Direct Training Method for Continuous-time Nonlinear Optimal Feedback Controller 1 2

The solutions of most nonlinear optimal control problems are given in the form of open-loop optimal control which is computed from a given xed initial condition. Optimal feedback control can in principle be obtained by solving the corresponding Hamilton-Jacobi-Bellman (HJB) dynamic programming equation, though in general this is a di cult task. We propose a practical and e ective alternative for constructing an approximate optimal feedback controller in the form of a feedforward neural network, and we justify this choice by several reasons. The controller is capable of approximately minimizing an arbitrary performance index for a nonlinear dynamical system for initial conditions arising from a nontrivial bounded subset of the state-space. A direct training algorithm is proposed and several illustrative examples are given.