Game-Theoretic Approximate Solution of Nonlinear Optimal Control Problems

This paper proposes an approximate solution to minimum-energy nonlinear optimal control problems employing theory of differential games. Nonlinear systems whose state dynamics are separable into linear and nonlinear subdynamics are considered. We propose to cast this optimal control problem as a two-player zero-sum game in which linear subdynamics minimize a quadratic cost functional whereas nonlinear sub-dynamics maximize this cost functional. As a result of this game-theoretic rendering of nonlinear optimal control problem, the Hamilton-Jacobi-Bellman equation is replaceable with a functional equation similar in form to the state-dependent Riccati equation. We have shown through simulation that our proposed game-theoretic functional equation out performs the state-dependent Riccati equation which is the most popular alternative of the Hamilton-Jacobi-Bellman equation.