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September  2017, 6(3): 381-407. doi: 10.3934/eect.2017020

Null controllability for parabolic equations with dynamic boundary conditions

1. 

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

2. 

Institut für Mathematik, Martin-Luther-Universität Halle-Wittenberg, 06099 Halle, Germany

3. 

Department of Mathematics, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany

Received  August 2015 Revised  March 2017 Published  July 2017

Fund Project: We thank the Deutsche Forschungsgemeinschaft which supported this research within the grants ME 3848/1-1 and SCHN 570/4-1. M.M. thanks L.M. for a very pleasant stay in Marrakesh, where parts of this work originated.

We prove null controllability for linear and semilinear heat equations with dynamic boundary conditions of surface diffusion type. The results are based on a new Carleman estimate for this type of boundary conditions.

Citation: Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
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]

I. Bejenaru, J. I. Díaz and I. I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions, Electron. J. Differential Equations, (2001), 19 pp.  Google Scholar

[3]

D. BotheJ. Prüss and G. Simonett, Well-posedness of a two-phase flow with soluble surfactant, In: Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Basel, 64 (2005), 37-62.  doi: 10.1007/3-7643-7385-7_3.  Google Scholar

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D. ChaeO. Yu. 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 Lp–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. (4), 193 (2014), 1295-1318.  doi: 10.1007/s10231-013-0329-7.  Google Scholar

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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

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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

[10]

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

[11]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.  Google Scholar

[12]

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

[13]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Note Series 34 Research Institute of Mathematics, Seoul National University, Seoul, 1996.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

O. Yu. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

[20]

J. Jost, Riemannian Geometry and Geometric Analysis, Fifth edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Google Scholar

[22]

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

[23]

O. A. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence (RI), 1968. Reprinted with corrections, 1988.  Google Scholar

[24]

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

[25]

A. Lunardi, Interpolation Theory, Second edition, Edizioni della Normale, Pisa, 2009.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Analysis, 72 (2010), 3028-3048.  doi: 10.1016/j.na.2009.11.043.  Google Scholar

[31]

M. E. Taylor, Partial Differential Equations. Basic theory, Springer–Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[32]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, , J. A. Barth, Heidelberg, 1995.  Google Scholar

[33]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[34]

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

[35]

J. Zabczyk, Mathematical Control Theory, Birkhäuser 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]

I. Bejenaru, J. I. Díaz and I. I. Vrabie, An abstract approximate controllability result and applications to elliptic and parabolic systems with dynamic boundary conditions, Electron. J. Differential Equations, (2001), 19 pp.  Google Scholar

[3]

D. BotheJ. Prüss and G. Simonett, Well-posedness of a two-phase flow with soluble surfactant, In: Nonlinear Elliptic and Parabolic Problems. Progress in Nonlinear Differential Equations and Their Applications, Birkhäuser, Basel, 64 (2005), 37-62.  doi: 10.1007/3-7643-7385-7_3.  Google Scholar

[4]

D. ChaeO. Yu. 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 Lp–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. (4), 193 (2014), 1295-1318.  doi: 10.1007/s10231-013-0329-7.  Google Scholar

[8]

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

[9]

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

[10]

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

[11]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.  Google Scholar

[12]

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

[13]

A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Note Series 34 Research Institute of Mathematics, Seoul National University, Seoul, 1996.  Google Scholar

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

O. Yu. Imanuvilov, Controllability of parabolic equations, Sb. Math., 186 (1995), 879-900.  doi: 10.1070/SM1995v186n06ABEH000047.  Google Scholar

[20]

J. Jost, Riemannian Geometry and Geometric Analysis, Fifth edition, Springer-Verlag, Berlin, 2008.  Google Scholar

[21]

T. Kato, Perturbation Theory for Linear Operators, Reprint of the 1980 edition. Classics in Mathematics. Springer-Verlag, Berlin, 1995.  Google Scholar

[22]

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

[23]

O. A. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasi-linear Equations of Parabolic Type, American Mathematical Society, Providence (RI), 1968. Reprinted with corrections, 1988.  Google Scholar

[24]

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

[25]

A. Lunardi, Interpolation Theory, Second edition, Edizioni della Normale, Pisa, 2009.  Google Scholar

[26]

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

[27]

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

[28]

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

[29]

J. Simon, Compact sets in the space Lp(0, T; B), Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.  Google Scholar

[30]

J. Sprekels and H. Wu, A note on parabolic equation with nonlinear dynamical boundary condition, Nonlinear Analysis, 72 (2010), 3028-3048.  doi: 10.1016/j.na.2009.11.043.  Google Scholar

[31]

M. E. Taylor, Partial Differential Equations. Basic theory, Springer–Verlag, New York, 1996. doi: 10.1007/978-1-4684-9320-7.  Google Scholar

[32]

H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, , J. A. Barth, Heidelberg, 1995.  Google Scholar

[33]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[34]

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

[35]

J. Zabczyk, Mathematical Control Theory, Birkhäuser Boston, Inc., Boston, MA, 2008. doi: 10.1007/978-0-8176-4733-9.  Google Scholar

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