Controllability of nonlinear PDE’s: Agrachev–Sarychev approach
Journées équations aux dérivées partielles (2007), article no. 4, 11 p.

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.

DOI : https://doi.org/10.5802/jedp.43
Classification:  35Q35,  93B05,  93C20
Keywords: Burgers equation, approximate controllability, exact controllability in projection, Agrachev–Sarychev method
@article{JEDP_2007____A4_0,
author = {Shirikyan, Armen},
title = {Controllability of nonlinear PDE's: Agrachev--Sarychev approach},
journal = {Journ\'ees \'equations aux d\'eriv\'ees partielles},
publisher = {Groupement de recherche 2434 du CNRS},
year = {2007},
doi = {10.5802/jedp.43},
language = {en},
url = {http://www.numdam.org/item/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://www.numdam.org/item/JEDP_2007____A4_0/

[AS04] A. A. Agrachev and Yu. L. Sachkov, Control Theory from Geometric Viewpoint, Springer-Verlag, Berlin, 2004. | MR 2062547

[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 2127744 | Zbl 1075.93014

[AS06] —, Controllability of 2D Euler and Navier–Stokes equations by degenerate forcing, Commun. Math. Phys. 265 (2006), no. 3, 673–697. | MR 2231685 | Zbl 1105.93008

[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 2459254

[Lio69] J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969. | Zbl 0189.40603

[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 2253360 | Zbl 1105.35085

[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 2231968 | Zbl 1105.93016

[Shi07] —, Exact controllability in projections for three-dimensional Navier-Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire 24 (2007), no. 4, 521–537. | Numdam | MR 2334990 | Zbl 1119.93021

[Tay97] M. E. Taylor, Partial Differential Equations. I–III, Springer-Verlag, New York, 1996-1997. | MR 1395148 | Zbl 0869.35003