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Partial asymptotic stability of abstract differential equations

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Abstract

We consider the problem of partial asymptotic stability with respect to a continuous functional for a class of abstract dynamical processes with multivalued solutions on a metric space. This class of processes includes finite-and infinite-dimensional dynamical systems, differential inclusions, and delay equations. We prove a generalization of the Barbashin-Krasovskii theorem and the LaSalle invariance principle under the conditions of the existence of a continuous Lyapunov functional. In the case of the existence of a differentiable Lyapunov functional, we obtain sufficient conditions for the partial stability of continuous semigroups in a Banach space.

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Authors and Affiliations

  1. Institute of Applied Mathematics and Mechanics, Ukrainian Academy of Sciences, Donetsk

    A. L. Zuev

  2. Technische Universitat Ilmenau, Ilmenau, Germany

    A. L. Zuev

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  1. A. L. Zuev
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 5, pp. 629–637, May, 2006.

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Zuev, A.L. Partial asymptotic stability of abstract differential equations. Ukr Math J 58, 709–717 (2006). https://doi.org/10.1007/s11253-006-0096-3

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  • DOI: https://doi.org/10.1007/s11253-006-0096-3

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