Numerical methods for solving differential games with nonterminal payoff

Linear convex positional differential games with geometric constraints on control actions and nonterminal payoff which evaluates a norm of a set of motion deviations at given instants of time from given target points are considered. Cases when the saddle point in a small game is either present or absent together with possible presence of additional integral constraints on control actions are studied. In each of these cases numerical methods for calculating the game value in appropriate classes of strategies and for constructing corresponding optimal control laws are elaborated. Numerical methods are based on backward constructions of upper convex hulls of auxiliary program functions. Domains of these functions are approximated by a pixel method, functions are stored as tables, upper convex hull is computed approximately as a lower envelope of a finite family of supporting hyperplanes to subgraphs of these functions. Details of software implementation for modern computational systems are discussed. Results of simulations in model examples are given.