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

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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

Abstract
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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

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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

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