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 Izv. IMI UdGU, 2016, Issue 1(47), Pages 44–53 (Mi iimi327)

Characteristics of invariancy for the attainability set of a control system

L. I. Rodina, A. H. Hammady

Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: We study characteristics associated with invariancy or weak invariancy of a given set $\mathfrak M\doteq\{(t,x)\in [0,+\infty)\times\mathbb R^n: x\in M(t)\}$ with respect to a control system $\dot x=f(t,x,u)$ on a finite time interval. One of such characteristics is relative frequency $freq_{[\tau,\tau+\vartheta]}(D,M)$ of containing the attainability set $D(t,X)$ of this system in the set $\mathfrak M$ on a segment $[\tau,\tau+\vartheta]$. This characteristic is equal to the quotient of the Lebegues measure of those $t$ from $[\tau,\tau+\vartheta]$ at which $D(t,X)\subseteq M(t)$ to the length of the given segment. Other characteristic, $freq_{\vartheta}(D,M)\doteq\inf\limits_{\tau\geqslant 0} freq_{[\tau,\tau+\vartheta]}(D,M)$ displays uniformity of containing the attainability set $D(t,X)$ in the set $\mathfrak M$ on a segment of the fixed length $\vartheta$. We prove theorems about estimation and calculation of these characteristics for various multivalued functions $M(t)$ and $D(t,X)$. In particular, we receive equalities for $freq_{T}(D, M)$ if the function $M(t)$ is periodic with a period $T$ and the function $D(t, X)$ satisfies the inclusion $D(t+T,X)\subseteq D(t,X)$ for all $t\geqslant 0$. We consider examples of calculation and estimations of these characteristics.

Keywords: control systems, differential inclusions, attainability set.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00346_à Ministry of Education and Science of the Russian Federation 2003

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UDC: 517.958, 530.145.6
MSC: 34H05, 34H99, 93C10

Citation: L. I. Rodina, A. H. Hammady, “Characteristics of invariancy for the attainability set of a control system”, Izv. IMI UdGU, 2016, no. 1(47), 44–53

Citation in format AMSBIB
\Bibitem{RodHam16} \by L.~I.~Rodina, A.~H.~Hammady \paper Characteristics of invariancy for the attainability set of a control system \jour Izv. IMI UdGU \yr 2016 \issue 1(47) \pages 44--53 \mathnet{http://mi.mathnet.ru/iimi327} \elib{http://elibrary.ru/item.asp?id=25980776}