A solution to Hamilton-Jacobi equation by neural networks and optimal state feedback control law of nonlinear systems

Abstract

The paper is concerned with a state feedback controller using neural networks for a nonlinear optimal regulator problem. A nonlinear optimal feedback control law can be synthesized by solving the Hamilton-Jacobi equation with three layered neural networks. The Hamilton-Jacobi equation solves the value function by which the optimal feedback law is synthesized. To obtain an approximate solution of the Hamilton-Jacobi equation, we solve an optimization problem which determines connection weights and thresholds in the neural networks. Gradient functions with respect to the connection weights and thresholds are calculated explicitly by the Lagrange multiplier method and used in the learning algorithm of the networks. We propose also a device such that an approximate solution to the Hamilton-Jacobi equation converges to the true value function. The effectiveness of the proposed method was confirmed with simulations for various plants.