Identification and Construction of Singular Surfaces in Pursuit-Evasion Games

The existence of singular surfaces in pursuit-evasion games is a rule rather than an exception. Identification and construction of these surfaces, which indicate some discontinuity or other singular phenomenon, is an inherent part of the game solution. Recently, a general approach, based on the method of singular characteristics, was developed for constructing singular paths and surfaces in optimal control and differential games, as well as for the generalized (viscosity) solutions of first-order PDEs. In the present paper a brief description of this technique is given and the procedure of identification and construction for several main singularities is demonstrated using two-parametric first-order PDEs. The existence of the different types of singular manifolds, which depend on the parameters, are identified, including dispersal, equivocal, and focal surfaces. The method above is also applied to a linear pursuit-evasion game with elliptical vectograms.