Abstract
Consideration was given to the question of asymptotic (exponential) stability of the maximum periodic solutions of the integrodifferential equations which have an asymptotically stable linear part and small periodic (exponential maximum periodic) perturbation. Under the unlimitedly increasing time, these solutions tend to the periodic modes. The sufficient conditions for asymptotic stability were indicated. In the resonance case where the linearized equation has a pair of purely imaginary roots with the corresponding oscillation frequency coinciding with the oscillation frequency of the periodic part of small perturbation (time function) and the coefficients of the power series expansion of the nonlinear terms, consideration was given to the problem of existence for the maximum periodic solutions of the integrodifferential equation. Conditions were established for existence of such solutions representable by the power series in the fractional degrees of the small parameter characterizing the value of small perturbation in the equation.
Similar content being viewed by others
References
Bykov, Ya.V., O nekotorykh zadachakh teorii integro-differentsial’nykh uravnenii (On Some Problems of the Theory of Integrodifferential Equations), Frunze: Kirgiz. Gos. Univ., 1957.
Sergeev, V.S., On Maximum Periodic Motions in Some Systems with Aftereffect, Prikl. Mat. Mekh., 2004, vol. 68, no. 5, pp. 857–869.
Lyapunov, A.M., Obshchaya zadacha ob ustoichivosti dvizheniya (General Problem of Motion Stability), Moscow: Akad. Nauk SSSR, 1956, vol. 2, pp. 7–263.
Sergeev, V.S., On Instability in the Critical Case of a Pair of Purely Imaginary Roots for One Class of Systems with Aftereffect, Prikl. Mat. Mekh., 1998, vol. 62, no. 1, pp. 79–86.
Sergeev, V.S., Stability of Solutions of Volterra Integro-differential Equations, Math. Comput. Model., 2007, vol. 45, nos. 11–12, pp. 1376–1394.
Sergeev, V.S., On Stability in Critical Cases for the Integrodifferential Equations of the Volterra Type, Mat. Zh., 2003, vol. 3, no. 3(9), pp. 91–105.
Sergeev, V.S., Maximum Periodic Solutions of the Volterra Integrodifferential Equations in the Critical Case of a Pair of Pure Imaginary Roots, Autom. Remote Control, 2011, vol. 72, no. 9, pp. 1876–1886.
Lika, D.K. and Ryabov, Yu.A., Metody iteratsii i mazhoriruyushchee uravnenie Lyapunova v teorii nelineinykh kolebanii (Methods of Iterations and the Majorizing Lyapunov Equation in the Theory of Nonlinear Oscillations), Kishinev: Shtiintsa, 1974.
Grebenikov, E.A. and Ryabov, Yu.A., Konstruktivnye metody analiza nelineinykh sistem (Constructive Methods of Analysis of the Nonlinear Systems), Moscow: Nauka, 1979.
Sergeev, V.S., On Maximum Periodic Solutions of Integrodifferential Equations of the Volterra Type, in Zadachi issledovaniya ustoichivosti i stabilizatsii dvizheniya (Problems of Stydying Stability and Motion Stabilization), Moscow: Vychisl. Tsentr Ross. Akad. Nauk, 2011, pp. 4–24.
Sergeev, V.S., Maximum Periodic Motions in System with Aftereffect Obeying the Integrodifferential Equations of the Volterra Type, in Tr. X Mezhd. Chetaev. konf. “Analiticheskaya mekhanika, ustoichivost’ i upravlenie,” tom 2: Analiticheskaya mekhanika, ustoichivost’ dvizheniya (Proc. X Int. Chetaev Conf. “Analytical Mechanics, Stability and Control”, vol 2: Analytical Mechanics, Motion Stability), Kazan, June 12–16, 2012, Kazan: Kazan. Gos. Tekhn. Univ., 2012, pp. 447–454.
Author information
Authors and Affiliations
Additional information
Original Russian Text © V.S. Sergeev, 2013, published in Avtomatika i Telemekhanika, 2013, No. 8, pp. 148–159.
Rights and permissions
About this article
Cite this article
Sergeev, V.S. On maximum periodic solutions of integrodifferential equations of volterra type and their stability. Autom Remote Control 74, 1356–1365 (2013). https://doi.org/10.1134/S0005117913080122
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0005117913080122