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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 and 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, 2000), Internat. Ser. Numer. Math. 139.. Birkh..auser, Basel, (2002), 31-42. doi: 10.1007/978-3-0348-8148-7_3.

[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), 973-978. doi: 10.1016/S0764-4442(00)88588-2.

[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), 161-204. doi: 10.1007/s00028-006-0222-6.

[4]

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.

[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), 607-635. doi: 10.3934/cpaa.2004.3.607.

[6]

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

[7]

T. Cazenave and A. Haraux, Introduction aux problemes d' évolution semi-lineaires, in "Mathe'matiques et Applications,'' Ellipses, Paris, 1990.

[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), 272-292. doi: 10.1007/BF00250466.

[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), 45-69.

[10]

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

[11]

A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996.

[12]

P. Martinez and J. Vancostenoble, Carleman estimates for one - dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 161-204. doi: 10.1007/s00028-006-0214-6.

[13]

P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability for a linearized Crocco-type equation, SIAM J. Control Optim., 42 (2003), 709-728. doi: 10.1137/S0363012902403547.

[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), 255-259. doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2.

[15]

T. I. Seidman, Exact boundary control for some evolution equations, SIAM J. Control Optim., 16 (1978), 979-999. doi: 10.1137/0316066.

[16]

N. Shimakura, "Partial Differential Operatots of Elliptic Type," Translations of Mathematical Monographs. 99, American Mathematical Society, Providence, RI, 1992.

[17]

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

[18]

J. Vancostenoble, Improved Hardy-Poincare inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst.Ser. S, 4 (2011), 761-790. doi: 10.3934/dcdss.2011.4.761.

[19]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1287-1317. doi: 10.1080/03605302.2011.587491.

[20]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, Journal of Functional Analysis, 254 (2008), 1864-1902. doi: 10.1016/J.Jfa.2007.12.015.

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, 2000), Internat. Ser. Numer. Math. 139.. Birkh..auser, Basel, (2002), 31-42. doi: 10.1007/978-3-0348-8148-7_3.

[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), 973-978. doi: 10.1016/S0764-4442(00)88588-2.

[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), 161-204. doi: 10.1007/s00028-006-0222-6.

[4]

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.

[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), 607-635. doi: 10.3934/cpaa.2004.3.607.

[6]

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

[7]

T. Cazenave and A. Haraux, Introduction aux problemes d' évolution semi-lineaires, in "Mathe'matiques et Applications,'' Ellipses, Paris, 1990.

[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), 272-292. doi: 10.1007/BF00250466.

[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), 45-69.

[10]

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

[11]

A. V. Fursikov and O. Yu Imanuvilov, "Controllability of Evolution Equations," Lecture Notes Series 34, Seoul National University, Seoul, Korea, 1996.

[12]

P. Martinez and J. Vancostenoble, Carleman estimates for one - dimensional degenerate heat equations, J. Evol. Equ., 6 (2006), 161-204. doi: 10.1007/s00028-006-0214-6.

[13]

P. Martinez, J. P. Raymond and J. Vancostenoble, Regional null controllability for a linearized Crocco-type equation, SIAM J. Control Optim., 42 (2003), 709-728. doi: 10.1137/S0363012902403547.

[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), 255-259. doi: 10.1175/1520-0469(1979)036<0255:VFOBSC>2.0.CO;2.

[15]

T. I. Seidman, Exact boundary control for some evolution equations, SIAM J. Control Optim., 16 (1978), 979-999. doi: 10.1137/0316066.

[16]

N. Shimakura, "Partial Differential Operatots of Elliptic Type," Translations of Mathematical Monographs. 99, American Mathematical Society, Providence, RI, 1992.

[17]

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

[18]

J. Vancostenoble, Improved Hardy-Poincare inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discrete Contin. Dyn. Syst.Ser. S, 4 (2011), 761-790. doi: 10.3934/dcdss.2011.4.761.

[19]

J. Vancostenoble, Lipschitz stability in inverse source problems for singular parabolic equations, Communications in Partial Differential Equations, 36 (2011), 1287-1317. doi: 10.1080/03605302.2011.587491.

[20]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, Journal of Functional Analysis, 254 (2008), 1864-1902. doi: 10.1016/J.Jfa.2007.12.015.

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