn an attainable set

We apply a modification of the viscosity solution concept introduced in [8] to the Isaacs equation defined on the set attainable from a given set of initial conditions. We extend the notion of a lower strategy introduced by us in [17] to a more general setting to prove that the lower and upper values of a differential game are subsolutions (resp. supersolutions) in our sense to the upper (resp. lower) Isaacs equation of the differential game. Our basic restriction is that the variable duration time of the game is bounded above by a certain number T>0. In order to obtain our results, we prove the Bellman optimality principle of dynamic programming for differential games.