A Representation of Properly Relaxed Controls with Two Delays

The search of a proper relaxation procedure in the sense that relaxed minimiz- ers can be approximated with ordinary controls. for optimal control problems involving delays in the control variables has been the main topic of several recent papers. Three models of relaxation, which we call the weak, strong, and D procedures, have been proposed and, in all cases, the existence of minimizers has been established. It has been proved that the D-model is a proper relaxation procedure, but determining the set of D-relaxed controls for specific problems is very difficult and perhaps even a hopeless task. Thus there is a need to find more concrete characterizations of the closure of the space of ordinary delayed controls. In the event of commensurate delays this is solved through the strong model: the space of strongly relaxed controls coincides with the space of D-relaxed controls. For the noncommensurate case, the problem of how to characterize D-relaxed controls has remained unsolved and, although a natural candidate had been the space of weakly relaxed controls, now we know that for either commensurate or noncommensurate delays it may fail to be proper. In this paper we are finally able to characterize D-relaxed controls for two noncommensurate delays by the introduction of a new model of relaxation, the P-model, which turns out to be a space strictly contained in that of weakly relaxed controls.