Algorithms for Differential Games with Bounded Control and States

摘要

Pursuit and Evasion problems are probably the most natural application of differential game theory and have been treated by many authors as such. Very few problems of this class can be solved analytically. Fast and efficient numerical algorithm is needed to solve for an optimal or near optimal solution of a realistic pursuit and evasion differential game. Some headways have been made in the development of numerical algorithm for this purpose. Most researchers, however, worked under an assumption that a saddle point exists for their differential game. Here, it is shown via two examples and a nonlinear stochastic differential game that such is not the case. A first-order algorithm for computing an optimal control for each player, subject to control and/or state constraints, is developed without the assumption of saddle point existence. It is shown that a linear quadratic differential game with control and/or state constraints generally cannot be solved analytically. One such problem is developed and solved by the above algorithm. A new rationalization is offered in formulating a missile anti-missile problem as a nonlinear stochastic differential game. The algorithm developed here together with a convergence control method introduced by Jarmark is used to solve the missile anti-missile problem with fast computation time. (Author)