Local null controllability of a model system for strong interaction between internal solitary waves

Jon Asier Bárcena-Petisco (1, 2, 3) , Sergio Guerrero (1) , Ademir F. Pazoto (4)

1 LJLL - Laboratoire Jacques-Louis Lions
2 UAM - Universidad Autónoma de Madrid
3 Fundación Deusto
4 IM / UFRJ - Instituto de Matemática da Universidade Federal do Rio de Janeiro

In this paper we prove the local null controllability property for a nonlinear coupled system of two Korteweg-de Vries equations posed on bounded interval and with a source term decaying exponentially on $t = T$. The system was introduced by Gear and Grimshaw to model the interactions of two dimensional, long, internal gravity waves propagation in a stratified fluid. We address the controllability problem by means of a control supported on an interior open subset of the domain and acting on one equation only. The proof consists mainly on proving the controllability of the linearized system, which is done by getting a Carleman estimate for the adjoint system. While doing the Carleman we improve the techniques for dealing with the fact that the solutions of dispersive and parabolic equations with a source term in $L^2$ have a limited regularity. A local inversion theorem is applied to get the result for the nonlinear system.

Mots clés: Carleman inequalities; internal control; Korteweg-de Vries systems; null controllability

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Jon Asier Bárcena-Petisco, Sergio Guerrero, Ademir F. Pazoto. Local null controllability of a model system for strong interaction between internal solitary waves. 2021. ⟨hal-02376059v3⟩