Guaranteed strategies with memory for alternative pursuit

Abstract

A pursuit game is called alternative if, at any state of the game, a pursuer can terminate it on any of two given target sets and the Boltz payoff function differs on these sets only in the terminal part. By assumption, the optimal universal feedback strategies of the pursuer are known for each fixed terminal alternative. Endeavoring to decrease its payoff via the choice of an alternative, the pursuer may compare guaranteed results (e.g., at the initial and current states) and apply one of two fixed-alternative strategies. If both players optimize the choice of their alternatives depending on the position, the state of the pursuit game can enter some neighborhood of a hypersurface with equivalent alternatives (in the sense of payoffs) and stay there for a finite period of time. In different formalizations, the solutions generated by an appropriate differential equation with discontinuous right-hand side (ideal solutions, limits of Euler polygonal lines, generalized solutions) lead to different outcomes. For certain states, they even increase the payoff in comparison with the pursuit strategy based on the preferable alternative chosen at the initial state (due to sliding mode occurrence). The author introduces a method of constructing pursuit strategies with memory and a finite number of admissible switchings for the target alternative. The proposed method allows estimating the guaranteed result of the pursuer by confining to the solution concept of the limits of Euler polygonal lines.