Controllability of 2D Euler and Navier-Stokes Equations by Forcing 4 Modes
Agrachev, Andrey ; Sarychev, Andrey
arXiv, 0507365 / Harvested from arXiv
Text on arXiv
We study controllability issues for the 2D Euler and Navier-Stokes (NS) systems under periodic boundary conditions. These systems describe motion of homogeneous ideal or viscous incompressible fluid on a two-dimensional torus $\mathbb{T}^2$. We assume the system to be controlled by a degenerate forcing applied to fixed number of modes. In our previous work \cite{ASpb,AS43,ASDAN} we studied global controllability by means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing viscosity ($\nu >0$). Methods of differential geometric/Lie algebraic control theory have been used for that study. In the present contribution we improve and extend the controllability results in several aspects: 1) we obtain a stronger sufficient condition for controllability of 2D NS system in an observed component and for $L_2$-approximate controllability; 2) we prove that these criteria are valid for the case of ideal incompressible fluid ($\nu=0)$; 3) we study solid controllability in projection on any finite-dimensional subspace and establish a sufficient criterion for such controllability.
Publié le : 2005-07-18
Classification:  Mathematics - Optimization and Control,  Mathematical Physics,  35Q30, 93C20, 93B05, 93B29 
@article{0507365,
     author = {Agrachev, Andrey and Sarychev, Andrey},
     title = {Controllability of 2D Euler and Navier-Stokes Equations by Forcing 4
  Modes},
     journal = {arXiv},
     volume = {2005},
     number = {0},
     year = {2005},
     language = {en},
     url = {http://dml.mathdoc.fr/item/0507365}
}

Agrachev, Andrey; Sarychev, Andrey. Controllability of 2D Euler and Navier-Stokes Equations by Forcing 4
  Modes. arXiv, Tome 2005 (2005) no. 0, . http://gdmltest.u-ga.fr/item/0507365/