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 Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 2017, Volume 27, Issue 3, Pages 326–343 (Mi vuu592)

MATHEMATICS

On the definition of uniform complete controllability

E. K. Makarova, S. N. Popovabc

a Institute of Mathematics, National Academy of Sciences of Belarus, ul. Surganova, 11, Minsk, 220072, Belarus
b N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620990, Russia
c Udmurt State University, ul. Universitetskaya, 1, Izhevsk, 426034, Russia

Abstract: We consider a linear control system
$$\dot x=A(t)x+B(t)u,\quad t\in\mathbb R,\quad x\in\mathbb R^{n},\quad u\in\mathbb R^{m}, \tag{1}$$
under the assumption that the transition matrix $X(t,s)$ of the free system $\dot x = A(t)x$ is continuous with respect to $t$ and $s$ separately. We also suppose that on each interval $[\tau, \tau + \vartheta]$ of fixed length $\vartheta$ the normed space $Z_{\tau}$ of functions defined on this interval is given. A control $u$ on the interval $[\tau, \tau+\vartheta]$ is called admissible if $u\in Z_{\tau}$ and there exists the integral $\mathcal Q_{\tau}(u):=\int_{\tau}^{\tau+\vartheta}X(\tau,s)B(s)u(s) ds$. The vector subspace $U_{\tau}$ of the space $Z_{\tau}$ where the operator $\mathcal Q_{\tau}$ is defined is called the space of admissible controls for the system (1) on the interval $[\tau,\tau +\vartheta]$. We propose a definition of uniform complete controllability of the system (1) for the case of an arbitrary dependence of the space of admissible controls on the moment of the beginning of the control process. In this situation direct and dual necessary and sufficient conditions for uniform complete controllability of a linear system are obtained. It is shown that with proper choice of the space of admissible controls, the resulting conditions are equivalent to the classical definitions of uniform complete controllability.

Keywords: linear control systems, uniform complete controllability.

 Funding Agency Grant Number Russian Foundation for Basic Research 16-01-00346_à

DOI: https://doi.org/10.20537/vm170304

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

Citation: E. K. Makarov, S. N. Popova, “On the definition of uniform complete controllability”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 27:3 (2017), 326–343

Citation in format AMSBIB
\Bibitem{MakPop17} \by E.~K.~Makarov, S.~N.~Popova \paper On the definition of uniform complete controllability \jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki \yr 2017 \vol 27 \issue 3 \pages 326--343 \mathnet{http://mi.mathnet.ru/vuu592} \crossref{https://doi.org/10.20537/vm170304} \elib{http://elibrary.ru/item.asp?id=30267244}