Controllability to trajectories of a Ladyzhenskaya model for a viscous incompressible fluid

Sergio Guerrero (1) , Takéo Takahashi (2, 3)

1 LJLL (UMR_7598) - Laboratoire Jacques-Louis Lions
2 IECL - Institut Élie Cartan de Lorraine
3 SPHINX - Systems with physical heterogeneities : inverse problems, numerical simulation, control and stabilization

We consider the controllability of a viscous incompressible fluid modeled by the Navier-Stokes systemwith a nonlinear viscosity. To prove the controllability to trajectories, we linearize around a trajectory andthe corresponding linear system includes a nonlocal spatial term. Our main result is a Carleman estimatefor the adjoint of this linear system. This estimate yields in a standard way the null controllability of thelinear system and the local controllability to trajectories. Our method to obtain the Carleman estimate iscompletely general and can be adapted to other parabolic systems when a Carleman estimate is available.

Mots clés: Carleman estimates; controllability to trajectories; viscous incompressible fluid; nonlocal spatial terms

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

Sergio Guerrero, Takéo Takahashi. Controllability to trajectories of a Ladyzhenskaya model for a viscous incompressible fluid. Comptes Rendus. Mathématique, 2021. ⟨hal-03140709⟩