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Measurable multifunctions and their applications to convex integral functionals. (English) Zbl 0659.28008
The purpose of this paper is to establish some new properties of set- valued measurable functions and of their sets of integrable selectors and to use them to study convex integral functionals defined on Lebesgue- Bochner spaces. In this process we also obtain a characterization of separable dual Banach spaces using multifunctions and we present some generalizations of the classical “bang-bang” principle to infinite dimensional linear control systems with time dependent control constraints.

##### MSC:
 28B20 Set-valued set functions and measures; integration of set-valued functions; measurable selections 46G10 Vector-valued measures and integration 28B05 Vector-valued set functions, measures and integrals 46E30 Spaces of measurable functions ($$L^p$$-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) 54C60 Set-valued maps in general topology 49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
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