Abstract
We obtain sufficient conditions for the Lyapunov stability of the trivial solution of a nonautonomous differential system of a special form ast ↑ ω, ω ≤ + ∞. For this system, the coefficient matrix of a differential system of the first approximation has almost Jordan form with triangular blocks. We indicate methods that enable one to reduce certain classes of differential systems of the general form to special differential systems.
References
O. Perron, “Die Stabilitatsfrage bei Differentialgleichungen,”Math. Z.,32, 703–728 (1930).
A. V. Kostin,Stability and Asymptotics of Almost Triangular Systems [in Russian], Candidate Degree Thesis (Physics and Mathematics), Kiev (1962).
I. E. Vitrichenko, “On the stability of one zero root and a pair of pure imaginary roots of a nonautonomous quasilinear equation of ordern in a critical case,”Differents. Uravn.,26, No. 12, 2027–2046 (1990).
I. M. Rappoport,On Some Asymptotic Methods in the Theory of Differential Equations [in Russian], Ukrainian Academy of Sciences, Kiev (1954).
S. A. Chaplygin,Selected Works [in Russian], Nauka, Moscow (1976).
I. E. Vitrichenko and V. V. Nikonenko, “On the reduction of a linear nonautonomous system to almost block-triangular diagonal form in the case of a multiple zero eigenvalue of the limit matrix of coefficients,”Proc. Razmadze Math. Inst.,110, 59–65 (1994).
K. P. Persidskii, “On eigenvalues of differential equations,”Izv. Akad. Nauk Kaz. SSR, Ser. Mat. Mekh.,12, Issue 1, 5–47 (1947).
A. V. Kostin, “Asymptotic formulas for solutions of linear systems of ordinary differential equations,”Dokl. Akad. Nauk Ukr. SSR,10, 1293–1297 (1962).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 8, pp. 1072–1079, August, 1994.
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Vitrichenko, I.E. On stability of the trivial solution of a nonautonomous quasilinear system whose characteristic equation has multiple roots. Ukr Math J 46, 1178–1187 (1994). https://doi.org/10.1007/BF01056178
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DOI: https://doi.org/10.1007/BF01056178