Guaranteed control in differential games with ellipsoidal payoff

Abstract

A class of antagonistic linear differential games (DGs) in a fixed time interval with ellipsoidal payoff functional is considered. This class of DGs includes problems which assume both rigid constraints on the players' controls and requirements to minimize control expenses. Other known classes of differential games, such as linear DGs with a quadratic performance index and linear DGs with ellipsoidal terminal sets and admissible sets of controls for the players, considered in Kurzhanskii's ellipsoidal technique, are limiting cases of DGs of this class. The concept of a u-strategic function, which expresses the property of u-stability for ellipsoidal functions, is introduced. An effective algorithm is presented for computing a u-strategic function, based on Kurzhanskii's ellipsoidal technique. The main result of this paper is that a guaranteed positional strategy for player u is defined by a certain explicit formula in terms of a u-strategic function. The proof of this result is based on a viability theorem for differential equations.