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October  2020, 25(10): 3963-3981. doi: 10.3934/dcdsb.2020136

Null controllability of one dimensional degenerate parabolic equations with first order terms

J. Carmelo Flores 1, and  Luz De Teresa 2,,

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

* Corresponding author: Luz de Teresa

Received  July 2019 Revised  January 2020 Published  October 2020 Early access  April 2020

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.

Keywords: Degenerate parabolic systems, controllability, Carleman inequality.
Mathematics Subject Classification: Primary: 35K65, 93B05; Secondary: 93C20.
Citation: J. Carmelo Flores, Luz De Teresa. Null controllability of one dimensional degenerate parabolic equations with first order terms. Discrete & Continuous Dynamical Systems - B, 2020, 25 (10) : 3963-3981. doi: 10.3934/dcdsb.2020136
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.  Google Scholar

[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.  Google Scholar

[4]

P. Cannarsa and L. de Teresa, Insensitizing controls for one dimensional degenerate parabolic equations, Electron. J. Differential Equations, 2009 (2009), 21 pp.  Google Scholar

[5]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 20 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[4]

P. Cannarsa and L. de Teresa, Insensitizing controls for one dimensional degenerate parabolic equations, Electron. J. Differential Equations, 2009 (2009), 21 pp.  Google Scholar

[5]

P. Cannarsa and G. Fragnelli, Null controllability of semilinear degenerate parabolic equations in bounded domains, Electron. J. Differential Equations, 2006 (2006), 20 pp.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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