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.