Abstract
A criterion (in terms of the singular numbers of Cauchy operators) is obtained for the reducibility in the mean, i.e., for the approximability (in the mean, on a half-line), of a given system to any degree of accuracy by means of Lyapunov transformations of another given system. It is proved that any two triangular systems with identical diagonals are reducible in the mean to each other.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Literature cited
B. F. Bylov, P. É. Vinograd, D. M. Grobman, and V. V. Nemytskii, Theory of Lyapunov Exponents and Its Application to Stability Problems [in Russian], Nauka, Moscow (1966).
S. A. Grishin, “The solution-rotation method and certain problems of motion control,” Dissertation, Moscow State Univ. (1975).
V. M. Millionshchikov, “On the instability of singular exponents and the asymmetry of the relation of almost reducibility of linear systems of differential equations,” Diff. Uravn.,5, No. 4, 749–750 (1969).
I. N. Sergeev, “Sharp bounds of the mobility of the Lyapunov exponents of linear systems under small average perturbations,” Trudy Seminara im. I. G. Petrovskogo,11, 32–73, Moscow State Univ. (1986).
I. N. Sergeev, “The problems of the stabilizability and destabilizability of linear systems by small average perturbations,” An abstract of the report in the article: “On the seminar on the qualitative theory of differential equations at Moscow University” Diff. Uravn.,18, No. 11, 2012–2013 (1982).
Additional information
Translated from Trudy Seminara im. I. G. Petrovskogo, No. 12, pp. 218–228, 1987.
Rights and permissions
About this article
Cite this article
Sergeev, I.N. Criterion for the reducibility in the mean of linear differential systems. J Math Sci 47, 2651–2660 (1989). https://doi.org/10.1007/BF01105915
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF01105915