Pontryagin First Direct Method for Differential Inclusions

摘要

Pontryagin direct methods are of great importance in the development of the theory of differential games and its application to specific applied problems. It turned out to be useful in control theory under conditions of uncertainty, also in solving the problem of control synthesis. Numerous studies deal with the corresponding theory. Direct methods have proved themselves as an effective means for solving problems of pursuit and control. Pontryagin direct methods consider integrals, having a number of significant differences from the classical integral. One of the differences consists in the use of multi-valued mapping. The second difference is connected with the application of the geometric difference (the Minkowski difference) and the intersection of sets in this operation. In this connection, some difficulties arise in these integrals calculation. In this paper, we consider a differential game described by differential inclusions z ∈ −F(t, ν), where F is continuous compactvalued mapping. The first direct method deals with such classes of games. In particular, the class of stroboscopic strategies of the pursuer, the trajectory of the system are determined. For these classes of games, it is proved that if the starting point belongs to the first integral (the integral from the multivalued (compact-valued) mapping, which is present in the definition of the first direct method, then this is the necessary and sufficient condition for completing the game at a fixed point in time in the class of stroboscopic strategies. Schemes for the approximate calculation of the integral of the first direct method are proposed. The approximation properties of this integral are studied and the stability of these integrals with respect to initial data of the differential game is proved. It is shown that the first integral is stable for unilateral perturbations.