Gamkrelidze, R. V. Sliding regimes in the theory of optimal control. (Russian) Zbl 0599.49010 Tr. Mat. Inst. Steklova 169, 180-193 (1985). Generalized controls and their efficiency in the control theory have been worked out by the author since 1962. In this paper, a time optimal problem described by the differential equation \(\dot x(\)t)\(=f(t,x(t),u(t))\) where the admissible control \(t\to u(t)\) has values in a subset U of \(R^ r\), the convexified problem described by the differential equation \(\dot x(t)=\int_{R^ r}f(t,x(t),u)\eta_ t(du)\) where the generalized control \(t\to \eta_ t\) has values in the set of Radon probability measures concentrated on U, and some of their properties are discussed. Weak convergence as well as strong convergence are defined for sequences of generalized controls and the main result of the paper states that strongly continuous functions from a metric space \(\Sigma\) into \({\mathfrak M}_ U\) (the set of generalized controls) can be uniformly weakly approximated by strongly continuous functions from \(\Sigma\) into \(\Omega_ U\) (the set of admissible controls). Reviewer: C.Ursescu Cited in 2 Documents MSC: 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.) 49J15 Existence theories for optimal control problems involving ordinary differential equations 93C15 Control/observation systems governed by ordinary differential equations 34A34 Nonlinear ordinary differential equations and systems 93C10 Nonlinear systems in control theory 34H05 Control problems involving ordinary differential equations Keywords:Generalized controls; time optimal problem; Weak convergence; strong convergence PDFBibTeX XMLCite \textit{R. V. Gamkrelidze}, Tr. Mat. Inst. Steklova 169, 180--193 (1985; Zbl 0599.49010)