APPROXIMATE SOLUTIONS TO THE TIME-INVARIANT HAMILTON-JACOBI-BELLMAN EQUATION

In this paper, we develop a new method to approximate the solution to the Hamilton-Jacobi-Bellman (HJB) equation which arises in optimal con trol when the plant is modeled by nonlinear dynamics. The approximatio n is comprised of two steps. First, successive approximation is used t o reduce the HJB equation to a sequence of linear partial differential equations. These equations are then approximated via the Galerkin spe ctral method. The resulting algorithm has several important advantages over previously reported methods. Namely, the resulting control is in feedback form and its associated region of attraction is well defined . In addition, all computations are performed off-line and the control can be made arbitrarily close to optimal. Accordingly, this paper pre sents a new tool for designing nonlinear control systems that adhere t o a prescribed integral performance criterion.