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Successive Galerkin approximation algorithms for nonlinear optimal and robust control. (English) Zbl 0945.49021
Summary: Nonlinear optimal control and nonlinear ${ℋ}_{\infty }$ control are two of the most significant paradigms in nonlinear systems theory. Unfortunately, these problems require the solution of Hamilton-Jacobi equations, which are extremely difficult to solve in practice. To make matters worse, 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-defined stability and robustness have remained elusive. 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-defined stability regions. In addition, we show how the structure of the algorithm can be exploited to reduce the amount of computation from exponential to polynomial growth in the dimension of the state space. The algorithms are illustrated with several examples.
##### MSC:
 49L20 Dynamic programming method (infinite-dimensional problems) 93B36 ${H}^{\infty }$-control 49M15 Newton-type methods in calculus of variations 65M60 Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods (IVP of PDE) 93B40 Computational methods in systems theory