Neural Network -based Nearly Optimal Hamilton-Jacobi-Bellman Solution for Affine Nonlinear Discrete-Time Systems

In this paper, we consider the use of nonlinear networks towards obtaining nearly optimal solutions to the control of nonlinear discrete-time systems. The method is based on least-squares successive approximation solution of the Generalized Hamilton-Jacobi-Bellman (HJB) equation. Since successive approximation using the GHJB has not been applied for nonlinear discrete-time systems, the proposed recursive method solves the GHJB equation in discrete-time on a well-defined region of attraction. The definition of GHJB, Pre-Hamiltonian function, HJB equation and method of updating the control function for the affine nonlinear discrete time systems are proposed. A neural network is used to approximate the GHJB solution. It is shown that the result is a closed-loop control based on a neural network that has been tuned a priori in off-line mode. Numerical example show that for nonlinear discrete-time systems, the updated control laws will converge to the suboptimal control.