Optimal control of terminal processes using neural networks

Feedforward neural networks are capable of approximating continuous multivariate functions and, as such, can implement nonlinear state-feedback controllers. Training methods such as backpropagation-through-time (BPTT), however, do not deal with terminal control problems in which the specified cost function includes the elapsed trajectory-time. In this paper, an extension to BPTT is proposed which addresses this limitation. The controller design is reformulated as a constrained optimization problem defined over the entire field of extremals and in which the set of trajectory times is incorporated into the cost function. Necessary first-order stationary conditions are derived which correspond to standard BPTT with the addition of certain transversality conditions. The new gradient algorithm based on these conditions, called time-optimal backpropagation through time, is tested on two benchmark minimum-time control problems.