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March  2019, 9(1): 191-219. doi: 10.3934/mcrf.2019011

Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential

Umberto Biccari

Facultad Ingeniería, Universidad de Deusto, Avda Universidades 24, 48007 Bilbao, Basque Country, Spain

Received  May 2016 Revised  May 2017 Published  November 2018

Fund Project: This project has received funding from the European Research Council (ERC) under the European Union's Horizon 2020 research and innovation programme (grant agreement No. 694126DyCon). Moreover, this work was partially supported by the Grants MTM2014-52347, MTM201792996 of MINECO (Spain), and AFOSR Grant FA9550-18-1-0242.

We analyze controllability properties for the one-dimensional heat equation with singular inverse-square potential
$\begin{align*} u_t-u_{xx}-\frac{\mu}{x^2}u = 0, \;\;\; (x, t)\in(0, 1)\times(0, T).\end{align*}$
For any
$\mu<1/4$
, we prove that the equation is null controllable through a boundary control
$f\in H^1(0, T)$
 acting at the singularity point x = 0. This result is obtained employing the moment method by Fattorini and Russell.
Keywords: Heat equation, singular potential, boundary controllability, moment method.
Mathematics Subject Classification: 35K05, 35K67, 93B05, 93B07.
Citation: Umberto Biccari. Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential. Mathematical Control and Related Fields, 2019, 9 (1) : 191-219. doi: 10.3934/mcrf.2019011
References:
[1]

F. Araruna, E. Fernández-Cara and M. Santos, Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM: COCV, 21 (2015), 835-856.  doi: 10.1051/cocv/2014052.

[2]

P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  doi: 10.1090/S0002-9947-1984-0742415-3.

[3]

H. Berestycki and M. J. Esteban, Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations, 134 (1997), 1-25.  doi: 10.1006/jdeq.1996.3165.

[4]

U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.

[5]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Diff. Eq., 10 (2005), 153-190. 

[6]

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.

[7]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman Estimates for Degenerate Parabolic Operators with Applications, vol. 239. American Mathematical Society, 2016. doi: 10.1090/memo/1133.

[8]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Mat. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.

[9]

P. Cannarsa, P. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, arXiv preprint, arXiv: 1706.02435, (2017).

[10]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, Indiana Univ. Math. J., 49 (2000), 815-818, arXiv: 1801.01380. doi: 10.1512/iumj.2000.49.2110.

[11]

P. Cannarsa, J. Tort and M. Yamamoto, Unique continuation and approximate controllability for a degenerate parabolic equation, Appl. Anal., 91 (2012), 1409-1425.  doi: 10.1080/00036811.2011.639766.

[12]

C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results, J. Funct. Anal., 263 (2012), 3741-3783.  doi: 10.1016/j.jfa.2012.09.006.

[13]

C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM J. Control Optim., 52 (2014), 2055-2089.  doi: 10.1137/120862557.

[14]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Part. Diff. Eq., 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.

[15]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[16]

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, Quarterly of Applied Mathematics, 32 (1974), 45-69.  doi: 10.1090/qam/510972.

[17]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.

[18]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differ. Equ., 5 (2000), 465-514. 

[19]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, In Ann. Inst. H. Poincarè Anal. Non Linèaire, 17 (2000), 583-616. doi: 10.1016/S0294-1449(00)00117-7.

[20]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.

[21]

P. R. Giri, K. S. Gupta, S. Meljanac and A. Samsarov, Electron capture and scaling anomaly in polar molecules, Phys. Lett. A, 372 (2008), 2967-2970. 

[22]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.

[23]

V. Hernández-Santamaría and L. de Teresa, Robust Stackelberg controllability for linear and semilinear heat equations, Evol. Equ. Control Theory, 7 (2018), 247-273.  doi: 10.3934/eect.2018012.

[24]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Science & Business Media, 2005.

[25]

L. Landau, Bessel functions: Monotonicity and bounds, Journal of the London Mathematical Society, 61 (2000), 197-215.  doi: 10.1112/S0024610799008352.

[26]

N. N. Lebedev and R. A. Silverman, Special Functions and Their Applications, Englewood Cliffs, N.J. 1965

[27]

L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numerical Algorithms, 49 (2008), 221-233.  doi: 10.1007/s11075-008-9189-4.

[28]

P. Martinez and J. Vancostenoble, The cost of boundary controllability for a parabolic equation with inverse square potential, Submitted.

[29]

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.

[30]

J. Vancostenoble, Improved Hardy-Poincaré 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.

[31]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015.

[32]

J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials, SIAM J. Math. Anal., 41 (2009), 1508-1532.  doi: 10.1137/080731396.

[33]

J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.

[34]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge university press, 1995.

show all references

References:
[1]

F. Araruna, E. Fernández-Cara and M. Santos, Stackelberg-Nash exact controllability for linear and semilinear parabolic equations, ESAIM: COCV, 21 (2015), 835-856.  doi: 10.1051/cocv/2014052.

[2]

P. Baras and J. A. Goldstein, The heat equation with a singular potential, Trans. Amer. Math. Soc., 284 (1984), 121-139.  doi: 10.1090/S0002-9947-1984-0742415-3.

[3]

H. Berestycki and M. J. Esteban, Existence and bifurcation of solutions for an elliptic degenerate problem, J. Differential Equations, 134 (1997), 1-25.  doi: 10.1006/jdeq.1996.3165.

[4]

U. Biccari and E. Zuazua, Null controllability for a heat equation with a singular inverse-square potential involving the distance to the boundary function, J. Differential Equations, 261 (2016), 2809-2853.  doi: 10.1016/j.jde.2016.05.019.

[5]

P. Cannarsa, P. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Diff. Eq., 10 (2005), 153-190. 

[6]

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.

[7]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman Estimates for Degenerate Parabolic Operators with Applications, vol. 239. American Mathematical Society, 2016. doi: 10.1090/memo/1133.

[8]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling weakly degenerate parabolic equations by boundary controls, Mat. Control Relat. Fields, 7 (2017), 171-211.  doi: 10.3934/mcrf.2017006.

[9]

P. Cannarsa, P. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, arXiv preprint, arXiv: 1706.02435, (2017).

[10]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, Indiana Univ. Math. J., 49 (2000), 815-818, arXiv: 1801.01380. doi: 10.1512/iumj.2000.49.2110.

[11]

P. Cannarsa, J. Tort and M. Yamamoto, Unique continuation and approximate controllability for a degenerate parabolic equation, Appl. Anal., 91 (2012), 1409-1425.  doi: 10.1080/00036811.2011.639766.

[12]

C. Cazacu, Schrödinger operators with boundary singularities: Hardy inequality, Pohozaev identity and controllability results, J. Funct. Anal., 263 (2012), 3741-3783.  doi: 10.1016/j.jfa.2012.09.006.

[13]

C. Cazacu, Controllability of the heat equation with an inverse-square potential localized on the boundary, SIAM J. Control Optim., 52 (2014), 2055-2089.  doi: 10.1137/120862557.

[14]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Part. Diff. Eq., 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.

[15]

H. O. Fattorini and D. L. Russell, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292.  doi: 10.1007/BF00250466.

[16]

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, Quarterly of Applied Mathematics, 32 (1974), 45-69.  doi: 10.1090/qam/510972.

[17]

E. Fernández-Cara, M. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 259 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.

[18]

E. Fernández-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differ. Equ., 5 (2000), 465-514. 

[19]

E. Fernández-Cara and E. Zuazua, Null and approximate controllability for weakly blowing up semilinear heat equations, In Ann. Inst. H. Poincarè Anal. Non Linèaire, 17 (2000), 583-616. doi: 10.1016/S0294-1449(00)00117-7.

[20]

J. Garcia Azorero and I. Peral Alonso, Hardy inequalities and some critical elliptic and parabolic problems, J. Differential Equations, 144 (1998), 441-476.  doi: 10.1006/jdeq.1997.3375.

[21]

P. R. Giri, K. S. Gupta, S. Meljanac and A. Samsarov, Electron capture and scaling anomaly in polar molecules, Phys. Lett. A, 372 (2008), 2967-2970. 

[22]

M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim., 52 (2014), 2037-2054.  doi: 10.1137/120901374.

[23]

V. Hernández-Santamaría and L. de Teresa, Robust Stackelberg controllability for linear and semilinear heat equations, Evol. Equ. Control Theory, 7 (2018), 247-273.  doi: 10.3934/eect.2018012.

[24]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Science & Business Media, 2005.

[25]

L. Landau, Bessel functions: Monotonicity and bounds, Journal of the London Mathematical Society, 61 (2000), 197-215.  doi: 10.1112/S0024610799008352.

[26]

N. N. Lebedev and R. A. Silverman, Special Functions and Their Applications, Englewood Cliffs, N.J. 1965

[27]

L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numerical Algorithms, 49 (2008), 221-233.  doi: 10.1007/s11075-008-9189-4.

[28]

P. Martinez and J. Vancostenoble, The cost of boundary controllability for a parabolic equation with inverse square potential, Submitted.

[29]

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.

[30]

J. Vancostenoble, Improved Hardy-Poincaré 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.

[31]

J. Vancostenoble and E. Zuazua, Null controllability for the heat equation with singular inverse-square potentials, J. Funct. Anal., 254 (2008), 1864-1902.  doi: 10.1016/j.jfa.2007.12.015.

[32]

J. Vancostenoble and E. Zuazua, Hardy inequalities, observability, and control for the wave and Schrödinger equations with singular potentials, SIAM J. Math. Anal., 41 (2009), 1508-1532.  doi: 10.1137/080731396.

[33]

J. L. Vázquez and E. Zuazua, The Hardy inequality and the asymptotic behaviour of the heat equation with an inverse-square potential, J. Funct. Anal., 173 (2000), 103-153.  doi: 10.1006/jfan.1999.3556.

[34]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge university press, 1995.

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