On the asymptotic stability for nonautonomous functional differential equations by Lyapunov functionals
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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–Krasovskiĭ 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 ${x(t)}$, the norm of the instantaneous value of the solutions or $\|x_t\|_2$, the $L_2$-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.References
- A. S. Andreev, On the stability of a nonautonomous functional-differential equation, Dokl. Akad. Nauk 356 (1997), no. 2, 151–153 (Russian). MR 1600838
- Zhivko S. Athanassov, Families of Liapunov-Krasovskiĭ functionals and stability for functional differential equations, Ann. Mat. Pura Appl. (4) 176 (1999), 145–165. MR 1746540, DOI 10.1007/BF02505994
- T. A. Burton, Uniform asymptotic stability in functional differential equations, Proc. Amer. Math. Soc. 68 (1978), no. 2, 195–199. MR 481371, DOI 10.1090/S0002-9939-1978-0481371-5
- Theodore Burton and László Hatvani, Stability theorems for nonautonomous functional-differential equations by Liapunov functionals, Tohoku Math. J. (2) 41 (1989), no. 1, 65–104. MR 985304, DOI 10.2748/tmj/1178227868
- T. A. Burton and L. Hatvani, On nonuniform asymptotic stability for nonautonomous functional-differential equations, Differential Integral Equations 3 (1990), no. 2, 285–293. MR 1025179
- T. A. Burton and G. Makay, Asymptotic stability for functional-differential equations, Acta Math. Hungar. 65 (1994), no. 3, 243–251. MR 1281434, DOI 10.1007/BF01875152
- Stavros N. Busenberg and Kenneth L. Cooke, Stability conditions for linear nonautonomous delay differential equations, Quart. Appl. Math. 42 (1984), no. 3, 295–306. MR 757167, DOI 10.1090/S0033-569X-1984-0757167-1
- I. V. Gaĭshun and L. B. Knyazhishche, Nonmonotone Lyapunov functionals. Conditions for the stability of equations with delay, Differentsial′nye Uravneniya 30 (1994), no. 8, 1291–1298, 1468 (Russian, with Russian summary); English transl., Differential Equations 30 (1994), no. 8, 1195–1200 (1995). MR 1334842
- Jack K. Hale and Sjoerd M. Verduyn Lunel, Introduction to functional-differential equations, Applied Mathematical Sciences, vol. 99, Springer-Verlag, New York, 1993. MR 1243878, DOI 10.1007/978-1-4612-4342-7
- L. Hatvani, On the asymptotic stability of the solutions of functional-differential equations, Qualitative theory of differential equations (Szeged, 1988) Colloq. Math. Soc. János Bolyai, vol. 53, North-Holland, Amsterdam, 1990, pp. 227–238. MR 1062649
- L. Hatvani, On the asymptotic stability for functional differential equations by Lyapunov functionals, Nonlinear Anal. 40 (2000), no. 1-8, Ser. A: Theory Methods, 251–263. Lakshmikantham’s legacy: a tribute on his 75th birthday. MR 1768411, DOI 10.1016/S0362-546X(00)85014-2
- László Hatvani, Annulus arguments in the stability theory for functional differential equations, Differential Integral Equations 10 (1997), no. 5, 975–1002. MR 1741762
- V. Lakshmikantham (ed.), World Congress of Nonlinear Analysts ’92. Vol. I–IV, Walter de Gruyter & Co., Berlin, 1996. MR 1389056, DOI 10.1515/9783110883237
- Younhee Ko, An asymptotic stability and a uniform asymptotic stability for functional-differential equations, Proc. Amer. Math. Soc. 119 (1993), no. 2, 535–545. MR 1169036, DOI 10.1090/S0002-9939-1993-1169036-6
- Katsumasa Kobayashi, Asymptotic stability in functional-differential equations, Nonlinear Anal. 20 (1993), no. 4, 359–364. MR 1206425, DOI 10.1016/0362-546X(93)90140-N
- V. B. Kolmanovskiĭ, L. Torelli, and R. Vermiglio, Stability of some test equations with delay, SIAM J. Math. Anal. 25 (1994), no. 3, 948–961. MR 1271319, DOI 10.1137/S0036141092238023
- N. N. Krasovskiĭ, Stability of motion. Applications of Lyapunov’s second method to differential systems and equations with delay, Stanford University Press, Stanford, Calif., 1963. Translated by J. L. Brenner. MR 0147744
- V. Lakshmikantham, S. Leela, and A. A. Martynyuk, Stability analysis of nonlinear systems, Monographs and Textbooks in Pure and Applied Mathematics, vol. 125, Marcel Dekker, Inc., New York, 1989. MR 984861
- V. Lakshmikantham, S. Leela, and S. Sivasundaram, Lyapunov functions on product spaces and stability theory of delay differential equations, J. Math. Anal. Appl. 154 (1991), no. 2, 391–402. MR 1088639, DOI 10.1016/0022-247X(91)90045-2
- Xinzhi Liu and Dao Yi Xu, Uniform asymptotic stability of abstract functional-differential equations, J. Math. Anal. Appl. 216 (1997), no. 2, 626–643. MR 1489602, DOI 10.1006/jmaa.1997.5694
- G. Makay, On the asymptotic stability in terms of two measures for functional-differential equations, Nonlinear Anal. 16 (1991), no. 7-8, 721–727. MR 1097328, DOI 10.1016/0362-546X(91)90178-4
- Géza Makay, An example on the asymptotic stability for functional-differential equations, Nonlinear Anal. 23 (1994), no. 3, 365–368. MR 1291576, DOI 10.1016/0362-546X(94)90176-7
- Géza Makay, On the asymptotic stability of the solutions of functional-differential equations with infinite delay, J. Differential Equations 108 (1994), no. 1, 139–151. MR 1268355, DOI 10.1006/jdeq.1994.1029
- P. Borne and V. Matrosov (eds.), The Lyapunov functions method and applications, IMACS Annals on Computing and Applied Mathematics, vol. 8, J. C. Baltzer A.G., Basel, 1990. MR 1148107
- Ting Xiu Wang, Stability in abstract functional-differential equations. I. General theorems, J. Math. Anal. Appl. 186 (1994), no. 2, 534–558. MR 1293010, DOI 10.1006/jmaa.1994.1316
- Ting Xiu Wang, Stability in abstract functional-differential equations. II. Applications, J. Math. Anal. Appl. 186 (1994), no. 3, 835–861. MR 1293858, DOI 10.1006/jmaa.1994.1336
- Bo Zhang, Asymptotic stability in functional-differential equations by Liapunov functionals, Trans. Amer. Math. Soc. 347 (1995), no. 4, 1375–1382. MR 1264834, DOI 10.1090/S0002-9947-1995-1264834-9
Additional Information
- László Hatvani
- Affiliation: Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, H-6720 Szeged, Hungary
- MR Author ID: 82460
- Email: hatvani@math.u-szeged.hu
- 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).
- © Copyright 2002 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 354 (2002), 3555-3571
- MSC (2000): Primary 34K20
- DOI: https://doi.org/10.1090/S0002-9947-02-03029-5
- MathSciNet review: 1911511