On a method of extremal control

Abstract

An encounter game problem /1,2/ is analyzed on a prescribed time interval for controlled objects whose dynamics are described by nonlinear differential equations. It is assumed that the game's payoff is a convex function, differentiable in some domain, of the difference between the objects' final states. Under specific conditions a procedure is justified for the formation of an extremal strategy of one of the players, guaranteeing him a game result no worse than in the corresponding programmed maximin problem for the initial position. By example it is shown that in the case of nonlinear systems the procedure described in the paper for constructing the optimal strategy covers certain irregular situations in which the extremal aiming rule developed for linear /1/ and nonlinear /3/ controlled systems is inapplicable. In the case of linear systems the conditions found in the paper ensure the regularity of the encounter game problem and, as shown in /4/, the method proposed for solving the encounter problem occupies an intermediate position between the extremal aiming rule /1,2/ and the direct methods in differential game theory /2,5/.