on Breakthroughs in the Control of Nonlinear Systems

Nonlinear optimal control and nonlinear H 1 control are two of the most signiicant paradigms in nonlinear systems theory. Unfortunately , these problems require the solution of Hamilton-Jacobi equations which are extremely diicult to solve in practice. To make matters worst, approximation techniques for these equations are inherently prone to the so-called \curse of dimensionality." While there have been many attempts to approximate these equations, solutions resulting in closed-loop control with well deened stability and ro-bustness problems have remained illusive. This paper describes a recent breakthrough in approximating the Hamilton-Jacobi-Bellman and Hamilton-Jacobi-Isaacs equations. Successive approximation and Galerkin approximation methods are combined to derive a novel algorithm that produces stabilizing, closed-loop control laws with well deened stability regions. In addition, we show how the structure of the algorithm can be exploited to reduce the amount of computations from exponential to polynomial growth in the dimension of the state space. The algorithms are illustrated with several examples.