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Sliding regimes in the theory of optimal control. (Russian) Zbl 0599.49010

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

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
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