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Russia Izhevsk
Section  Mathematics 
Title  The characteristics of attainability set connected with invariancy of control systems on the finite time interval 
Author(s)  Rodina L.I.^{a}, Hammady A.H.^{a} 
Affiliations  Udmurt State University^{a} 
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 than $\lambda_0.$ 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 
UDC  517.935, 517.938 
MSC  34A60, 37N35, 49J15, 93B03 
DOI  10.20537/vm130105 
Received  12 November 2012 
Language  Russian 
Citation  Rodina L.I., Hammady A.H. The characteristics of attainability set connected with invariancy of control systems on the finite time interval, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2013, issue 1, pp. 3548. 
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