On the controllability and stabilization of the Benjamin equation on a periodic domain
Panthee, M.1 ; Vielma Leal, F.1
1 Department of Mathematics, Statistics & Computing Science, Campinas University, Sao Paulo, Brazil
Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1605-1652.
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The aim of this paper is to study the controllability and stabilization for the Benjamin equation on a periodic domain T. We show that the Benjamin equation is globally exactly controllable and globally exponentially stabilizable in Hps(T), with s≥0. The global exponential stabilizability corresponding to a natural feedback law is first established with the aid of certain properties of solution, viz., propagation of compactness and propagation of regularity in Bourgain's spaces. The global exponential stability of the system combined with a local controllability result yields the global controllability as well. Using a different feedback law, the resulting closed-loop system is shown to be locally exponentially stable with an arbitrarily large decay rate. A time-varying feedback law is further designed to ensure a global exponential stability with an arbitrary large decay rate. The results obtained here extend the ones we proved for the linearized Benjamin equation in [32].

Reçu le :
Révisé le :
Accepté le :
DOI : 10.1016/j.anihpc.2020.12.004
Classification : 93B05, 93D15, 35Q53
Mots-clés : Dispersive equations, Benjamin equation, Well-posedness, Controllability, Stabilization
Panthee, M. 1 ; Vielma Leal, F. 1

1 Department of Mathematics, Statistics & Computing Science, Campinas University, Sao Paulo, Brazil 
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Panthee, M.; Vielma Leal, F. On the controllability and stabilization of the Benjamin equation on a periodic domain. Annales de l'I.H.P. Analyse non linéaire, septembre – octobre 2021, Tome 38 (2021) no. 5, pp. 1605-1652. doi : 10.1016/j.anihpc.2020.12.004. http://www.numdam.org/articles/10.1016/j.anihpc.2020.12.004/

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This work was partially supported by FAPESP , Brazil, with grants 2016/25864-6 and 2015/06131-5 and CNPq , Brazil with grants (308131/2017-7 and 402849/2016-7).