Navier-Stokes Equation on the Rectangle

Details Navier-Stokes Equation on the Rectangle Navier-Stokes Equation on the Rectangle

2005-04-15

arxiv.org/abs/math/0504323v1

J. Dyn. Cont. Systems, vol 12, no 4, 2006 (517-562)

Mathematics

Optimization and Control

Scientific paper

We study controllability issues for the Navier-Stokes Equation on a two dimensional rectangle with so-called Lions boundary conditions. Rewriting the Equation using a basis of harmonic functions we arrive to an infinite-dimensional system of ODEs. Methods of Geometric/Lie Algebraic Control Theory are used to prove controllability by means of low modes forcing of finite-dimensional Galerkin approximations of that system. Proving the continuity of the ``control $\mapsto$ solution'' map in the so-called relaxation metric we use it to prove both solid controllability on observed component and $\mathbf L^2$-approximate controllability of the Equation (full system) by low modes forcing.

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