Internal null controllability of the generalized Hirota-Satsuma system

Nicolás Carreño (1) , Eduardo Cerpa (2) , Emmanuelle Crépeau (3, 4)

1 Departamento de Matematicas, Univesidad Tecnica Federico Santa-Maria (Valparaiso)
2 Departamento de Matematica, Universidad Tecnica Federico Santa Maria
3 LMV - Laboratoire de Mathématiques de Versailles
4 EDP - Equations aux Dérivées Partielles

The generalized Hirota-Satsuma system consists of three coupled nonlinear Korteweg-de Vries (KdV) equations. By using two distributed controls it is proven in this paper that the local null controllability property holds when the system is posed on a bounded interval. First, the system is linearized around the origin obtaining two decoupled subsystems of third order dispersive equations. This linear system is controlled with two inputs, which is optimal. This is done with a duality approach and some appropriate Carleman estimates. Then, by means of an inverse function theorem, the local null controllability of the nonlinear system is proven.

Mots clés: Korteweg-de Vries equation; null controllability; Carleman estimates

Domaines: Optimisation et contrôle [math.OC]

Nicolás Carreño, Eduardo Cerpa, Emmanuelle Crépeau. Internal null controllability of the generalized Hirota-Satsuma system. 2019. ⟨hal-02060951v1⟩