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IMA Journal of Mathematical Control and Information 1997 14(3):289-298; doi:10.1093/imamci/14.3.289
© 1997 by Institute of Mathematics and its Applications
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Is the weak relaxation procedure proper for noncommensurately delayed controls?

JAVIER F. ROSENBLUETH

IIMAS–UNAM Apartado Postal 20–726, México DF01000, Mexico

Looking for a proper relaxation procedure for optimal control problems with time delays in the control variables, Warga introduced in a privately circulated paper the notion of ‘weakly’ relaxed controls. It has been proved that, for systems involving two delays with a rational quotient (the commensurate case), one can find weakly relaxed controls that cannot be approximated with original controls, showing that this model may fail to be proper. The same conclusion, for the noncommensurate case, was never established. In fact, for this case, we proved in a recent paper that (assuming the underlying control compact space to have two points) all constant weakly relaxed controls belong to the (weak-star) closure of the space of ordinary controls. The purpose of this paper is to extend this result to the general case of time-dependent weakly relaxed controls.


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