Computational Method for a Class of Optimal Switching Control Problems

Abstract

In this paper, we consider a class of optimal control problems in which the system is to be determined in an optimal way. This class of problems involves the choice of a fixed number of switching time points which divide the system’s time horizon into a number of time periods. For each of these time periods, a subsystem is selected, from a finite number of given candidate subsystems, to run during that time periods. The choice of the switching points and the selection of the subsystems are carried out in such a way that a given cost functional is minimized. We consider only problems involving ordinary differential equations over a finite time horizon. A computational method is developed for solving these problems. In our method, the candidate subsystems are combined into a single system by considering their ‘linear combination’. By introducing suitable constraints on the coefficients of the linear combination and using a time rescaling technique, the original problem is transformed into an equivalent optimal control problem with system parameters. An algorithm is proposed for solving this transformed problem and the required gradient formulae are derived. To show the effectiveness of the method, a numerical example is solved.