Stability Conditions of a Class of Linear Retarded Differential Systems

Deger, Serbun Ufuk; Bolat, Yasar

doi:10.28924/2291-8639-16-2018-454

Research Article

Stability Conditions of a Class of Linear Retarded Differential Systems

Serbun Ufuk Deger*
Yasar Bolat

* Corresponding author

Abstract

In this paper, we give some new necessary and sufficient conditions for the asymptotic stability of a linear retarded differential system with two delays
x”²(t)+(1-a)x(t)+A(x(t-k)+x(t-l))=0, t≥0,
where a<1 is a real number, A is a 2×2 real constant matrix, and k, l are positive numbers such that k>l.

How to Cite

S. U. Deger, Y. Bolat, Stability Conditions of a Class of Linear Retarded Differential Systems, Int. J. Anal. Appl. 16 (2018), 454–461. https://doi.org/10.28924/2291-8639-16-2018-454

BibTeX

@article{Deger2018,
  author  = {Deger, Serbun Ufuk and Bolat, Yasar},
  title   = {Stability Conditions of a Class of Linear Retarded Differential Systems},
  journal = {Int. J. Anal. Appl.},
  volume  = {16},
  year    = {2018},
  pages   = {454--461},
  doi     = {10.28924/2291-8639-16-2018-454}
}

References

[1] H. Matsunaga, Delay Dependent and Delay Independent Stability Criteria For A Delay Differential System, American Mathematical Society, 136 Fields Inst. Commun. 42 (2008), 4305-4312.
[2] K. L. Cooke and P. van den Driessche, On zeroes of some transcendental equations, Funkcial.Ekvacioj, 29 (1986), 77-90.
[3] Y. Kuang, Delay Differential Equations with Applications in Population Dynamics, Academic Press, Boston, 1993.
[4] K. L. Cooke and Z. Grossman, Discrete delay, distributed delay and stability switches, J. Math. Anal. Appl. 86 (1982), 592-627.
[5] S.Ruan and J.Wei, On The Zeros Of Transcendental Functions With Applications To Stability Of Delay Differential Equations With Two Delays, Dynamic of Continuous, Discr. impuls. Syst. (2003), 863-874.
[6] J.K.Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993.
[7] S. Elaydi, An Introduction to Difference Equations, 3rd ed., Springer-Verlag, New York, 2005.
[8] T. Khokhlovaa, M. Kipnis, V. Malygina, Discrete delay, The stability cone for a delay differential matrix equation, Appl. Math. Lett. 24 (2011), 742-745.
[9] J. Cermák, J.Jánsky, Stability switches in linear delay difference equations, Appl. Math. Comput. 243 (2014) 755-766.
[10] Jana Hrabalova, Stability Properties of a Discrrtized Neutral Delay Differential Equation, Tatra Mt. Math. Publ. 54 (2013), 83-92
[11] H. Nakajima, On the Stability of a linear Retarded Differential-Difference Equation, Funkcialaj Ekvacioj. 57 (2014), 43-56
[12] T. Hara, S. Sakata, An application of the Hurwitz theorem to the root analysis of the characteristic equation, Appl. Math. Lett. 24 (2011) 12-15
[13] H. Smith, An Introduction to Delay Differential Equations with Applications to The Life Science, Springer, New York 2010.
[14] H. I .Freedman, Y.Kuang, Stability switches in linear scalar neutral delay equation, Funkcial. Ekvac. 34 (1991), 187-209.
[15] R. Bellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.