Numerical Methods for Solving Differential Games, Prospective Applications to Technical Problems

Abstract

The majority of numerical methods for solving differential games is based on the backward procedure for computing the solvability sets [1–3]. The solvability set comprises all initial positions of a game such that one player can bring the state vector to a terminal target set for any possible actions of the opposite player. In the case of linear differential games, each step of the backward procedure consists of two operations with convex polyhedra: finding both an algebraic sum and a geometric difference of polyhedra. For two-or three-dimensional state space, these operations were implemented by the authors in terms of computing the convex hull of some piecewise-linear “almost” convex function. For this low-dimensional case, especially effective algorithms, which use information about local violation of convexity, have been created [4]. These algorithms were applied to various problems of an aircraft guidance in the presence of wind disturbances [5–7]. The way used for solving the above mentioned problems includes such steps as linearization of the corresponding nonlinear dynamic system with respect to some nominal trajectory, solving an auxiliary linear differential game, computing optimal strategies, and simulating the nonlinear dynamic system, using the strategies obtained from the auxiliary linear differential game.