SUCCESSIVE GALERKIN APPROXIMATION ALGORITHMS FOR NONLINEAR OPTIMAL AND ROBUST-CONTROL

Nonlinear optimal control and nonlinear H-infinity control are two of the most significant paradigms in nonlinear systems theory. Unfortunat ely, these problems require the solution of Hamilton-Jacobi equations, which are extremely difficult to solve in practice. To make matters w orse, approximation techniques for these equations are inherently pron e to the so-called 'curse of dimensionality'. While there have been ma ny attempts to approximate these equations, solutions resulting in clo sed-loop control with well-defined stability and robustness have remai ned elusive. This paper describes a recent breakthrough in approximati ng the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations. S uccessive approximation and Galerkin approximation methods are combine d to derive a novel algorithm that produces stabilizing, closed-loop c ontrol laws with well-defined stability regions. In addition, we show how the structure of the algorithm can be exploited to reduce the amou nt of computation from exponential to polynomial growth in the dimensi on of the state space. The algorithms are illustrated with several exa mples.