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Title: Exponential stability for nonautonomous functional differential equations with state-dependent delay
Authors: Maroto, Ismael
Núñez, Carmen
Obaya, Rafael
Issue Date: 2017
Publisher: American Institute of Mathematical Sciences
Citation: Discrete and Continuous Dynamical Systems, Series B 22 (8) 2017, 3167-3197
Abstract: The properties of stability of a compact set $K$ which is positively invariant for a semiflow $(\W\times W^{1,\infty}([-r,0],\mathbb{R}^n),\Pi,\mathbb{R}^+)$ determined by a family of nonautonomous FDEs with state-dependent delay taking values in $[0,r]$ are analyzed. The solutions of the variational equation through the orbits of $K$ induce linear skew-product semiflows on the bundles $K\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and $\mK\times C([-r,0],\mathbb{R}^n)$. The coincidence of the upper-Lyapunov exponents for both semiflows is checked, and it is a fundamental tool to prove that the strictly negative character of this upper-Lyapunov exponent is equivalent to the exponential stability of $K$ in $\W\times W^{1,\infty}([-r,0],\mathbb{R}^n)$ and also to the exponential stability of this compact set when the supremum norm is taken in $W^{1,\infty}([-r,0],\mathbb{R}^n)$. In particular, the existence of a uniformly exponentially stable solution of a uniformly almost periodic FDE ensures the existence of exponentially stable almost periodic solutions.
ISSN: 1531-3492
Peer Review: SI
DOI: 10.3934/dcdsb.2017169
Sponsor: Ministerio de Economía, Industria y Competitividad (MTM2015-66330-P)
Programme: info:eu-repo/grantAgreement/EC/H2020/643073
Language: eng
Rights: info:eu-repo/semantics/openAccess
Appears in Collections:Documentos OpenAire(Open Access Infrastructure for Research in Europe)
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