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## Archive of Issues

Belarus; Russia Izhevsk; Minsk; Yekaterinburg
Year
2017
Volume
27
Issue
3
Pages
326-343
 Section Mathematics Title On the definition of uniform complete controllability Author(-s) Makarov E.K.a, Popova S.N.bc Affiliations Institute of Mathematics, National Academy of Sciences of Belarusa, Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciencesb, Udmurt State Universityc Abstract We consider a linear control system $$\dot x = A(t)x + B(t)u,\quad t\in\mathbb{R},\quad x\in\mathbb{R}^{n},\quad u\in\mathbb{R}^{m}, \qquad \qquad(1)$$ under the assumption that the transition matrix $X(t,s)$ of the free system $\dot x = A(t)x$ is continuous with respect to $t$ and $s$ separately. We also suppose that on each interval $[\tau, \tau + \vartheta]$ of fixed length $\vartheta$ the normed space $Z_{\tau}$ of functions defined on this interval is given. A control $u$ on the interval $[\tau, \tau+\vartheta]$ is called admissible if $u\in Z_{\tau}$ and there exists the integral $\mathcal Q_{\tau}(u):=\int_{\tau}^{\tau+\vartheta}X(\tau,s)B(s)u(s)\,ds$. The vector subspace $U_{\tau}$ of the space $Z_{\tau}$ where the operator $\mathcal Q_{\tau}$ is defined is called the space of admissible controls for the system $(1)$ on the interval $[\tau,\tau +\vartheta]$. We propose a definition of uniform complete controllability of the system $(1)$ for the case of an arbitrary dependence of the space of admissible controls on the moment of the beginning of the control process. In this situation direct and dual necessary and sufficient conditions for uniform complete controllability of a linear system are obtained. It is shown that with proper choice of the space of admissible controls, the resulting conditions are equivalent to the classical definitions of uniform complete controllability. Keywords linear control systems, uniform complete controllability UDC 517.977.1, 517.926 MSC 93B05, 93C05 DOI 10.20537/vm170304 Received 22 June 2017 Language Russian Citation Makarov E.K., Popova S.N. On the definition of uniform complete controllability, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 3, pp. 326-343. References Krasovskii N.N. Teoriya upravleniya dvizheniem (Theory of motion control), Moscow: Nauka, 1968, 476 p. Kalman R.E. Contribution to the theory of optimal control, Boletin de la Sociedad Matematiсa Mexicana, 1960, vol. 5, no. 1, pp. 102-119. Popova S.N. Problems of control over Lyapunov exponents, Cand. Sci. (Phys.-Math.) Dissertation, Izhevsk, 1992, 103 p. (In Russian). Zaitsev V.A. Criteria for uniform complete controllability of a linear system, Vestn. Udmurt. Univ. Mat. Mech. Komp'yt. Nauki, 2015, vol. 25, issue 2, pp. 157-179 (in Russian). DOI: 10.20537/vm150202 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. Nashed M.Z., Votruba G.F. A unified operator theory of generalized inverses, Generalized Inverses and Applications, 1976, pp. 1-109. DOI: 10.1016/B978-0-12-514250-2.50005-6 Tonkov E.L. A criterion of uniform controllability and stabilization of a linear recurrent system, Differential Equations, 1979, vol. 15, pp. 1285-1292. Krein S.G. Lineinye uravneniya v banakhovykh prostranstvakh (Linear equations in Banach spaces), Moscow: Nauka, 1971, 104 p. Makarov E.K., Popova S.N. On the global controllability of a complete set of Lyapunov invariants of two-dimensional linear systems, Differential Equations, 1999, vol. 35, no. 1, pp. 97-107. Kozlov A.A. On the control of Lyapunov exponents of two-dimensional linear systems with locally integrable coefficients, Differential Equations, 2008, vol. 44, no. 10, pp. 1375-1392. DOI: 10.1134/S0012266108100042 Kozlov A.A., Burak A.D. On the control of characteristic exponents of three-dimensional linear differential systems with discontinuous and fast oscillating coefficients, Vesnik Vitsebsk. Dzyarzh. Univ., 2013, no. 5 (77), pp. 11-31 (in Russian). Kozlov A.A., Ints I.V. On the global Lyapunov reducibility of two-dimensional linear systems with locally integrable coefficients, Differential Equations, 2016, vol. 52, issue 6, pp. 699-721. DOI: 10.1134/S0012266116060021 Gabasov R.F., Kirillova F.M. Optimizatsiya lineinykh sistem (Optimization of linear systems), Minsk: Belarusian State University, 1973, 248 p. Demidovich V.B. A certain criterion for the stability of difference equations, Differ. Uravn., 1969, vol. 5, no. 7, pp. 1247-1255 (in Russian). Latushkin Y., Randolph T., Schnaubelt R. Exponential dichotomy and mild solutions of non autonomous equations in Banach spaces, Journal of Dynamics and Differential Equations, 1998, vol. 10, no. 3, pp. 489-510. DOI: 10.1023/A:1022609414870 Full text