Internal null controllability of the generalized Hirota-Satsuma system

Carreño, Nicolás; Cerpa, Eduardo; Crépeau, Emmanuelle

ESAIM: Control, Optimisation and Calculus of Variations, Tome 26 (2020), article no. 75.

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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.

Classification : 35Q35, 93B05, 93C20, 93C10
Mots-clés : Korteweg-de Vries equation, null controllability, Carleman estimates

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