December  2012, 2(4): 361-382. doi: 10.3934/mcrf.2012.2.361

Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain

1. 

Université Pierre et Marie Curie, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France

Received  March 2012 Revised  April 2012 Published  October 2012

In this paper we deal with the local exact controllability to a particular class of trajectories of the $N$-dimensional Boussinesq system with internal controls having $2$ vanishing components. The main novelty of this work is that no condition is imposed on the control domain.
Citation: Nicolás Carreño. Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain. Mathematical Control & Related Fields, 2012, 2 (4) : 361-382. doi: 10.3934/mcrf.2012.2.361
References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control,'', Translated from the Russian by V. M. Volosov, (1987).   Google Scholar

[2]

N. Carreño and S. Guerrero, Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain,, to appear in Journal of Mathematical Fluid Mechanics, ().   Google Scholar

[3]

J.-M. Coron and S. Guerrero, Null controllability of the N-dimensional Stokes system with N-1 scalar controls,, J. Differential Equations, 246 (2009), 2908.  doi: 10.1016/j.jde.2008.10.019.  Google Scholar

[4]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system,, J. Math. Pures Appl., 83 (2004), 1501.   Google Scholar

[5]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Some controllability results for the N-dimensional Navier-Stokes system and Boussinesq systems with N-1 scalar controls,, SIAM J. Control Optim., 45 (2006), 146.  doi: 10.1137/04061965X.  Google Scholar

[6]

A. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,'', Lecture Notes 34, (1996).   Google Scholar

[7]

A. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the Boussinesq equation,, SIAM J. Control Optim., 36 (1998), 391.  doi: 10.1137/S0363012996296796.  Google Scholar

[8]

A. Fursikov and O. Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations,, Russian Math. Surveys, 54 (1999), 565.  doi: 10.1070/RM1999v054n03ABEH000153.  Google Scholar

[9]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. I. H. Poincaré, 23 (2006), 29.   Google Scholar

[10]

O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 6 (2001), 39.   Google Scholar

[11]

O. Yu. Imanuvilov, J.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions,, Chin. Ann. Math B., 30 (2009), 333.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

[12]

O. A. Ladyzenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Revised English Edition, (1963).   Google Scholar

[13]

J.-L. Lions and E. Magenes, "Problèmes aux Limites non Homogènes et Applications,'', Volume 2, (1968).   Google Scholar

[14]

R. Temam, "Navier-Stokes Equations: Theory ans Numerical Analysis,'', Stud. Math. Appl., (1977).   Google Scholar

show all references

References:
[1]

V. M. Alekseev, V. M. Tikhomirov and S. V. Fomin, "Optimal Control,'', Translated from the Russian by V. M. Volosov, (1987).   Google Scholar

[2]

N. Carreño and S. Guerrero, Local null controllability of the N-dimensional Navier-Stokes system with N-1 scalar controls in an arbitrary control domain,, to appear in Journal of Mathematical Fluid Mechanics, ().   Google Scholar

[3]

J.-M. Coron and S. Guerrero, Null controllability of the N-dimensional Stokes system with N-1 scalar controls,, J. Differential Equations, 246 (2009), 2908.  doi: 10.1016/j.jde.2008.10.019.  Google Scholar

[4]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Local exact controllability of the Navier-Stokes system,, J. Math. Pures Appl., 83 (2004), 1501.   Google Scholar

[5]

E. Fernández-Cara, S. Guerrero, O. Yu. Imanuvilov and J.-P. Puel, Some controllability results for the N-dimensional Navier-Stokes system and Boussinesq systems with N-1 scalar controls,, SIAM J. Control Optim., 45 (2006), 146.  doi: 10.1137/04061965X.  Google Scholar

[6]

A. Fursikov and O. Yu. Imanuvilov, "Controllability of Evolution Equations,'', Lecture Notes 34, (1996).   Google Scholar

[7]

A. Fursikov and O. Yu. Imanuvilov, Local exact boundary controllability of the Boussinesq equation,, SIAM J. Control Optim., 36 (1998), 391.  doi: 10.1137/S0363012996296796.  Google Scholar

[8]

A. Fursikov and O. Yu. Imanuvilov, Exact controllability of the Navier-Stokes and Boussinesq equations,, Russian Math. Surveys, 54 (1999), 565.  doi: 10.1070/RM1999v054n03ABEH000153.  Google Scholar

[9]

S. Guerrero, Local exact controllability to the trajectories of the Boussinesq system,, Ann. I. H. Poincaré, 23 (2006), 29.   Google Scholar

[10]

O. Yu. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations,, ESAIM Control Optim. Calc. Var., 6 (2001), 39.   Google Scholar

[11]

O. Yu. Imanuvilov, J.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions,, Chin. Ann. Math B., 30 (2009), 333.  doi: 10.1007/s11401-008-0280-x.  Google Scholar

[12]

O. A. Ladyzenskaya, "The Mathematical Theory of Viscous Incompressible Flow,'', Revised English Edition, (1963).   Google Scholar

[13]

J.-L. Lions and E. Magenes, "Problèmes aux Limites non Homogènes et Applications,'', Volume 2, (1968).   Google Scholar

[14]

R. Temam, "Navier-Stokes Equations: Theory ans Numerical Analysis,'', Stud. Math. Appl., (1977).   Google Scholar

[1]

Zongyuan Li, Weinan Wang. Norm inflation for the Boussinesq system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020353

[2]

Xuhui Peng, Rangrang Zhang. Approximations of stochastic 3D tamed Navier-Stokes equations. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5337-5365. doi: 10.3934/cpaa.2020241

[3]

Zhilei Liang, Jiangyu Shuai. Existence of strong solution for the Cauchy problem of fully compressible Navier-Stokes equations in two dimensions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020348

[4]

Leanne Dong. Random attractors for stochastic Navier-Stokes equation on a 2D rotating sphere with stable Lévy noise. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020352

[5]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[6]

Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049

[7]

Helmut Abels, Andreas Marquardt. On a linearized Mullins-Sekerka/Stokes system for two-phase flows. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020467

[8]

Neng Zhu, Zhengrong Liu, Fang Wang, Kun Zhao. Asymptotic dynamics of a system of conservation laws from chemotaxis. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 813-847. doi: 10.3934/dcds.2020301

[9]

Craig Cowan, Abdolrahman Razani. Singular solutions of a Lane-Emden system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 621-656. doi: 10.3934/dcds.2020291

[10]

Abdelghafour Atlas, Mostafa Bendahmane, Fahd Karami, Driss Meskine, Omar Oubbih. A nonlinear fractional reaction-diffusion system applied to image denoising and decomposition. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020321

[11]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[12]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, 2021, 20 (1) : 389-404. doi: 10.3934/cpaa.2020273

[13]

Yuxin Zhang. The spatially heterogeneous diffusive rabies model and its shadow system. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020357

[14]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[15]

Hao Wang. Uniform stability estimate for the Vlasov-Poisson-Boltzmann system. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 657-680. doi: 10.3934/dcds.2020292

[16]

Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002

[17]

Yichen Zhang, Meiqiang Feng. A coupled $ p $-Laplacian elliptic system: Existence, uniqueness and asymptotic behavior. Electronic Research Archive, 2020, 28 (4) : 1419-1438. doi: 10.3934/era.2020075

[18]

Youshan Tao, Michael Winkler. Critical mass for infinite-time blow-up in a haptotaxis system with nonlinear zero-order interaction. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 439-454. doi: 10.3934/dcds.2020216

[19]

Jianquan Li, Xin Xie, Dian Zhang, Jia Li, Xiaolin Lin. Qualitative analysis of a simple tumor-immune system with time delay of tumor action. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020341

[20]

Denis Bonheure, Silvia Cingolani, Simone Secchi. Concentration phenomena for the Schrödinger-Poisson system in $ \mathbb{R}^2 $. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020447

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (30)
  • HTML views (0)
  • Cited by (8)

Other articles
by authors

[Back to Top]