Control and stabilization for the dispersion generalized Benjamin equation on the circle
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Published:2022
Issue:
Volume:28
Page:54
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ISSN:1292-8119
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Container-title:ESAIM: Control, Optimisation and Calculus of Variations
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language:
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Short-container-title:ESAIM: COCV
Author:
Vielma Leal Francisco J.,
Pastor AdemirORCID
Abstract
This paper is concerned with controllability and stabilization properties of the dispersion generalized Benjamin equation on the periodic domain T. First, by assuming the control input acts on all the domain, the system is proved to be globally exactly controllable in the Sobolev space Hsp(𝕋), with s ≥ 0. Second, by providing a locally-damped term added to the equation as a feedback law, it is shown that the resulting equation is globally well-posed and locally exponentially stabilizable in the space L2p(𝕋). The main ingredient to prove the global well-posedness is the introduction of the dissipation-normalized Bourgain spaces which allows one to gain smoothing properties simultaneously from the dissipation and dispersion present in the equation. Finally, the local exponential stabilizability result is accomplished taking into account the decay of the associated semigroup combined with the fixed point argument.
Funder
Fundação de Amparo à Pesquisa do Estado de São Paulo
Conselho Nacional de Desenvolvimento Científico e Tecnológico
Subject
Computational Mathematics,Control and Optimization,Control and Systems Engineering
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