GALERKIN APPROXIMATIONS OF THE GENERALIZED HAMILTON-JACOBI-BELLMAN EQUATION

In this paper we study the convergence of the Galerkin approximation m ethod applied to the generalized Hamilton-Jacobi-Bellman (GHJB) equati on over a compact set containing the origin. The GHJB equation gives t he cost of an arbitrary control law and can be used to improve the per formance of this control. The GHJB equation can also be used to succes sively approximate the Hamilton-Jacobi-Bellman equation. We state suff icient conditions that guarantee that the Galerkin approximation conve rges to the solution of the GHJB equation and that the resulting appro ximate control is stabilizing on the same region as the initial contro l. The method is demonstrated on a simple nonlinear system and is comp ared to a result obtained by using exact feedback linearization in con junction with the LQR design method. (C) 1997 Elsevier Science Ltd. Al l rights reserved.