Abstract
Given a closed convex pointed cone
which is positively invariant with respect to motions of the differential equation\(\dot x = Ax\) (A being a real (n × n) matrix), it is proven that a necessary and sufficient condition for asymptotic stability of
(and therefore of the linear span of
) is
In case
, this result yields a known equivalence from the theory ofM-matrices.
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References
Berman A, Plemmons RJ (1979) Nonnegative matrices in the mathematical sciences. Academic Press
Hájek O (unpublished manuscript) A short proof of Brammer's theorem.
Hirsch MW, Smale S (1974) Differential equations, dynamical systems, and linear algebra. Academic Press
Nagumo M (1942) Uber die Lage der Integralkurven gewohnlicher Differentialgleichungen. Proc Phys—Math Soc Japan 24:551–559
Ostrowski AM (1937) Uber die Determinaten mit uberwiegender Hauptdiagonale. Comment Math Helv 10:69–96
Robert F (1966) Recherche d'uneM-matrice, parmi les minorantes d'un operateur lineaire. Numer Math 9:189–199
Schneider H, Vidyasagar M (1970) Cross-positive matrices. SIAM J Numer Anal 9:508–519
Yorke JA (1967) Invariance for ordinary differential equations. Math Systems Theory 1:353–372
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Communicated by A. V. Balakrishnan
This research was supported by the Natural Sciences and Engineering Council Canada under grant A4641.
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Stern, R.J. A note on positively invariant cones. Appl Math Optim 9, 67–72 (1982). https://doi.org/10.1007/BF01460118
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DOI: https://doi.org/10.1007/BF01460118