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doi: 10.3934/mcrf.2021038
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Local null controllability of the penalized Boussinesq system with a reduced number of controls

1. 

Departamento de Matemáticas, Universidad Autónoma de Madrid, 28049 Madrid, Spain

2. 

Chair of Computational Mathematics, Fundación Deusto, Avenida de las Universidades 24, 48007 Bilbao, Basque Country, Spain

3. 

Institut de Mathématiques de Bordeaux, 351 Cours de la Libération, 33400 Bordeaux, France

* Corresponding author: Jon Asier Bárcena-Petisco

Received  August 2020 Revised  June 2021 Early access July 2021

In this paper we consider the Boussinesq system with homogeneous Dirichlet boundary conditions, defined in a regular domain $ \Omega\subset\mathbb R^N $ for $ N = 2 $ and $ N = 3 $. The incompressibility condition of the fluid is replaced by its approximation by penalization with a small parameter $ \varepsilon > 0 $. We prove that our system is locally null controllable using a control with a restricted number of components, localized in an open set $ \omega $ contained in $ \Omega $. We also show that the control cost is bounded uniformly with respect to $ \varepsilon \rightarrow 0 $. The proof is based on a linearization argument. The null controllability of the linearized system is obtained by proving a new Carleman estimate for the adjoint system. This inequality is derived by exploiting the coercivity of some second order differential operator involving crossed derivatives.

Citation: Jon Asier Bárcena-Petisco, Kévin Le Balc'h. Local null controllability of the penalized Boussinesq system with a reduced number of controls. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021038
References:
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F. Ammar-KhodjaA. BenabdallahM. González Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[2]

M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations, Math. Control Relat. Fields, 1 (2011), 149-175.  doi: 10.3934/mcrf.2011.1.149.

[3]

J. A. Bárcena-Petisco, Null controllability of a penalized Stokes problem in dimension two with one scalar control, Asymptot. Anal., 117 (2020), 161-198.  doi: 10.3233/ASY-191550.

[4]

J. A. Bárcena-Petisco, Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit, SIAM J. Control Optim., 58 (2020), 1597-1625.  doi: 10.1137/19M1252004.

[5]

K. Beauchard and F. Marbach, Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations, J. Math. Pures Appl. (9), 136 (2020), 22-91.  doi: 10.1016/j.matpur.2020.02.001.

[6]

M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO Anal. Numér., 12 (1978), 211-236.  doi: 10.1051/m2an/1978120302111.

[7]

N. Carreño, Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain, Math. Control Relat. Fields, 2 (2012), 361-382.  doi: 10.3934/mcrf.2012.2.361.

[8]

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, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.

[9]

N. CarreñoS. Guerrero and M. Gueye, Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system, ESAIM Control Optim. Calc. Var., 21 (2015), 73-100.  doi: 10.1051/cocv/2014020.

[10]

F. W. Chaves-Silva, E. Fernández-Cara, K. Le Balc'h, J. L. F. Machado and D. A. Souza, Small-time global exact controllability to the trajectories for the viscous Boussinesq system, preprint, arXiv: 2006.01682.

[11]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[12]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.

[13]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.

[14]

J.-M. CoronF. Marbach and F. Sueur, Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc. (JEMS), 22 (2020), 1625-1673.  doi: 10.4171/JEMS/952.

[15]

M. Duprez and P. Lissy, Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control, preprint, arXiv: 1909.02831.

[16]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[17]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.

[18]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier–Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.

[19]

E. Fernádez-CaraJ. Limaco and S. B. de Menezes, Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control, ESAIM Control Optim. Calc. Var., 22 (2016), 1017-1039.  doi: 10.1051/cocv/2016031.

[20]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[21]

B. Geshkovski and E. Zuazua, Controllability of one-dimensional viscous free boundary flows, SIAM J. Control Optim., 59 (2021), 1830-1850.  doi: 10.1137/19M1285354.

[22]

S. Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029-1054.  doi: 10.1016/j.anihpc.2006.11.001.

[23]

S. Guerrero and C. Montoya, Local null controllability of the $N$-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and $N-1$ scalar controls, J. Math. Pures Appl. (9), 113 (2018), 37-69.  doi: 10.1016/j.matpur.2018.03.004.

[24]

V. Hernández-Santamaría and K. Le Balc'h, Local null-controllability of a nonlocal semilinear heat equation, Appl. Math. Optim., (2020), 1–49.

[25]

O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[26]

O. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.

[27]

K. Le Balc'h, Local controllability of reaction-diffusion systems around nonnegative stationary states, ESAIM Control Optim. Calc. Var., 26 (2020), 32pp. doi: 10.1051/cocv/2019033.

[28]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988.

[29]

J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, 1996,221–235.

[30]

Y. LiuT. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19 (2013), 20-42.  doi: 10.1051/cocv/2011196.

[31]

S. Micu and T. Takahashi, Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity, J. Differential Equations, 264 (2018), 3664-3703.  doi: 10.1016/j.jde.2017.11.029.

[32]

J. T. Oden and O.-P. Jacquotte, Stability of some mixed finite element methods for Stokesian flows, Comput. Methods Appl. Mech. Engrg., 43 (1984), 231-247.  doi: 10.1016/0045-7825(84)90007-0.

[33]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[34]

J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations, in Proceedings of the 1994 Beijing Symposium on Nonlinear Evolution Equations and Infinite Dynamical Systems, 1997, 68–78. Available from: https://www.math.purdue.edu/shen7/pub/Pseudo-c.pdf.

[35]

T. Takahashi, Boundary local null-controllability of the Kuramoto–Sivashinsky equation, Math. Control Signals Systems, 29 (2017), 21pp. doi: 10.1007/s00498-016-0182-5.

[36]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[37]

R. Temam, Une méthode d'approximation de la solution des équations des Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152.  doi: 10.24033/bsmf.1662.

[38]

E. Zuazua, A uniqueness result for the linear system of elasticity and its control theoretical consequences, SIAM J. Control Optim., 34 (1996), 1473-1495.  doi: 10.1137/S0363012993260070.

show all references

References:
[1]

F. Ammar-KhodjaA. BenabdallahM. González Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[2]

M. Badra, Global Carleman inequalities for Stokes and penalized Stokes equations, Math. Control Relat. Fields, 1 (2011), 149-175.  doi: 10.3934/mcrf.2011.1.149.

[3]

J. A. Bárcena-Petisco, Null controllability of a penalized Stokes problem in dimension two with one scalar control, Asymptot. Anal., 117 (2020), 161-198.  doi: 10.3233/ASY-191550.

[4]

J. A. Bárcena-Petisco, Uniform controllability of a Stokes problem with a transport term in the zero-diffusion limit, SIAM J. Control Optim., 58 (2020), 1597-1625.  doi: 10.1137/19M1252004.

[5]

K. Beauchard and F. Marbach, Unexpected quadratic behaviors for the small-time local null controllability of scalar-input parabolic equations, J. Math. Pures Appl. (9), 136 (2020), 22-91.  doi: 10.1016/j.matpur.2020.02.001.

[6]

M. Bercovier, Perturbation of mixed variational problems. Application to mixed finite element methods, RAIRO Anal. Numér., 12 (1978), 211-236.  doi: 10.1051/m2an/1978120302111.

[7]

N. Carreño, Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain, Math. Control Relat. Fields, 2 (2012), 361-382.  doi: 10.3934/mcrf.2012.2.361.

[8]

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, J. Math. Fluid Mech., 15 (2013), 139-153.  doi: 10.1007/s00021-012-0093-2.

[9]

N. CarreñoS. Guerrero and M. Gueye, Insensitizing controls with two vanishing components for the three-dimensional Boussinesq system, ESAIM Control Optim. Calc. Var., 21 (2015), 73-100.  doi: 10.1051/cocv/2014020.

[10]

F. W. Chaves-Silva, E. Fernández-Cara, K. Le Balc'h, J. L. F. Machado and D. A. Souza, Small-time global exact controllability to the trajectories for the viscous Boussinesq system, preprint, arXiv: 2006.01682.

[11]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.

[12]

J.-M. Coron and S. Guerrero, Null controllability of the $N$-dimensional Stokes system with $N-1$ scalar controls, J. Differential Equations, 246 (2009), 2908-2921.  doi: 10.1016/j.jde.2008.10.019.

[13]

J.-M. Coron and P. Lissy, Local null controllability of the three-dimensional Navier-Stokes system with a distributed control having two vanishing components, Invent. Math., 198 (2014), 833-880.  doi: 10.1007/s00222-014-0512-5.

[14]

J.-M. CoronF. Marbach and F. Sueur, Small-time global exact controllability of the Navier-Stokes equation with Navier slip-with-friction boundary conditions, J. Eur. Math. Soc. (JEMS), 22 (2020), 1625-1673.  doi: 10.4171/JEMS/952.

[15]

M. Duprez and P. Lissy, Bilinear local controllability to the trajectories of the Fokker-Planck equation with a localized control, preprint, arXiv: 1909.02831.

[16]

L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[17]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Null controllability of the heat equation with boundary Fourier conditions: The linear case, ESAIM Control Optim. Calc. Var., 12 (2006), 442-465.  doi: 10.1051/cocv:2006010.

[18]

E. Fernández-CaraS. GuerreroO. Y. Imanuvilov and J.-P. Puel, Some controllability results for the $N$-dimensional Navier–Stokes and Boussinesq systems with $N-1$ scalar controls, SIAM J. Control Optim., 45 (2006), 146-173.  doi: 10.1137/04061965X.

[19]

E. Fernádez-CaraJ. Limaco and S. B. de Menezes, Controlling linear and semilinear systems formed by one elliptic and two parabolic PDEs with one scalar control, ESAIM Control Optim. Calc. Var., 22 (2016), 1017-1039.  doi: 10.1051/cocv/2016031.

[20]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[21]

B. Geshkovski and E. Zuazua, Controllability of one-dimensional viscous free boundary flows, SIAM J. Control Optim., 59 (2021), 1830-1850.  doi: 10.1137/19M1285354.

[22]

S. Guerrero, Controllability of systems of Stokes equations with one control force: Existence of insensitizing controls, Ann. Inst. H. Poincaré Anal. Non Linéaire, 24 (2007), 1029-1054.  doi: 10.1016/j.anihpc.2006.11.001.

[23]

S. Guerrero and C. Montoya, Local null controllability of the $N$-dimensional Navier–Stokes system with nonlinear Navier-slip boundary conditions and $N-1$ scalar controls, J. Math. Pures Appl. (9), 113 (2018), 37-69.  doi: 10.1016/j.matpur.2018.03.004.

[24]

V. Hernández-Santamaría and K. Le Balc'h, Local null-controllability of a nonlocal semilinear heat equation, Appl. Math. Optim., (2020), 1–49.

[25]

O. Y. Imanuvilov, Remarks on exact controllability for the Navier-Stokes equations, ESAIM Control Optim. Calc. Var., 6 (2001), 39-72.  doi: 10.1051/cocv:2001103.

[26]

O. Y. ImanuvilovJ.-P. Puel and M. Yamamoto, Carleman estimates for parabolic equations with nonhomogeneous boundary conditions, Chin. Ann. Math. Ser. B, 30 (2009), 333-378.  doi: 10.1007/s11401-008-0280-x.

[27]

K. Le Balc'h, Local controllability of reaction-diffusion systems around nonnegative stationary states, ESAIM Control Optim. Calc. Var., 26 (2020), 32pp. doi: 10.1051/cocv/2019033.

[28]

J.-L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systemes Distribués. Tome 1, Recherches en Mathématiques Appliquées, 8, Masson, Paris, 1988.

[29]

J.-L. Lions and E. Zuazua, A generic uniqueness result for the Stokes system and its control theoretical consequences, in Partial Differential Equations and Applications, Lecture Notes in Pure and Appl. Math., 177, Dekker, New York, 1996,221–235.

[30]

Y. LiuT. Takahashi and M. Tucsnak, Single input controllability of a simplified fluid-structure interaction model, ESAIM Control Optim. Calc. Var., 19 (2013), 20-42.  doi: 10.1051/cocv/2011196.

[31]

S. Micu and T. Takahashi, Local controllability to stationary trajectories of a Burgers equation with nonlocal viscosity, J. Differential Equations, 264 (2018), 3664-3703.  doi: 10.1016/j.jde.2017.11.029.

[32]

J. T. Oden and O.-P. Jacquotte, Stability of some mixed finite element methods for Stokesian flows, Comput. Methods Appl. Mech. Engrg., 43 (1984), 231-247.  doi: 10.1016/0045-7825(84)90007-0.

[33]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[34]

J. Shen, Pseudo-compressibility methods for the unsteady incompressible Navier-Stokes equations, in Proceedings of the 1994 Beijing Symposium on Nonlinear Evolution Equations and Infinite Dynamical Systems, 1997, 68–78. Available from: https://www.math.purdue.edu/shen7/pub/Pseudo-c.pdf.

[35]

T. Takahashi, Boundary local null-controllability of the Kuramoto–Sivashinsky equation, Math. Control Signals Systems, 29 (2017), 21pp. doi: 10.1007/s00498-016-0182-5.

[36]

R. Temam, Navier-Stokes Equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, 2, North-Holland Publishing Co., Amsterdam-New York-Oxford, 1977.

[37]

R. Temam, Une méthode d'approximation de la solution des équations des Navier-Stokes, Bull. Soc. Math. France, 96 (1968), 115-152.  doi: 10.24033/bsmf.1662.

[38]

E. Zuazua, A uniqueness result for the linear system of elasticity and its control theoretical consequences, SIAM J. Control Optim., 34 (1996), 1473-1495.  doi: 10.1137/S0363012993260070.

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