A numerical method based on the complementarity and optimal control formulations for solving a family of zero-sum pursuit-evasion differential games

Abstract

This paper presents an efficient numerical method for a class of Zero-Sum Pursuit-Evasion Differential Games (ZSPEDGs). The aim of the presented method is to resolve the drawbacks of the indirect methods in solving ZSPEDGs. In the indirect methods, the solution of ZSPEDG is found by solving a Two-Point Boundary Value Problem (TPBVP) derived from the necessary conditions. The indirect methods are accurate and fast. However, they are very sensitive to initial guess, and when the control is bounded, some non-smooth equations appear in the resulting TPBVP. These drawbacks restrict the use of indirect methods for solving ZSPEDGs. To overcome these drawbacks, at first, we reformulate the discontinuous equations of TPBVP by complementarity conditions, and as a result a Differential Complementarity System (DCS) is obtained. Then, the resulting DCS is reformulated to an optimal control problem, which can be solved by a well-developed direct or indirect method. The efficiency and robustness of the method are reported by means of two benchmarks and two real-life ZSPEDG problems.