The existence of optimal control on the basis of Weierstrass's theorem.

The problem of existence of an optimal control is solved on the basis of Weierstrass's classical theorem if the set of admissible controls belongs to the class of piecewise continuous functions. In the process of describing admissible controls, the main assumption is that the number of switchings (points of discontinuity) is uniformly bounded and not just finite, as in the main problem of optimal control theory. On the one hand, this assumption does not restrict the spectrum of optimal control applications. On the other hand, it fits the Weierstrass's theorem owing to the convenience in characterizing the sequential compactness. The formulation of Weierstrass's theorem, which asserts the existence of continuous function extrema on sequentially compact sets, is customary, and its proof complies with the traditional scheme, whereas the concepts (convergent sequences and some others) are adapted to the peculiarity of optimal problems. [ABSTRACT FROM AUTHOR]