On the controllability of the BBM equation

Melek Jellouli

doi:10.3934/mcrf.2022002

Article Contents

2023, Volume 13, Issue 1: 415-430. Doi: 10.3934/mcrf.2022002

This issue Previous Article Constructive exact control of semilinear 1D heat equations Next Article Optimal control of parameterized stationary Maxwell's system: Reduced basis, convergence analysis, and a posteriori error estimates

On the controllability of the BBM equation

Melek Jellouli^1,2, ,

1.
Laboratoire de Mathématiques de Versailles, Université Paris-Saclay, UVSQ, CNRS, 78000 Versailles, France
2.
Laboratoire de Mathématiques: Modélisation Déterministe et Aléatoire, Université de Sousse, 4011 Hammam Sousse, Tunisie

^*Corresponding author: Melek Jellouli
^*Corresponding author: Melek Jellouli

Received: June 2021

Revised: November 2021

Early access: January 2022

Published: March 2023

Abstract Full Text(HTML) Related Papers Cited by

Abstract

In this article, we consider the nonlinear BBM equation on the torus. We use controls taking values in a finite dimensional space to show that the equation is approximately controllable in $ H^1(\mathbb{T}) $. We also show that the equation is not exactly controllable in $ H^s(\mathbb{T}) $ for $ s\in[1,2[ $.

Keywords:

Mathematics Subject Classification: Primary: 93B05; Secondary: 37L50.

Citation:

Full Text(HTML)

Related Papers

Cited by

References

[1]	A. A. Agrachev and A. V. Sarychev, Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing, Comm. Math. Phys., 265 (2006), 673-697. doi: 10.1007/s00220-006-0002-8.
[2]	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. doi: 10.1007/s00021-004-0110-1.
[3]	T. B. Benjamin, J. L. Bona and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London Ser. A, 272 (1972), 47-78. doi: 10.1098/rsta.1972.0032.
[4]	J. L. Bona and N. Tzvetkov, Sharp well-posedness results for the BBM equation, Discrete Contin. Dyn. Syst., 23 (2009), 1241-1252. doi: 10.3934/dcds.2009.23.1241.
[5]	D. E. Edmunds and H. Triebel, Function Spaces, Entropy Numbers, Differential Operators, Cambridge Tracts in Mathematics, 120, Cambridge University Press, Cambridge, 1996. doi: 10.1017/CBO9780511662201.
[6]	E. Giusti, Minimal Surfaces and Functions of Bounded Variation, Monographs in Mathematics, 80, Birkhäuser Verlag, Basel, 1984. doi: 10.1007/978-1-4684-9486-0.
[7]	H. Leiva, Controllability of the impulsive functional BBM equation with nonlinear term involving spatial derivative, Systems Control Lett., 109 (2017), 12-16. doi: 10.1016/j.sysconle.2017.09.001.
[8]	G. G. Lorentz, Approximation of Functions, 2$^{nd}$ edition, Chelsea Publishing Co., New York, 1986.
[9]	S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control Optim., 39 (2001), 1677-1696. doi: 10.1137/S0363012999362499.
[10]	H. Nersisyan, Controllability of 3D incompressible Euler equations by a finite-dimensional external force, ESAIM Control Optim. Calc. Var., 16 (2010), 677-694. doi: 10.1051/cocv/2009017.
[11]	H. Nersisyan, Controllability of the 3D compressible Euler system, Comm. Partial Differential Equations, 36 (2011), 1544-1564. doi: 10.1080/03605302.2011.596605.
[12]	V. Nersesyan, Approximate controllability of Lagrangian trajectories of the 3D Navier–Stokes system by a finite-dimensional force, Nonlinearity, 28 (2015), 825-848. doi: 10.1088/0951-7715/28/3/825.
[13]	V. Nersesyan, Approximate controllability of nonlinear parabolic PDEs in arbitrary space dimension, Math. Control Relat. Fields, 11 (2021), 237-251. doi: 10.3934/mcrf.2020035.
[14]	V. Nersesyan and H. Nersisyan, Global exact controllability in infinite time of Shrödinger equation, J. Math. Pures Appl., 97 (2012), 295-317. doi: 10.1016/j.matpur.2011.11.005.
[15]	M. Panthee, On the ill-posedness result for the BBM equation, Discrete Contin. Dyn. Syst., 30 (2011), 253-259. doi: 10.3934/dcds.2011.30.253.
[16]	L. Rosier and B.-Y. Zhang, Unique continuation property and control for the Benjamin–Bona–Mahony equation on a periodic domain, J. Differential Equations, 254 (2013), 141-178. doi: 10.1016/j.jde.2012.08.014.
[17]	A. Shirikyan, Approximate controllability of the viscous Burgers equation on the real line, in Geometric Control Theory and Sub-Riemannian Geometry, Springer INdAM Ser., 5, Springer, Cham, 2014,351–370. doi: 10.1007/978-3-319-02132-4_20.
[18]	A. Shirikyan, Approximate controllability of three-dimensional Navier–Stokes equations, Comm. Math. Phys., 266 (2006), 123-151. doi: 10.1007/s00220-006-0007-3.
[19]	A. Shirikyan, Control theory for the Burgers equation: Agrachev-Sarychev approach, Pure Appl. Funct. Anal., 3 (2018), 219-240.
[20]	A. Shirikyan, Euler equations are not exactly controllable by a finite-dimensional external force, Phys. D, 237 (2008), 1317-1323. doi: 10.1016/j.physd.2008.03.021.
[21]	A. Shirikyan, Exact controllability in projections for three-dimensional Navier–Stokes equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 521-537. doi: 10.1016/j.anihpc.2006.04.002.
[22]	X. Zhang and E. Zuazua, Unique continuation for the linearized Benjamin–Bona–Mahony equation with space-dependent potential, Math. Ann., 325 (2003), 543-582. doi: 10.1007/s00208-002-0391-8.