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Navier-Stokes Equations: Controllability by Means of Low Modes Forcing

Summary: Navier-Stokes Equations: Controllability by
Means of Low Modes Forcing
Andrey A. Agrachev

Andrey V. Sarychev

We study controllability issues for 2D and 3D Navier-Stokes (NS) sys-
tems with periodic boundary conditions. The systems are controlled by
a degenerate (applied to few low modes) forcing. Methods of dierential
geometric/Lie algebraic control theory are used to establish global control-
lability of nite-dimensional Galerkin approximations of 2D and 3D NS
and Euler systems, global controllability in nite-dimensional projection
of 2D NS system and L2-approximate controllability for 2D NS system.
Beyond these main goals we obtain results on boundedness and contin-
uous dependence of trajectories of 2D NS system on degenerate forcing,
when the space of forcings is endowed with so called relaxation metric.
Keywords: Navier-Stokes equation, controllability, geometric control
AMS Subject Classication: 35Q30, 93C20, 93B05, 93B29
1 Introduction


Source: Agrachev, Andrei - Functional Analysis Sector, Scuola Internazionale Superiore di Studi Avanzati (SISSA)


Collections: Engineering; Mathematics