Abstract
Considering nonautonomous differential inclusions we introduce the concept of limit differential inclusions, study their properties and invariance-type properties of the ω-limit sets of solutions, and establish an analog of La Salle’s invariance principle using Lyapunov functions with the derivatives of constant sign. The method is equally applicable to differential equations and, under appropriate assumptions, yields some previously-available results.
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References
Barbashin E. A., The Lyapunov Functions [in Russian], Nauka, Moscow (1970).
Rouche N., Habets P., and Laloy M., Stability Theory by Lyapunov’s Direct Method [Russian translation], Mir, Moscow (1980).
Andreev A. S., “The Lyapunov functionals method in stability problems for functional differential equations,” Automation and Remote Control, 70, No. 9, 1438–1486 (2009).
Artstein Z., “Topological dynamics of an ordinary differential equation,” J. Differential Equations, 23, 216–223 (1977).
Artstein Z., “Topological dynamics of ordinary differential equations,” J. Differential Equations, 23, 224–243 (1977).
Artstein Z., “The limiting equations of nonautonomous ordinary differential equations,” J. Differential Equations, 25, 184–202 (1977).
Artstein Z., “Uniform asymptotic stability via limiting equations,” J. Differential Equations, 27, 172–189 (1978).
Davy J. L., “Properties of solution set of a generalized differential equation,” Bull. Austral. Math. Soc., 6, 379–398 (1972).
Kantorovich L. V. and Akilov G. P., Functional Analysis in Normed Spaces [in Russian], Nauka, Moscow (1977).
Rudin W., Functional Analysis [Russian translation], Mir, Moscow (1975).
Logemann H. and Ryan E. P., “Non-autonomous systems: asymptotic behaviour and weak invariance principles,” J. Differential Equations, 189, 440–460 (2003).
Kuratowski K., Topology. Vol. 1 [Russian translation], Mir, Moscow (1966).
Borisovich Yu. G., Gel’man B. D., Myshkis A. D., and Obukhovskiĭ V. V., Introduction to the Theory of Multivalued Maps and Differential Inclusions [in Russian], KomKniga, Moscow (2005).
Kuratowski K., Topology. Vol. 2 [Russian translation], Mir, Moscow (1969).
Filippov A. F., Differential Equations with a Discontinuous Right-Hand Side [in Russian], Nauka, Moscow (1985).
Natanson I. P., Theory of Functions of Real Variable [in Russian], Gostekhizdat, Moscow (1957).
Edwards R. D., Functional Analysis. Theory and Applications [Russian translation], Mir, Moscow (1969).
Tolstonogov A. A. and Finogenko I. A., “On solutions of a differential inclusion with lower semicontinuous nonconvex right-hand side in a Banach space,” Math. USSR-Sb., 53, No. 1, 203–231 (1986).
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Original Russian Text Copyright © 2014 Finogenko I.A.
The author was partially supported by the Interdisciplinary Project of the Siberian Division of the Russian Academy of Sciences (Grant No. 80), the Presidium of the Russian Academy of Sciences (Fundamental Research Program No.17), and the Russian Foundation for Basic Research (Grant 13-01-00287-a).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 2, pp. 454–471, March–April, 2014.
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Finogenko, I.A. Limit differential inclusions and the invariance principle for nonautonomous systems. Sib Math J 55, 372–386 (2014). https://doi.org/10.1134/S0037446614020190
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DOI: https://doi.org/10.1134/S0037446614020190