Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations

Viorel Barbu (1) , Sergio S. Rodrigues (2) , Armen Shirikyan (2)

1 Romanian Academy [IASI]
2 AGM - UMR 8088 - Analyse, Géométrie et Modélisation

We consider the Navier-Stokes system in a bounded domain with a smooth boundary. Given a sufficiently regular time-dependent global solution, we construct a finite-dimensional feedback control that is supported by a given open set and stabilizes the linearized equation. The proof of this fact is based on a truncated observability inequality, the regularizing property for the linearized equation, and some standard techniques of the optimal control theory. We then show that the control constructed for the linear problem stabilizes locally also the full Navier-Stokes system.

Mots clés: Navier-Stokes system; exponential stabilization; feedback control

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

Viorel Barbu, Sergio S. Rodrigues, Armen Shirikyan. Internal exponential stabilization to a non-stationary solution for 3D Navier-Stokes equations. 2010. ⟨hal-00455250v2⟩