June  2020, 9(2): 535-559. doi: 10.3934/eect.2020023

Null controllability for a heat equation with dynamic boundary conditions and drift terms

Cadi Ayyad University, Faculty of Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC) B.P. 2390, Marrakesh, Morocco

* Corresponding author: Lahcen Maniar

Received  December 2018 Revised  September 2019 Published  June 2020 Early access  December 2019

We consider the heat equation in a bounded domain of $ \mathbb{R}^N $ with distributed control (supported on a small open subset) subject to dynamic boundary conditions of surface diffusion type and involving drift terms on the bulk and on the boundary. We prove that the system is null controllable at any time. The result is based on new Carleman estimates for this type of boundary conditions.

Citation: Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations and Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023
References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[2]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. Gonzales-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.

[3]

D. BotheJ. Prüss and G. Simonett, Well-posedness of a two-phase flow with soluble surfactant, Nonlinear elliptic and parabolic problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 64 (2005), 37-61.  doi: 10.1007/3-7643-7385-7_3.

[4]

D. ChaeO. Y. Imanuvilov and S. M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions, J. Dynam. Control Systems, 2 (1996), 449-483.  doi: 10.1007/BF02254698.

[5]

R. DenkJ. Prüss and R. Zacher, Maximal $L_p$-regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187.  doi: 10.1016/j.jfa.2008.07.012.

[6]

A. DoubovaE. Fernández-CaraM. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.

[7]

A. F. M. ter ElstM. Meyries and J. Rehberg, Parabolic equations with dynamical boundary conditions and source terms on interfaces, Ann. Mat. Pura Appl., 193 (2014), 1295-1318.  doi: 10.1007/s10231-013-0329-7.

[8]

L. C. Evans, Partial Differential Equations, Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.

[9]

J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[10]

A. FaviniJ. A. GoldsteinG. R. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.  doi: 10.1007/s00028-002-8077-y.

[11]

E. Fernndez-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696.

[12]

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

[13]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: The semilinear case, ESAIM Control Optim. Calc. Var., 12 (2006), 466-483.  doi: 10.1051/cocv:2006011.

[14]

E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Lináire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[15]

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.

[16]

C. G. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[17]

C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166.  doi: 10.1016/j.jde.2012.02.010.

[18]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[19]

C. G. Gal and M. Meyries, Nonlinear elliptic problems with dynamical boundary conditions of reactive and reactive-diffusive type, Proc. Lond. Math. Soc., 108 (2014), 1351-1380.  doi: 10.1112/plms/pdt057.

[20]

A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM J. Math. Anal., 44 (2012), 3874-3900.  doi: 10.1137/110858847.

[21]

A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. Angew. Math. Phys., 64 (2013), 29-52.  doi: 10.1007/s00033-012-0207-y.

[22]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[23]

G. R. Goldstein, Derivation of dynamical boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[24]

D. HömbergK. Krumbiegel and J. Rehberg, Optimal control of a parabolic equation with dynamic boundary condition, Appl. Math. Optim., 67 (2013), 3-31.  doi: 10.1007/s00245-012-9178-9.

[25]

O. Y. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900. 

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Control of Nonlinear Distributed Parameter Systems (College Station, TX, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York, 218 (2001), 113-137. 

[27]

J. B. Kennedy, On the isoperimetric problem for the laplacian with robin and wentzell boundary conditions, Bull. Aust. Math. Soc., 82 (2010), 348-350.  doi: 10.1017/S0004972710000456.

[28]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equation with dynamic boundary conditions and drift terms, preprint, arXiv: 1909.02377.pdf.

[29]

M. Kumpf and G. Nickel, Dynamic boundary conditions and boundary control for the one-dimensional heat equation, J. Dynam. Control Systems, 10 (2004), 213-225.  doi: 10.1023/B:JODS.0000024122.71407.83.

[30]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., 35 (1932), 260-275. 

[31]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[32]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic problems with dynamic boundary conditions of reactive-diffusive type, Evol. Equ. Control Theory, 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[33]

K. Mauffrey, Contrôlabilité de Systèmes Gouvernés par Des équations aux Dérivées Partielles, Ph. D thesis, University of Franche-Comté, 2013.

[34]

M. Meyries, Maximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors, Ph. D. thesis, Karlsruhe Institute of Technology, 2010.

[35]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[36]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

J. Prüss and R. Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl., 256 (2001), 405-430.  doi: 10.1006/jmaa.2000.7247.

[38]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Anal., 72 (2006), 3028-3048.  doi: 10.1016/j.na.2009.11.043.

[39]

M. E. Taylor, Partial Differential Equations. Basic Theory, Texts in Applied Mathematics, Springer-Verlag, New York, 1996.

[40]

H. Triebel, Interpolation Theory, Function Spaces, and Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

[41]

J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012.

[42]

J. Zabczyk, Mathematical Control Theory: An Introduction, Modern Birkhäuser Classics. Birkhüuser Boston Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

show all references

References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.

[2]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. Gonzales-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.

[3]

D. BotheJ. Prüss and G. Simonett, Well-posedness of a two-phase flow with soluble surfactant, Nonlinear elliptic and parabolic problems, Progr. Nonlinear Differential Equations Appl., Birkhäuser, Basel, 64 (2005), 37-61.  doi: 10.1007/3-7643-7385-7_3.

[4]

D. ChaeO. Y. Imanuvilov and S. M. Kim, Exact controllability for semilinear parabolic equations with Neumann boundary conditions, J. Dynam. Control Systems, 2 (1996), 449-483.  doi: 10.1007/BF02254698.

[5]

R. DenkJ. Prüss and R. Zacher, Maximal $L_p$-regularity of parabolic problems with boundary dynamics of relaxation type, J. Funct. Anal., 255 (2008), 3149-3187.  doi: 10.1016/j.jfa.2008.07.012.

[6]

A. DoubovaE. Fernández-CaraM. González-Burgos and E. Zuazua, On the controllability of parabolic systems with a nonlinear term involving the state and the gradient, SIAM J. Control Optim., 41 (2002), 798-819.  doi: 10.1137/S0363012901386465.

[7]

A. F. M. ter ElstM. Meyries and J. Rehberg, Parabolic equations with dynamical boundary conditions and source terms on interfaces, Ann. Mat. Pura Appl., 193 (2014), 1295-1318.  doi: 10.1007/s10231-013-0329-7.

[8]

L. C. Evans, Partial Differential Equations, Vol. 19 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI, second edition, 2010.

[9]

J. Z. Farkas and P. Hinow, Physiologically structured populations with diffusion and dynamic boundary conditions, Math. Biosci. Eng., 8 (2011), 503-513.  doi: 10.3934/mbe.2011.8.503.

[10]

A. FaviniJ. A. GoldsteinG. R. Goldstein and S. Romanelli, The heat equation with generalized Wentzell boundary condition, J. Evol. Equ., 2 (2002), 1-19.  doi: 10.1007/s00028-002-8077-y.

[11]

E. Fernndez-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1399-1446.  doi: 10.1137/S0363012904439696.

[12]

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

[13]

E. Fernández-CaraM. González-BurgosS. Guerrero and J.-P. Puel, Exact controllability to the trajectories of the heat equation with Fourier boundary conditions: The semilinear case, ESAIM Control Optim. Calc. Var., 12 (2006), 466-483.  doi: 10.1051/cocv:2006011.

[14]

E. Fernandez-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, Ann. Inst. H. Poincaré Anal. Non Lináire, 17 (2000), 583-616.  doi: 10.1016/S0294-1449(00)00117-7.

[15]

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.

[16]

C. G. Gal and L. Tebou, Carleman inequalities for wave equations with oscillatory boundary conditions and application, SIAM J. Control Optim., 55 (2017), 324-364.  doi: 10.1137/15M1032211.

[17]

C. G. Gal, On a class of degenerate parabolic equations with dynamic boundary conditions, J. Differential Equations, 253 (2012), 126-166.  doi: 10.1016/j.jde.2012.02.010.

[18]

C. G. Gal and M. Grasselli, The non-isothermal Allen-Cahn equation with dynamic boundary conditions, Discrete Contin. Dyn. Syst., 22 (2008), 1009-1040.  doi: 10.3934/dcds.2008.22.1009.

[19]

C. G. Gal and M. Meyries, Nonlinear elliptic problems with dynamical boundary conditions of reactive and reactive-diffusive type, Proc. Lond. Math. Soc., 108 (2014), 1351-1380.  doi: 10.1112/plms/pdt057.

[20]

A. Glitzky, An electronic model for solar cells including active interfaces and energy resolved defect densities, SIAM J. Math. Anal., 44 (2012), 3874-3900.  doi: 10.1137/110858847.

[21]

A. Glitzky and A. Mielke, A gradient structure for systems coupling reaction-diffusion effects in bulk and interfaces, Z. Angew. Math. Phys., 64 (2013), 29-52.  doi: 10.1007/s00033-012-0207-y.

[22]

G. R. Goldstein, Derivation and physical interpretation of general boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[23]

G. R. Goldstein, Derivation of dynamical boundary conditions, Adv. Differential Equations, 11 (2006), 457-480. 

[24]

D. HömbergK. Krumbiegel and J. Rehberg, Optimal control of a parabolic equation with dynamic boundary condition, Appl. Math. Optim., 67 (2013), 3-31.  doi: 10.1007/s00245-012-9178-9.

[25]

O. Y. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900. 

[26]

O. Y. Imanuvilov and M. Yamamoto, Carleman estimate for a parabolic equation in a Sobolev space of negative order and its applications, Control of Nonlinear Distributed Parameter Systems (College Station, TX, 1999), Lecture Notes in Pure and Appl. Math., Dekker, New York, 218 (2001), 113-137. 

[27]

J. B. Kennedy, On the isoperimetric problem for the laplacian with robin and wentzell boundary conditions, Bull. Aust. Math. Soc., 82 (2010), 348-350.  doi: 10.1017/S0004972710000456.

[28]

A. Khoutaibi, L. Maniar, D. Mugnolo and A. Rhandi, Parabolic equation with dynamic boundary conditions and drift terms, preprint, arXiv: 1909.02377.pdf.

[29]

M. Kumpf and G. Nickel, Dynamic boundary conditions and boundary control for the one-dimensional heat equation, J. Dynam. Control Systems, 10 (2004), 213-225.  doi: 10.1023/B:JODS.0000024122.71407.83.

[30]

R. E. Langer, A problem in diffusion or in the flow of heat for a solid in contact with a fluid, Tohoku Math. J., 35 (1932), 260-275. 

[31]

G. Lebeau and L. Robbiano, Contrôle exact de l'équation de la chaleur, Comm. Partial Differential Equations, 20 (1995), 335-356.  doi: 10.1080/03605309508821097.

[32]

L. ManiarM. Meyries and R. Schnaubelt, Null controllability for parabolic problems with dynamic boundary conditions of reactive-diffusive type, Evol. Equ. Control Theory, 6 (2017), 381-407.  doi: 10.3934/eect.2017020.

[33]

K. Mauffrey, Contrôlabilité de Systèmes Gouvernés par Des équations aux Dérivées Partielles, Ph. D thesis, University of Franche-Comté, 2013.

[34]

M. Meyries, Maximal Regularity in Weighted Spaces, Nonlinear Boundary Conditions, and Global Attractors, Ph. D. thesis, Karlsruhe Institute of Technology, 2010.

[35]

A. Miranville and S. Zelik, Exponential attractors for the Cahn-Hilliard equation with dynamic boundary conditions, Math. Methods Appl. Sci., 28 (2005), 709-735.  doi: 10.1002/mma.590.

[36]

A. Pazy, Semigroups of Linear Operators and Application to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[37]

J. Prüss and R. Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl., 256 (2001), 405-430.  doi: 10.1006/jmaa.2000.7247.

[38]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Anal., 72 (2006), 3028-3048.  doi: 10.1016/j.na.2009.11.043.

[39]

M. E. Taylor, Partial Differential Equations. Basic Theory, Texts in Applied Mathematics, Springer-Verlag, New York, 1996.

[40]

H. Triebel, Interpolation Theory, Function Spaces, and Differential Operators, North-Holland Mathematical Library, 18. North-Holland Publishing Co., Amsterdam-New York, 1978.

[41]

J. L. Vázquez and E. Vitillaro, Heat equation with dynamical boundary conditions of reactive-diffusive type, J. Differential Equations, 250 (2011), 2143-2161.  doi: 10.1016/j.jde.2010.12.012.

[42]

J. Zabczyk, Mathematical Control Theory: An Introduction, Modern Birkhäuser Classics. Birkhüuser Boston Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.

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