
-
Previous Article
On a final value problem for a class of nonlinear hyperbolic equations with damping term
- EECT Home
- This Issue
-
Next Article
Infimal convolution and duality in convex optimal control problems with second order evolution differential inclusions
Boundary null-controllability of coupled parabolic systems with Robin conditions
1. | Institut de Mathématiques de Toulouse, UMR 5219, Université Paul Sabatier |
2. | Institut Universitaire de France, 31062 Toulouse Cedex 09, France |
The main goal of this paper is to investigate the boundary controllability of some coupled parabolic systems in the cascade form in the case where the boundary conditions are of Robin type. In particular, we prove that the associated controls satisfy suitable uniform bounds with respect to the Robin parameters, that let us show that they converge towards a Dirichlet control when the Robin parameters go to infinity. This is a justification of the popular penalisation method for dealing with Dirichlet boundary data in the framework of the controllability of coupled parabolic systems.
References:
[1] |
F. Alabau-Boussouira and M. Léautaud,
Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl., 99 (2013), 544-576.
doi: 10.1016/j.matpur.2012.09.012. |
[2] |
D. Allonsius and F. Boyer, Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries, Mathematical Control & Related Fields.
doi: 10.3934/mcrf.2019037. |
[3] |
D. Allonsius, F. Boyer and M. Morancey,
Spectral analysis of discrete elliptic operators and applications in control theory, Numer. Math., 140 (2018), 857-911.
doi: 10.1007/s00211-018-0983-1. |
[4] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa,
The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.
doi: 10.1016/j.matpur.2011.06.005. |
[5] |
F. Ammar-Khodja, A. Benabdallah, M. 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. |
[6] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa,
Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151.
doi: 10.1016/j.jfa.2014.07.024. |
[7] |
F. V. Atkinson, Discrete and continuous boundary problems, in Mathematics in Science and Engineering, 8, Academic Press, New York-London, 1964. |
[8] |
I. Babuška,
The finite element method with penalty, Math. Comp., 27 (1973), 221-228.
doi: 10.1090/S0025-5718-1973-0351118-5. |
[9] |
F. B. Belgacem, H. E. Fekih and H. Metoui,
Singular perturbation for the dirichlet boundary control of elliptic problems, M2AN Math. Model. Numer. Anal., 37 (2003), 833-850.
doi: 10.1051/m2an:2003057. |
[10] |
F. B. Belgacem, H. E. Fekih and J. P. Raymond,
A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions, Asymptot. Anal., 34 (2003), 121-136.
|
[11] |
A. Benabdallah, F. Boyer, M. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970–3001.
doi: 10.1137/130929680. |
[12] |
A. Benabdallah, F. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Annales Henri Lebesgue, preprint, URL https://hal.archives-ouvertes.fr/hal-01949391. Google Scholar |
[13] |
F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
[14] |
E. Casas, M. Mateos and J.-P. Raymond, Penalization of Dirichlet optimal control problems, ESAIM Control Optim. Calc. Var., 15 (2009), 782–809.
doi: 10.1051/cocv:2008049. |
[15] |
J.-M. Coron, Control and nonlinearity, in Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. |
[16] |
K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, in Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. |
[17] |
H. O. Fattorini,
Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.
doi: 10.1137/0304048. |
[18] |
H. O. Fattorini and D. L. Russell,
Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.
doi: 10.1090/qam/510972. |
[19] |
E. Fernández-Cara, M. González-Burgos and L. de Teresa,
Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758.
doi: 10.1016/j.jfa.2010.06.003. |
[20] |
H. Hochstadt,
Asymptotic estimates for the Sturm-Liouville spectrum, Comm. Pure Appl. Math., 14 (1961), 749-764.
doi: 10.1002/cpa.3160140408. |
[21] |
Q. Kong and A. Zettl,
Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19.
doi: 10.1006/jdeq.1996.0154. |
[22] |
Q. Lü,
A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.
doi: 10.1051/cocv/2012008. |
[23] |
R. Nittka,
Inhomogeneous parabolic Neumann problems, Czechoslovak Math. J., 64 (2014), 703-742.
doi: 10.1007/s10587-014-0127-4. |
[24] |
G. Olive,
Boundary approximate controllability of some linear parabolic systems, Evol. Equ. Control Theory, 3 (2014), 167-189.
doi: 10.3934/eect.2014.3.167. |
[25] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
show all references
References:
[1] |
F. Alabau-Boussouira and M. Léautaud,
Indirect controllability of locally coupled wave-type systems and applications, J. Math. Pures Appl., 99 (2013), 544-576.
doi: 10.1016/j.matpur.2012.09.012. |
[2] |
D. Allonsius and F. Boyer, Boundary null-controllability of semi-discrete coupled parabolic systems in some multi-dimensional geometries, Mathematical Control & Related Fields.
doi: 10.3934/mcrf.2019037. |
[3] |
D. Allonsius, F. Boyer and M. Morancey,
Spectral analysis of discrete elliptic operators and applications in control theory, Numer. Math., 140 (2018), 857-911.
doi: 10.1007/s00211-018-0983-1. |
[4] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa,
The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.
doi: 10.1016/j.matpur.2011.06.005. |
[5] |
F. Ammar-Khodja, A. Benabdallah, M. 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. |
[6] |
F. Ammar-Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa,
Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151.
doi: 10.1016/j.jfa.2014.07.024. |
[7] |
F. V. Atkinson, Discrete and continuous boundary problems, in Mathematics in Science and Engineering, 8, Academic Press, New York-London, 1964. |
[8] |
I. Babuška,
The finite element method with penalty, Math. Comp., 27 (1973), 221-228.
doi: 10.1090/S0025-5718-1973-0351118-5. |
[9] |
F. B. Belgacem, H. E. Fekih and H. Metoui,
Singular perturbation for the dirichlet boundary control of elliptic problems, M2AN Math. Model. Numer. Anal., 37 (2003), 833-850.
doi: 10.1051/m2an:2003057. |
[10] |
F. B. Belgacem, H. E. Fekih and J. P. Raymond,
A penalized Robin approach for solving a parabolic equation with nonsmooth Dirichlet boundary conditions, Asymptot. Anal., 34 (2003), 121-136.
|
[11] |
A. Benabdallah, F. Boyer, M. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and application to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970–3001.
doi: 10.1137/130929680. |
[12] |
A. Benabdallah, F. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Annales Henri Lebesgue, preprint, URL https://hal.archives-ouvertes.fr/hal-01949391. Google Scholar |
[13] |
F. Boyer and P. Fabrie, Mathematical tools for the study of the incompressible Navier-Stokes equations and related models, Applied Mathematical Sciences, 183, Springer, New York, 2013.
doi: 10.1007/978-1-4614-5975-0. |
[14] |
E. Casas, M. Mateos and J.-P. Raymond, Penalization of Dirichlet optimal control problems, ESAIM Control Optim. Calc. Var., 15 (2009), 782–809.
doi: 10.1051/cocv:2008049. |
[15] |
J.-M. Coron, Control and nonlinearity, in Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. |
[16] |
K.-J. Engel and R. Nagel, One-parameter semigroups for linear evolution equations, in Graduate Texts in Mathematics, 194, Springer-Verlag, New York, 2000. |
[17] |
H. O. Fattorini,
Some remarks on complete controllability, SIAM J. Control, 4 (1966), 686-694.
doi: 10.1137/0304048. |
[18] |
H. O. Fattorini and D. L. Russell,
Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974/75), 45-69.
doi: 10.1090/qam/510972. |
[19] |
E. Fernández-Cara, M. González-Burgos and L. de Teresa,
Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758.
doi: 10.1016/j.jfa.2010.06.003. |
[20] |
H. Hochstadt,
Asymptotic estimates for the Sturm-Liouville spectrum, Comm. Pure Appl. Math., 14 (1961), 749-764.
doi: 10.1002/cpa.3160140408. |
[21] |
Q. Kong and A. Zettl,
Eigenvalues of regular Sturm-Liouville problems, J. Differential Equations, 131 (1996), 1-19.
doi: 10.1006/jdeq.1996.0154. |
[22] |
Q. Lü,
A lower bound on local energy of partial sum of eigenfunctions for Laplace-Beltrami operators, ESAIM Control Optim. Calc. Var., 19 (2013), 255-273.
doi: 10.1051/cocv/2012008. |
[23] |
R. Nittka,
Inhomogeneous parabolic Neumann problems, Czechoslovak Math. J., 64 (2014), 703-742.
doi: 10.1007/s10587-014-0127-4. |
[24] |
G. Olive,
Boundary approximate controllability of some linear parabolic systems, Evol. Equ. Control Theory, 3 (2014), 167-189.
doi: 10.3934/eect.2014.3.167. |
[25] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |

[1] |
Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHUM approach. Numerical Algebra, Control & Optimization, 2020 doi: 10.3934/naco.2020055 |
[2] |
Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 399-414. doi: 10.3934/dcds.2009.23.399 |
[3] |
Vandana Sharma. Global existence and uniform estimates of solutions to reaction diffusion systems with mass transport type boundary conditions. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021001 |
[4] |
Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249 |
[5] |
Felix Finster, Jürg Fröhlich, Marco Oppio, Claudio F. Paganini. Causal fermion systems and the ETH approach to quantum theory. Discrete & Continuous Dynamical Systems - S, 2020 doi: 10.3934/dcdss.2020451 |
[6] |
Gabrielle Nornberg, Delia Schiera, Boyan Sirakov. A priori estimates and multiplicity for systems of elliptic PDE with natural gradient growth. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3857-3881. doi: 10.3934/dcds.2020128 |
[7] |
Kai Zhang, Xiaoqi Yang, Song Wang. Solution method for discrete double obstacle problems based on a power penalty approach. Journal of Industrial & Management Optimization, 2020 doi: 10.3934/jimo.2021018 |
[8] |
Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 |
[9] |
Liupeng Wang, Yunqing Huang. Error estimates for second-order SAV finite element method to phase field crystal model. Electronic Research Archive, 2021, 29 (1) : 1735-1752. doi: 10.3934/era.2020089 |
[10] |
Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021004 |
[11] |
Md. Masum Murshed, Kouta Futai, Masato Kimura, Hirofumi Notsu. Theoretical and numerical studies for energy estimates of the shallow water equations with a transmission boundary condition. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1063-1078. doi: 10.3934/dcdss.2020230 |
[12] |
Kazunori Matsui. Sharp consistency estimates for a pressure-Poisson problem with Stokes boundary value problems. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 1001-1015. doi: 10.3934/dcdss.2020380 |
[13] |
Alessandro Fonda, Rodica Toader. A dynamical approach to lower and upper solutions for planar systems "To the memory of Massimo Tarallo". Discrete & Continuous Dynamical Systems - A, 2021 doi: 10.3934/dcds.2021012 |
[14] |
Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020103 |
[15] |
Nitha Niralda P C, Sunil Mathew. On properties of similarity boundary of attractors in product dynamical systems. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021004 |
[16] |
Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062 |
[17] |
Duy Phan, Lassi Paunonen. Finite-dimensional controllers for robust regulation of boundary control systems. Mathematical Control & Related Fields, 2021, 11 (1) : 95-117. doi: 10.3934/mcrf.2020029 |
[18] |
Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349 |
[19] |
Kung-Ching Chang, Xuefeng Wang, Xie Wu. On the spectral theory of positive operators and PDE applications. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3171-3200. doi: 10.3934/dcds.2020054 |
[20] |
Gervy Marie Angeles, Gilbert Peralta. Energy method for exponential stability of coupled one-dimensional hyperbolic PDE-ODE systems. Evolution Equations & Control Theory, 2020 doi: 10.3934/eect.2020108 |
2019 Impact Factor: 0.953
Tools
Metrics
Other articles
by authors
[Back to Top]