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Stability analysis of Volterra functional equations by realization theory

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Abstract

The method of realization of input–output operators in the form of abstract discrete-time control systems and the frequency-domain method are used for the stability/instability analysis of a class of nonlinear Volterra functional equations. To this end, we construct an associated time-invariant abstract discrete-time control system in some weighted function spaces.

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References

  1. Brusin, V.A., Technique of Abstract Differential Equations in the Study of Integral Equations of Volterra Type, Sibirsk Mat. Zh., 1977, vol. 18, no. 6, pp. 1246–1258.

    MathSciNet  Google Scholar 

  2. Gelig, A.Kh. and Churilov, A.N., Kolebaniya i ustoichivost’ nelineinykh impul’snykh sistem (Oscillations and Stability of Nonlinear Impulsive Systems), St. Petersburg: Izd. St. Petersburg. Univ., 1993.

    Google Scholar 

  3. Zharkov, A.V., Criterion for the Absolute Instability, Dichotomy, and Dissipativity of Nonlinear Control Systems, Cand. Sci. (Phys.–Math.) Dissertation, Leningrad, 1978.

    Google Scholar 

  4. Krein, S.G., Lineinye differentsial’nye uravneniya v banakhovykh prostranstvakh (Linear Differential Equations in Banach Spaces), Moscow: Nauka, 1967.

    Google Scholar 

  5. Mal’tseva, A.A and Reitmann, V., Stability in the Whole and Bifurcations of Invariant Measures for Discrete Cocycles of the Equation of the Cardiac Conduction System, Differ. Uravn. Protses. Upravl., 2014, no. 3, pp. 32–54.

    Google Scholar 

  6. Peller, V.V., Operatory Gankelya i ikh prilozheniya (Hankel Operators and Their Applications), Moscow–Izhevsk: R & C Dynamics, 2005.

    Google Scholar 

  7. Yakubovich, V.A., On Abstract Theory of Absolute Stability of Nonlinear Systems, Vestnik St. Petersburg Univ. Ser. 1 Mat. Mekh. Astron., 1977, no. 13, pp. 99–118.

    MathSciNet  MATH  Google Scholar 

  8. Baras, J. S. and Brockett, R.W., H2-Functions and Infinite-Dimensional Realization Theory, SIAM J. Control, 1975, vol. 13, no. 1.

    Google Scholar 

  9. Fuhrmann, P.A., On Realizations of Linear Systems and Applications to Some Questions of Stability, Math. Systems Theory, 1974, vol. 8, pp. 132–141.

    Article  MathSciNet  MATH  Google Scholar 

  10. Helton, J. W., Discrete Time Systems, Operator Models and Scattering Theory, J. Funct. Anal., 1974, vol. 16, pp. 15–38.

    Article  MathSciNet  MATH  Google Scholar 

  11. Kalman, R.E., Arbib, M., and Falb, P., Topics in Mathematical Systems Theory, New York: McGraw-Hill, 1969.

    MATH  Google Scholar 

  12. Reitmann, V., Realization Theory Methods for the Stability Investigation of Nonlinear Infinite- Dimensional Input-Output Systems, MATH. BOHEM., 2011, vol. 136, no. 2, pp. 185–194.

    MathSciNet  MATH  Google Scholar 

  13. Reitmann, V. and Kantz, H., Stability Investigation of Volterra Integral Equations by Realization Theory and Frequency-Domain Methods, Preprint-Series DFG-SPP 1114, 2004, no. 61.

  14. Salamon, D., Realization Theory in Hilbert Space, Math. Systems Theory, 1989, vol. 21, pp. 147–164.

    Article  MathSciNet  MATH  Google Scholar 

  15. Staffans, O.J., Well-Posed Linear Systems, Cambridge: Cambridge Univ., 2005.

    Book  MATH  Google Scholar 

  16. Yamamoto, Y., Realization Theory of Infinite-Dimensional Linear Systems, Parts I and II. Math. Systems Theory, 1981/2, vol. 15, pp. 55–77; pp. 169–190.

    Google Scholar 

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Authors and Affiliations

  1. St. Petersburg State University, St. Petersburg, Russia

    Yu. A. Abdalova & V. Reitmann

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  1. Yu. A. Abdalova
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  2. V. Reitmann
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Correspondence to V. Reitmann.

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Abdalova, Y.A., Reitmann, V. Stability analysis of Volterra functional equations by realization theory. Diff Equat 51, 1683–1691 (2015). https://doi.org/10.1134/S0012266115130029

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