The Solution to the Conjecture on Properness of Weakly Relaxed Delayed Controls

In 1986 Warga proposed a “weak” relaxation procedure applicable to fully nonlinear problems with delays in the control variables and showed that the resulting relaxed problem has a solution. However, in the event of commensurate delays, several examples were found for which weakly relaxed controls cannot be approximated with original controls, so that this extension fails to be “proper.” Although the case of commensurate delays was solved by the introduction of a “strong” model, the question of how to properly relax noncommensurately delayed controls has remained open, and a natural candidate has been precisely that of weakly relaxed controls. In this paper, a general counterexample is constructed which rules out the weak relaxation as a proper relaxation when there are two or more delays. It is hoped that this result will provide some insight into the problem of finding a general representation of properly relaxed controls.