Geodesic
Method of Lyapunov functions in problems of stability of solutions
Sbornik. Mathematics, Tome 194 (2003) no. 10, pp. 1543-1558
Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

A system of ordinary differential equations with impulse action at fixed moments of time is considered. The system is assumed to have the zero solution. It is shown that the existence of a corresponding Lyapunov function is a necessary and sufficient condition for the uniform asymptotic stability of the zero solution. Restrictions on perturbations of the right-hand sides of differential equations and impulse actions are obtained under which the uniform asymptotic stability of the zero solution of the “unperturbed” system implies the uniform asymptotic stability of the zero solution of the “perturbed” system. 
@article{SM_2003_194_10_a5,
     author = {A. O. Ignatyev},
     title = {Method of {Lyapunov} functions in problems of stability of solutions},
     journal = {Sbornik. Mathematics},
     pages = {1543--1558},
     year = {2003},
     volume = {194},
     number = {10},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2003_194_10_a5/}
}
TY  - JOUR
AU  - A. O. Ignatyev
TI  - Method of Lyapunov functions in problems of stability of solutions
JO  - Sbornik. Mathematics
PY  - 2003
SP  - 1543
EP  - 1558
VL  - 194
IS  - 10
UR  - http://geodesic.mathdoc.fr/item/SM_2003_194_10_a5/
LA  - en
ID  - SM_2003_194_10_a5
ER  - 
%0 Journal Article
%A A. O. Ignatyev
%T Method of Lyapunov functions in problems of stability of solutions
%J Sbornik. Mathematics
%D 2003
%P 1543-1558
%V 194
%N 10
%U http://geodesic.mathdoc.fr/item/SM_2003_194_10_a5/
%G en
%F SM_2003_194_10_a5
A. O. Ignatyev. Method of Lyapunov functions in problems of stability of solutions. Sbornik. Mathematics, Tome 194 (2003) no. 10, pp. 1543-1558. http://geodesic.mathdoc.fr/item/SM_2003_194_10_a5/

[1] Samoilenko A. M., Perestyuk N. A., Impulsive differential equations, World Scientific, Singapore, 1995 | MR | Zbl

[2] Milman V. D., Myshkis A. D., “Ob ustoichivosti dvizheniya pri nalichii tolchkov”, Sib. matem. zhurn., 1:2 (1960), 233–237 | MR

[3] Myshkis A. D., Samoilenko A. M., “Sistemy s tolchkami v zadannye momenty vremeni”, Matem. sb., 74:2 (1967), 202–208

[4] Khalanai A., Veksler D., Kachestvennaya teoriya impulsnykh sistem, Mir, M., 1971 | MR

[5] Samoilenko A. M., Perestyuk N. A., “Ustoichivost reshenii differentsialnykh uravnenii s impulsnym vozdeistviem”, Differents. uravneniya, 13:11 (1977), 1981–1992 | MR | Zbl

[6] Samoilenko A. M., Perestyuk N. A., “Ob ustoichivosti reshenii sistem s impulsnym vozdeistviem”, Differents. uravneniya, 17:11 (1981), 1995–2001 | MR | Zbl

[7] Samoilenko A. M., Perestyuk N. A., “Ustoichivost reshenii differentsialnykh uravnenii s impulsnym vozdeistviem”, Differents. uravneniya, 13:11 (1977), 1981–1992 | MR | Zbl

[8] Samoilenko A. M., Perestyuk N. A., “Periodicheskie resheniya slabo nelineinykh sistem s impulsnym vozdeistviem”, Differents. uravneniya, 14:6 (1978), 1034–1045 | MR | Zbl

[9] Perestyuk N. A., Shovkoplyas V. N., “Periodicheskie resheniya nelineinykh differentsialnykh uravnenii s impulsnym vozdeistviem”, Ukr. matem. zhurn., 31:5 (1979), 517–524 | MR | Zbl

[10] Samoilenko A. M., Perestyuk N. A., “Ob ustoichivosti reshenii sistem s impulsnym vozdeistviem”, Differents. uravneniya, 17:11 (1981), 1995–2001 | MR | Zbl

[11] Samoilenko A. M., Perestyuk N. A., “Periodicheskie i pochti periodicheskie resheniya differentsialnykh uravnenii s impulsnym vozdeistviem”, Ukr. matem. zhurn., 34:1 (1982), 66–73 | MR | Zbl

[12] Bainov D. D., Simeonov P. S., Systems with impulse effect: stability, theory and applications, Halsted Press, New York, 1989 | MR | Zbl

[13] Lakshmikantham V., Bainov D. D., Simeonov P. S., Theory of impulsive differential equations, World Scientific, Singapore, 1989 | MR | Zbl

[14] Hu S., Lakshmikantham V., “Periodic boundary value problems for second order impulsive differential systems”, Nonlinear Anal., 13:1 (1989), 75–87 | DOI | MR

[15] Lakshmikantham V., Liu X., “On quasi stability for impulsive differential systems”, Nonlinear Anal., 13:7 (1989), 819–829 | DOI | MR

[16] Cabada A., Liz E., “Discontinuous impulsive differential equations with nonlinear boundary conditions”, Nonlinear Anal., 28:9 (1997), 1491–1497 | DOI | MR | Zbl

[17] Gurgula S. I., Perestyuk N. A., “Ob ustoichivosti polozheniya ravnovesiya impulsnykh sistem”, Matem. fizika, 1982, no. 31, 9–14 | MR | Zbl

[18] Rush N., Abets P., Lalua M., Pryamoi metod Lyapunova v teorii ustoichivosti, Mir, M., 1980 | MR

[19] Savchenko A. Ya., Ignatev A. O., Nekotorye zadachi ustoichivosti neavtonomnykh dinamicheskikh sistem, Naukova dumka, Kiev, 1989 | MR | Zbl

[20] Malkin I. G., Teoriya ustoichivosti dvizheniya, Nauka, M., 1966 | MR | Zbl

[21] Yoshizawa T., Stability theory and the existence of periodic solutions and almost periodic solutions, Springer-Verlag, New York, 1975 | MR | Zbl