Discrete-time optimal control using continuous neural networks

Artificial neural networks offer an attractive approach to synthesis of near-optimal feedback control policies. Open-loop optimal trajectories, obtained with standard optimization techniques, may be used as training data for a neural controller. The training objective is that the trajectories of the closed-loop system approximate those resulting from open-loop optimization. This paper investigates the question of existence of an appropriate neural controller, so that the approximation problem is well posed. The optimal control problem is considered in discrete time for a class of additive quality criterions. An associated relaxed sub-optimal control problem is introduced to simplify the analysis. The controller structure is that of static state feedback realized by feedforward multilayered networks with continuous neuron activation functions. Under mild assumptions about the system, the obtained results guarantee existence of an appropriate network, such that the closed-loop system closely follows the open-loop optimal trajectories, except for a set of initial conditions of arbitrarily small measure.