A design of strategies in alternative pursuit games

In this work we consider the games where $P$ can terminate pursuit at will on any of two terminal manifolds. If the optimal feedback strategies for every variant of termination are known, an obvious pursuit strategy assigns the control that corresponds to the alternative with less value at every state. On the manifold with equal alternative values, this strategy may become discontinuous even when the value functions themselves are smooth. We describe smooth approximations for the minimum functions that allow to construct smooth alternative strategies and to deal with generalized solutions for differential equations with discontinuous right-hand sides. However, as shown by an example, the state may stay on a equivalued manifold and the game never terminates.