Abstract:
The concept of αα-set in a finite-dimensional Euclidean space, which is one of generalizations of the notion of a convex set, is introduced. The emergence of this concept is connected with the study of properties of attainability sets of nonlinear controlled systems. The numerical characteristic of nonconvexity degree of a set on the basis of which a classification of sets is carried out is defined in the paper. Analogs of basic concepts from the convex analysis are introduced into consideration and their properties are studied. Statements in the spirit of such theorems from the convex analysis as the theorem of existence of basic hyperplane to a convex set and theorems of separability of convex sets in Euclidean space are formulated and proved. The concept of magoriums of nonconvex sets is studied. Property of a magoriums is a sufficient condition for representation of a closed nonconvex set in the form of crossing of half-spaces in the sense of definitions entered in this work. The obtained results of the theory of separability of nonconvex sets can be extended on a case of hypograph and epigraph of the scalar functions with Lipschitz condition.
Citation:
V. N. Ushakov, A. A. Uspenskii, “αα-sets in finite dimensional Euclidean spaces and their properties”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:1 (2016), 95–120
\Bibitem{UshUsp16}
\by V.~N.~Ushakov, A.~A.~Uspenskii
\paper $\alpha$-sets in finite dimensional Euclidean spaces and their properties
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2016
\vol 26
\issue 1
\pages 95--120
\mathnet{http://mi.mathnet.ru/vuu522}
\crossref{https://doi.org/10.20537/vm160109}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3485577}
\elib{https://elibrary.ru/item.asp?id=25681789}
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This publication is cited in the following 16 articles:
P. D. Lebedev, A. A. Uspenskii, “Metod Nyutona pri postroenii singulyarnogo mnozhestva minimaksnogo resheniya v odnom klasse kraevykh zadach dlya uravnenii Gamiltona — Yakobi”, Chelyab. fiz.-matem. zhurn., 9:1 (2024), 63–76
A. A. Uspenskii, P. D. Lebedev, “Alfa-mnozhestva i ikh obolochki:analiticheskie vzaimosvyazi v ploskom sluchae”, Vestnik rossiiskikh universitetov. Matematika, 29:146 (2024), 204–217
P. D. Lebedev, A. A. Uspenskii, “Numerical-analytic construction of a generalized solution to the eikonal equation in the plane case”, Sb. Math., 215:9 (2024), 1224–1248
O. A. Kuvshinov, “O geometrii ovala Kassini, ego mere nevypuklosti i εε-sloe”, Izv. IMI UdGU, 60 (2022), 34–57
A. A. Uspenskii, P. D. Lebedev, “On singularity structure of minimax solution to Dirichlet problem for eikonal type equation with discontinuous curvature of boundary of boundary set”, Ufa Math. J., 13:3 (2021), 126–151
V. N. Ushakov, A. A. Ershov, A. R. Matviychuk, “On Estimating the Degree of Nonconvexity of Reachable Sets of Control Systems”, Proc. Steklov Inst. Math., 315 (2021), 247–256
A. A. Ershov, O. A. Kuvshinov, “O svoistvakh peresecheniya αα-mnozhestv”, Izv. IMI UdGU, 55 (2020), 79–92
P. D. Lebedev, A. A. Uspenskii, “Elementy analiticheskogo konstruktora reshenii v klasse zadach upravleniya po bystrodeistviyu s tselevym mnozhestvom s razryvnoi kriviznoi granitsy”, Vestnik rossiiskikh universitetov. Matematika, 25:132 (2020), 370–386
A. A. Uspenskii, P. D. Lebedev, “Svoistva nestatsionarnykh psevdovershin kraevogo mnozhestva pri razryve gladkosti krivizny ego granitsy v zadache Dirikhle dlya uravneniya tipa eikonala”, Sib. elektron. matem. izv., 17 (2020), 2028–2044
Vladimir Ushakov, Aleksandr Ershov, Maksim Pershakov, Communications in Computer and Information Science, 1090, Mathematical Optimization Theory and Operations Research, 2019, 329
V. N. Ushakov, A. A. Ershov, “An estimate of the Hausdorff distance between a set and its convex hull in Euclidean spaces of small dimension”, Proc. Steklov Inst. Math. (Suppl.), 305, suppl. 1 (2019), S178–S190
A. A. Uspenskii, P. D. Lebedev, “Vyyavlenie singulyarnosti u obobschennogo resheniya zadachi Dirikhle dlya uravneniya tipa eikonala v usloviyakh minimalnoi gladkosti granitsy kraevogo mnozhestva”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:1 (2018), 59–73
V. N. Ushakov, A. A. Uspenskii, A. A. Ershov, “Alfa-mnozhestva v konechnomernykh evklidovykh prostranstvakh
i ikh prilozheniya v teorii upravleniya”, Vestn. S.-Peterburg. un-ta. Ser. 10. Prikl. matem. Inform. Prots. upr., 14:3 (2018), 261–272
P.D. Lebedev, A.A. Uspenskii, “Construction of Singular Sets in a Velocity Control Problem with Nonconvex Target”, IFAC-PapersOnLine, 51:32 (2018), 681
A. V. Seliverstov, “O kasatelnykh pryamykh k affinnym giperpoverkhnostyam”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 248–256
Vladimir N. Ushakov, Aleksandr A. Uspenskii, Aleksandr A. Ershov, 2017 Constructive Nonsmooth Analysis and Related Topics (dedicated to the memory of V.F. Demyanov) (CNSA), 2017, 1
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