Abstract:
We study αα-sets in Euclidean space Rn. The notion of α-set is introduced as a generalization of a convex closed set in Rn. This notion appeared in the study of reachable sets and integral funnels of nonlinear control systems in Euclidean spaces. Reachable sets of nonlinear dynamic systems are usually nonconvex, and the degree of their nonconvexity is different in different systems. This circumstance prompted the introduction of a classification of sets in Rn according to the degree of their nonconvexity. Such a classification stems from control theory and is presented here as the notion of α-set in Rn.
Keywords:α-set, convex set in Rn, convex hull in Rn, α-hyperplane, α-separability, Bouligand cone, normal cone.
Citation:
V. N. Ushakov, A. A. Uspenskii, “Theorems on the separability of α-sets in Euclidean space”, Trudy Inst. Mat. i Mekh. UrO RAN, 22, no. 2, 2016, 277–291; Proc. Steklov Inst. Math. (Suppl.), 299, suppl. 1 (2017), 231–245
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\by V.~N.~Ushakov, A.~A.~Uspenskii
\paper Theorems on the separability of $\alpha$-sets in Euclidean space
\serial Trudy Inst. Mat. i Mekh. UrO RAN
\yr 2016
\vol 22
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\pages 277--291
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\jour Proc. Steklov Inst. Math. (Suppl.)
\yr 2017
\vol 299
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\pages 231--245
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Linking options:
https://www.mathnet.ru/eng/timm1313
https://www.mathnet.ru/eng/timm/v22/i2/p277
This publication is cited in the following 4 articles:
V. N. Ushakov, A. A. Ershov, “O sootnoshenii mezhdu α-mnozhestvami i slabo vypuklymi mnozhestvami”, Tr. IMM UrO RAN, 30, no. 4, 2024, 276–285
O. A. Kuvshinov, “O geometrii ovala Kassini, ego mere nevypuklosti i ε-sloe”, Izv. IMI UdGU, 60 (2022), 34–57
V. N. Ushakov, A. A. Ershov, “On Guaranteed Estimates of the Area of Convex Subsets of Compact Sets on the Plane”, Autom Remote Control, 82:11 (2021), 1976
Vladimir Ushakov, Aleksandr Ershov, Maksim Pershakov, Communications in Computer and Information Science, 1090, Mathematical Optimization Theory and Operations Research, 2019, 329