Abstract
We present a method for the investigation of the stability and positivity of systems of linear differential equations of arbitrary order. Conditions for the invariance of classes of cones of circular and ellipsoidal types are established. We propose algebraic conditions for the exponential stability of linear positive systems based on the notion of maximal eigenpairs of a matrix polynomial.
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M. A. Krasnosel’skii, E. A. Lifshits, and A. V. Sobolev, Positive Linear Systems [in Russian], Nauka, Moscow (1985).
M. W. Hirsch and H. Smith, “Competitive and cooperative systems: mini-review. Positive systems,” Lect. Notes Control Inform. Sci., 294, 183–190 (2003).
A. A. Martynyuk, Qualitative Methods in Nonlinear Dynamics: Novel Approaches to Liapunov’s Matrix Functions, Marcel Dekker, New York (2002).
A. G. Mazko, Localization of Spectrum and Stability of Dynamical Systems [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1999).
A. G. Mazko, “Stability and comparison of states of dynamical systems with respect to a time-varying cone,” Ukr. Mat. Zh., 57, No. 2, 198–213 (2005).
R. J. Stern and H. Wolkowicz, “Exponential nonnegativity on the ice cream cone,” SIAM J. Matrix Anal. Appl., 12, No. 1, 160–165 (1991).
A. G. Mazko, “Semiinversion and properties of invariants of matrices,” Ukr. Mat. Zh., 40, No. 4, 525–528 (1988).
M. W. Hirsch, “Stability and convergence in strongly monotone dynamical systems,” J. Reine Angew. Math., 383, 1–53 (1988).
A. M. Aliluiko and O. H. Mazko, “Invariant cones and stability of multiply connected systems,” Zb. Prats’ Inst. Mat. NAN Ukr., 2, No. 1, 28–45 (2005).
J. S. Vandergraft, “Spectral properties of matrices which have invariant cones,” SIAM J. Matrix Appl. Math., 16, 1208–1222 (1968).
L. Elsner, “Monotone und Randspektrum bei vollstetigen Operatoren,” Arch. Ration. Mech. Anal., 36, 356–365 (1970).
R. J. Stern and H. Wolkowicz, “Invariant ellipsoidal cones,” Linear Algebra Appl., 150, 81–106 (1991).
G. N. Mil’shtein, “Exponential stability of positive semigroups in a linear topological space,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat., No. 9, 35–42 (1975).
A. G. Mazko, “Stability of linear positive systems,” Ukr. Mat. Zh., 53, No. 3, 323–330 (2001).
A. G. Mazko, “Positive and monotone systems in a semiordered space,” Ukr. Mat. Zh., 55, No. 2, 164–173 (2003).
R. Loewy and H. Schneider, “Positive operators on the ice-cream cone,” J. Math. Anal. Appl., 49, 375–392 (1975).
A. G. Mazko, “Positive stabilization of multiply connected systems,” Zb. Prats’ Inst. Mat. NAN Ukr., 1, No. 2, 130–142 (2004).
F. R. Gantmakher, Theory of Matrices [in Russian], Nauka, Moscow (1988).
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 11, pp. 1446–1461, November, 2006.
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Aliluiko, A.M., Mazko, O.H. Invariant cones and stability of linear dynamical systems. Ukr Math J 58, 1635–1655 (2006). https://doi.org/10.1007/s11253-006-0159-5
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DOI: https://doi.org/10.1007/s11253-006-0159-5