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Localization of limit sets and stability of systems of the neutral type with nonmonotone Lyapunov functionals

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Abstract

We suggest new approaches to the study of the asymptotic stability of equilibria for equations of the neutral type. Nonmonotone indefinite Lyapunov functionals are used. We investigate the localization of solutions with respect to the level surfaces of a Lyapunov functional and a functional estimating the derivative of the Lyapunov functional along the solutions. By using solution localization tests, we obtain new conditions for the asymptotic stability of equilibria for equations of the neutral type with bounded right-hand side. We present asymptotic stability tests that do not impose any a priori stability condition on the difference operator. A generalization of the Barbashin–Krasovskii theorem for nonmonotone indefinite Lyapunov functionals is proved for autonomous equations.

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Authors and Affiliations

  1. Institute of Mathematics, National Academy of Sciences, Minsk, Belarus

    L. B. Knyazhishche

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  1. L. B. Knyazhishche
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Correspondence to L. B. Knyazhishche.

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Original Russian Text © L.B. Knyazhishche, 2016, published in Differentsial’nye Uravneniya, 2016, Vol. 52, No. 10, pp. 1327–1334.

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Knyazhishche, L.B. Localization of limit sets and stability of systems of the neutral type with nonmonotone Lyapunov functionals. Diff Equat 52, 1274–1281 (2016). https://doi.org/10.1134/S0012266116100037

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  • DOI: https://doi.org/10.1134/S0012266116100037

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