Stabilization to trajectories for equations of fluid mechanics

The project is concerned with the stabilization to trajectories for the equations of fluid mechanics. It will give contributions by establishing the stabilization to a nonstationary solution for the equations on the basis of controls given in feedback form, taking values on a finite-dimensional space and supported in a (possibly small) open subset of the boundary and, on the other side, it aims to establish new results concerning the space-dimension of the controls. The design of a stabilizing controller is important for applications, because such a controller plays a crucial role in the suppression of instabilities that can occur in the dynamics of a fluid. The feedback form of the controller and its finite-dimensional range make it robust and appropriate for applications.

In the course of this project, it has been shown that a control can be constructed which locally stabilizes the NavierStokes system and a semilinear parabolic-like equation to a given time dependent trajectory. This class of equations model some real world phenomena like fluid/gas dynamics, traffic flow, population dynamics, and temperature in a room. The control is localized either in the interior of the domain (room) or in its boundary (wall). Previous works were mostly concerned with stabilization to a time independent trajectory.For the first time, estimates were found on the number of actuators which are needed to stabilize the system to the given nonstationary target trajectory.The control can be taken in (dynamical) feedback form, which is a property highly demanded in applications, because feedback controllers can respond to small disturbances. Numerical simulations have been performed whose results suggest the applicability of the constructed internal and boundary controllers to real world problems.