Relaxation procedures for time delay systems

Abstract

Despite the attention that optimal control problems involving time delays have received over the last two decades, little attention has been paid to the important question of how such problems should be relaxed in order to assure existence of minimizers. Recently, Warga has proposed a relaxation procedure for fully nonlinear problems with delays in the state and control variables and showed that the resulting relaxed problem has a solution. Nevertheless, the effect of this relaxation may reduce the infimum cost as we show through an example which belongs to a class of problems with “commensurate” delays; i.e., the quotient of any two delays is rational. For problems of this nature we provide a new relaxation procedure for which the extension is “proper”; i.e., the infimum costs coincide. We also present an abstract relaxation procedure applicable to problems with possibly noncommensurate delays.