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. Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the mathematical stability theory), Moscow: Moscow State University, 1998.
- Zaitsev V.A., Tonkov E.L. Attainability, compatibility and V.M. Millionshchikov's method of rotations, Russian Mathematics, 1999, no. 2, pp. 42-52. https://zbmath.org/?q=an:1049.93504
- 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, no. 1, pp. 97-107.
- 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.
- Izobov N.A. Linear systems of ordinary differential equations, Journal of Soviet Mathematics, 1976, vol. 5, no. 1, pp. 46-96. https://doi.org/10.1007/BF01091661
- Bogdanov Yu.S. On asymptotically equivalent linear differential systems, Differ. Uravn., 1965, vol. 1, no. 6, pp. 707-716 (in Russian). http://mi.mathnet.ru/eng/de8686
- Kalman R.E. Contribution to the theory of optimal control, Boletín de la Sociedad Matemática Mexicana, 1960, vol. 5, no. 1, pp. 102-119.
- Zaitsev V.A. Global attainability and global reducibility of two- and tree-dimensional linear control systems with constant coefficients, Vestnik Udmurtskogo Universiteta. Matematika, 2003, no. 1, pp. 31-62 (in Russian). https://www.elibrary.ru/item.asp?id=22419350
- Gabdrakhimov A.F., Zaitsev V.A. Lyapunov's reducibility for four-dimensional linear stationary control systems in the class of piecewise-constant controls, Vestnik Udmurtskogo Universiteta. Matematika, 2006, no. 1, pp. 25-40 (in Russian). http://mi.mathnet.ru/vuu244
- Sergeev I.N. Controlling solutions to a linear differential equation, Moscow University Mathematics Bulletin, 2009, vol. 64, issue 3, pp. 113-120. https://doi.org/10.3103/S0027132209030048
- Smirnov E.Ya. Stabilizatsiya programmnykh dvizhenii (Stabilization of program motion), Saint Petersburg: Saint Petersburg State University, 1997.
- Gaishun I.V. Vvedenie v teoriyu lineinykh nestatsionarnykh sistem (Introduction to the theory of linear nonstationary systems), Moscow: URSS, 2004.
- Zaitsev V.A. Uniform global attainability and global Lyapunov reducibility of linear control systems in the Hessenberg form, Journal of Mathematical Sciences, 2018, vol. 230, issue 5, pp. 677-682. https://doi.org/10.1007/s10958-018-3768-2
- Zaitsev V.A. To the theory of stabilization of control systems, Dr. Sci. (Phys.–Math.) Dissertation, Izhevsk, 2015, 293 p. (In Russian).
- Popova S.N. Control over asymptotical invariants of linear systems, Dr. Sci. (Phys.–Math.) Dissertation, Izhevsk, 2004, 264 p. (In Russian).
- 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).
- Kozlov A.A. Control of Lyapunov exponents of differential systems with discontinuous and fast oscillated coefficients, Cand. of Sci. (Phys.–Math.) Dissertation, Minsk, 2008, 111 p. (In Russian).
- Kozlov A.A., Makarov E.K. On the control of Lyapunov exponents of linear systems in the nondegenerate case, Differential Equations, 2007, vol. 43, no. 5, pp. 636-642. https://doi.org/10.1134/S0012266107050072
- 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 (in Russian). https://doi.org/10.20537/vm170203
- 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). http://mi.mathnet.ru/eng/de3820
- Horn R., Johnson C. Matrix analysis, Cambridge: Cambridge University Press, 1988.
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
|
|