A numerical method for solving optimal control problems with unspecified terminal time†

This paper presents a numerical technique which automatically computes the unspecified terminal-time of an optimal control problem as a systematic part of the whole procedure for finding the optimal control inputs. In the proposed technique, a given unspecified terminal-time problem is transformed into a fixed terminal-time problem by a time-sealing method. Then, by considering the time-scaling parameter as either an extra control input or as an additional variable, and by applying the necessary conditions of optimality, a fixed terminal-time two-point boundary-value problem is obtained. Three algorithms are proposed to solve such a class of two-point boundary-value problems : two of them are based on the gradient methods, and one on the Newton-Raphson method in function space. Several numerical examples are included. A comparison of the rate of convergence of different algorithms used in the solution of each example problem is given.