Two evasion problems in a plane with separate controls and non-convex terminal sets

Abstract

Two different pursuit-evasion games are considered from the evader's point of view. The phase space is a plane, each of the two players controlling the motion of a point only along its own coordinate. The terminal sets are not convex; in the first problem, the set is an arc of a circle, in the second, the union of tow segments. In both games evasion cannot the achieved by means of programmed controls, but it can be achieved using feedback control. However, the strategies, which are continuous functions of the phase vector, have different properties in each problem. In the first, they cannot guarantee evasion (which is typical for the linear-convex case as well), but in the second they can (which is impossible in linear-convex games with a fixed final time). Verification that evasion is unachievable using such strategies reduces here to proving the solvability of a certain initial-value problem for an advanced differential equation, to which the Schauder principle is applicable.