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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2013, Issue 1, Pages 35–48 (Mi vuu362)

MATHEMATICS

The characteristics of attainability set connected with invariancy of control systems on the finite time interval

L. I. Rodina, A. H. Hammady

Department of Mathematical Analysis, Udmurt State University, Izhevsk, Russia

Abstract: We study the statistical characteristics of the attainability set $A(t,\sigma,X)$ of the control system which is parametrized by means of a topological dynamical system $(\Sigma,h^t).$ We obtain the lower estimates for characteristics connected with invariance of given set on a finite time interval. We also consider the following problem arising in many applications. Let numbers $\lambda_0\in (0,1]$ and $\vartheta>0$ are given. It is necessary to find the conditions which the control system and set $X$ should satisfy providing that for given $\sigma\in\Sigma$ relative frequency of containing of the attainability set $A(t,\sigma,X)$ in the given set $M$ on any interval of time length $\vartheta$ would be not less then $\lambda_0$. Let's notice, that the characteristic $\vartheta$ is assumed given depending on an applying problems. In particular, if control process is periodic, then $\vartheta$ is the period of the process. Results are illustrated by examples of the control systems which describe different models of population growth.

Keywords: control systems, dynamical systems, differential inclusions, statistically invariant sets.

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UDC: 517.935+517.938
MSC: 34A60, 37N35, 49J15, 93B03

Citation: L. I. Rodina, A. H. Hammady, “The characteristics of attainability set connected with invariancy of control systems on the finite time interval”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2013, no. 1, 35–48

Citation in format AMSBIB
\Bibitem{RodHam13} \by L.~I.~Rodina, A.~H.~Hammady \paper The characteristics of attainability set connected with invariancy of control systems on the finite time interval \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2013 \issue 1 \pages 35--48 \mathnet{http://mi.mathnet.ru/vuu362} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. A. Kh. Khammadi, “O svoistvakh kharakteristik mnozhestva dostizhimosti upravlyaemoi sistemy”, Izv. IMI UdGU, 2015, no. 2(46), 216–227
2. L. I. Rodina, A. Kh. Khammadi, “Kharakteristiki invariantnosti mnozhestva dostizhimosti upravlyaemoi sistemy”, Izv. IMI UdGU, 2016, no. 1(47), 44–53
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