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]

Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743

[2]

Vena Pearl Bongolan-walsh, David Cheban, Jinqiao Duan. Recurrent motions in the nonautonomous Navier-Stokes system. Discrete & Continuous Dynamical Systems - B, 2003, 3 (2) : 255-262. doi: 10.3934/dcdsb.2003.3.255

[3]

Mehdi Badra. Abstract settings for stabilization of nonlinear parabolic system with a Riccati-based strategy. Application to Navier-Stokes and Boussinesq equations with Neumann or Dirichlet control. Discrete & Continuous Dynamical Systems - A, 2012, 32 (4) : 1169-1208. doi: 10.3934/dcds.2012.32.1169

[4]

Viorel Barbu, Ionuţ Munteanu. Internal stabilization of Navier-Stokes equation with exact controllability on spaces with finite codimension. Evolution Equations & Control Theory, 2012, 1 (1) : 1-16. doi: 10.3934/eect.2012.1.1

[5]

Hammadi Abidi, Taoufik Hmidi, Sahbi Keraani. On the global regularity of axisymmetric Navier-Stokes-Boussinesq system. Discrete & Continuous Dynamical Systems - A, 2011, 29 (3) : 737-756. doi: 10.3934/dcds.2011.29.737

[6]

Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311

[7]

Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665

[8]

Grzegorz Karch, Xiaoxin Zheng. Time-dependent singularities in the Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2015, 35 (7) : 3039-3057. doi: 10.3934/dcds.2015.35.3039

[9]

Igor Kukavica. Interior gradient bounds for the 2D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2001, 7 (4) : 873-882. doi: 10.3934/dcds.2001.7.873

[10]

Atanas Stefanov. On the Lipschitzness of the solution map for the 2 D Navier-Stokes system. Discrete & Continuous Dynamical Systems - A, 2010, 26 (4) : 1471-1490. doi: 10.3934/dcds.2010.26.1471

[11]

Mehdi Badra. Global Carleman inequalities for Stokes and penalized Stokes equations. Mathematical Control & Related Fields, 2011, 1 (2) : 149-175. doi: 10.3934/mcrf.2011.1.149

[12]

Stanislaw Migórski, Anna Ochal. Navier-Stokes problems modeled by evolution hemivariational inequalities. Conference Publications, 2007, 2007 (Special) : 731-740. doi: 10.3934/proc.2007.2007.731

[13]

A. V. Fursikov. Stabilization for the 3D Navier-Stokes system by feedback boundary control. Discrete & Continuous Dynamical Systems - A, 2004, 10 (1&2) : 289-314. doi: 10.3934/dcds.2004.10.289

[14]

Yinghua Li, Shijin Ding, Mingxia Huang. Blow-up criterion for an incompressible Navier-Stokes/Allen-Cahn system with different densities. Discrete & Continuous Dynamical Systems - B, 2016, 21 (5) : 1507-1523. doi: 10.3934/dcdsb.2016009

[15]

Pedro Marín-Rubio, Antonio M. Márquez-Durán, José Real. Three dimensional system of globally modified Navier-Stokes equations with infinite delays. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 655-673. doi: 10.3934/dcdsb.2010.14.655

[16]

Kuanysh A. Bekmaganbetov, Gregory A. Chechkin, Vladimir V. Chepyzhov, Andrey Yu. Goritsky. Homogenization of trajectory attractors of 3D Navier-Stokes system with randomly oscillating force. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2375-2393. doi: 10.3934/dcds.2017103

[17]

Donatella Donatelli, Eduard Feireisl, Antonín Novotný. On incompressible limits for the Navier-Stokes system on unbounded domains under slip boundary conditions. Discrete & Continuous Dynamical Systems - B, 2010, 13 (4) : 783-798. doi: 10.3934/dcdsb.2010.13.783

[18]

Lucas C. F. Ferreira, Elder J. Villamizar-Roa. On the existence of solutions for the Navier-Stokes system in a sum of weak-$L^{p}$ spaces. Discrete & Continuous Dynamical Systems - A, 2010, 27 (1) : 171-183. doi: 10.3934/dcds.2010.27.171

[19]

Debanjana Mitra, Mythily Ramaswamy, Jean-Pierre Raymond. Largest space for the stabilizability of the linearized compressible Navier-Stokes system in one dimension. Mathematical Control & Related Fields, 2015, 5 (2) : 259-290. doi: 10.3934/mcrf.2015.5.259

[20]

Reinhard Farwig, Paul Felix Riechwald. Regularity criteria for weak solutions of the Navier-Stokes system in general unbounded domains. Discrete & Continuous Dynamical Systems - S, 2016, 9 (1) : 157-172. doi: 10.3934/dcdss.2016.9.157

2018 Impact Factor: 1.292

Metrics

  • PDF downloads (5)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]