Volume 29, 2023
|Number of page(s)||47|
|Published online||11 January 2023|
Global control aspects for long waves in nonlinear dispersive media
Departamento de Matemática, Universidade Federal de Pernambuco (UFPE),
2 Universidade Federal do Delta do Parnaíba, Campus Ministro Reis Velloso, Coordenação de Matemática, 64202-020 Parnaíba (PI), Brazil
* Corresponding author: firstname.lastname@example.org
Accepted: 8 December 2022
A class of models of long waves in dispersive media with coupled quadratic nonlinearities on a periodic domain T are studied. We used two distributed controls, supported in ω ⊂ T and assumed to be generated by a linear feedback law conserving the “mass” (or “volume”), to prove global control results. The first result, using spectral analysis, guarantees that the system in consideration is locally controllable in Hs(T), for s ≥ 0. After that, by certain properties of Bourgain spaces, we show a property of global exponential stability. This property together with the local exact controllability ensures for the first time in the literature that long waves in nonlinear dispersive media are globally exactly controllable in large time. Precisely, our analysis relies strongly on the bilinear estimates using the Fourier restriction spaces in two different dispersions that will guarantee a global control result for coupled systems of the Korteweg—de Vries type. This result, of independent interest in the area of control of coupled dispersive systems, provides a necessary first step for the study of global control properties to the coupled dispersive systems in periodic domains.
Mathematics Subject Classification: 35Q53 / 35L56 / 93B05 / 93D15
Key words: Long waves systems / global well-posedness / Bourgain spaces / global control properties
© The authors. Published by EDP Sciences, SMAI 2023
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (https://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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