Abstract
We establish criteria of asymptotic stability for positive differential systems in the form of conditions of monotone invertibility of linear operators. The structure of monotone and monotonically invertible operators in the space of matrices is investigated.
Similar content being viewed by others
REFERENCES
M. A. Krasnosel'skii, E. A. Lifshits, and A. V. Sobolev, Positive Linear Systems [in Russian], Nauka, Moscow (1985).
M. G. Krein and M. A. Rutman, “Linear operators leaving invariant a cone in a Banach space” Usp. Mat. Nauk 3, No. 1, 3–95 (1948).
P. Clément, H. J. A. M. Heijmans, S. Angenent, C. J. van Duijn, and B. de Pagter, One-Parameter Semigroups North-Holland, Amsterdam (1987).
M. Fielder and V. Pták, “On matrices with nonpositive off-diagonal elements and positive principal minors” Czech. Math. J. 12 (87), 382–400 (1962).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in a Banach Space [in Russian], Nauka, Moscow (1970).
P. Lankaster, Theory of Matrices Academic Press, New York (1969).
F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).
I. I. Gikhman, “On the stability of solutions of stochastic differential equations” in: Limit Theorems and Statistical Conclusions [in Russian], Fan, Tashkent (1966), pp. 14–45.
K. G. Valeev, O. L. Karelova, and V. I. Gorelov, Optimization of Linear Systems with Random Coefficients [in Russian], Russian University for Friendship of Peoples, Moscow (1996).
D. G. Korenevskii, Stability of Solutions of Deterministic and Stochastic Differential-Difference Equations (Algebraic Criteria) [in Russian], Naukova Dumka, Kiev (1992).
A. G. Mazko, “Matrix equations and inequalities in problems of localization of the spectrum” in: Problems of Analytic Mechanics and Its Applications [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1998), pp. 203–218.
A. G. Mazko, “Localization of the spectrum and representation of solutions of linear dynamical systems” Ukr. Mat. Zh. 50, No. 10, 1341–1351 (1998).
A. G. Mazko, Localization of the Spectrum and Stability of Dynamical Systems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1999).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Mazko, A.G. Stability of Linear Positive Systems. Ukrainian Mathematical Journal 53, 368–376 (2001). https://doi.org/10.1023/A:1012336103731
Issue Date:
DOI: https://doi.org/10.1023/A:1012336103731