Strongly and Weakly Relaxed Controls for Time Delay Systems

Abstract

In recent articles, two different relaxation procedures for time delay problems in optimal control have been proposed. A “weak” procedure, due to Warga [Nonadditively coupled delayed controls, privately circulated, 1986], for which the existence of minimizers is assured, applies to fully nonlinear problems with delays in the state and control variables. Nevertheless, an example is found for which the effect of weak relaxation reduces the infimum cost. The example involves two delays, one being twice the value of the other, and it belongs to a class, of problems that is called “commensurate,” where the quotient of any two delays is rational. For these problems a “strong” procedure is proposed and it is proved that the extended problem has a solution and the extension is “proper”; i.e., the infimum costs coincide. In this paper it is shown through several examples why, in general, the weak relaxation technique may fail to provide a proper extension and, for noncommensurate delay problems, an approximation r...