A solution to Hamilton-Jacobi equation by neural networks and optimal state feedback control

This paper is concerned with state feedback controller design using neural networks for nonlinear optimal regulator problem. Nonlinear optimal feedback control law can be synthesized by solving the Hamilton-Jacobi equation with three layered neural networks. The Hamilton-Jacobi equation generates 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 by the gradient method, which determines connection weights and thresholds in the neural networks. Gradient functions 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.