A Fixed-Point Theoretic Solution of the Hamilton-Jacobi Equation and Its Application to Nonlinear H∞ Control

This paper proposes a novel approach to the Hamilton-Jacobi equation (HJE). Through a functional analytic formulation, a stabilizing solution of the HJE is derived from a fixed-point of a nonlinear mapping on a function space. A sufficient condition for the existence of such a fixed-point is derived quantitatively. For computing the fixed-point, a numerical method based on optimization techniques is introduced. Based on these results, the nonlinear H∞ control problem is reconsidered. Simple numerical examples show that the resulting controllers have enough performance and computed solutions can approximate their analytical counterparts almost uniformly in the entire region under consideration.