Abstract
For an autonomous linear homogeneous asymptotically stable differential system, we obtain sufficient conditions for the smallness of perturbations in the class of autonomous linear homogeneous systems under which a quadratic form that is a Lyapunov function of the original system remains a Lyapunov function of the perturbed system.
This is a preview of subscription content, log in via an institution to check access.
REFERENCES
Lyapunov, A.M., Obshchaya zadacha ob ustoichivosti dvizheniya (General Problem on the Stability of Motion), Moscow–Leningrad: Gos. Izd. Tekh.-Teor. Lit, 1950.
Chetaev, N.G., Ustoichivost’ dvizheniya (Stability of Motion), Moscow: Nauka, 1966.
Kosyakin, A.A. and Shamrikov, B.M., Kolebaniya v tsifrovykh avtomaticheskikh sistemakh (Oscillations in Digital Automatic Systems), Moscow: Nauka, 1983.
Barbashin, E.A., Funktsii Lyapunova (Lyapunov Functions), Moscow: Nauka, 1970.
Khusainov, D.Ya. and Yun’kova, E.A., On a method for finding a solution of the matrix Lyapunov equation with a given spectrum, Ukr. Mat. Zh., 1984, vol. 36, no. 4, pp. 528–531.
Sarybekov, R.A., Extremal quadratic Lyapunov functions of systems of second-order equations, Sib. Math. J., 1977. Т. 18. № 5, pp. 823–829.
Komarov, Yu.A. and Khusainov, D.Ya., Some remarks on the Lyapunov extremal function for linear systems, Ukr. Mat. Zh., 1983, vol. 35, no. 6, pp. 750–753.
Propoi, A.I., On the motion stability problem, Autom. Remote Control, 2000, vol. 61, no. 4, pp. 585–594.
Antonovskaya, O.G. and Goryunov, V.I., On a method of evaluation of attraction domain for fixed point of nonlinear point mapping of arbitrary dimension, Russ. Math., 2016, vol. 60, no. 12, pp. 9–14.
Antonovskaya, O.G., On the construction of a quadratic Lyapunov function with given properties, Differ. Equations, 2013, vol. 49, no. 9, pp. 1187–1191.
Antonovskaya, O.G., Determination of the coefficients of a quadratic Lyapunov function with given properties, Differ. Equations, 2016, vol. 52, no. 3, pp. 275–281.
Antonovskaya, O.G., On the maximum possible negativity margin for the first derivative (first difference) of a quadratic Lyapunov function, Differ. Equations, 2003, vol. 39, no. 11, pp. 1645–1647.
Antonovskaya, O.G., Construction of Lyapunov quadratic functions satisfying given constraints for continuous and discrete dynamical systems, Russ. Math., 2004, vol. 48, no. 2, pp. 16–20.
Gantmakher, F.R., Teoriya matrits (Theory of Matrices), Moscow: Nauka, 1967.
Horn, R. and Johnson, C., Matrix Analysis, New York: Cambridge Univ. Press, 1985. Translated under the title: Matrichnyi analiz, Moscow: Mir, 1989.
Demidovich, B.P., Lektsii po matematicheskoi teorii ustoichivosti (Lectures on the Mathematical Theory of Stability), Moscow: Nauka, 1967.
Polyak, B.T., Khlebnikov, M.V., and Shcherbakov, P.S., Upravlenie lineinymi sistemami pri vneshnikh vozmushcheniyakh: tekhnika lineinykh matrichnykh neravenstv (Control of Linear Systems under Exogenous Disturbances: Technique of Linear Matrix Inequalities), Moscow: URSS, 2014.
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
Antonovskaya, O.G. On the Preservation of a Quadratic Lyapunov Function of a Linear Differential Autonomous System under Constant Perturbations of the Coefficients. Diff Equat 59, 293–300 (2023). https://doi.org/10.1134/S0012266123030011
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266123030011