Global Carleman inequalities for the Oseen system. Application to the cost control in small times
Résumé
We consider the local exact controllability to trajectories of the Navier-Stokes system with distributed controls. Such a property was already obtained in previous works but here we improve the "cost" of the control by refining the observability inequality associated with the Oseen system. More precisely, our main result corresponds to a Carleman inequality for the Oseen system with weight functions similar to the ones for the heat equation. In our proof, we need in particular to estimate precisely a pressure term on the boundary and this is done by treating differently the low-frequency part and the high-frequency part of the solutions. For the low-frequency part of the solution, we can obtain a Carleman estimate without any boundary condition by using standard methods of micro-local analysis whereas the high-frequency part of the solution can be estimated by using energy inequality of the Stokes system.
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