Two crucial examples related to properness of weakly relaxed delayed controls Original Research Article

In this paper, we deal with the problem of finding a proper relaxation procedure (in the sense that relaxed minimizers can be approximated with ordinary controls) for optimal control problems involving delays in the control variables. For systems involving commensurate delays it is well known that a ‘strong’ model provides a proper relaxation procedure. For the noncommensurate case, however, the question of how to properly relax delayed controls has remained open and a natural candidate has been a modified version of the strong model, which we call the ‘weak’ procedure, first introduced by Warga. In this paper, we show by means of two examples that this procedure may fail to be proper for systems with either commensurate or noncommensurate delays. It is expected that these examples will provide some insight into the problem of finding a proper relaxation procedure applicable to the general case.