- Previous Article
- DCDS-S Home
- This Issue
-
Next Article
Riesz systems, spectral controllability and a source identification problem for heat equations with memory
Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems
1. | Université de Toulouse; Université Paul Sabatier Toulouse III, Institut de Mathématiques de Toulouse, 118 route de Narbonne, F-31062 Toulouse, France |
$Pu=u_t-(x^\a u_x)_x-$λ$ u$/($x^$β) , $x\in (0,1)$,
associated to homogeneous boundary conditions of Dirichlet and/or Neumann type. Under optimal conditions on the parameters $\a\geq 0$, β, λ$ \in \mathbb R$, we derive sharp global Carleman estimates. As an application, we deduce observability and null controllability results for the corresponding evolution problem. A key step in the proof of Carleman estimates is the correct choice of the weight functions and a key ingredient in the proof takes the form of special Hardy-Poincaré inequalities
References:
show all references
References:
[1] |
El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations and Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441 |
[2] |
Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete and Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687 |
[3] |
Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks and Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695 |
[4] |
Thuy N. T. Nguyen. Carleman estimates for semi-discrete parabolic operators with a discontinuous diffusion coefficient and applications to controllability. Mathematical Control and Related Fields, 2014, 4 (2) : 203-259. doi: 10.3934/mcrf.2014.4.203 |
[5] |
Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force. Evolution Equations and Control Theory, 2021, 10 (3) : 545-573. doi: 10.3934/eect.2020080 |
[6] |
J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete and Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136 |
[7] |
Abdelhakim Belghazi, Ferroudja Smadhi, Nawel Zaidi, Ouahiba Zair. Carleman inequalities for the two-dimensional heat equation in singular domains. Mathematical Control and Related Fields, 2012, 2 (4) : 331-359. doi: 10.3934/mcrf.2012.2.331 |
[8] |
Lin Yan, Bin Wu. Null controllability for a class of stochastic singular parabolic equations with the convection term. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3213-3240. doi: 10.3934/dcdsb.2021182 |
[9] |
Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133 |
[10] |
André da Rocha Lopes, Juan Límaco. Local null controllability for a parabolic equation with local and nonlocal nonlinearities in moving domains. Evolution Equations and Control Theory, 2022, 11 (3) : 749-779. doi: 10.3934/eect.2021024 |
[11] |
Morteza Fotouhi, Leila Salimi. Controllability results for a class of one dimensional degenerate/singular parabolic equations. Communications on Pure and Applied Analysis, 2013, 12 (3) : 1415-1430. doi: 10.3934/cpaa.2013.12.1415 |
[12] |
Brahim Allal, Abdelkarim Hajjaj, Jawad Salhi, Amine Sbai. Boundary controllability for a coupled system of degenerate/singular parabolic equations. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021055 |
[13] |
Brahim Allal, Genni Fragnelli, Jawad Salhi*. Controllability for degenerate/singular parabolic systems involving memory terms. Discrete and Continuous Dynamical Systems - S, 2022 doi: 10.3934/dcdss.2022071 |
[14] |
Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control and Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1 |
[15] |
Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete and Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51 |
[16] |
Lingyang Liu, Xu Liu. Controllability and observability of some coupled stochastic parabolic systems. Mathematical Control and Related Fields, 2018, 8 (3&4) : 829-854. doi: 10.3934/mcrf.2018037 |
[17] |
El Mustapha Ait Ben Hassi, Mohamed Fadili, Lahcen Maniar. Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix. Mathematical Control and Related Fields, 2020, 10 (3) : 623-642. doi: 10.3934/mcrf.2020013 |
[18] |
Alhabib Moumni, Jawad Salhi. Exact controllability for a degenerate and singular wave equation with moving boundary. Numerical Algebra, Control and Optimization, 2022 doi: 10.3934/naco.2022001 |
[19] |
Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations and Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020 |
[20] |
Xinchi Huang, Atsushi Kawamoto. Inverse problems for a half-order time-fractional diffusion equation in arbitrary dimension by Carleman estimates. Inverse Problems and Imaging, 2022, 16 (1) : 39-67. doi: 10.3934/ipi.2021040 |
2020 Impact Factor: 2.425
Tools
Metrics
Other articles
by authors
[Back to Top]