Lower closure for orientor fields by lower semicontinuity of outer integral functionals
Basic classical results in optimal control theory rely on the lower semicontinuity of the cost functional and measurability of the control functions. Corresponding assumptions of measurability of set-valued functions appear in the formulation of control systems by differential inclusions. The proof of measurability of these related set-valued functions is usually technically quite involved. The present paper makes such proof unnecessary in many cases, by proving that the lower semicontinuity of the outer integral (which always exists) already guarantees the closure properties which are the usual goal in existence theorems of optimal controls. The proof (done in a very general setting in abstract spaces) is strongly related to papers by \textit{C. Castaing} and \textit{M. Valadier} [Convex analysis and measurable multifunctions, Lect. Notes Math. 580 (1977; Zbl 0346.46038)], \textit{T. Rockafellar} [Lect. Notes Math. 543, 157-207 (1976; Zbl 0374.49001)], and \textit{I. Ekeland} and \textit{R. Temam} [Analyse convexe et problèmes variationnels (1974; Zbl 0281.49001)], among others. In this light, Cesari's ''Property Q'' [see e.g. \textit{L. Cesari} and \textit{M. B. Suryanarayana}, SIAM J. Control Optimization 12, 705-720 (1974; Zbl 0299.49018)] is analyzed in this paper.
scientific article; zbMATH DE number 3909203
scientific article