On game problems for second-order evolution equations

Abstract

1. In this paper, we consider certain problems of the theory of differential games in systems with distributed parameters. The players influence on the systemwith the use of control parameters contained in the right-hand side of the equation. Controls of players are chosen in the form of functions on which various constraints are imposed, so-called geometric, integral, and mixed constraints. Note papers [1]– [14] devoted to this research area. In the first three games, the goal of the first player is to bring the system into an unperturbed state. In the fourth game, the goal of the first player is to bring the system and its velocity into an arbitrary neighborhood of zero. The second player in all the games has the opposite goal. We present conditions (see below) which are sufficient in order that the first player can reach the goal in a finite time. For the third game, we also consider the encounter-evasion problem (see Proposition in Item 4). Consider in the space L2(Ω), where Ω is a bounded domain with piece-wise smooth boundary inRn, a differential operator A of the form