Complex differential games of pursuit-evasion type with state constraints,part 2: Numerical computation of optimal open-loop strategies

In Part 1 of this paper (Ref. 1), necessary conditions for optimal open-loop strategies in differential games of pursuit-evasion type have been developed for problems which involve state variable inequality constraints and nonsmooth data. These necessary conditions lead to multipoint boundary-value problems with jump conditions. These problems can be solved very efficiently and accurately by the well-known multiple-shooting method. By this approach, optimal open-loop strategies and their associated saddle-point trajectories can be computed for the entire capture zone of the game. This also includes the computation of optimal open-loop strategies and saddle-point trajectories on the barrier of the pursuit-evasion game. The open-loop strategies provide an open-loop representation of the optimal feedback strategies. Numerical results are obtained for a special air combat scenario between one medium-range air-to-air missile and one high-performance aircraft in a vertical plane. A dynamic pressure limit for the aircraft imposes a state variable inequality constraint of the first order. Special emphasis is laid on realistic approximations of the lift, drag, and thrust of both vehicles and the atmospheric data. In particular, saddle-point trajectories on the barrier are computed and discussed. Submanifolds of the barrier which separate the initial values of the capture zone from those of the escape zone are computed for two representative launch positions of the missible. By this way, the firing range of the pursuing missile is determined and visualized.This paper is dedicated to the memory of Professor John V. Breakwell.The authors would like to express their sincere and grateful appreciation to Professors R. Bulirsch and K. H. Well for their encouraging interest in this work.