On exponential stability of second order delay differential equations

Ravi P. Agarwal; Alexander Domoshnitsky; Abraham Maghakyan

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 1047-1068
  • ISSN: 0011-4642

Abstract

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We propose a new method for studying 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.

How to cite

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Agarwal, Ravi P., Domoshnitsky, Alexander, and Maghakyan, Abraham. "On exponential stability of second order delay differential equations." Czechoslovak Mathematical Journal 65.4 (2015): 1047-1068. <http://eudml.org/doc/276112>.

@article{Agarwal2015,
abstract = {We propose a new method for studying 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.},
author = {Agarwal, Ravi P., Domoshnitsky, Alexander, Maghakyan, Abraham},
journal = {Czechoslovak Mathematical Journal},
keywords = {delay equations; uniform exponential stability; exponential estimates of solutions; Cauchy function},
language = {eng},
number = {4},
pages = {1047-1068},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On exponential stability of second order delay differential equations},
url = {http://eudml.org/doc/276112},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Agarwal, Ravi P.
AU - Domoshnitsky, Alexander
AU - Maghakyan, Abraham
TI - On exponential stability of second order delay differential equations
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 1047
EP - 1068
AB - We propose a new method for studying 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.
LA - eng
KW - delay equations; uniform exponential stability; exponential estimates of solutions; Cauchy function
UR - http://eudml.org/doc/276112
ER -

References

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  1. Azbelev, N. V., Maksimov, V. P., Rakhmatullina, L. F., Introduction to the Theory of Functional-Differential Equations, Nauka, Moskva Russian. English summary (1991). (1991) Zbl0725.34071
  2. Berezansky, L., Braverman, E., Domoshnitsky, A., 10.1007/s12591-008-0012-4, Differ. Equ. Dyn. Syst. 16 (2008), 185-205. (2008) Zbl1180.34077MR2534439DOI10.1007/s12591-008-0012-4
  3. Burton, T. A., Stability by Fixed Point Theory for Functional Differential Equations, Dover Publications, Mineola (2006). (2006) Zbl1160.34001MR2281958
  4. Burton, T. A., Stability and Periodic Solutions of Ordinary and Functional-Differential Equations, Mathematics in Science and Engineering 178 Academic Press, Orlando (1985). (1985) Zbl0635.34001
  5. Burton, T. A., Furumochi, T., Asymptotic behavior of solutions of functional differential equations by fixed point theorems, Dyn. Syst. Appl. 11 (2002), 499-519. (2002) Zbl1044.34033MR1946140
  6. Burton, T. A., Hatvani, L., 10.1006/jmaa.1993.1212, J. Math. Anal. Appl. 176 (1993), 261-281. (1993) Zbl0779.34042DOI10.1006/jmaa.1993.1212
  7. Cahlon, B., Schmidt, D., 10.1016/j.cam.2003.12.043, J. Comput. Appl. Math. 170 (2004), 79-102. (2004) Zbl1064.34060MR2075825DOI10.1016/j.cam.2003.12.043
  8. Cahlon, B., Schmidt, D., Stability criteria for certain second order delay differential equations, Dyn. Contin. Discrete Impuls. Syst., Ser. A, Math. Anal. 10 (2003), 593-621. (2003) Zbl1036.34085MR1978592
  9. Domoshnitsky, A., Nonoscillation, maximum principles, and exponential stability of second order delay differential equations without damping term, J. Inequal. Appl. (2014), 2014:361, 26 pages. (2014) MR3347683
  10. Domoshnitsky, A., Unboundedness of solutions and instability of differential equations of the second order with delayed argument, Differ. Integral Equ. 14 (2001), 559-576. (2001) Zbl1023.34061MR1824743
  11. Domoshnitsky, A., Componentwise applicability of Chaplygin’s 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 Russian. (1990) MR1089738
  12. Došlá, Z., Kiguradze, I., On boundedness and stability of solutions of second order linear differential equations with advanced arguments, Adv. Math. Sci. Appl. 9 (1999), 1-24. (1999) Zbl0926.34061
  13. Erbe, L. H., Kong, Q., Zhang, B. G., Oscillation Theory for Functional Differential Equations, Pure and Applied Mathematics 190 Marcel Dekker, New York (1995). (1995) 
  14. Erneux, T., Applied Delay Differential Equations, Surveys and Tutorials in the Applied Mathematical Sciences 3 Springer, New York (2009). (2009) Zbl1201.34002MR2498700
  15. Fomin, V. N., Fradkov, A. L., Yakubovich, V. A., Adaptive Control of Dynamical Objects, Nauka, Moskva Russian (1981). (1981) Zbl0522.93002
  16. Izyumova, D. V., On the boundedness and stability of the solutions of nonlinear second order functional-differential equations, Soobshch. Akad. Nauk Gruz. SSR 100 Russian (1980), 285-288. (1980) Zbl0457.34050
  17. Kolmanovskii, V., Myshkis, A., Introduction to the Theory and Applications of Functional-Differential Equations, Mathematics and Its Applications 463 Kluwer Academic Publishers, Dordrecht (1999). (1999) Zbl0917.34001MR1680144
  18. Ladde, G. S., Lakshmikantham, V., Zhang, B. G., Oscillation Theory of Differential Equations with Deviating Arguments, Pure and Applied Mathematics 110 Marcel Dekker, New York (1987). (1987) Zbl0832.34071MR1017244
  19. Minorsky, N., Nonlinear Oscillations, D. Van Nostrand Company, Princeton (1962). (1962) Zbl0102.30402
  20. Myshkis, A. D., Linear Differential Equations with Retarded Argument, Izdat. Nauka, Moskva Russian (1972). (1972) Zbl0261.34040
  21. Pinto, M., 10.1016/S0362-546X(96)00024-7, Nonlinear Anal., Theory Methods Appl. 28 (1997), 1729-1740. (1997) Zbl0871.34045MR1430514DOI10.1016/S0362-546X(96)00024-7
  22. Pontryagin, L. S., 10.1090/trans2/001/06, Am. Math. Soc., Transl., II. Ser. 1 (1955), 95-110; Izv. Akad. Nauk SSSR, Ser. Mat. 6 Russian (1942), 115-134. (1942) Zbl0068.05803DOI10.1090/trans2/001/06
  23. Zhang, B., On the retarded Liénard equation, Proc. Am. Math. Soc. 115 (1992), 779-785. (1992) Zbl0756.34075

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