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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. Global attainability and global Lyapunov reducibility of two-dimensional and three-dimensional linear control systems with the constant coefficients, Vestn. Udmurt. Univ. Mat., 2003, no. 1, pp. 31-62 (in Russian). 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, 407 p. Demidovich B.P. Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the mathematical stability theory), Moscow: Moscow State University, 1998, 624 p. Bogdanov Yu.S. About the asymptotically equivalent linear differential systems, Differ. Uravn., 1965, vol. 1, no. 6, pp. 707-716 (in Russian). Makarov E.K., Popova S.N. The global controllability of a complete set of Lyapunov invariants for two-dimensional linear systems, Differential Equations, 1999, vol. 35, issue 1, pp. 97-107. Zaitsev V.A. Tonkov E.L. Attainability, compatibility and V.M. Millionshchikov’s method of rotations, Russian Mathematics, 1999, vol. 43, no. 2, pp. 42-52. Popova S.N., Tonkov E.L. Control over the Lyapunov exponents of consistent systems. I, Differential Equations, 1994, vol. 30, no. 10, pp. 1556-1564. Popova S.N., Tonkov E.L. Control of the Lyapunov exponents of consistent systems. II, Differential Equations, 1994, vol. 30, no. 11, pp. 1800-1807. Popova S.N., Tonkov E.L. Control over Lyapunov exponents of consistent systems. III, Differential Equations, 1995, vol. 31, no. 2, pp. 209-218. Popova S.N., Tonkov E.L. Uniform consistency of linear systems, Differential Equations, 1995, vol. 31, no. 4, pp. 672-674. Popova S.N., Tonkov E.L. Consistent systems and control of Lyapunov exponents, Differential Equations, 1997, vol. 33, no. 2, pp. 226-235. Popova S.N. Equivalence between local attainability and complete controllability of linear systems, Russian Mathematics, 2002, vol. 46, no. 6, pp. 48-51. Popova S.N. Global controllability of the complete set of Lyapunov invariants of periodic systems, Differential Equations, 2003, vol. 39, issue 12, pp. 1713-1723. DOI: 10.1023/B:DIEQ.0000023551.43484.e5 Popova S.N. On the global controllability of Lyapunov exponents of linear systems, Differential Equations, 2007, vol. 43, issue 8, pp. 1072-1078. DOI: 10.1134/S0012266107080058 Kozlov A.A., Makarov E.K. About uniform global attainability of linear control systems in the non-degenerate case, Vestn. Vitsebsk. Dzyarzh. Univ., 2007, no. 3 (45), pp. 100-109 (in Russian). Izobov N.A. Linear systems of ordinary differential equations, Journal of Soviet Mathematics, 1976, vol. 5, issue 1, pp. 46-96. DOI: 10.1007/BF01091661 Kalman R.E. Contribution to the theory of optimal control, Boletin de la Sociedad Matematica Mexicana, 1960, vol. 5, no. 1, pp. 102-119. Gabdrakhimov A.F., Zaitsev V.A. Lyapunov reducibility for four-dimensional linear stationary control systems in the class of the piecewise-constant control functions, Vestn. Udmurt. Univ. Mat., 2006, no. 1, pp. 25-40 (in Russian). Zaitsev V.A. Criteria for uniform complete controllability of a linear system, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 2015, vol. 25, issue 2, pp. 157-179 (in Russian). DOI: 10.20537/vm150202 Tonkov E.L. A criterion of uniform controllability and stabilization of a linear recurrent system, Differ. Uravn., 1979, vol. 15, no. 10, pp. 1804-1813 (in Russian). Horn R., Johnson C. Matrix analysis, Cambridge: Cambridge University Press, 1988. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989, 655 p. Kozlov A.A. On the partial case of global Lyapunov's reducibility of two-dimensional systems, Vestn. Vitsebsk. Dzyarzh. Univ., 2008, no. 3 (49), pp. 105-110 (in Russian). Kozlov A.A. 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