Volume 28, 2022
|Number of page(s)
|02 August 2022
Control and stabilization for the dispersion generalized Benjamin equation on the circle
IMECC-UNICAMP, Rua Sérgio Buarque de Holanda, 651, 13083-859 Campinas, SP, Brasil
* Corresponding author: firstname.lastname@example.org
Accepted: 27 June 2022
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.
Mathematics Subject Classification: 93B05 / 93D15 / 93D23 / 35Q53
Key words: Dispersive equations / dispersion generalized Benjamin equation / controllability / well-posedness / stabilization
© The authors. Published by EDP Sciences, SMAI 2022
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