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Home » Issues » Archive of Issues » 2015 - 2 issue » pp. 157-179

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Russia Izhevsk
Year
2015
Volume
25
Issue
2
Pages
157-179
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Section Mathematics
Title Criteria for uniform complete controllability of a linear system
Author(-s) Zaitsev V.A.a
Affiliations Udmurt State Universitya
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
UDC 517.977.1, 517.926
MSC 93B05, 93C05
DOI 10.20537/vm150202
Received 15 March 2015
Language Russian
Citation Zaitsev V.A. Criteria for uniform complete controllability of a linear system, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2015, vol. 25, issue 2, pp. 157-179.
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