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Positive invertibility of matrices and stability of Itô delay differential equations

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Abstract

We study the global exponential p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations of a special form using the theory of positively invertible matrices. To this end, we apply a method developed by N.V. Azbelev and his students for the stability analysis of deterministic functional-differential equations. We obtain sufficient conditions for the global exponential 2p-stability (1 ≤ p < ∞) of systems of Itô nonlinear delay differential equations in terms of the positive invertibility of a matrix constructed from the original system. We verify these conditions for specific equations.

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References

  1. Kolmanovskii, V.B. and Nosov, V.R., Ustoichivost’ i periodicheskie rezhimy reguliruemykh sistem s posledeistviem (Stability and Periodic Modes of Control Systems with Aftereffect), Moscow: Nauka, 1981.

    Google Scholar 

  2. Tsar’kov, E.F., Sluchainye vozmushcheniya differentsial’no-funktsional’nykh uravnenii (Stochastic Perturbations of Functional-Differential Equations), Riga: Zinatne, 1989.

    MATH  Google Scholar 

  3. Mao, X., Stochastic Differential Equations and Applications, Chichester, 1997.

    Google Scholar 

  4. Mohammed, S.-E.F., Stochastic functional differential equations with memory. Theory, examples and applications, Proceeding of the Sixth Oslo–Silivri Workshop on Stochastic Analysis, Geilo, Norway, 1996, pp. 1–91.

    Google Scholar 

  5. Kadiev, R.I., On the stability of stochastic functional-differential equations with respect to the first approximation, Russ. Math., 1999, vol. 43, no. 10, pp. 1–6.

    MathSciNet  MATH  Google Scholar 

  6. Kadiev, R.I., Solutions of nonlinear impulsive Itô functional-differential equations: stability by the linear approximation, Differ. Equations, 2013, vol. 49, no. 8, pp. 933–940.

    Article  MathSciNet  MATH  Google Scholar 

  7. Berezansky, L., Braverman, E., and Idels, L., Nev global exponential stability criteria for nonlinear delay differential systems with applications to BAM neural networks, Appl. Math. Comput., 2014, vol. 243, pp. 899–910.

    MathSciNet  MATH  Google Scholar 

  8. Horn, R.A. and Johnson, C.R., Matrix Analysis, Cambridge: Cambridge Univ. Press, 1985. Translated under the title Matrichnyi analiz, Moscow: Mir, 1989.

    Book  MATH  Google Scholar 

  9. Bellman, R., Introduction to Matrix Analysis, New York–Toronto–London: McGraw-Hill Book Co., 1960. Translated under the title Vvedenie v teoriyu matrits, Moscow: Nauka, 1969.

    MATH  Google Scholar 

  10. Liptser, R.Sh. and Shiryaev, A.N., Teoriya martingalov (Martingale Theory), Moscow: Nauka, 1986.

    MATH  Google Scholar 

  11. Kadiev, R.I., Existence and uniqueness of the solution of the Cauchy problem for functional-differential equations with respect to a semimartingale, Russ. Math., 1996, vol. 39, no. 10, pp. 33–37.

    MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. Dagestan Scientific Center of Russian Academy of Sciences, Makhachkala, 367025, Russia

    R. I. Kadiev & A. V. Ponosov

  2. Dagestan State University, Makhachkala, 367025, Russia

    R. I. Kadiev & A. V. Ponosov

  3. Norwegian University of Life Sciences, Ås, NO-1432, Norway

    R. I. Kadiev & A. V. Ponosov

Authors
  1. R. I. Kadiev
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  2. A. V. Ponosov
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Correspondence to R. I. Kadiev.

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Original Russian Text © R.I. Kadiev, A.V. Ponosov, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 5, pp. 579–590.

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Kadiev, R.I., Ponosov, A.V. Positive invertibility of matrices and stability of Itô delay differential equations. Diff Equat 53, 571–582 (2017). https://doi.org/10.1134/S0012266117050019

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  • DOI: https://doi.org/10.1134/S0012266117050019

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