A value function criterion in a linear differential game

The paper considers linear differential games with a convex terminal payoff function. A value function criterion is obtained in the form of a single equality that can be interpreted as the Bellman-Isaacs equation in terms of directional derivatives. Two properties lie at the basis of the result: the u-stability of a level set of the value function, and the ‘repulsion’ property, which is stronger than the v-stability of the complement of a level set. The indicated properties also enable a relation to be obtained which characterizes the dynamics of the evolution of the sections of a level set of the value function (of a maximal stable bridge). Namely, an expression is written for the right-hand time derivative of the support function of sections of the bridge.