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Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 2018, Volume 14, Issue 3, Pages 261–272 (Mi vspui375)  

Control processes

Alpha-sets in finite-dimensional Euclidean spaces and their applications in control theory

V. N. Ushakov, A. A. Uspenskii, A. A. Ershov

Krasovskii Institute of Mathematics and Mechanics Ural Branch of Russian Academy Sciences, 16, S. Kovalevskaya ul., Yekaterinburg, 620990, Russian Federation

Abstract: In this paper, a technique for investigating nonconvex sets that occur when describing the evolution of wave fronts, in the construction of generalized solutions of boundary value problems for equations of Hamilton–Jacobi type, in the formation of resolving structures in the problems of dynamic control is developed. An estimate is obtained for the Hausdorff distance between such sets and their convex hulls. The estimate is based on the concept of a measure of nonconvexity $\alpha$. It is shown that for small $\alpha$, nonconvex $\alpha$-sets are close to convex. An example of a solution of the optimal control problem on the basis of $\alpha$-sets is give.

Keywords: $\alpha$-set, convex hull, Hausdorff distance, control, performance, Hamilton–Jacobi equation.

DOI: https://doi.org/10.21638/11702/spbu10.2018.307

Full text: PDF file (491 kB)
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UDC: 514.74
MSC: 52A27, 49J15, 49J52
Received: January 21, 2018
Accepted: June 14, 2018

Citation: V. N. Ushakov, A. A. Uspenskii, A. A. Ershov, “Alpha-sets in finite-dimensional Euclidean spaces and their applications in control theory”, Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 14:3 (2018), 261–272

Citation in format AMSBIB
\Bibitem{UshUspErs18}
\by V.~N.~Ushakov, A.~A.~Uspenskii, A.~A.~Ershov
\paper Alpha-sets in finite-dimensional Euclidean spaces
and their applications in control theory
\jour Vestnik S.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr.
\yr 2018
\vol 14
\issue 3
\pages 261--272
\mathnet{http://mi.mathnet.ru/vspui375}
\crossref{https://doi.org/10.21638/11702/spbu10.2018.307}
\elib{http://elibrary.ru/item.asp?id=35572248}


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  • Вестник Санкт-Петербургского университета. Серия 10. Прикладная математика. Информатика. Процессы управления
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