Abstract
We establish a criterion for exponential stability in the H1-topology in terms of operator inequalities for a linear FDE system of retarded type by Lyapunov’s direct method. As a corollary, some sufficient condition of exponential stability in terms of the matrix specifying the Stieltjes integral is obtained in the autonomous case. A few examples illustrating the results are exhibited.
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Vorobëva E. V. and Romanovskiĭ R. K., “On stability of solutions to the Cauchy problem for hyperbolic systems in two independent variables,” Siberian Math. J., 39, No. 6, 1112–1114 (1998).
Alekseenko N. V., “Stability of solutions of nonlinear almost periodic systems of functional-differential equations of delay type,” Russian Math., 44, No. 2, 3–6 (2000).
Aleksenko N. V. and Romanovskiĭ R. K., “The method of Lyapunov functionals for linear difference-differential systems with almost periodic coefficients,” Differential Equations, 37, No. 2, 159–165 (2001).
Romanovskiĭ R. K. and Trotsenko G. A., “The method of Lyapunov functionals for neutral type linear difference-differential systems with almost periodic coefficients,” Siberian Math. J., 44, No. 2, 355–362 (2003).
Trotsenko G. A., “On the stability of solutions of an almost periodic system of functional-differential equations of neutral type,” Russian Math. (Iz. VUZ), 47, No. 6, 73–77 (2003).
Romanovskiĭ R. K., Vorobëva E. V., and Makarova I. D., “On the stability of solutions of a mixed problem for an almost linear parabolic system on a plane,” Sibirsk. Zh. Industr. Mat., 6, No. 1, 118–124 (2003).
Dobrovol’skiĭ S. M. and Rogozin A. V., “The direct Lyapunov method for an almost periodic difference system on a compactum,” Siberian Math. J., 46, No. 1, 77–82 (2005).
Rogozin A. V., “On stability of solutions of a linear differential equation in a Hilbert space with an almost-periodic operator,” Dokl. Akad. Nauk VSh. RF, 6, No. 1, 24–32 (2006).
Romanovskiĭ R. K. and Mendziv M. V., “Stability of solutions to the mixed problem for a plane hyperbolic system with time-periodic coefficients,” Dokl. Akad. Nauk VSh. RF, 6, No. 1, 78–85 (2006).
Romanovskiĭ R. K. and Mendziv M. V., “Stability of solutions to the Cauchy problem for a plane hyperbolic system with time-periodic coefficients,” Siberian Math. J., 48, No. 5, 913–918 (2007).
Mendziv M. V. and Romanovskiĭ R. K., “Direct Lyapunov method for hyperbolic systems on the plane with time-periodic coefficients,” Differential Equations, 44, No. 2, 267–273 (2008).
Romanovskiĭ R. K. and Bel’gart L. V., “On the exponential dichotomy of solutions of the Cauchy problem for a hyperbolic system on a plane,” Differential Equations, 46, No. 8, 1135–1144 (2010).
Romanovskiĭ R. K. and Bel’gart L. V., “Dichotomy of solutions to the Cauchy problem with an almost periodic periodic system on a plane,” Dokl. Akad. Nauk VSh. RF, 15, No. 2, 14–24 (2010).
Bel’gart L. V., “On a class of indefinite Lyapunov functionals,” Omsk Nauch. Vestnik Ser. Pribory, Mashiny, i Tekhnologii, 93, No. 3, 11–13 (2010).
Vlasov V. V., “On the solvability and estimates of solutions to functional differential equations in Sobolev spaces,” Proc. Steklov Inst. Math., 227, 104–115 (1999).
Vlasov V. V. and Medvedev D. A., “Functional-differential equations in Sobolev spaces and related problems of spectral theory,” J. Math. Sci., 164, No. 5, 659–841 (2010).
Zhabko A. P. and Kharitonov V. L., Methods of Linear Algebra in Control Problems [in Russian], St. Petersburg Univ., St. Petersburg (1993).
Andreev A. S. and Pavlikov S. V., “The method of Lyapunov functionals in stability analysis of functional-differential equations,” Math. Notes, 68, No. 3, 281–288 (2000).
Pavlikov S. V., “Lyapunov functionals of constant signs in the stability problem for a functional-differential equation of neutral type,” Math. Notes, 83, No. 3, 378–388 (2008).
Martynyuk A. A., Kato D., and Shestakov A. A., Stability of Motion: The Method of Limited Equations [in Russian], Naukova Dumka, Kiev (1990).
Zubov V. I., Lectures on Control Theory [in Russian], Nauka, Moscow (1975).
Myshkis A. D., Linear Differential Equations with Retarded Argument [in Russian], Nauka, Moscow (1972).
Kolmogorov A. N. and Fomin S. V., Elements of the Theory of Functions and Functional Analysis [in Russian], Nauka, Moscow (1976).
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Original Russian Text Copyright © 2014 Romanovsky R.K. and Nazaruk E.M.
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 4, pp. 851–862, July–August, 2014.
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Romanovsky, R.K., Nazaruk, E.M. Lyapunov’s direct method for linear systems of functional-differential equations in Sobolev space. Sib Math J 55, 696–705 (2014). https://doi.org/10.1134/S0037446614040119
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DOI: https://doi.org/10.1134/S0037446614040119