Computation of open-loop solutions for zero-sum differential games by parametrization

Abstract

Given a dynamical system controlled by two parties aiming at diametrically opposite goals — mathematically such a conflict situation is formulated as zero-sum differential game — and an initial state, regard the time histories of the controls when both parties do the best they can do. In this paper an algorithm is presented to compute them numerically. For this purpose the search for the “optimal” open-loop controls corresponding to a given initial state is restricted to a finite dimensional class of parametrized control functions. To solve the resulting saddlepoint problem in the parameter space an implementation of Wilson-Han-Powell's sequential quadratic programming method for solving general nonlinear optimization problems is used doing some iteration steps alternating between the parameter vectors of both parties. This procedure is continued until both optimizations are completed and thus ends up with a solution of the parametrized problem. To demonstrate the feasibility of the algorithm it is used to operate solutions for the “Homicidal Chauffeur”, a classical pursuit-evasion problem in the plane, which can be solved by hand. A second example will be a rather complex and realistic aircraft pursuit-evasion problem where an exact analytical solution is not found.