Certain classes of properly relaxed delayed controls

Optimal control problems involving time delays in the control variables can be formulated by treating the control functions as independent variables which satisfy certain compatibility conditions in terms of the delays. By generalizing these conditions in the corresponding space of relaxed controls, Warga proposed a ‘weak’ model of relaxation and proved existence of weakly relaxed minimizers. However, several examples were found, involving commensurate delays, for which weakly relaxed controls cannot be approximated with original controls, so that this extension fails to be ‘proper’. The case of commensurate delays was solved by the introduction of a ‘strong’ model, but the question of how to properly relax noncommensurately delayed controls has remained unsolved, and a natural candidate has been precisely that of weakly relaxed controls. In this paper we show that, for the noncommensurate case, certain classes of weakly relaxed controls belong to the (weak star) closure of the space of ordinary delayed controls. The proof is based on previous results obtained for a different, larger space of controls for which the notion of weak relaxation can also be applied, and on a recent result related to periodic measurable functions. Also, we extend those previous results obtained for the larger space of weakly relaxed controls, which we call the ‘R-weak’ procedure, and exhibit, for the first time, certain cases of delays for which both the space of R-weakly relaxed controls and the closure of the space of ordinary controls coincide. In other words, we prove that the R-weak model is, for certain delays, a proper relaxation procedure.