Equivalence of regularizations in abstract problems with different classes of admissible controls (Q1286310)

The authors consider a finite dimensional control system with a single input. Admissible control functions are constrained by integral equality and integral inclusion. It is demonstrated on examples that there are situations where the attainable set for a traditional class of piecewise continuous admissible controls is different from the attainable set for the class of pulse controls, i.e. controls expressed as weighted sums of Dirac's \(\delta\)-functions. In the considered example both attainable sets are closed. Using a technique based on finite additive measures developed earlier by the second author [see e.g. Autom. Remote Control 53, No. 5, pt. 1, 646-655 (1992; Zbl 0794.93041)], the class of admissible controls is extended. The conventional controls in the new settings are step-function with respect to the introduced measure whereas pulse controls are analogs of weighted sums of \(\delta\)-functions. It is obtained that in the relaxed problem the respective analogs of attainable sets both for conventional admissible controls and for purely impulse controls are equal.