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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 and 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.

[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.

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