On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals

Author:
László Hatvani

Journal:
Trans. Amer. Math. Soc. **354** (2002), 3555-3571

MSC (2000):
Primary 34K20

DOI:
https://doi.org/10.1090/S0002-9947-02-03029-5

Published electronically:
April 30, 2002

MathSciNet review:
1911511

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Sufficient conditions are given for the asymptotic stability and uniform asymptotic stability of the zero solution of the nonautonomous FDE's whose right-hand sides can be unbounded functions of the time. The theorems are based upon Lyapunov-Krasovski functionals whose derivatives with respect to the equations are negative semidefinite and can vanish at long intervals. The functionals and their derivatives are estimated by either , the norm of the instantaneous value of the solutions or , the -norm of the phase segment of the solutions. Examples are given to show that the conditions are sharp, and the main theorems with the two different types of estimates are independent and improve earlier results. The theorems are applied to linear and nonlinear retarded FDE's with one delay and with distributed delays.

**1.**A. S. Andreev,*On the stability of a nonautonomous functional-differential equation.*Dokl. Akad. Nauk**356**(1997), 151-153. (Russian) MR**98m:34149****2.**Z. S. Athanassov,*Families of Liapunov-Krasovski functionals and stability for functional differential equations.*Ann. Mat. Pura Appl. (4)**176**(1999), 145-165. MR**2000m:34164****3.**T. A. Burton,*Uniform asymptotic stability in functional differential equations.*Proc. Amer. Math. Soc.**68**(1978), 195-199. MR**58:1489****4.**T. A. Burton and L. Hatvani,*Stability theorems for nonautonomous functional-differential equations by Liapunov functionals.*Tôhoku Math. J. (2)**41**(1989), 65-104. MR**90d:34147****5.**T. A. Burton and L. Hatvani,*On nonuniform asymptotic stability for nonautonomous functional-differential equations.*Differential Integral Equations**3**(1990), 285-293. MR**90k:34098****6.**T. A. Burton and G. Makay,*Asymptotic stability for functional-differential equations.*Acta Math. Hungar.**65**(1994), 243-251. MR**95d:34126****7.**S. N. Busenberg and K. L. Cooke,*Stability conditions for linear nonautonomous delay differential equations.*Quart. Appl. Math.**42**(1984), 295-306. MR**85j:34156****8.**I. V. Gashun and L. B. Knyazhishche,*Nonmonotone Lyapunov functionals. Conditions for the stability of equations with delay.*Differential Equations**30**(1994), 1195-1200. MR**96g:34118****9.**J. K. Hale and S. M. Verduyn Lunel,*Introduction to functional-differential equations*. Springer-Verlag, New York, 1993. MR**94m:34169****10.**L. Hatvani,*On the asymptotic stability of the solutions of functional-differential equations.*Qualitative theory of differential equations (Szeged, 1988), North-Holland, Amsterdam, 1990, pp. 227-238. MR**92f:34072****11.**L. Hatvani,*On the asymptotic stability for functional differential equations by Lyapunov functionals.*Nonlinear Anal. (Ser. A: Theory Methods)**40**(2000), 251-263. MR**2001c:34158****12.**L. Hatvani,*Annulus arguments in the stability theory for functional differential equations.*Differential Integral Equations**10**(1997), 975-1002. MR**2000m:34169****13.**J. Kato,*A conjecture in Lyapunov method for functional-differential equations.*World Congress of Nonlinear Analysts 1992, Vols. I-IV (Tampa, FL, 1992), de Gruyter, Berlin, 1996, pp. 1239-1246. MR**97a:00029****14.**Y. Ko,*An asymptotic stability and a uniform asymptotic stability for functional-differential equations.*Proc. Amer. Math. Soc.**119**(1993), 535-545. MR**93k:34166****15.**K. Kobayashi,*Asymptotic stability in functional-differential equations.*Nonlinear Anal.**20**(1993), 359-364. MR**93m:34127****16.**V. B. Kolmanovski, L. Torelli, and R. Vermiglio,*Stability of some test equations with delay.*SIAM J. Math. Anal.**25**(1994), 948-961. MR**95c:34135****17.**N. N. Krasovski,*Stability of motion. Applications of Lyapunov's second method to differential systems and equations with delay*. Stanford University Press, Stanford, Calif., 1963. MR**26:5258****18.**V. Lakshmikantham, S. Leela, and A. A. Martynyuk,*Stability analysis of nonlinear systems*. Marcel Dekker Inc., New York, 1989. MR**90h:34073****19.**V. Lakshmikantham, S. Leela, and S. Sivasundaram,*Liapunov functions on product spaces and stability theory of delay differential equations.*J. Math. Anal. Appl.**154**(1991), 391-402. MR**92a:34086****20.**X. Liu and D. Y. Xu,*Uniform asymptotic stability of abstract functional-differential equations.*J. Math. Anal. Appl.**216**(1997), 626-643. MR**98j:34159****21.**G. Makay,*On the asymptotic stability in terms of two measures for functional-differential equations.*Nonlinear Anal.**16**(1991), 721-727. MR**92b:34093****22.**G. Makay,*An example on the asymptotic stability for functional-differential equations.*Nonlinear Anal.**23**(1994), 365-368. MR**95i:34147****23.**G. Makay,*On the asymptotic stability of the solutions of functional-differential equations with infinite delay.*J. Differential Equations**108**(1994), 139-151. MR**95d:34130****24.**J. Terjéki,*Power-asymptotic stability for functional-differential equations.*The Lyapunov functions method and applications, Baltzer, Basel, 1990, pp. 161-164. MR**92i:34003****25.**T. X. Wang,*Stability in abstract functional-differential equations. I. General theorems.*J. Math. Anal. Appl.**186**(1994), 534-558. MR**95k:34112****26.**T. X. Wang,*Stability in abstract functional-differential equations. II. Applications.*J. Math. Anal. Appl.**186**(1994), 835-861. MR**96a:34155****27.**B. Zhang,*Asymptotic stability in functional-differential equations by Liapunov functionals.*Trans. Amer. Math. Soc.**347**(1995), 1375-1382. MR**95g:34116**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
34K20

Retrieve articles in all journals with MSC (2000): 34K20

Additional Information

**László Hatvani**

Affiliation:
Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary

Email:
hatvani@math.u-szeged.hu

DOI:
https://doi.org/10.1090/S0002-9947-02-03029-5

Received by editor(s):
November 5, 2001

Published electronically:
April 30, 2002

Additional Notes:
The author was supported by the Hungarian National Foundation for Scientific Research (OTKA T/029188).

Article copyright:
© Copyright 2002
American Mathematical Society