Infinite time problems with shifted and delayed controls Original Research Article

In a recent paper, Warga and Zhu studied optimal control problems defined by functional-integral equations with variable shifts in the controls and with the shifted controls nonadditively coupled. Properness (approximation of relaxed minimizers with original controls) of a certain relaxation procedure was established, leading to a “basic theory” for such problems. A fundamental aspect in this theory relies on the fact that the underlying time interval is assumed to be a compact metric space. The purpose of this note is to prove that, applying a transformation similar to that introduced by Warga for delay free problems, one can reformulate an infinite time problem with shifted and delayed controls as one defined on a compact interval, thus showing that the basic theory remains valid also for this case.