×
Berezansky, Leonid; Braverman, Elena

New stability conditions for linear differential equations with several delays. (English) Zbl 1222.34089

Abstr. Appl. Anal. 2011, Article ID 178568, 19 p. (2011).
Summary: New explicit conditions of asymptotic and exponential stability are obtained for the scalar nonautonomous linear delay differential equation
\[ \dot x (t) + \sum^m_{k=1} a_k(t)x(h_k(t))=0 \]
with measurable delays and coefficients. These results are compared with known stability tests.

Cited in 11 Documents

MSC:

34K20 Stability theory of functional-differential equations
34K06 Linear functional-differential equations

Keywords:

asymptotic and exponential stability; linear differential equations with several delays
PDFBibTeX XMLCite
\textit{L. Berezansky} and \textit{E. Braverman}, Abstr. Appl. Anal. 2011, Article ID 178568, 19 p. (2011; Zbl 1222.34089)
Full Text: DOI arXiv
OA License

References:

[1] L. Berezansky and E. Braverman, “On stability of some linear and nonlinear delay differential equations,” Journal of Mathematical Analysis and Applications, vol. 314, no. 2, pp. 391-411, 2006. · Zbl 1101.34057 · doi:10.1016/j.jmaa.2005.03.103
[2] L. Berezansky and E. Braverman, “On exponential stability of linear differential equations with several delays,” Journal of Mathematical Analysis and Applications, vol. 324, no. 2, pp. 1336-1355, 2006. · Zbl 1112.34055 · doi:10.1016/j.jmaa.2006.01.022
[3] L. Berezansky and E. Braverman, “Explicit exponential stability conditions for linear differential equations with several delays,” Journal of Mathematical Analysis and Applications, vol. 332, no. 1, pp. 246-264, 2007. · Zbl 1118.34069 · doi:10.1016/j.jmaa.2006.10.016
[4] T. Yoneyama and J. Sugie, “On the stability region of scalar delay-differential equations,” Journal of Mathematical Analysis and Applications, vol. 134, no. 2, pp. 408-425, 1988. · Zbl 0663.34059 · doi:10.1016/0022-247X(88)90031-5
[5] T. Yoneyama, “The 3/2 stability theorem for one-dimensional delay-differential equations with unbounded delay,” Journal of Mathematical Analysis and Applications, vol. 165, no. 1, pp. 133-143, 1992. · Zbl 0755.34074 · doi:10.1016/0022-247X(92)90071-K
[6] T. Krisztin, “On stability properties for one-dimensional functional-differential equations,” Funkcialaj Ekvacioj, vol. 34, no. 2, pp. 241-256, 1991. · Zbl 0746.34045
[7] J. W.-H. So, J. S. Yu, and M.-P. Chen, “Asymptotic stability for scalar delay differential equations,” Funkcialaj Ekvacioj, vol. 39, no. 1, pp. 1-17, 1996. · Zbl 0930.34056
[8] I. Gy\Hori, F. Hartung, and J. Turi, “Preservation of stability in delay equations under delay perturbations,” Journal of Mathematical Analysis and Applications, vol. 220, no. 1, pp. 290-312, 1998. · Zbl 0914.34070 · doi:10.1006/jmaa.1997.5883
[9] I. Gy\Hori and F. Hartung, “Stability in delay perturbed differential and difference equations,” in Topics in Functional Differential and Difference Equations (Lisbon, 1999), vol. 29 of Fields Inst. Commun., pp. 181-194, American Mathematical Society, Providence, RI, USA, 2001. · Zbl 0990.34066
[10] S. A. Gusarenko and A. I. Domoshnitskiĭ, “Asymptotic and oscillation properties of first-order linear scalar functional-differential equations,” Differential Equations, vol. 25, no. 12, pp. 1480-1491, 1989. · Zbl 0726.45011
[11] T. Wang, “Inequalities and stability for a linear scalar functional differential equation,” Journal of Mathematical Analysis and Applications, vol. 298, no. 1, pp. 33-44, 2004. · Zbl 1064.34062 · doi:10.1016/j.jmaa.2004.01.048
[12] J. H. Shen and J. S. Yu, “Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 195, no. 2, pp. 517-526, 1995. · Zbl 0844.34078 · doi:10.1006/jmaa.1995.1371
[13] X. Wang and L. Liao, “Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients,” Journal of Mathematical Analysis and Applications, vol. 279, no. 1, pp. 326-338, 2003. · Zbl 1054.34128 · doi:10.1016/S0022-247X(03)00021-0
[14] Z. Zhang and Z. Wang, “Asymptotic behavior of solutions of neutral differential equations with positive and negative coefficients,” Annals of Differential Equations, vol. 17, no. 3, pp. 295-305, 2001. · Zbl 0997.34075
[15] Z. Zhang and J. Yu, “Asymptotic behavior of solutions of neutral difference equations with positive and negative coefficients,” Mathematical Sciences Research Hot-Line, vol. 2, no. 6, pp. 1-12, 1998. · Zbl 0960.39503
[16] N. V. Azbelev and P. M. Simonov, Stability of Differential Equations with Aftereffect, vol. 20 of Stability and Control: Theory, Methods and Applications, Taylor & Francis, London, UK, 2003. · Zbl 1049.34090
[17] A. Ivanov, E. Liz, and S. Trofimchuk, “Halanay inequality, Yorke 3/2 stability criterion, and differential equations with maxima,” The Tohoku Mathematical Journal, vol. 54, no. 2, pp. 277-295, 2002. · Zbl 1025.34078 · doi:10.2748/tmj/1113247567
[18] E. Liz, V. Tkachenko, and S. Trofimchuk, “A global stability criterion for scalar functional differential equations,” SIAM Journal on Mathematical Analysis, vol. 35, no. 3, pp. 596-622, 2003. · Zbl 1069.34109 · doi:10.1137/S0036141001399222
[19] X. H. Tang, “Asymptotic behavior of delay differential equations with instantaneously terms,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 342-359, 2005. · Zbl 1068.34076 · doi:10.1016/j.jmaa.2003.12.048
[20] V. V. Malygina, “Some criteria for stability of equations with retarded argument,” Differential Equations, vol. 28, no. 10, pp. 1398-1405, 1992. · Zbl 0801.34076
[21] V. V. Malygina, “Stability of solutions of some linear differential equations with aftereffect,” Russian Mathematics, vol. 37, no. 5, pp. 63-75, 1993. · Zbl 0836.34083
[22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. · Zbl 0787.34002
[23] N. V. Azbelev, L. Berezansky, and L. F. Rahmatullina, “A linear functional-differential equation of evolution type,” Differential Equations, vol. 13, no. 11, pp. 1331-1339, 1977. · Zbl 0396.34005
[24] I. Gy\Hori and G. Ladas, Oscillation Theory of Delay Differential Equations with Applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, NY, USA, 1991. · Zbl 0780.34048
[25] L. Berezansky and E. Braverman, “On non-oscillation of a scalar delay differential equation,” Dynamic Systems and Applications, vol. 6, no. 4, pp. 567-580, 1997. · Zbl 0890.34059
[26] L. Berezansky and E. Braverman, “Preservation of exponential stability for linear non-autonomous functional differential systems,” Automatica, vol. 46, no. 12, pp. 2077-2081, 2010. · Zbl 1214.34060 · doi:10.1016/j.automatica.2010.09.007
[27] L. Berezansky and E. Braverman, “Preservation of the exponential stability under perturbations of linear delay impulsive differential equations,” Zeitschrift für Analysis und ihre Anwendungen, vol. 14, no. 1, pp. 157-174, 1995. · Zbl 0821.34072 · doi:10.4171/ZAA/668
[28] G. Ladas, Y. G. Sficas, and I. P. Stavroulakis, “Asymptotic behavior of solutions of retarded differential equations,” Proceedings of the American Mathematical Society, vol. 88, no. 2, pp. 247-253, 1983. · Zbl 0521.34070 · doi:10.2307/2044710
[29] A. D. Myshkis, Differential Equations with Retarded Argument, Nauka, Moscow, Russia, 1951. · Zbl 0043.30904
[30] N. Krasovskii, Stability of Motion, Nauka, Moscow, Russia, 1959, translation, Stanford University Press, 1963.
[31] E. Liz and M. Pituk, “Exponential stability in a scalar functional differential equation,” Journal of Inequalities and Applications, Article ID 37195, 10 pages, 2006. · Zbl 1155.34037 · doi:10.1155/JIA/2006/37195
[32] I. Gy\Hori, “Interaction between oscillations and global asymptotic stability in delay differential equations,” Differential and Integral Equations, vol. 3, no. 1, pp. 181-200, 1990. · Zbl 0726.34054
[33] T. Faria and W. Huang, “Special solutions for linear functional differential equations and asymptotic behaviour,” Differential and Integral Equations, vol. 18, no. 3, pp. 337-360, 2005. · Zbl 1212.34233
[34] I. Gy\Hori and M. Pituk, “Stability criteria for linear delay differential equations,” Differential and Integral Equations, vol. 10, no. 5, pp. 841-852, 1997. · Zbl 0894.34064
[35] L. Berezansky and E. Braverman, “On exponential stability of a linear delay differential equation with an oscillating coefficient,” Applied Mathematics Letters, vol. 22, no. 12, pp. 1833-1837, 2009. · Zbl 1187.34096 · doi:10.1016/j.aml.2009.07.007
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.