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On exponential stability of second order delay differential equations. (English) Zbl 1363.34250

Summary: We propose a new method for studying the stability of second-order delay differential equations. Results we obtained are of the form: the exponential stability of ordinary differential equation implies the exponential stability of the corresponding delay differential equation if the delays are small enough. We estimate this smallness through the coefficients of this delay equation. Examples demonstrate that our tests of the exponential stability are essentially better than the known ones. This method works not only for autonomous equations but also for equations with variable coefficients and delays.

MSC:

34K20 Stability theory of functional-differential equations
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[1] N. V. Azbelev, V. P. Maksimov, L. F. Rakhmatullina: Introduction to the Theory of Functional-Differential Equations. Nauka, Moskva, 1991. (In Russian. English summary.)
[2] L. Berezansky, E. Braverman, A. Domoshnitsky: Stability of the second order delay differential equations with a damping term. Differ. Equ. Dyn. Syst. 16 (2008), 185–205. · Zbl 1180.34077 · doi:10.1007/s12591-008-0012-4
[3] T. A. Burton: Stability by Fixed Point Theory for Functional Differential Equations. Dover Publications, Mineola, 2006.
[4] T. A. Burton: Stability and Periodic Solutions of Ordinary and Functional-Differential Equations. Mathematics in Science and Engineering 178, Academic Press, Orlando, 1985. · Zbl 0635.34001
[5] T. A. Burton, T. Furumochi: Asymptotic behavior of solutions of functional differential equations by fixed point theorems. Dyn. Syst. Appl. 11 (2002), 499–519. · Zbl 1044.34033
[6] T. A. Burton, L. Hatvani: Asymptotic stability of second order ordinary, functional, and partial differential equations. J. Math. Anal. Appl. 176 (1993), 261–281. · Zbl 0779.34042 · doi:10.1006/jmaa.1993.1212
[7] B. Cahlon, D. Schmidt: Stability criteria for certain second-order delay differential equations with mixed coefficients. J. Comput. Appl. Math. 170 (2004), 79–102. · Zbl 1064.34060 · doi:10.1016/j.cam.2003.12.043
[8] B. Cahlon, D. Schmidt: Stability criteria for certain second order delay differential equations. Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10 (2003), 593–621. · Zbl 1036.34085
[9] A. Domoshnitsky: Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term. J. Inequal. Appl. (2014), 2014:361, 26 pages. · Zbl 1337.34069
[10] A. Domoshnitsky: Unboundedness of solutions and instability of differential equations of the second order with delayed argument. Differ. Integral Equ. 14 (2001), 559–576. · Zbl 1023.34061
[11] A. Domoshnitsky: Componentwise applicability of Chaplygins theorem to a system of linear differential equations with time-lag. Differ. Equations 26 (1990), 1254–1259; translation from Differ. Uravn. 26 (1990), 1699–1705. (In Russian.) · Zbl 0726.34061
[12] Z. Došlá, I. Kiguradze: On boundedness and stability of solutions of second order linear differential equations with advanced arguments. Adv. Math. Sci. Appl. 9 (1999), 1–24. · Zbl 0926.34061
[13] L. H. Erbe, Q. Kong, B. G. Zhang: Oscillation Theory for Functional Differential Equations. Pure and Applied Mathematics 190, Marcel Dekker, New York, 1995.
[14] T. Erneux: Applied Delay Differential Equations. Surveys and Tutorials in the Applied Mathematical Sciences 3, Springer, New York, 2009. · Zbl 1201.34002
[15] V. N. Fomin, A. L. Fradkov, V. A. Yakubovich: Adaptive Control of Dynamical Objects. Nauka, Moskva, 1981. (In Russian.) · Zbl 0522.93002
[16] D. V. Izyumova: On the boundedness and stability of the solutions of nonlinear second order functional-differential equations. Soobshch. Akad. Nauk Gruz. SSR 100 (1980), 285–288. (In Russian.) · Zbl 0457.34050
[17] V. Kolmanovskii, A. Myshkis: Introduction to the Theory and Applications of Functional-Differential Equations. Mathematics and Its Applications 463, Kluwer Academic Publishers, Dordrecht, 1999. · Zbl 0917.34001
[18] G. S. Ladde, V. Lakshmikantham, B. G. Zhang: Oscillation Theory of Differential Equations with Deviating Arguments. Pure and Applied Mathematics 110, Marcel Dekker, New York, 1987. · Zbl 0832.34071
[19] N. Minorsky: Nonlinear Oscillations. D. Van Nostrand Company, Princeton, 1962.
[20] A. D. Myshkis: Linear Differential Equations with Retarded Argument. Izdat. Nauka, Moskva, 1972. (In Russian.)
[21] M. Pinto: Asymptotic solutions for second order delay differential equations. Nonlinear Anal., Theory Methods Appl. 28 (1997), 1729–1740. · Zbl 0871.34045 · doi:10.1016/S0362-546X(96)00024-7
[22] L. S. Pontryagin: On the zeros of some elementary transcendental functions. Am. Math. Soc., Transl., II. Ser. 1 (1955), 95–110; Izv. Akad. Nauk SSSR, Ser. Mat. 6 (1942), 115–134. (In Russian.) · Zbl 0068.05803
[23] B. Zhang: On the retarded Liénard equation. Proc. Am. Math. Soc. 115 (1992), 779–785. · Zbl 0756.34075
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