# American Institute of Mathematical Sciences

September  2007, 8(2): 279-314. doi: 10.3934/dcdsb.2007.8.279

## Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller

 1 Department of Mathematics, University of Virginia, P.O. Box 400137, Charlottesville, VA 22904

Received  January 2006 Revised  June 2006 Published  June 2007

Consider a 2-D, linearized Navier-Stokes channel flow with periodic boundary conditions in the streamwise direction and subject to a wall-normal control on the top wall. There exists an infinite-dimensional subspace $E^0$, where the normal component $v$ of the velocity vector, as well as the vorticity $\omega$, are not influenced by the control. The corresponding control-free dynamics for $v$ and $\omega$ on $E^0$ are inherently exponentially stable, though with limited decay rate. In the case of the linear 2-D channel, the stability margin of the component $v$ on the complementary space $Z$ can be enhanced by a prescribed decay rate, by means of an explicit, 2-D wall-normal controller acting on the top wall, whose space component is subject to algebraic rank conditions. Moreover, its support may be arbitrarily small. Corresponding optimal decays, by the same 2-D wall-normal controller, of the tangential component $u$ of the velocity vector; of the pressure $p$; and of the vorticity $\omega$ over $Z$ are also obtained, to complete the optimal analysis.
Citation: Roberto Triggiani. Stability enhancement of a 2-D linear Navier-Stokes channel flow by a 2-D, wall-normal boundary controller. Discrete and Continuous Dynamical Systems - B, 2007, 8 (2) : 279-314. doi: 10.3934/dcdsb.2007.8.279
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