The criterion of uniform global attainability of linear systems

A. A. Kozlov

Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta, Tome 52 (2018), pp. 47-58

Résumé

In this paper, we consider a linear time-varying control system with locally integrable and integrally bounded coefficients \begin{equation} \dot x =A(t)x+ B(t)u, \quad x\in\mathbb{R}^n,\quad u\in\mathbb{R}^m,\quad t\geqslant 0. \tag{1} \end{equation} We construct control of the system $(1)$ as a linear feedback $u=U(t)x$ with a measurable and bounded function $U(t)$, $t\geqslant 0$. For the closed-loop system \begin{equation} \dot x =(A(t)+B(t)U(t))x, \quad x\in\mathbb{R}^n, \quad t\geqslant 0, \tag{2} \end{equation} the criterion for its uniform global attainability is established. The latter property means the existence of $T>0$ such that for any positive $\alpha$ and $\beta$ there exists a $d=d(\alpha,\beta)>0$ such that for any $t_0\geqslant 0$ and for any $(n\times n)$-matrix $H$, $\|H\|\leqslant\alpha$, $\det H\geqslant\beta$, there exists a measurable on $[t_0,t_0+T]$ gain matrix function $U(\cdot)$ such that $\sup\limits_{t\in [t_0,t_0+T]}\|U(t)\|\leqslant d$ and $X_U(t_0+T,t_0)=H$, where $X_U$ is the state transition matrix for the system (2). The proof of the criterion is based on the theorem on the representation of an arbitrary $(n\times n)$-matrix with a positive determinant in the form of a product of nine upper and lower triangular matrices with positive diagonal elements and additional conditions on the norm and determinant of these matrices.

Détail

Keywords: linear control system, uniform global attainability.
Mots-clés : state-transition matrix

Bibliographie

[1] Bylov B. F., Vinograd R. E., Grobman D. M., Nemytskii V. V., Theory of Lyapunov exponents and its application to problems of stability, Nauka, M., 1966, 576 pp. | MR

[2] Zaitsev V. A., Tonkov E. L., “Attainability, compatibility and V. M. Millionshchikov's method of rotations”, Russian Mathematics, 43:2, 42–52 | MR | MR | Zbl

[3] Makarov E. K., Popova S. N., Controllability of asymptotic invariants of non-stationary linear systems, Belarus. Navuka, Minsk, 2012, 407 pp.

[4] Makarov E. K., Popova S. N. The global controllability of a complete set of Lyapunov invariants for two-dimensional linear systems, Differential Equations, 35:1 (1999), 97–107 | MR | Zbl

[5] Bogdanov Yu.S., “On asymptotically equivalent linear differential systems”, Differ. Uravn., 1:6 (1965), 707–716 (in Russian) | MR | Zbl

[6] Demidovich B. P., Lectures on the mathematical stability theory, Moscow State University, M., 1990

[7] Kalman R. E., “Contribution to the theory of optimal control”, Boletin de la Sociedad Matematica Mexicana, 5:1 (1960), 102–119 | MR | Zbl

[8] Tonkov E. L., “A criterion of uniform controllability and stabilization of a linear recurrent system”, Differ. Uravn., 15:10 (1979), 1804–1813 (in Russian) | MR | Zbl

[9] Zaitsev V. A., “Uniform global attainability and global Lyapunov reducibility of linear control systems in the Hessenberg form”, Journal of Mathematical Sciences, 230:5 (2018), 677–682 | DOI | Zbl

[10] Zaitsev V. A., To the theory of stabilization of control systems, Dr. Sci. (Phys.–Math.) Dissertation, Izhevsk, 2015, 293 pp. (In Russian)

[11] Horn R., Johnson C., Matrix analysis, Cambridge University Press, Cambridge, 1988 | MR | MR

[12] 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, 27:2 (2017), 178–192 (in Russian) | DOI | MR | Zbl

[13] Kozlov A. A., “On a factorization of square matrices with positive determinant”, Trudy Instituta Matematiki, 25:1 (2017), 51–61 (in Russian) | MR

[14] Kolmogorov A. N., Fomin S. V., Elements of the theory of functions and functional analysis, Nauka, M., 1976, 543 pp. | MR

[15] Bortakovskii A. S., Linear algebra in examples and problems, Vysshaya shkola, M., 2005, 591 pp.