Optimal control problems with stopping constraints

We consider a novel optimal control problem in which the terminal time is governed by a stopping constraint. This stopping constraint is a nonlinear equality constraint depending on the state variables, and the terminal time is defined as the first time at which this constraint is satisfied. Since the stopping constraint causes the terminal time to be an implicit function of the control, the optimal control problem we consider cannot be solved using conventional techniques. We propose a new computational approach that involves approximating the original problem by a standard optimal control problem with fixed terminal time. Our main result shows that this approximation, which depends on two adjustable parameters, can be made to arbitrarily high accuracy. On this basis, the original optimal control problem with stopping constraints can be transformed into a sequence of approximate problems, each of which can be solved readily using conventional optimal control techniques. We conclude the paper by demonstrating this approach with numerical simulations in three application areas: range maximization of a hang glider, range maximization of a hypersonic re-entry vehicle, and time-optimal control of a nuclear reactor.