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Belarus Novopolotsk
Year
2020
Volume
30
Issue
2
Pages
221-236
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Section Mathematics
Title The criterion of uniform global attainability of periodic systems
Author(-s) Kozlov A.A.a
Affiliations Polotsk State Universitya
Abstract We consider a linear time-varying control system $$\dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\in \mathbb{R},\qquad \qquad (1)$$ with piecewise continuous and bounded $\omega$-periodic coefficient matrices $A(\cdot)$ and $B(\cdot).$ We construct control of the system (1) as a linear feedback $u=U(t)x$ with piecewise continuous and bounded matrix function $U(t)$, $t\in \mathbb{R}$. For the closed-loop system $$\dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\in \mathbb{R}, \qquad \qquad (2)$$ the conditions of its uniform global attainability are studied. The latest property of the system (2) means existence of matrix $U(t)$, $t\in \mathbb{R}$, ensuring equalities $X_U((k+1)T,kT)=H_k$ for the state-transition matrix $X_U(t,s)$ of the system (2) with fixed $T>0$ and arbitrary $k\in\mathbb{Z}$, $\det H_k>0$. The problem is solved under the assumption of uniform complete controllability (by Kalman) of the system (1), corresponding to the closed-loop system (2), i.e. assuming the existence of such numbers $\sigma>0$ and $\alpha_i>0$, $i=\overline{1,4}$, that for any number $t_0\in\mathbb{R}$ and vector $\xi\in \mathbb{R}^n$ the following inequalities hold: $$\alpha_1\|\xi\|^2\leqslant\xi^*\int\nolimits_{t_0}^{t_0+\sigma}X(t_0,s)B(s)B^*(s)X^*(t_0,s)\,ds\,\xi\leqslant\alpha_2\|\xi\|^2,$$ $$\alpha_3\|\xi\|^2\leqslant\xi^*\int\nolimits_{t_0}^{t_0+\sigma}X(t_0+\sigma,s)B(s)B^*(s)X^*(t_0+\sigma,s)\,ds\,\xi\leqslant\alpha_4 \|\xi\|^2,$$ where $X(t,s)$ is the state-transition matrix of linear system (1) with $u(t)\equiv0.$ It is proved that the property of uniform complete controllability (by Kalman) of the periodic system (1) is a necessary and sufficient condition of uniform global attainability of the corresponding system (2).
Keywords linear control system with periodic coefficients, uniform complete controllability, uniform global attainability
UDC 517.926, 517.977
MSC 34D08, 34H05, 93C15
DOI 10.35634/vm200206
Received 30 August 2019
Language Russian
Citation Kozlov A.A. The criterion of uniform global attainability of periodic systems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2020, vol. 30, issue 2, pp. 221-236.
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