Level Sets Of The Value Function In Differential Games With The Homicidal Chauffeur Dynamics

Three differential games with the dynamics of the homicidal chauffeur are considered. The first problem is the Isaacs' homicidal chauffeur differential game. In this game, a pursuerPminimises the capture time of an evaderE. The objective of the evader is to prevent the capture or to maximise the capture time. The magnitude of the velocity is constant for the pursuer, and his maneuverability is bounded through a minimal turn radius. The maneuverability of the evader is not bounded. The pursuer's control is the rate of turn; the evader steers by choosing directions of his velocity. The main difference of the second problem is that the size of the constraint on the control parameter of the evader depends on the position of the game. The idea of such a modification was suggested by Bernhard. The third problem is a conic surveillance-evasion game studied by Lewin and Olsder. In this game, the dynamics is the same as in the Isaacs' problem, but the goals of the players differ from the classic formulation: an evaderEminimises the time of escaping from a detection set that is a two-dimensional semiinfinite cone. The detection set is attached to the velocity vector of a pursuerPwhose objective is to keep the evader within the detection set for maximal time. The paper describes the computation of level sets of the value functions for these games. The algorithm proposed by the authors is used. An analysis of families of semipermeable curves is carried out. The results of this analysis are used to check the correctness of the computation of level sets and to explain the appearance of holes in victory domains of the pursuer in the second problem.