Journal of Dynamical and Control Systems

, Volume 12, Issue 4, pp 517-562

Navier-Stokes Equation on the Rectangle: Controllability by Means of Low Mode Forcing

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We study controllability issues for the Navier-Stokes equation on a two-dimensional rectangle under so-called Lions boundary conditions. The Navier-Stokes equation is controlled by forcing applied to a small number of harmonic modes. Methods of Geometric/Lie Algebraic Control Theory are used to prove controllability by means of low mode forcing of finite-dimensional Galerkin approximations of this system. Proving the continuity of the “control ↦ solution” mapping in the so-called relaxation metric we use it to prove both solid controllability on the observed component and L 2-approximate controllability of the Navier-Stokes equation (full system) by low mode forcing.

Key words and phrases.

Incompressible fluid two-dimensional Navier-Stokes system controllability