Near-time-optimal neural control of discrete-time systems

Investigates theoretical foundations for the use of artificial neural networks for closed-loop approximation of time-optimal control policies. The results are obtained using elementary properties of continuous mappings, with almost no assumptions on smoothness of state transition function f. The developments are greatly simplified by assumption of certain uniqueness properties satisfied for sampled continuous time systems controlled in zero-hold fashion. The results presented guarantee existence of time-optimal feedback approximation using networks with discontinuous Heaviside activation functions. Unfortunately, while such networks are attractive from the theoretical point of view, there are hardly any efficient training algorithms available for them. The architecture dominating in practice is that with smooth sigmoidal activation functions, and most training methods are derivatives of the celebrated backpropagation algorithm. Therefore, there is a need to investigate exactly for what kind of control problems it is possible to apply the sigmoidal networks. The presented results constitute a step in this direction, but further studies are necessary. By strengthening the assumptions on the form of the controlled system, it should be possible to improve the results presented here, and to eliminate the residual set on which the approximate controller may fail