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Section
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Mathematics
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Title
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On the definition of uniform complete controllability
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Author(-s)
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Makarov E.K.a,
Popova S.N.bc
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Affiliations
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Institute of Mathematics, National Academy of Sciences of Belarusa,
Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesb,
Udmurt State Universityc
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Abstract
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We consider a linear control system
$$\dot x = A(t)x + B(t)u,\quad t\in\mathbb{R},\quad x\in\mathbb{R}^{n},\quad u\in\mathbb{R}^{m}, \qquad \qquad(1)$$
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.
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Keywords
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linear control systems, uniform complete controllability
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UDC
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517.977.1, 517.926
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MSC
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93B05, 93C05
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DOI
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10.20537/vm170304
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Received
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22 June 2017
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Language
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Russian
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Citation
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Makarov E.K., Popova S.N. On the definition of uniform complete controllability, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 3, pp. 326-343.
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