This short note is devoted to a discussion of a general approach to controllability of PDE’s introduced by Agrachev and Sarychev in 2005. We use the example of a 1D Burgers equation to illustrate the main ideas. It is proved that the problem in question is controllable in an appropriate sense by a two-dimensional external force. This result is not new and was proved earlier in the papers [AS05, AS07] in a more complicated situation of 2D Navier–Stokes equations.
Mots clés : Burgers equation, approximate controllability, exact controllability in projection, Agrachev–Sarychev method
@article{JEDP_2007____A4_0,
author = {Shirikyan, Armen},
title = {Controllability of nonlinear {PDE{\textquoteright}s:} {Agrachev{\textendash}Sarychev} approach},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
eid = {4},
pages = {1--11},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2007},
doi = {10.5802/jedp.43},
language = {en},
url = {http://archive.numdam.org/articles/10.5802/jedp.43/}
}
TY - JOUR AU - Shirikyan, Armen TI - Controllability of nonlinear PDE’s: Agrachev–Sarychev approach JO - Journées équations aux dérivées partielles PY - 2007 SP - 1 EP - 11 PB - Groupement de recherche 2434 du CNRS UR - http://archive.numdam.org/articles/10.5802/jedp.43/ DO - 10.5802/jedp.43 LA - en ID - JEDP_2007____A4_0 ER -
%0 Journal Article %A Shirikyan, Armen %T Controllability of nonlinear PDE’s: Agrachev–Sarychev approach %J Journées équations aux dérivées partielles %D 2007 %P 1-11 %I Groupement de recherche 2434 du CNRS %U http://archive.numdam.org/articles/10.5802/jedp.43/ %R 10.5802/jedp.43 %G en %F JEDP_2007____A4_0
Shirikyan, Armen. Controllability of nonlinear PDE’s: Agrachev–Sarychev approach. Journées équations aux dérivées partielles (2007), article no. 4, 11 p. doi : 10.5802/jedp.43. http://archive.numdam.org/articles/10.5802/jedp.43/
[AS04] A. A. Agrachev and Yu. L. Sachkov, Control Theory from Geometric Viewpoint, Springer-Verlag, Berlin, 2004. | MR
[AS05] A. A. Agrachev and A. V. Sarychev, Navier–Stokes equations: controllability by means of low modes forcing, J. Math. Fluid Mech. 7 (2005), 108–152. | MR | Zbl
[AS06] —, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys. 265 (2006), no. 3, 673–697. | MR | Zbl
[AS07] —, Solid controllability in fluid dynamics, Instabilities in Models Connected with Fluid Flow. I (C. Bardos and A. Fursikov, eds.), Springer, 2007, pp. 1–35. | MR
[Lio69] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. | Zbl
[Rod06] S. S. Rodrigues, Navier-Stokes equation on the rectangle: controllability by means of low mode forcing, J. Dyn. Control Syst. 12 (2006), no. 4, 517–562. | MR | Zbl
[Rod07] —, Controllability of nonlinear PDE’s on compact Riemannian manifolds, Workshop on Mathematical Control Theory and Finance, vol. Lisbon, 10–14 April, 2007, pp. 462–493.
[Shi06] A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Commun. Math. Phys. 266 (2006), no. 1, 123–151. | MR | Zbl
[Shi07] —, Exact controllability in projections for three-dimensional Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 4, 521–537. | EuDML | Numdam | MR | Zbl
[Tay97] M. E. Taylor, Partial Differential Equations. I–III, Springer-Verlag, New York, 1996-1997. | MR | Zbl
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