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## Archive of Issues

Belarus Novopolotsk
Year
2017
Volume
27
Issue
2
Pages
178-192
 Section Mathematics Title On uniform global attainability of two-dimensional linear systems with locally integrable coefficients Author(-s) Kozlov A.A.a, Ints I.V.a Affiliations Polotsk State Universitya Abstract We consider a linear time-varying control system with locally integrable and integrally bounded coefficients $$\dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\geqslant 0. \qquad\qquad (1)$$ We construct control of the system $(1)$ as a linear feedback $u=U(t)x$ with measurable and bounded function $U(t),$ $t\geqslant 0.$ For the closed-loop system $$\dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\geqslant 0, \qquad \qquad (2)$$ we study a question about the conditions for its uniform global attainability. The last property of the system $(2)$ means existence of a matrix $U(t),$ $t\geqslant 0,$ that ensure 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 $k\in\mathbb N,$ $\det H_k>0.$ The problem is solved under the assumption of uniform complete controllability of the system $(1),$ corresponding to the closed-loop system $(2),$ i.e. assuming the existence of such $\sigma>0$ and $\gamma>0,$ that for any initial time $t_0\geqslant 0$ and initial condition $x(t_0)=x_0\in \mathbb{R}^n$ of the system $(1)$ on the segment $[t_0,t_0+\sigma]$ there exists a measurable and bounded vector control $u=u(t),$ $\|u(t)\|\leqslant\gamma\|x_0\|,$ $t\in[t_0,t_0+\sigma],$ that transforms a vector of the initial state of the system into zero on that segment. It is proved that in two-dimensional case, i.e. when $n=2,$ the property of uniform complete controllability of the system $(1)$ is a sufficient condition of uniform global attainability of the corresponding system $(2).$ Keywords linear control system, uniform complete controllability, uniform global attainability UDC 517.926, 517.977 MSC 34D08, 34H05, 93C15 DOI 10.20537/vm170203 Received 30 May 2017 Language Russian Citation Kozlov A.A., Ints I.V. On uniform global attainability of two-dimensional linear systems with locally integrable coefficients, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 2, pp. 178-192. References Bylov B.F., Vinograd R.E., Grobman D.M., Nemytskii V.V. Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti (Theory of Lyapunov exponents and its application to problems of stability), Moscow: Nauka, 1966, 576 p. Zaitsev V.A. 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Control over Lyapunov's exponents of a differential systems with break and fast oscillated coefficients, Abstract of Cand. Sci. (Phys.–Math.) Dissertation, Minsk, 2008, 20 p. (In Russian). Kozlov A.A., Ints I.V. On the global Lyapunov reducibility of two-dimensional linear systems with locally integrable coefficients, Differential Equation, 2016, vol. 52, issue 6, pp. 699-721. DOI: 10.1134/S0012266116060021 Full text