Relative optimality in nonlinear differential games with discrete control

Abstract

Two control problems with an obstacle that is the second player in a differential game are considered. The dynamics in the first problem is described by a nonlinear system of differential equations of the first order, whereas the dynamics in the second is described by a nonlinear system of differential equations of the second order. A piecewise constant control with finite set of values is used. The control is aimed at moving arbitrarily closely to a finite trajectory described by an auxiliary control system of simple form, for any actions of the obstacle. For both problems phase constraints on the auxiliary system under which the control of the auxiliary system can be arbitrary are obtained. For any neighbourhood and any control of the auxiliary system satisfying these constraints, there are admissible controls in the original problems ensuring that at each moment of time the phase point of the original system is in the indicated neighbourhood of the corresponding phase point of the auxiliary system. Thus, in view of the above constraints, when the control of the auxiliary system is chosen to be optimal in a certain sense, the original system can move arbitrarily closely to such a solution of the auxiliary system for any actions of the obstacle. Bibliography: 29 titles.