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Multidimensional graph completions and Cellina approximable multifunctions. (English) Zbl 1222.49008

Summary: Relying on the continuous approximate selection method of Cellina, ideas and techniques from Sobolev spaces can be applied to the theory of multifunctions and differential inclusions. The first part of this paper introduces a concept of graph completion, which extends the earlier construction in [A. Bressan and F. Rampao, Boll. Unione Mat. Ital., VII. Ser., B 2, No. 3, 641–656 (1988; Zbl 0653.49002)] to functions of several space variables. The second part introduces the notion of Cellina’s \(W^{1,p}\)-approximable multifunction. To show its relevance, we consider the Cauchy problem on the plane \(\dot x\in F(x)\), \(x(0)= 0\in\mathbb R^2\). If \(F\) is an upper semicontinuous multifunction with compact but possibly non-convex values, this problem may not have any solution, even if \(F\) is Cellina-approximable in the usual sense. However, we prove that a solution exists under the assumption that \(F\) is Cellina \(W^{1,1}\)-approximable.

MSC:

49J30 Existence of optimal solutions belonging to restricted classes (Lipschitz controls, bang-bang controls, etc.)
34H10 Chaos control for problems involving ordinary differential equations

Citations:

Zbl 0653.49002
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