The absolute maximum payoff in differential games and optimal control

This paper deals with a differential game or optimal control problem in which the payoff is the maximum (or minimum), during play, of some scalar functionK of the statex. This unconventional payoff has many practical applications. By defining certain auxiliary games for a significant class of problems, one can show how to solve the general case where more than one maximum ofk(t)=K[x(t)] occurs under optimal play. For a subclass of such problems, it is found thatclosed optimal solutions can exist on certain surfaces in the playing space. As the playing interval becomes indefinitely long, the open optimal trajectories converge to (or diverge from) such surfaces. In particular, for two-dimensional problems of this subclass, the closed optimal trajectories are periodic and called periodic barriers. They are analogous to limit cycles in uncontrolled nonlinear systems.The writer is indebted to Dr. R. Isaacs for many stimulating conversations in connection with this study.