An Iterative Algorithm for Solving Hamilton--Jacobi Type Equations

Abstract

Solutions of the optimal control and $H_\infty$-control problems for nonlinear affine systems can be found by solving Hamilton--Jacobi equations. However, these first-order nonlinear partial differential equations can, in general, not be solved analytically. This paper introduces an iterative algorithm which solves these equations numerically for points near the origin. The procedure converges to the stabilizing solution exponentially with respect to the iteration variable. The algorithm is implemented on both illustrative and comparative examples.