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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2015, Volume 25, Issue 2, Pages 157–179 (Mi vuu474)

MATHEMATICS

Criteria for uniform complete controllability of a linear system

V. A. Zaitsev

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

Abstract: The notion of uniform complete controllability of linear system introduced by R. Kalman plays a key role in problems of control of asymptotic properties for linear systems closed by linear feedback control. E. L. Tonkov has found a necessary and sufficient condition of uniform complete controllability for systems with piecewise continuous and bounded coefficients. The Tonkov criterion can be considered as the definition of uniform complete controllability. If the coefficients of the system satisfy weak conditions then the definitions of Kalman and Tonkov are not coincide. We obtain necessary conditions and sufficient conditions for uniform complete controllability in the sense of Kalman and Tonkov for systems with measurable and locally integrable coefficients. We introduce a new definition of uniform complete controllability that extends the definition of Tonkov and coincides with the definition of Kalman providing the matrix $B(\cdot)$ is bounded. We prove some known results on the controllability of linear systems that allow the weakening of the requirements on the coefficients. We prove that if a linear control system $\dot x=A(t)x+B(t)u$, $x\in\mathbb{R}^n$, $u\in\mathbb{R}^m$, with measurable and bounded matrix $B(\cdot)$ is uniformly completely controllable in the sense of Kalman then for any measurable and integrally bounded $m\times n$-matrix function $Q(\cdot)$ the system $\dot x=(A(t)+B(t)Q(t))x+B(t)u$ is also uniformly completely controllable in the sense of Kalman.

Keywords: linear control system, uniform complete controllability.

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Document Type: Article
UDC: 517.977.1, 517.926
MSC: 93B05, 93C05

Citation: V. A. Zaitsev, “Criteria for uniform complete controllability of a linear system”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 25:2 (2015), 157–179

Citation in format AMSBIB
\Bibitem{Zai15} \by V.~A.~Zaitsev \paper Criteria for uniform complete controllability of a linear system \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2015 \vol 25 \issue 2 \pages 157--179 \mathnet{http://mi.mathnet.ru/vuu474} \elib{http://elibrary.ru/item.asp?id=23681099} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Zaitsev, “Ravnomernaya polnaya upravlyaemost i globalnoe upravlenie asimptoticheskimi invariantami lineinoi sistemy v forme Khessenberga”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 25:3 (2015), 318–337
2. A. A. Kozlov, “O dostatochnom uslovii globalnoi skalyarizuemosti lineinykh upravlyaemykh sistem s lokalno integriruemymi koeffitsientami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 26:2 (2016), 221–230
3. A. A. Kozlov, I. V. Ints, “O ravnomernoi globalnoi dostizhimosti dvumernykh lineinykh sistem s lokalno integriruemymi koeffitsientami”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:2 (2017), 178–192
4. E. K. Makarov, S. N. Popova, “Ob opredelenii ravnomernoi polnoi upravlyaemosti”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:3 (2017), 326–343
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