Geodesic
Investigating stability using nonlinear quasihomogeneous approximation to differential equations with impulsive action
Sbornik. Mathematics, Tome 205 (2014) no. 6, pp. 862-891 Cet article a éte moissonné depuis la source Math-Net.Ru

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Inverse theorems to Lyapunov's direct method are established for quasihomogeneous systems of differential equations with impulsive action. Conditions for the existence of Lyapunov functions satisfying typical bounds for quasihomogeneous functions are obtained. Using these results, we establish conditions for an equilibrium of a nonlinear system with impulsive action to be stable, using the properties of a quasihomogeneous approximation to the system. The results are illustrated by an example of a large-scale system with homogeneous subsystems. Bibliography: 30 titles.
Keywords: quasihomogeneous system, Lyapunov stability, Lyapunov's direct method.
Mots-clés : impulsive action 
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A. I. Dvirnyj; V. I. Slyn'ko. Investigating stability using nonlinear quasihomogeneous approximation to differential equations with impulsive action. Sbornik. Mathematics, Tome 205 (2014) no. 6, pp. 862-891. http://geodesic.mathdoc.fr/item/SM_2014_205_6_a4/

[1] A. M. Lyapunov, Obschaya zadacha ob ustoichivosti dvizheniya, Gostekhizdat, M.–L., 1950, 471 pp. ; A. M. Lyapunov, The general problem of the stability of motion, Taylor Francis, Ltd., London, 1992, x+270 pp. | MR | Zbl | MR | Zbl

[2] L. G. Khazin, E. E. Shnol, Ustoichivost kriticheskikh polozhenii ravnovesiya, Izd-vo NTsBI AN SSSR, Puschino, 1985, 216 pp. ; L. G. Khazin, È. È. Shnol', Stability of critical equilibrium states, Nonlinear Sci. Theory Appl., Manchester Univ. Press, Manchester, 1991, xii+208 pp. | MR | Zbl | MR | Zbl

[3] A. Ya. Savchenko, A. O. Ignatev, Nekotorye zadachi ustoichivosti neavtonomnykh dinamicheskikh sistem, Naukova dumka, K., 1989, 304 pp. | MR | Zbl

[4] G. V. Kamenkov, Izbrannye trudy, v. I, II, Nauka, M., 1971, 258 pp., 214 pp. | MR | MR | Zbl

[5] F. Khartman, Obyknovennye differentsialnye uravneniya, Mir, M., 1970, 720 pp. ; Ph. Hartman, Ordinary differential equations, John Wiley Sons, Inc., New York–London–Sydney, 1964, xiv+612 pp. | MR | Zbl | MR | Zbl

[6] V. A. Pliss, “Printsip svedeniya v teorii ustoichivosti dvizheniya”, Izv. AN SSSR. Seriya matem., 28:6 (1964), 1297–1324 | MR | Zbl

[7] V. I. Zubov, Ustoichivost dvizheniya (metody Lyapunova i ikh primenenie), 2-e izd., Vysshaya shkola, M., 1984, 232 pp. | MR | Zbl

[8] J. G. Malkin,, Theorie der Stabilität einer Bewegung, R. Oldenbourg, München, 1959, xiii+402 pp. | MR | MR | Zbl | Zbl

[9] N. N. Krasovskii, Nekotorye zadachi teorii ustoichivosti dvizheniya, Fizmatgiz, M., 1959, 211 pp. ; N. N. Krasovskii, Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay, Stanford Univ. Press, Stanford, Calif., 1963, vi+188 pp. | MR | Zbl | MR | Zbl

[10] N. N. Krasovskii, “Ob ustoichivosti po pervomu priblizheniyu”, PMM, 19 (1955), 516–530 | MR | Zbl

[11] V. I. Zubov, Matematicheskie metody issledovaniya sistem avtomaticheskogo regulirovaniya, Sudpromgiz, L., 1959, 324 pp. ; V. I. Zubov, Mathematical methods for the study of automatic control systems, Pergamon Press, New York–Oxford–London–Paris; Jerusalem Academic Press, Jerusalem, 1962, vii+327 pp. | MR | Zbl | MR | Zbl

[12] O. S. Chernikova, “Printsip svedeniya dlya sistem differentsialnykh uravnenii s impulsnym vozdeistviem”, Ukr. matem. zhurn., 34:5 (1982), 601–607 | MR | Zbl

[13] A. I. Dvirnyi, V. I. Slynko, “Analog kriticheskogo sluchaya ustoichivosti G. V. Kamenkova dlya nestatsionarnykh sistem differentsialnykh uravnenii s impulsnym vozdeistviem”, Ukr. matem. vestnik, 9:3 (2012), 318–340 ; A. I. Dvirny, V. I. Slyn'ko, “An analog to Kamenkov's critical stability case for nonstationary systems of impulsive differential equations”, J. Math. Sci., 188:2 (2013), 85–101 | MR | Zbl | DOI

[14] A. I. Dvirnyi, V. I. Slynko, “Ustoichivost reshenii differentsialnykh uravnenii s impulsnym vozdeistviem v kriticheskikh sluchayakh”, Sib. matem. zhurn., 52:1 (2011), 70–80 | MR | Zbl

[15] A. M. Samoilenko, N. A. Perestyuk, Differentsialnye uravneniya s impulsnym vozdeistviem, Vischa shkola, Kiev, 1987, 288 pp.; A. M. Samoilenko, N. A. Perestyuk, Impulsive differential equations, World Sci. Ser. Nonlinear Sci. Ser. A Monogr. Treatises, 14, World Scientific Publishing Co., Inc., River Edge, NJ, 1995, ix+462 pp. | MR | Zbl

[16] A. O. Ignatyev, “On the stability of invariant sets of systems with impulse effect”, Nonlinear Anal., 69:1 (2008), 53–72 | DOI | MR | Zbl

[17] V. V. Kozlov, S. D. Furta, Asimptotiki reshenii silno nelineinykh sistem differentsialnykh uravnenii, 2-e izd., NITs “Regulyarnaya i khaoticheskaya dinamika”, Izhevskii Institut kompyuternykh issledovanii, M.–Izhevsk, 2009, 312 pp. ; V. V. Kozlov, S. D. Furta, Asymptotic solutions of strongly nonlinear systems of differential equations, Springer Monogr. Math., Springer, Heidelberg, 2013, xx+262 pp. | MR | Zbl | DOI | MR | Zbl

[18] A. O. Ignatev, “Metod funktsii Lyapunova v zadachakh ustoichivosti reshenii sistem differentsialnykh uravnenii s impulsnym vozdeistviem”, Matem. sb., 194:10 (2003), 117–132 | DOI | MR | Zbl

[19] R. I. Gladilina, A. O. Ignatev, “O neobkhodimykh i dostatochnykh usloviyakh asimptoticheskoi ustoichivosti impulsnykh sistem”, Ukr. matem. zhurnal, 55:8 (2003), 1035–1043 | MR | Zbl

[20] V. V. Rumyantsev, A. S. Oziraner, Ustoichivost i stabilizatsiya dvizheniya po otnosheniyu k chasti peremennykh, Nauka, M., 1987, 255 pp. | MR | Zbl

[21] V. Lakshmikantham, D. D. Bainov, P. S. Simeonov, Theory of impulsive differential equations, Ser. Modern Appl. Math., 6, World Scientific Publishing Co., Inc., Teaneck, NJ, 1989, xii+273 pp. | MR | Zbl

[22] A. I. Dvirnyi, V. I. Slynko, “Ustoichivost reshenii monotonnykh differentsialnykh uravnenii s impulsnym vozdeistviem”, Differents. uravneniya, 48:3 (2012), 316–325 ; A. I. Dvirnyi, V. I. Slyn'ko, “Stability of solutions of monotone systems of impulsive differential equations”, Differ. Equ., 48:3, 318–327 | Zbl | DOI

[23] A. I. Dvirnyi, V. I. Slynko, “Ob ustoichivosti reshenii kvaziodnorodnykh monotonnykh impulsnykh sistem”, Tr. In-ta prikl. matematiki i mekhaniki, 24 (2012), 99–115 | MR

[24] G. Pólya, G. Szegö, Aufgaben und Lehrsätze aus der Analysis, v. 1, 2, Grundlehren Math. Wiss., XIX, XX, 2te Aufl., Springer-Verlag, Berlin–Göttingen–Heidelberg, 1954, xvi+338 pp., x+407 pp. | MR | MR | Zbl

[25] M. A. Krasnoselskii, E. A. Lifshits, A. V. Sobolev, Pozitivnye lineinye sistemy: metod polozhitelnykh operatorov, Nauka, M., 1985, 256 pp. ; M. A. Krasnosel'skij, Je. A. Lifshits, A. V. Sobolev, Positive linear systems. The method of positive operators, Sigma Ser. Appl. Math., 5, Heldermann Verlag, Berlin, 1989, viii+354 pp. | MR | Zbl | MR | Zbl

[26] A. A. Martynyuk, A. Yu. Obolenskii, Issledovanie ustoichivosti avtonomnykh sistem sravneniya, Preprint 78.28, In-t matematiki AN USSR, Kiev, 1978

[27] A. Yu. Obolenskii, “Ob ustoichivosti sistem sravneniya”, Dop. AN URSR, 8 (1979), 607–611 | MR | Zbl

[28] A. Yu. Obolenskii, “Ob ustoichivosti lineinykh sistem sravneniya”, Mat. fizika i nelinein. mekhanika, 1 (1984), 51–55 | MR | Zbl

[29] A. Yu. Obolenskii, Kriterii ustoichivosti dvizheniya nekotorykh nelineinykh sistem, Feniks, K., 2010, 228 pp.

[30] A. Yu. Obolenskii, “Ob ustoichivosti reshenii avtonomnykh sistem Vazhevskogo s zapazdyvaniem”, Ukr. matem. zhurn., 35:5 (1983), 574–579 | MR | Zbl