American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Erratum on: Controllability of the heat and wave equations and their finite difference approximations by the shape of the domain
  • MCRF Home
  • This Issue
  • Next Article
    Extension of the strong law of large numbers for capacities
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

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

[25]

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

[26]

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

[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.  Google Scholar

[28]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

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

[1]

Víctor Hernández-Santamaría, Liliana Peralta. Some remarks on the Robust Stackelberg controllability for the heat equation with controls on the boundary. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 161-190. doi: 10.3934/dcdsb.2019177
[2]

Patrick Martinez, Judith Vancostenoble. The cost of boundary controllability for a parabolic equation with inverse square potential. Evolution Equations & Control Theory, 2019, 8 (2) : 397-422. doi: 10.3934/eect.2019020
[3]

Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042
[4]

Giacomo Dimarco. The moment guided Monte Carlo method for the Boltzmann equation. Kinetic & Related Models, 2013, 6 (2) : 291-315. doi: 10.3934/krm.2013.6.291
[5]

Vladimir Georgiev, Sandra Lucente. Focusing nlkg equation with singular potential. Communications on Pure & Applied Analysis, 2018, 17 (4) : 1387-1406. doi: 10.3934/cpaa.2018068
[6]

Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143
[7]

Matthias Eller. A remark on Littman's method of boundary controllability. Evolution Equations & Control Theory, 2013, 2 (4) : 621-630. doi: 10.3934/eect.2013.2.621
[8]

Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete & Continuous Dynamical Systems - A, 2009, 25 (3) : 991-1001. doi: 10.3934/dcds.2009.25.991
[9]

Laurence Cherfils, Stefania Gatti, Alain Miranville. A doubly nonlinear parabolic equation with a singular potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 51-66. doi: 10.3934/dcdss.2011.4.51
[10]

Kazuhiro Ishige, Y. Kabeya. Hot spots for the two dimensional heat equation with a rapidly decaying negative potential. Discrete & Continuous Dynamical Systems - S, 2011, 4 (4) : 833-849. doi: 10.3934/dcdss.2011.4.833
[11]

Kazuhiro Ishige, Asato Mukai. Large time behavior of solutions of the heat equation with inverse square potential. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4041-4069. doi: 10.3934/dcds.2018176
[12]

Sho Matsumoto, Jonathan Novak. A moment method for invariant ensembles. Electronic Research Announcements, 2018, 25: 60-71. doi: 10.3934/era.2018.25.007
[13]

Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305
[14]

Jamel Ben Amara, Hedi Bouzidi. Exact boundary controllability for the Boussinesq equation with variable coefficients. Evolution Equations & Control Theory, 2018, 7 (3) : 403-415. doi: 10.3934/eect.2018020
[15]

Abdelhakim Belghazi, Ferroudja Smadhi, Nawel Zaidi, Ouahiba Zair. Carleman inequalities for the two-dimensional heat equation in singular domains. Mathematical Control & Related Fields, 2012, 2 (4) : 331-359. doi: 10.3934/mcrf.2012.2.331
[16]

David Gómez-Castro, Juan Luis Vázquez. The fractional Schrödinger equation with singular potential and measure data. Discrete & Continuous Dynamical Systems - A, 2019, 39 (12) : 7113-7139. doi: 10.3934/dcds.2019298
[17]

Ning-An Lai, Jinglei Zhao. Potential well and exact boundary controllability for radial semilinear wave equations on Schwarzschild spacetime. Communications on Pure & Applied Analysis, 2014, 13 (3) : 1317-1325. doi: 10.3934/cpaa.2014.13.1317
[18]

Mi-Ho Giga, Yoshikazu Giga, Takeshi Ohtsuka, Noriaki Umeda. On behavior of signs for the heat equation and a diffusion method for data separation. Communications on Pure & Applied Analysis, 2013, 12 (5) : 2277-2296. doi: 10.3934/cpaa.2013.12.2277
[19]

Arnaud Heibig, Mohand Moussaoui. Exact controllability of the wave equation for domains with slits and for mixed boundary conditions. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 367-386. doi: 10.3934/dcds.1996.2.367
[20]

Abdelmouhcene Sengouga. Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints. Evolution Equations & Control Theory, 2020, 9 (1) : 1-25. doi: 10.3934/eect.2020014

2018 Impact Factor: 1.292

Tools

Metrics

  • PDF downloads (69)
  • HTML views (503)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences