No. 37 (254), issue 10Pages 30 - 39

ESTIMATES FOR SOLUTIONS AND ATTRACTION DOMAINS OF THE ZERO SOLUTION TO SYSTEMS OF QUASI-LINEAR EQUATIONS OF NEUTRAL TYPE

M.A. Skvortsova
The present paper is devoted to study a class of systems of differential equations of neutral type. We obtain attraction domains of the zero solution and establish estimates of exponential decay at infinity for solutions. In particular, asymptotic stability of the zero solution follows from these estimates. These results were derived by the use of a modified Lyapunov - Krasovskii functional.
Full text
Keywords
systems of quasi-linear equations of neutral type, asymptotic stability, attraction domains, uniform estimates for solutions, modified Lyapunov-Krasovskii functional.
References
1. Krasovskiy N.N. Nekotorye zadachi teorii ustoychivosti dvizheniya [Certain problems in the theory of stability of motion]. Moscow, Fizmatgiz, 1959.
2. El'sgol'ts L.E., Norkin S.B. Vvedeniye v teoriyu differentsial'nykh uravneniy s otklonyayushchimsya argumentom [Introduction to the theory of differential equations with deviating argument]. Moscow, Nauka, 1971.
3. Hale J. Theory of functional differential equations. New York, Heidelberg, Berlin, Springer-Verlag, 1977.
4. Korenevskiy D.G. Ustoychivost' dinamicheskikh sistem pri sluchaynykh vozmushcheniyakh parametrov. Algebraicheskiye kriterii [Stability of dinamical systems under random perturbations of parameters. Algebraic criteria]. Kiev, Naukova dumka, 1989.
5. Gu K., Kharitonov V.L., Chen J. Stability of time-delay systems. Control Engineering. Boston, MA: Birkh'auser, 2003.
6. Kharitonov V.L., Zhabko A.P. Lyapunov-Krasovskii approach to the robust stability analysis of time-delay systems. Automatica, 2003, vol. 39, no. 1, pp. 15 - 20.
7. Kharitonov V.L., Hinrichsen D. Exponential estimates for time delay systems. Systems Control Lett., 2004, vol. 53, no. 5, pp. 395 - 405.
8. Mondi'e, S., Kharitonov V.L. Exponential estimates for retarded time-delay systems: an LMI approach. IEEE Trans. Automat. Control, 2005, vol. 50, no. 2, pp. 268 - 273.
9. Kharitonov V.L., Mondi'e S., Collado J. Exponential estimates for neutral time-delay systems: an LMI approach. IEEE Trans. Automat. Control, 2005, vol. 50, no. 5, pp. 666 - 670.
10. Khusainov D.Ya., Ivanov A.F., Kozhametov A.T. Convergence estimates for solutions of linear stationary systems of differential-difference equations with constant delay [Otsenki skhodimosti resheniy lineynykh statsionarnykh sistem differentsial'no-raznostnykh uravneniy s postoyannym zapazdyvaniem]. Differ. Equ., 2005, vol. 41, no. 8, pp. 1196-1200.
11. Demidenko G.V., Matveeva I.I. Asymptotic properties of solutions to delay differential equations [Asimptoticheskiye svoystva resheniy differentsial'nykh uravneniy s zapazdyvayushchim argumentom]. Vestnik NGU. Ser.: Matematika, Mekhanika, Informatika, 2005, vol. 5, iss. 3, pp. 20 - 28.
12. Demidenko G.V., Matveeva I.I. Stability of solutions to delay differential equations with periodic coefficients of linear terms [Ustoychivost' resheniy differentsial'nykh uravneniy s zapazdyvayushchim argumentom i periodicheskimi koeffitsientami v lineynykh chlenakh]. Sib. Math. J., 2007, vol. 48, no. 5, pp. 824 - 836.
13. Demidenko G.V. Stability of solutions to linear differential equations of neutral type. J. Anal. Appl., 2009, vol. 7, no. 3, pp. 119 - 130.
14. Melchor-Aguilar D., Niculescu S.I. Estimates of the attraction region for a class of nonlinear time-delay systems. IMA J. Math. Control Inform., 2007, vol. 24, no. 4, pp. 523 - 550.
15. Demidenko G.V., Kotova T.V., Skvortsova M.A. Stability of solutions to differential equations of neutral type [Ustoychivost' resheniy differentsial'nykh uravneniy neytral'nogo tipa]. Vestnik NGU. Ser.: Matematika, Mekhanika, Informatika, 2010, vol. 10, iss. 3, pp. 17 - 29.
16. Skvortsova M.A. Asymptotic stability of the zero solution to quasi-linear systems of neutral type [Asimptoticheskaya ustoychivost' nulevogo resheniya kvazilineynykh sistem neytral'nogo tipa]. Trudy mat. tsentra im. N.I. Lobachevskogo [Proceedings of the Lobachevsky Mathematical Centre]. Kazan, 2010, vol. 40, pp. 307 - 311.
17. Skvortsova M.A. Quasi-linear systems of differential equations of neutral type [Kvazilineynye sistemy differentsial'nykh uravneniy neytral'nogo tipa]. Materialy XLVIII mezhdunar. nauch. stud. konf. 'Student i nauchno-tekhnicheskiy progress': Matematika [Materials of XLVIII International Scientific Students Conference 'Students and Progress in Science and Technology': Mathematics]. Novosibirsk, 2010, p. 64.