Feedback boundary stabilization of the three-dimensional incompressible Navier–Stokes equations
By Jean-Pierre Raymond. 2007
111 citations recorded by the source corpus.
We study the local stabilization of the three-dimensional Navier–Stokes equations around an unstable stationary solution w, by means of a feedback boundary control. We first determine a feedback law for the linearized system around w. Next, we show that this feedback provides a local stabilization of the Navier–Stokes equations. To deal with the nonlinear term, the solutions to the closed loop system must be in H3/2+e,3/4+e/2(Q), with 0<e. In [V. Barbu, I. Lasiecka, R. Triggiani, Boundary stabilization of Navier–Stokes equations, Mem. Amer. Math. Soc. 852 (2006); V. Barbu, I. Lasiecka, R. Triggiani, Abstract settings for tangential boundary stabilization of Navier–Stokes equations by high- and low-gain feedback controllers, Nonlinear Anal. 64 (2006) 2704–2746], such a regularity is achieved with a feedback obtained by minimizing a functional involving a norm of the state variable strong enough. In that case, the feedback controller cannot be determined by a well posed Riccati equation. Here, we choose a functional involving a very weak norm of the state variable. The compatibility condition between the initial state and the feedback controller at t=0, is achieved by choosing a time varying control operator in a neighbourhood of t=0.
Subfields: Applied Mathematics, Computational Theory and Mathematics, Control and Systems Engineering
View Paper: https://www.semanticscholar.org/paper/d14fbe54e5fe8a34102fe7de4871ff3c8c1060ea
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In Applied Mathematics
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Citations in the corpus
- Feedback stabilization of the 2-D and 3-D Navier-Stokes equations based on an extended system (2009, background): “This explains why the feedback law which is given in [21] cannot be used in the three dimensional case, and why the author overcomes this difficulty in [22] by introducing a feedback law which is time dependent in an initial transitory time interval.”