Abstract
Various tests for the exponential stability of a linear system of ordinary differential equations are obtained by the method of frozen coefficients. To this end, we prove and use an improved Gelfand–Shilov estimate for the matrix exponential. The cases in which the coefficient matrix of the system satisfies the Lipschitz condition or the Hölder condition on the positive real line are considered separately.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
REFERENCES
Bylov, B.F., Vinograd, R.E., Grobman, D.M., and Nemytskii, V.V., Teoriya pokazatelei Lyapunova i ee prilozheniya k voprosam ustoichivosti (Theory of Lyapunov Exponents and Its Applications to Stability Problems), Moscow: Nauka, 1966.
Daletskii, Yu.L. and Krein, M.G., Ustoichivost’ reshenii differentsial’nykh uravnenii v banakhovom prostranstve (Stability of Solutions of Differential Equations in a Banach Space), Moscow: Nauka, 1970.
Perov, A.I. and Kostrub, I.D., Priznaki ustoichivosti periodicheskikh reshenii sistem differentsial’nykh uravnenii, osnovannye na teorii vnediagonal’no neotritsatel’nykh matrits (Tests for Stability of Periodic Solutions of Systems of Differential Equations Based on the Theory of Off-Diagonally Nonnegative Matrices), Voronezh: Nauchn. Kniga, 2015.
Lozinskii, S.M., Estimate of errors in numerical integration of ordinary differential equations, Izv. VUZov. Mat., 1958, no. 5, pp. 52–90.
Gelfand, I.M. and Shilov, G.E., Nekotorye voprosy teorii differentsial’nykh uravnenii (Some Questions of the Theory of Differential Equations), Moscow: Fizmatgiz, 1958.
Levin, A.Yu., Kharitonov’s theorem for weakly non-stationary systems, Russ. Math. Surv., 1995, vol. 50, no. 6, pp. 1280–1281.
Alekseev, V.M., Estimate of numerical integration error, Dokl. Akad. Nauk SSSR, 1960, vol. 134, no. 2, pp. 247–250.
Fikhtengol’ts, G.M., Kurs differentsial’nogo i integral’nogo ischisleniya (A Course in Differential and Integral Calculus), Moscow: FML, 1970.
Perov, A.I., Kostrub, I.D., and Kaverina, V.K., Method of frozen coefficients in Hölder conditions. Modern methods in the theory of boundary value problems, Mater. mezhdunar. konf. Voronezhskaya vesennyaya mat. shkola “Pontryaginskie chteniya XXXI” (Proc. Int. Conf. Voronezh Spring Math. School “Pontryagin Readings XXXI”) (Voronezh, May 3–9, 2020), Voronezh, 2020, pp. 171–172.
Funding
The research by A.I. Perov was supported by the Russian Foundation for Basic Research, project no. 19–01–00732.
Author information
Authors and Affiliations
Corresponding authors
Additional information
Translated by V. Potapchouck
Rights and permissions
About this article
Cite this article
Perov, A.I., Kostrub, I.D. & Kaverina, V.K. Method of Frozen Coefficients in Hölder Conditions. Diff Equat 57, 587–593 (2021). https://doi.org/10.1134/S0012266121050037
Received:
Revised:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1134/S0012266121050037