On partial stability theory of nonlinear dynamic systems / Vorotnikov V.I., Martyshenko Yu.G. // Journal of Computer and Systems Sciences International. - 2010. - V. 49, l. 5. - P. 702-709.

ISSN:
10642307
Type:
Article
Abstract:
A stability problem with respect to a part of variables of the zero equilibrium position is considered for nonlinear non-stationary systems of ordinary differential equations with the continuous right-hand side. As compared to known assumptions, more general assumptions are made on the initial values of variables non-controlled in the course of studying stability. In addition, a stability problem is considered with respect to a part of variables of the "partial" equilibrium position, with similar assumptions made for initial values of variables that do not define the given equilibrium position. Conditions of stability and asymptotic stability of this type are obtained within the method of Lyapunov functions and generalize a number of existing results. The results are applied to the stability problem with respect to a part of variables of equilibrium positions of nonlinear holonomic mechanical systems. The problem of unification (to a certain extent) of the process of studying partial stability problems of stationary and non-stationary systems is discussed. © 2010 Pleiades Publishing, Ltd.
Author keywords:
Index keywords:
A-stability; Equilibrium positions; Holonomic mechanical system; Initial values; Method of Lyapunov functions; Non-linear dynamic systems; Nonstationary systems; Partial stability; Right-hand sides; S
DOI:
10.1134/S1064230710050047
Смотреть в Scopus:
https://www.scopus.com/inward/record.uri?eid=2-s2.0-78049520559&doi=10.1134%2fS1064230710050047&partnerID=40&md5=a6f9c4c911fd7033ba1cc2467e79e35d
Соавторы в МНС:
Другие поля
Поле Значение
Link https://www.scopus.com/inward/record.uri?eid=2-s2.0-78049520559&doi=10.1134%2fS1064230710050047&partnerID=40&md5=a6f9c4c911fd7033ba1cc2467e79e35d
Affiliations Nizhnii Tagil Technological Institute, Ural State Technical University, ul. Krasnogvardeiskaya 59, Nizhnii Tagil 622031, Russian Federation
References Rumyantsev, V.V., On Motion Stability with Respect to a Part of Variables (1957) Vestn. Mosk. Univ., Ser. Mat., Phiz., Astron., Khim., (4), pp. 9-16; Rumyantsev, V.V., Oziraner, A.S., (1987) Stability and Partial Motion Stabilization, , Nauka Moscow; Ya., S.A., Ignat'Ev, A.O., (1989) Certain Stability Problems of Nonautonomous Systems, , Naukova Dumka Kiev; Vorotnikov, V.I., (1998) Partial Stability and Control, , Birkhauser Boston 0891.93004; Vorotnikov, V.I., Rumyantsev, V.V., (2001) Stability and Control in A Part of Coordinate of the Phase Vector of Dynamic Systems: Theory, Methods, and Applications, , Nauchnyi Mir Moscow; Peiffer, K., Rouche, N., Liapounov's Second Method Applied to Partial Stability (1969) J. Mecanique, 8 (2), pp. 323-334. , 0183.36701 249746; Khapaev, M.M., (1993) Averaging in Stability Theory, , Kluwer Dordrecht 0782.34057; Fradkov, A.L., Miroshnik, I.V., Nikiforov, V.O., (1999) Nonlinear and Adaptive Control of Complex Systems, , Kluwer Dordrecht 0934.93002; Lin, Y., Sontag, E.D., Wang, Y., A Smooth Converse Lyapunov Theorem for Robust Stability (1996) SIAM J. Control Optim., 34 (1), pp. 124-160. , 0856.93070 10.1137/S0363012993259981 1372908; Efimov, D.V., (2005) Robust and Adaptive Control of Nonlinear Oscillations, , Nauka St. Petersburg; Corduneanu, C., Sur la Stabilite Partielle (1964) Rev. Roum. Math. Pure Et. Appl., 9 (3), pp. 229-236. , 0134.07104 180725; Andreev, A.S., On Investigation of Partial Asymptotic Stability (1991) Prikl. Mat. Mekh., 54 (4), pp. 539-547; Chellaboina, V., Haddad, W.M., A Unification between Partial Stability and Stability Theory for Time-Varying Systems (2002) IEEE Control Syst. Magazine, 22 (6), pp. 66-75. , 10.1109/MCS.2002.1077786 1967478; Vorotnikov, V.I., Two Classes of Problems of Partial Stability: To Unification of Concepts and Unified Solvability Conditions (2002) Dokl. Akad. Nauk, 384 (1), pp. 47-51. , 1932205; Teel, A.R., Zaccarian, L., On "uniformity" in definitions of global asymptotic stability for time-varying nonlinear systems (2006) Automatica, 42 (12), pp. 2219-2222. , DOI 10.1016/j.automatica.2006.07.012, PII S0005109806003086; Lagrange, J.L., (1788) Mecanique Analytique, , Veuve Desaint Paris; Lejeune-Dirichlet, G., Bedingungen der Stabilitat der Gleichgewichts Lagen (1846) J. Reine und Angew. Math, 2, pp. 85-88; Rouche, N., Habets, P., Laloy, M., (1977) Stability Theory by Liapounov's Direct Method, , Springer New York; Rumyantsev, V.V., Certain Problems on Motion Stability with Respect to a Part of Variables (1972) Continuum Mechanics and Similar Analysis Problems, pp. 429-436. , Nauka Moscow; Vorotnikov, V.I., On Stability and Partial Stability of Partial Equilibrium Position of Nonlinear Dynamic Systems (2003) Dokl. Akad. Nauk, 389 (3), pp. 332-337. , 2004487; Barbashin, E.A., Method of Sections in the Theory of Dynamic Systems (1951) Mat. Sb., 29 (712), pp. 233-337; Zubov, V.I., Lyapunov, A.M., (1957) Lyapunov Methods and Their Application, , Leningr. Gos. Univ. Leningrad; Bogolyubov Jr., N.N., (1974) A Method for Studying Model Hamiltonians, , Nauka Moscow; Galiullin, A.S., Mukhametzyanov, I.A., Mukharlyamov, R.G., (1971) Design of Systems of Open-Loop Motion, , Nauka Moscow; Samoilenko, A.M., (1987) Elements of Mathematical Theory of Multi-Frequency Oscillations, Invariant Tori, , Nauka Moscow; Teel, A., Praly, L., A Smooth Lyapounov Function from a Class-KL Estimate Involving Semidefinite Functions (2000) ESAIM: Control, Optimization, and Calculus of Variations, 5, pp. 313-367. , 0953.34042 10.1051/cocv:2000113 1765429; Massera, J.L., On Liapunouff's Condition of Stability Ann. Math., 50 (3), pp. 705-725
Correspondence Address Vorotnikov, V. I.; Nizhnii Tagil Technological Institute, Ural State Technical University, ul. Krasnogvardeiskaya 59, Nizhnii Tagil 622031, Russian Federation
CODEN JSSIE
Language of Original Document English
Abbreviated Source Title J. Comput. Syst. Sci. Int.
Source Scopus