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## Archive of Issues

Belarus Novopolotsk
Year
2020
Volume
30
Issue
2
Pages
221-236
 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. References Popova S.N. Global controllability of the complete set of Lyapunov invariants of periodic systems, Differential Equations, 2003, vol. 39, no. 12, pp. 1713-1723. https://doi.org/10.1023/B:DIEQ.0000023551.43484.e5 Makarov E.K., Popova S.N. Upravlyaemost' asimptoticheskikh invariantov nestatsionarnykh lineinykh sistem (Controllability of asymptotic invariants of non-stationary linear systems), Minsk: Belarus. Navuka, 2012. Demidovich B.P. 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Translated under the title Matrichnyi analiz, Moscow: Mir, 1989. Kozlov A.A. The criterion of uniform global attainability of linear systems, Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, 2018, vol. 52, pp. 47-58 (in Russian). https://doi.org/10.20537/2226-3594-2018-52-04 Full text