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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). Google Scholar |
[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). Google Scholar |
[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. |
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