-
Previous Article
Remarks on large time behavior of level-set mean curvature flow equations with driving and source terms
- DCDS-B Home
- This Issue
-
Next Article
Estimating the division rate from indirect measurements of single cells
Null controllability of one dimensional degenerate parabolic equations with first order terms
1. | Academia de Matemáticas, Universidad Autónoma de la Ciudad de México, Avenida la Corona 320, Col. Loma la Palma, Del. Gustavo A. Madero, CDMX, C.P. 07160. Mexico |
2. | Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., C. P. 04510 CDMX, Mexico |
In this paper we present a null controllability result for a degenerate semilinear parabolic equation with first order terms. The main result is obtained after the proof of a new Carleman inequality for a degenerate linear parabolic equation with first order terms.
References:
[1] |
F. Alabau-Boussouira, P. 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. D. Araruna, B. S. V. Araújo and E. Fernández-Cara, Stackelberg-Nash null controllability for some linear and semilinear degenerate parabolic equations, Math. Control Signals Systems, 30 (2018). |
[3] |
M. Campiti, G. Metafune and D. Pallara,
Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.
doi: 10.1007/PL00005959. |
[4] |
P. Cannarsa and L. de Teresa, Insensitizing controls for one dimensional degenerate parabolic equations, Electron. J. Differential Equations, 2009 (2009), 21 pp. |
[5] |
P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 20 pp. |
[6] |
P. Cannarsa, G. Fragnelli and D. Rocchetti,
Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715.
doi: 10.3934/nhm.2007.2.695. |
[7] |
P. Cannarsa, G. Fragnelli and J. Vancostenoble,
Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl., 320 (2006), 804-818.
doi: 10.1016/j.jmaa.2005.07.006. |
[8] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.
doi: 10.1137/04062062X. |
[9] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.
|
[10] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Persistent regional null controllability for a class of degenerate parabolic equations, Communications on Pure and Applied Analysis, 3 (2004), 607-635.
doi: 10.3934/cpaa.2004.3.607. |
[11] |
C. Flores and L. de Teresa,
Carleman estimates for degenerate parabolic equations with first order terms and applications, C. R. Acad. Sci. Paris, 348 (2010), 391-396.
doi: 10.1016/j.crma.2010.01.007. |
[12] |
A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. |
[13] |
O. Yu. Imanuvilov and M. Yamamoto,
Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Math. Sci., 39 (2003), 227-274.
doi: 10.2977/prims/1145476103. |
[14] |
P. Martinez and J. Vancostenoble,
Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.
doi: 10.1007/s00028-006-0214-6. |
[15] |
J. Simon,
Compact sets in the spaces $L^{p}(0, T;B)$, Annali di Matematica Puraed Applicata, 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[16] |
J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.
doi: 10.1007/978-0-8176-4733-9. |
[17] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅳ. Applications to Mathematical Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-5020-3. |
show all references
References:
[1] |
F. Alabau-Boussouira, P. 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. D. Araruna, B. S. V. Araújo and E. Fernández-Cara, Stackelberg-Nash null controllability for some linear and semilinear degenerate parabolic equations, Math. Control Signals Systems, 30 (2018). |
[3] |
M. Campiti, G. Metafune and D. Pallara,
Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36.
doi: 10.1007/PL00005959. |
[4] |
P. Cannarsa and L. de Teresa, Insensitizing controls for one dimensional degenerate parabolic equations, Electron. J. Differential Equations, 2009 (2009), 21 pp. |
[5] |
P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 20 pp. |
[6] |
P. Cannarsa, G. Fragnelli and D. Rocchetti,
Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715.
doi: 10.3934/nhm.2007.2.695. |
[7] |
P. Cannarsa, G. Fragnelli and J. Vancostenoble,
Regional controllability of semilinear degenerate parabolic equations in bounded domains, J. Math. Anal. Appl., 320 (2006), 804-818.
doi: 10.1016/j.jmaa.2005.07.006. |
[8] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19.
doi: 10.1137/04062062X. |
[9] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.
|
[10] |
P. Cannarsa, P. Martinez and J. Vancostenoble,
Persistent regional null controllability for a class of degenerate parabolic equations, Communications on Pure and Applied Analysis, 3 (2004), 607-635.
doi: 10.3934/cpaa.2004.3.607. |
[11] |
C. Flores and L. de Teresa,
Carleman estimates for degenerate parabolic equations with first order terms and applications, C. R. Acad. Sci. Paris, 348 (2010), 391-396.
doi: 10.1016/j.crma.2010.01.007. |
[12] |
A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996. |
[13] |
O. Yu. Imanuvilov and M. Yamamoto,
Carleman inequalities for parabolic equations in Sobolev spaces of negative order and exact controllability for semilinear parabolic equations, Publ. Res. Math. Sci., 39 (2003), 227-274.
doi: 10.2977/prims/1145476103. |
[14] |
P. Martinez and J. Vancostenoble,
Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 325-362.
doi: 10.1007/s00028-006-0214-6. |
[15] |
J. Simon,
Compact sets in the spaces $L^{p}(0, T;B)$, Annali di Matematica Puraed Applicata, 146 (1987), 65-96.
doi: 10.1007/BF01762360. |
[16] |
J. Zabczyk, Mathematical Control Theory: An Introduction, Systems & Control: Foundations & Applications, Birkhäuser Boston, Inc., Boston, MA, 1992.
doi: 10.1007/978-0-8176-4733-9. |
[17] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. Ⅳ. Applications to Mathematical Physics, Springer-Verlag, New York, 1988.
doi: 10.1007/978-1-4612-5020-3. |
[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] |
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 |
[4] |
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 |
[5] |
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 |
[6] |
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 |
[7] |
Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761 |
[8] |
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 |
[9] |
Mu-Ming Zhang, Tian-Yuan Xu, Jing-Xue Yin. Controllability properties of degenerate pseudo-parabolic boundary control problems. Mathematical Control and Related Fields, 2020, 10 (1) : 157-169. doi: 10.3934/mcrf.2019034 |
[10] |
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 |
[11] |
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 |
[12] |
Felipe Wallison Chaves-Silva, Sergio Guerrero, Jean Pierre Puel. Controllability of fast diffusion coupled parabolic systems. Mathematical Control and Related Fields, 2014, 4 (4) : 465-479. doi: 10.3934/mcrf.2014.4.465 |
[13] |
Guillaume Olive. Boundary approximate controllability of some linear parabolic systems. Evolution Equations and Control Theory, 2014, 3 (1) : 167-189. doi: 10.3934/eect.2014.3.167 |
[14] |
Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control and Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001 |
[15] |
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 |
[16] |
Brooke L. Hollingsworth, R.E. Showalter. Semilinear degenerate parabolic systems and distributed capacitance models. Discrete and Continuous Dynamical Systems, 1995, 1 (1) : 59-76. doi: 10.3934/dcds.1995.1.59 |
[17] |
Dung Le. Higher integrability for gradients of solutions to degenerate parabolic systems. Discrete and Continuous Dynamical Systems, 2010, 26 (2) : 597-608. doi: 10.3934/dcds.2010.26.597 |
[18] |
Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control and Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004 |
[19] |
Thuy N. T. Nguyen. Uniform controllability of semidiscrete approximations for parabolic systems in Banach spaces. Discrete and Continuous Dynamical Systems - B, 2015, 20 (2) : 613-640. doi: 10.3934/dcdsb.2015.20.613 |
[20] |
Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations and Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052 |
2021 Impact Factor: 1.497
Tools
Metrics
Other articles
by authors
[Back to Top]