Geodesic
Lyapunov's first method: estimates of characteristic numbers of functional matrices
Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 4, pp. 442-456 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

This paper contains the development of theoretical fundamentals of the first method of Lyapunov. We analyze the relations between characteristic numbers of functional matrices, their rows, and columns. We consider Lyapunov's results obtained to evaluate and calculate characteristic numbers for products of scalar functions and prove a theorem on the generalization of these results to the products of matrices. This theorem states necessary and sufficient conditions for the existence of rigorous estimates for characteristic numbers of matrix products. Also, we prove a theorem that establishes a relationship between the characteristic number of a square non-singular matrix and the characteristic number of its inverse matrix, and the determinant. The stated relations and properties of the characteristic numbers of square matrices we reformulate in terms of the Lyapunov exponents. Examples of matrices illustrate the proved theorems.
Keywords: Lyapunov's first method, stability theory, characteristic numbers, the Lyapunov exponent, functional matrices. 
@article{VSPUI_2019_15_4_a2,
     author = {V. S. Ermolin and T. V. Vlasova},
     title = {Lyapunov's first method: estimates of characteristic numbers of functional matrices},
     journal = {Vestnik Sankt-Peterburgskogo universiteta. Prikladna\^a matematika, informatika, processy upravleni\^a},
     pages = {442--456},
     year = {2019},
     volume = {15},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VSPUI_2019_15_4_a2/}
}
TY  - JOUR
AU  - V. S. Ermolin
AU  - T. V. Vlasova
TI  - Lyapunov's first method: estimates of characteristic numbers of functional matrices
JO  - Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
PY  - 2019
SP  - 442
EP  - 456
VL  - 15
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VSPUI_2019_15_4_a2/
LA  - en
ID  - VSPUI_2019_15_4_a2
ER  - 
%0 Journal Article
%A V. S. Ermolin
%A T. V. Vlasova
%T Lyapunov's first method: estimates of characteristic numbers of functional matrices
%J Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ
%D 2019
%P 442-456
%V 15
%N 4
%U http://geodesic.mathdoc.fr/item/VSPUI_2019_15_4_a2/
%G en
%F VSPUI_2019_15_4_a2
V. S. Ermolin; T. V. Vlasova. Lyapunov's first method: estimates of characteristic numbers of functional matrices. Vestnik Sankt-Peterburgskogo universiteta. Prikladnaâ matematika, informatika, processy upravleniâ, Tome 15 (2019) no. 4, pp. 442-456. http://geodesic.mathdoc.fr/item/VSPUI_2019_15_4_a2/

[1] A. M. Lyapunov, The general problem of the stability of motion, Gostekhizdat, M., 1950, 472 pp. (In Russian) | MR | Zbl

[2] A. V. Ekimov, Yu. E. Balykina, M. V. Svirkin, “Analysis of attainability sets of bilinear control systems”, ICNAAM 2016, AIP Conference Proceedings, 1863, American Institute of Physics Publ. LLC, Rodos, 2017, 170012 | DOI

[3] V. S. Ermolin, “Value sets of the discrete interval length in the problem of discrete stabilization”, Avtomatika, 1995, no. 3, 15–21 | MR | Zbl

[4] V. S. Ermolin, T. V. Vlasova, “Identification of the domain of attraction”, Proceedings of SCP 2015 Conference, IEEE Publ., St. Petersburg, 2015, 9–12

[5] A. V. Zubov, “Stabilization of program motion and kinematic trajectories in dynamic systems in case of systems of direct and indirect control”, Automation and Remote Control, 68:3 (2007), 386–398 | DOI | MR | Zbl

[6] V. I. Zubov, Fluctuations in nonlinear and control systems, Sudpromgiz Publ., L., 1962, 632 pp. (In Russian) | MR

[7] V. I. Zubov, Lectures on the theory of control, 2nd ed., Lan's Publ, St. Petersburg, 2009, 496 pp. (In Russian) | MR

[8] V. V. Kozlov, S. D. Furta, “Lyapunows first method for strongly non-linear systems”, Journal of Applied Mathematics and Mechanics, 60:1 (1996), 7–18 | DOI | MR | Zbl

[9] N. G. Chetaev, The stability of motion, Gostekhizdat, M., 1955, 176 pp. (In Russian) | MR

[10] I. G. Malkin, Theory of stability of motion, Nauka, M., 1966, 530 pp. (In Russian) | MR | Zbl

[11] B. F. Bylov, R. E. Vinograd, D. M. Grobman, V. V. Nemyckij, The theory of Lyapunov characteristic numbers and their application to the theory of stability, Nauka, M., 1966, 576 pp. (In Russian) | MR

[12] B. P. Demidovich, Lectures on the mathematical theory of stability, Lan's Publ, St. Petersburg, 2008, 480 pp. (In Russian) | MR

[13] T. M. Adami, E. Best, J. J. Zhu, “Stability assessment using Lyapunov's first method”, Proceedings of the Annual Southeastern Symposium on System Theory, IEEE Publ., Huntsville, 2002, 297–301

[14] P. Masarati, A. Tamer, “Sensitivity of trajectory stability estimated by Lyapunov characteristic exponents”, Aerospace Science and Technology, 47 (2015), 501–510 | DOI

[15] V. I. Oseledets, “A multiplicative ergodic theorem. Lyapunov characteristic numbers for dynamical systems”, Trans. Moscow Mathematical Society Journal, 19 (1968), 197–231 | MR | Zbl

[16] M. Cencini, F. Ginelli, “Lyapunov analysis: from dynamical systems theory to applications”, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 250301 | DOI | MR

[17] L. S. Young, “Mathematical theory of Lyapunov exponents”, Journal of Physics A: Mathematical and Theoretical, 46:254001 (2013) | MR

[18] V. S. Ermolin, “Invariant transformations in Lyapunov's first method”, Vestnik of Saint Petersburg University. Series 10. Applied Mathematics. Computer Science. Control Processes, 2014, no. 2, 36–48 (In Russian)

[19] V. S. Ermolin, T. V. Vlasova, “A group of invariant transformations in the stability problem via Lyapunov's first method”, Proceedings of ICCTPEA 2014 Conference, IEEE Publ., St. Petersburg, 2014, 48–49