Approximation of strongly relaxed minimizers with ordinary delayed controls

Abstract

In a recent paper, a procedure for relaxing optimal control problems with nonseparable commensurate delays in the controls was proposed. This model (strong relaxation) was shown to be proper in the sense that the minimum cost for the relaxed problem coincides with the infimum cost for the original one, and it solves some of the difficulties encountered when another procedure, based on a well-known technique for reducing a control system of this nature to a delay-free problem, is used. However, the proof of its being proper is unsatisfactory in two respects, not only from a theoretical point of view but also in order to construct numerical approximation algorithms. In this proof the use of the reduced-problem technique is still needed, so that the disadvantages of this procedure remain present. Moreover, the existence of a sequence of ordinary delayed controls which converges to a given strongly relaxed control is assured, but one may fail to exhibit it. In this paper we solve these questions by giving an explicit methodology, which is independent of the reduced-problem technique, for constructing such a sequence. Also, this constructive approach provides a new and simple proof of the fact that the space of strongly relaxed controls coincides with the weak-star closure of the space of ordinary delayed controls.