Geodesic
On stability with respect to the part of variables of a~non-autonomous system in a cylindrical phase space
Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 3, pp. 273-284.

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The stability problem of a system of differential equations with a right-hand side periodic with respect to the phase (angular) coordinates is considered. It is convenient to consider such systems in a cylindrical phase space which allows a more complete qualitative analysis of their solutions. The authors propose to investigate the dynamic properties of solutions of a non-autonomous system with angular coordinates by constructing its topological dynamics in such a space. The corresponding quasi-invariance property of the positive limit set of the system’s bounded solution is derived. The stability problem with respect to part of the variables is investigated basing of the vector Lyapunov function with the comparison principle and also basing on the constructed topological dynamics. Theorem like a quasi-invariance principle is proved on the basis of a vector Lyapunov function for the class of systems under consideration. Two theorems on the asymptotic stability of the zero solution with respect to part of the variables (to be more precise, non-angular coordinates) are proved. The novelty of these theorems lies in the requirement only for the stability of the comparison system, in contrast to the classical results with the condition of the corresponding asymptotic stability property. The results obtained in this paper make it possible to expand the usage of the direct Lyapunov method in solving a number of applied problems.
Keywords: non-autonomous system, cylindrical phase space, stability with respect to the part of variables, Lyapunov function. 
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J. I. Buranov; D. Kh. Khusanov. On stability with respect to the part of variables of a~non-autonomous system in a cylindrical phase space. Žurnal Srednevolžskogo matematičeskogo obŝestva, Tome 23 (2021) no. 3, pp. 273-284. http://geodesic.mathdoc.fr/item/SVMO_2021_23_3_a1/

[1] Barbashin E. A., Tabueva V. A., Dynamical systems with a cylindrical phase space, Nauka Publ., Moscow, 1969, 302 pp. (In Russ.)

[2] Leonov G. A., “A Class of Dynamical Systems with Cylindrical Phase Spaces”, Siberian Mathematical Journal, 17 (1976), 72–90 | DOI | Zbl | Zbl

[3] Krasovskii N. N., Stability of Motion: Applications of Lyapunov’s Second Method to Differential Systems and Equations with Delay, Stanford Univ. Press, Stanford, 1963, 194 pp. | Zbl

[4] La Salle J. P., “Stability theory for ordinary differential equations”, J. Differ. Equat., 4:1 (1968), 57–65 (In English) | DOI | Zbl

[5] Kayumov O. R., “Asymptotic Stability in the Large in Systems with a Cylindrical Phase Space”, Soviet Math. (Iz. VUZ), 31:10 (1987), 79–82 | Zbl

[6] Leonov G. A., Seledzhi S. M., Systems of Phase Synchronization in Analog and Digital Circuit Technology, Nevskii Dialekt, St. Petersburg, 2002, 112 pp. (In Russ.)

[7] Andreev A. S., Peregudova O. A., “On global trajectory tracking control of robot manipulators in cylindrical phase space”, International Journal of Control, 93:2 (2020), 3003–3015 (In English) | DOI | Zbl

[8] Matrosov V. M., Method of Lyapunov Vector Functions: Analysis of Dynamical Properties of Nonlinear Systems, Fizmatlit Publ., Moscow, 2001, 380 pp. (In Russ.)

[9] Rumyantsev V. V., “On Stability with Respect to Some of the Variables”, Vestnik MGU. Ser. Mat., Mechanics, Phys., Astron., Chem., 1957, no. 4, 9–16 (In Russ.)

[10] Corduneanu C., “Some problems, concerning partial stability”, Sympos Math., 1971, 141–154 (In English) | Zbl

[11] Rumyantsev V. V., Oziraner A. S., Stability and Stabilization of Motion in Relation to Some of the Variables, Nauka Publ., Moscow, 1987, 253 pp. (In Russ.)

[12] The Vector Lyapunov Functions in Stability Theory, eds. R. Z. Abdullin, R. Z. Kozlov, V. M. Matrosov, others, World Federation Publishers Inc., 1996, 394 pp.

[13] Vorotnikov V. I., Rumyantsev V. V., Stability and Control in a Part of Coordinate of the Phase Vector of Dynamic Systems: Theory, Methods, and Applications, Nauchnyy Mir Publ., Moscow, 2001, 320 pp. (In Russ.)

[14] Vorotnikov V. I., “Partial Stability and Control: The State-of-the-Art and Development Prospects”, Automation and Remote Control, 2005, no. 66, 511–561 | DOI | Zbl

[15] Peregudova O. A., Comparison Method in Problems of Stability and Motion Control of Mechanical Systems, UlGU Publ., Ulyanovsk, 2009, 253 pp. (In Russ.)

[16] Khusanov J. Kh., Berdierov A. Sh., Buranov Zh. I., “Comparison Method in Problems of Stability in Terms of Variables for Solutions of Ordinary Differential Equations”, Bulletin of the Institute of Mathematics, 2020, no. 4, 147–163 (In Russ.)

[17] Voskresenskiy E. V., Asymptotic Methods: Theory and Applications, SVMO Publ., Saransk, 2001, 300 pp. (In Russ.)

[18] Artstein Z., “Topological dynamics of ordinary differential equations”, J. Differ. Equat., 23 (1977), 216–223 (In English) | DOI | Zbl

[19] Sell G. R., “Nonautonomous differential equations and topological dynamics. I, II”, Trans. Amer. Math. Soc., 127:2 (1967), 241–283 (In English) | Zbl

[20] Andreyev A. S., Peregudova O. A., “The Comparison Method in Asymptotic Stability Problems”, Journal of Applied Mathematics and Mechanics, 70:6 (2006), 865–875 | DOI | Zbl

[21] Rouche N., Habets P., Laloy M., Stability Theory by Lyapunov’s Direct Method, Springer-Verlag, New York, 1977, 396 pp.

[22] Hatvani L., “On the application of differential inequalities to stability theory”, Moscow University Bulletin. Ser. Mathematics and Mechanics, 1975, no. 3, 83–89 (In English) | Zbl