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  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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.   Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

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

[23]

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

[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.  Google Scholar

[25]

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

[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.   Google Scholar

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

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

[34]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

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

[40]

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

[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.  Google Scholar

[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.  Google Scholar

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