American Institute of Mathematical Sciences

Advanced
May  2013, 12(3): 1415-1430. doi: 10.3934/cpaa.2013.12.1415

Controllability results for a class of one dimensional degenerate/singular parabolic equations

Morteza Fotouhi 1, and  Leila Salimi 1,

1. 

Department of Mathematical Sciences, Sharif University of Technology , P.O. Box 11365-9415, Tehran, Iran, Iran

Received  November 2011 Revised  June 2012 Published  September 2012

We study the null controllability properties of some degenerate/singular parabolic equations in a bounded interval of R. For this reason we derive a new Carleman estimate whose proof is based on Hardy inequalities.
Keywords: singular potential, null controllability, Carleman estimate, Degenerate parabolic equations, Hardy inequality..
Mathematics Subject Classification: Primary: 35K65, 93B07; Secondary: 53C3.
Citation: Morteza Fotouhi, Leila Salimi. Controllability results for a class of one dimensional degenerate/singular parabolic equations. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1415-1430. doi: 10.3934/cpaa.2013.12.1415
References:
[1]

J.-M. Buchot and J.-P. Raymond, A linearized model for boundary layer equations,, in Optimal Control of Complex Strcture (Oberwolfach, (2002), 31. doi: 10.1007/978-3-0348-8148-7_3. Google Scholar

[2]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1991), 973. doi: 10.1016/S0764-4442(00)88588-2. Google Scholar

[3]

F. Alabau- Boussouria, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability,, J. Evol. Equ., 6 (2006), 161. doi: 10.1007/s00028-006-0222-6. Google Scholar

[4]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM J. Control Optim., 47 (2008), 1. doi: 10.1137/04062062X. Google Scholar

[5]

P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Commun. Pure Appl. Anal., 3 (2004), 607. doi: 10.3934/cpaa.2004.3.607. Google Scholar

[6]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation in inverse problems,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/10/105003. Google Scholar

[7]

T. Cazenave and A. Haraux, Introduction aux problemes d' évolution semi-lineaires,, in, (1990). Google Scholar

[8]

H. O. Fattorini and D. L. Russsell, Exact controability theorems for linear parabolic equations in one space dimension,, Arch. Rational Mech. Anal., 4 (1971), 272. doi: 10.1007/BF00250466. Google Scholar

[9]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (1974), 45. Google Scholar

[10]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, , To appear in Journal of Dynamical Systems and Control., (). Google Scholar

[11]

A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Series {\bf 34}, 34 (1996). Google Scholar

[12]

P. Martinez and J. Vancostenoble, Carleman estimates for one - dimensional degenerate heat equations,, J. Evol. Equ., 6 (2006), 161. doi: 10.1007/s00028-006-0214-6. Google Scholar

[13]

P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability for a linearized Crocco-type equation,, SIAM J. Control Optim., 42 (2003), 709. doi: 10.1137/S0363012902403547. Google Scholar

[14]

G. R. North, L. Howard, D. Pollard and B. Wielicki, Variational formulation of Budyko-Sellers climate models,, Journal of the Atmospheric Sciences, 36 (1979), 255. doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2. Google Scholar

[15]

T. I. Seidman, Exact boundary control for some evolution equations,, SIAM J. Control Optim., 16 (1978), 979. doi: 10.1137/0316066. Google Scholar

[16]

N. Shimakura, "Partial Differential Operatots of Elliptic Type,", Translations of Mathematical Monographs. 99, (1992). Google Scholar

[17]

J. Tort and J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model,, in Annales, (). doi: 10.1016/j.anihpc.2012.03.003. Google Scholar

[18]

J. Vancostenoble, Improved Hardy-Poincare inequalities and sharp Carleman estimates for degenerate/singular parabolic problems,, Discrete Contin. Dyn. Syst.Ser. S, 4 (2011), 761. doi: 10.3934/dcdss.2011.4.761. Google Scholar

[19]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations,, Communications in Partial Differential Equations, 36 (2011), 1287. doi: 10.1080/03605302.2011.587491. Google Scholar

[20]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials,, Journal of Functional Analysis, 254 (2008), 1864. doi: 10.1016/J.Jfa.2007.12.015. Google Scholar

show all references

References:
[1]

J.-M. Buchot and J.-P. Raymond, A linearized model for boundary layer equations,, in Optimal Control of Complex Strcture (Oberwolfach, (2002), 31. doi: 10.1007/978-3-0348-8148-7_3. Google Scholar

[2]

X. Cabré and Y. Martel, Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier,, C. R. Acad. Sci. Paris, 329 (1991), 973. doi: 10.1016/S0764-4442(00)88588-2. Google Scholar

[3]

F. Alabau- Boussouria, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability,, J. Evol. Equ., 6 (2006), 161. doi: 10.1007/s00028-006-0222-6. Google Scholar

[4]

P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators,, SIAM J. Control Optim., 47 (2008), 1. doi: 10.1137/04062062X. Google Scholar

[5]

P. Cannarsa, P. Martinez and J. Vancostenoble, Persistent regional null controllability for a class of degenerate parabolic equations,, Commun. Pure Appl. Anal., 3 (2004), 607. doi: 10.3934/cpaa.2004.3.607. Google Scholar

[6]

P. Cannarsa, J. Tort and M. Yamamoto, Determination of source terms in a degenerate parabolic equation in inverse problems,, Inverse Problems, 26 (2010). doi: 10.1088/0266-5611/26/10/105003. Google Scholar

[7]

T. Cazenave and A. Haraux, Introduction aux problemes d' évolution semi-lineaires,, in, (1990). Google Scholar

[8]

H. O. Fattorini and D. L. Russsell, Exact controability theorems for linear parabolic equations in one space dimension,, Arch. Rational Mech. Anal., 4 (1971), 272. doi: 10.1007/BF00250466. Google Scholar

[9]

H. O. Fattorini and D. L. Russell, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations,, Quart. Appl. Math., 32 (1974), 45. Google Scholar

[10]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, , To appear in Journal of Dynamical Systems and Control., (). Google Scholar

[11]

A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations,", Lecture Notes Series {\bf 34}, 34 (1996). Google Scholar

[12]

P. Martinez and J. Vancostenoble, Carleman estimates for one - dimensional degenerate heat equations,, J. Evol. Equ., 6 (2006), 161. doi: 10.1007/s00028-006-0214-6. Google Scholar

[13]

P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability for a linearized Crocco-type equation,, SIAM J. Control Optim., 42 (2003), 709. doi: 10.1137/S0363012902403547. Google Scholar

[14]

G. R. North, L. Howard, D. Pollard and B. Wielicki, Variational formulation of Budyko-Sellers climate models,, Journal of the Atmospheric Sciences, 36 (1979), 255. doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2. Google Scholar

[15]

T. I. Seidman, Exact boundary control for some evolution equations,, SIAM J. Control Optim., 16 (1978), 979. doi: 10.1137/0316066. Google Scholar

[16]

N. Shimakura, "Partial Differential Operatots of Elliptic Type,", Translations of Mathematical Monographs. 99, (1992). Google Scholar

[17]

J. Tort and J. Vancostenoble, Determination of the insolation function in the nonlinear Sellers climate model,, in Annales, (). doi: 10.1016/j.anihpc.2012.03.003. Google Scholar

[18]

J. Vancostenoble, Improved Hardy-Poincare inequalities and sharp Carleman estimates for degenerate/singular parabolic problems,, Discrete Contin. Dyn. Syst.Ser. S, 4 (2011), 761. doi: 10.3934/dcdss.2011.4.761. Google Scholar

[19]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations,, Communications in Partial Differential Equations, 36 (2011), 1287. doi: 10.1080/03605302.2011.587491. Google Scholar

[20]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials,, Journal of Functional Analysis, 254 (2008), 1864. doi: 10.1016/J.Jfa.2007.12.015. Google Scholar

[1]

Genni Fragnelli. Null controllability of degenerate parabolic equations in non divergence form via Carleman estimates. Discrete & Continuous Dynamical Systems - S, 2013, 6 (3) : 687-701. doi: 10.3934/dcdss.2013.6.687
[2]

El Mustapha Ait Ben Hassi, Farid Ammar khodja, Abdelkarim Hajjaj, Lahcen Maniar. Carleman Estimates and null controllability of coupled degenerate systems. Evolution Equations & Control Theory, 2013, 2 (3) : 441-459. doi: 10.3934/eect.2013.2.441
[3]

Judith Vancostenoble. Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 761-790. doi: 10.3934/dcdss.2011.4.761
[4]

Piermarco Cannarsa, Genni Fragnelli, Dario Rocchetti. Null controllability of degenerate parabolic operators with drift. Networks & Heterogeneous Media, 2007, 2 (4) : 695-715. doi: 10.3934/nhm.2007.2.695
[5]

Farid Ammar Khodja, Cherif Bouzidi, Cédric Dupaix, Lahcen Maniar. Null controllability of retarded parabolic equations. Mathematical Control & Related Fields, 2014, 4 (1) : 1-15. doi: 10.3934/mcrf.2014.4.1
[6]

Lahcen Maniar, Martin Meyries, Roland Schnaubelt. Null controllability for parabolic equations with dynamic boundary conditions. Evolution Equations & Control Theory, 2017, 6 (3) : 381-407. doi: 10.3934/eect.2017020
[7]

Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control & Related Fields, 2019, 0 (0) : 0-0. doi: 10.3934/mcrf.2019031
[8]

Gary Lieberman. A new regularity estimate for solutions of singular parabolic equations. Conference Publications, 2005, 2005 (Special) : 605-610. doi: 10.3934/proc.2005.2005.605
[9]

Enrique Fernández-Cara, Luz de Teresa. Null controllability of a cascade system of parabolic-hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2004, 11 (2&3) : 699-714. doi: 10.3934/dcds.2004.11.699
[10]

Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. Persistent regional null contrillability for a class of degenerate parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (4) : 607-635. doi: 10.3934/cpaa.2004.3.607
[11]

Chunpeng Wang, Yanan Zhou, Runmei Du, Qiang Liu. Carleman estimate for solutions to a degenerate convection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4207-4222. doi: 10.3934/dcdsb.2018133
[12]

Farid Ammar Khodja, Franz Chouly, Michel Duprez. Partial null controllability of parabolic linear systems. Mathematical Control & Related Fields, 2016, 6 (2) : 185-216. doi: 10.3934/mcrf.2016001
[13]

Enrique Fernández-Cara, Manuel González-Burgos, Luz de Teresa. Null-exact controllability of a semilinear cascade system of parabolic-hyperbolic equations. Communications on Pure & Applied Analysis, 2006, 5 (3) : 639-658. doi: 10.3934/cpaa.2006.5.639
[14]

Boumediene Abdellaoui, Ahmed Attar. Quasilinear elliptic problem with Hardy potential and singular term. Communications on Pure & Applied Analysis, 2013, 12 (3) : 1363-1380. doi: 10.3934/cpaa.2013.12.1363
[15]

Ait Ben Hassi El Mustapha, Fadili Mohamed, Maniar Lahcen. On Algebraic condition for null controllability of some coupled degenerate systems. Mathematical Control & Related Fields, 2019, 9 (1) : 77-95. doi: 10.3934/mcrf.2019004
[16]

Xingwen Hao, Yachun Li, Qin Wang. A kinetic approach to error estimate for nonautonomous anisotropic degenerate parabolic-hyperbolic equations. Kinetic & Related Models, 2014, 7 (3) : 477-492. doi: 10.3934/krm.2014.7.477
[17]

Xumin Wang. Singular Hardy-Trudinger-Moser inequality and the existence of extremals on the unit disc. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2717-2733. doi: 10.3934/cpaa.2019121
[18]

Kim Dang Phung. Carleman commutator approach in logarithmic convexity for parabolic equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 899-933. doi: 10.3934/mcrf.2018040
[19]

Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183
[20]

Jong-Shenq Guo, Satoshi Sasayama, Chi-Jen Wang. Blowup rate estimate for a system of semilinear parabolic equations. Communications on Pure & Applied Analysis, 2009, 8 (2) : 711-718. doi: 10.3934/cpaa.2009.8.711

2018 Impact Factor: 0.925

Tools

Metrics

  • PDF downloads (7)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences