doi: 10.3934/mcrf.2022002
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On the controllability of the BBM equation

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

Received  June 2021 Revised  November 2021 Early access January 2022

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[ $.

Citation: Melek Jellouli. On the controllability of the BBM equation. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2022002
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. BenjaminJ. 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.

show all references

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. BenjaminJ. 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.

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