A nonlinear version of Halanay's inequality for the uniform convergence to the origin

Pierdomenico Pepe

doi:10.3934/mcrf.2021045

Article Contents

2022, Volume 12, Issue 3: 789-811. Doi: 10.3934/mcrf.2021045

This issue Previous Article Asymptotic gain results for attractors of semilinear systems Next Article Computation of open-loop inputs for uniformly ensemble controllable systems

A nonlinear version of Halanay's inequality for the uniform convergence to the origin

Pierdomenico Pepe^,

Center of Excellence for Research DEWS, Department of Information Engineering, Computer Science, and Mathematics, University of L'Aquila, Via Vetoio Coppito I, 67100 L'Aquila, Italy

^* Corresponding author: e-mail pierdomenico.pepe@univaq.it
^* Corresponding author: e-mail pierdomenico.pepe@univaq.it

This article was recruited by Fabian Wirth

Received: August 31, 2019

Revised: January 31, 2021

Early access: September 2021

Published: July 2022

The work has been supported in part by grant MIUR-FFABR 2017

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Abstract

A nonlinear version of Halanay's inequality is studied in this paper as a sufficient condition for the convergence of functions to the origin, uniformly with respect to bounded sets of initial values. The same result is provided in the case of forcing terms, for the uniform convergence to suitable neighborhoods of the origin. Related Lyapunov methods for the global uniform asymptotic stability and the input-to-state stability of systems described by retarded functional differential equations, with possibly nonconstant time delays, are provided. The relationship with the Razumikhin methodology is shown.

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Mathematics Subject Classification: Primary: 93C23, 93D05, 93D20, 93D25; Secondary: 39B72, 34D23, 34D05, 26A46, 26A48.

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