Sequential quadratic methods for differential games

Abstract

We present an iterative method for computing a Nash solution to a zero-sum differential game for a system of nonlinear differential equations. Given a solution estimate, we define a subproblem, which approximates the original problem around the solution estimate. This subproblem has a quadratic cost and a quadratic system dynamics. We propose to replace the subproblem with another subproblem which has a quadratic cost and a linear dynamics. Because the latter subproblem has only a linear dynamics, we can now apply a Riccati equation method and compute the Nash solution to the subproblem. We then add this Nash solution to the current solution estimate for the original game and the sum becomes a new solution estimate for the original game. We repeat this process and successively generate better solution estimates that converge to the Nash solution of the original differential game. Our experiments show that the solution estimates converge to the Nash solution of the original game. Furthermore, because the second subproblem still reflects the quadratic dynamics, we expect and observe faster convergence from the iterative method than other methods based on lower order approximations.