Remarks on Solving Algebraic Riccati Matrix Equations using a Hopfield Neural Network and Application to Optimal Control Problems

This paper discusses a method of solving algebraic Riccati matrix equations using a Hopfield neural network and presents its application to the optimal control of dynamic systems with a quadratic cost function. To solve an algebraic Riccati matrix equation using the optimization ability of a Hopfield neural network, an energy function is defined using the elements of the algebraic Riccati matrix equation and a penalty function of incorporating the positive definite constraint of the solution. The energy function is minimized through the dynamics of the Hopfield neural network, and the converged neuron states provide the solution of the algebraic Riccati matrix equation. Computational experiments using a linear second-order system confirm that the Hopfield neural network can solve the algebraic Riccati matrix equation with sufficient accuracy. The optimal control of an automotive vehicle is demonstrated as a practical application of controlling dynamic systems using the solution obtained by the Hopfield neural network; the simulation results indicate the feasibility and effectiveness of the proposed neural network–based optimal control.