An Approximation to Discrete Optimal Feedback Controls

We study discrete solutions of nonlinear optimal control problems. By value functions , we construct difference equations to approximate the optimal control on each interval of " small " time. We aim to find a discrete optimal feedback control. An algorithm is proposed for computing the solution of the optimal control problem. 1. Introduction and statement of the problem. For nonlinear analytic systems , there have been many works on control problems since the eighties of the last century. In [3, 4, 5, 6], Lie series was used in studying the controllability of nonlinear analytic systems. In [1], the discrete method in solving Hamilton-Jacobi-Bellman equations for value functions of nonlinear problems was discussed. In this paper, we use Lie series to construct difference equations by value functions in obtaining the discrete solutions of nonlinear optimal control problems. Taking advantage of the uniform convergence of Lie series on an interval of " small " time, we focus on the integral of the optimal control function. We aim to find a discrete optimal feedback control. We see that the optimal controls of a given problem can be constructed by these integral dates. We propose an algorithm which includes the process of pre-estimation and correction of an approximation to the solution of the optimal control problem. We begin by considering the following nonlinear control system: