On the sufficient condition of global scalarizability of linear control systems with locally integrable coefficients

A. A. Kozlov

Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 26 (2016) no. 2, pp. 221-230

Résumé

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 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} a definition of uniform global quasi-attainability is introduced. This notion is a weakening of the property of uniform global attainability. The last property means existence of matrix $U(t)$, $t\geqslant 0$, 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 $k\in\mathbb N$, $\det H_k>0$. We prove that uniform global quasi-attainability implies global scalarizability. The last property means that for any given locally integrable and integrally bounded scalar function $p=p(t)$, $t\geqslant0$, there exists a measurable and bounded function $U=U(t)$, $t\geqslant 0$, which ensures asymptotic equivalence of the system $(2)$ and the system of scalar type $\dot z=p(t)z$, $z\in\mathbb{R}^n$, $t\geqslant0$.

Détail

Keywords: linear control system, Lyapunov exponents, global scalarizability.

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] Demidovich B. P., Lectures on the mathematical stability theory, Moscow State University, M., 1990, 624 pp. | MR

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

[4] Popova S. N., “Global reducibility of linear control systems to systems of scalar type”, Differential Equations, 40:1 (2004), 43–49 | DOI | MR | Zbl

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

[6] Popova S. N., “On global controllability of the Lyapunov exponents for linear systems”, Differential Equations, 43:8 (2007), 1072–1078 | DOI | MR | Zbl

[7] Popova S. N., “Global controllability of the complete set of Lyapunov invariants of periodic systems”, Differential Equations, 39:12 (2003), 1713–1723 | DOI | MR | Zbl

[8] Kozlov A. A., Makarov E. K., “On the control of Lyapunov exponents of linear systems in the non-degenerate case”, Differential Equations, 43:5 (2007), 636–642 | DOI | MR | Zbl

[9] Kozlov A. A., “A control procedure for total set of Lyapunov invariants for linear systems in nondegenerate case”, Trudy Inst. Mat., 15:2 (2007), 33–37 (in Russian) | Zbl

[10] Kozlov A. A., Ints I. V., Burak A. D., “Global controllability of a separate asymptotic invariants of two-dimensional linear systems with locally integrable coefficients”, Vestsi Nats. Akad. Navuk Belarusi, Ser. Fiz.-Mat. Navuk, Suppl., 2014, 37–45 (in Russian) | MR

[11] Zaitsev V. A., “Uniform complete controllability and global control over asymptotic invariants of linear systems in the Hessenberg form”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 25:3 (2015), 318–337 (in Russian)

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

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

[14] Zaitsev V. A., “Criteria for uniform complete controllability of linear systems”, Vestn. Udmurt. Univ. Mat. Mekh. Komp'yut. Nauki, 25:2 (2015), 157–179 (in Russian) | Zbl

[15] Makarov E. K., Popova S. N., “Global controllability of central exponents of linear systems”, Russian Mathematics, 43:2 (1999), 56–63 | MR | Zbl

[16] Kozlov A. A., “On the control of Lyapunov exponents of two-dimensional linear systems with locally integrable coefficients”, Differential Equations, 44:10 (2008), 1375–1392 | DOI | MR | Zbl

[17] Kozlov A. A., Burak A. D., “About control over characteristic exponents of three-dimensional linear differential systems with a discontinuous and fast oscillated coefficients”, Vest. Vitseb. Dzyarzh. Univ., 2013, no. 5(77), 11–31 (in Russian)

[18] Tonkov E. L., “Uniform attainability and Lyapunov reducibility of bilinear control system”, Proceedings of the Steklov Institute of Mathematics, 2000, Suppl. 1, S228–S253 | MR | Zbl

[19] Zaitsev V. A., “Global Lyapunov reducibility of two-dimensional control systems with piecewise constant coefficients”, Vestn. Udmurt. Univ. Mat., 2002, no. 1, 3–12 (in Russian) | MR

[20] Zaitsev V. A., “Global attainability and global reducibility of two- and tree-dimensional linear control systems with constant coefficients”, Vestn. Udmurt. Univ. Mat., 2003, no. 1, 31–62 (in Russian)

[21] Zaitsev V. A., “Attainability and Lyapunov reducibility of linear control systems”, Optimization, control, intelligence: Transactions, 2(10), Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, 2005, 76–84 (in Russian) | Zbl

[22] 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), 97–106 | MR | Zbl

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