• 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

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

[6]

P. CannarsaP. 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. CannarsaP. 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. CannarsaJ. 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-CaraM. 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. GiriK. S. GuptaS. 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. ArarunaE. 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. CannarsaP. Martinez and J. Vancostenoble, Null controllability of degenerate heat equations, Adv. Diff. Eq., 10 (2005), 153-190.   Google Scholar

[6]

P. CannarsaP. 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. CannarsaP. 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. CannarsaJ. 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-CaraM. 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. GiriK. S. GuptaS. 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]

Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control & Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034

[2]

Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021004

[3]

Oleg Yu. Imanuvilov, Jean Pierre Puel. On global controllability of 2-D Burgers equation. Discrete & Continuous Dynamical Systems - A, 2009, 23 (1&2) : 299-313. doi: 10.3934/dcds.2009.23.299

[4]

Xinfu Chen, Huiqiang Jiang, Guoqing Liu. Boundary spike of the singular limit of an energy minimizing problem. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3253-3290. doi: 10.3934/dcds.2020124

[5]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[6]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[7]

Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi. Solvability and sliding mode control for the viscous Cahn–Hilliard system with a possibly singular potential. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020051

[8]

Kuntal Bhandari, Franck Boyer. Boundary null-controllability of coupled parabolic systems with Robin conditions. Evolution Equations & Control Theory, 2021, 10 (1) : 61-102. doi: 10.3934/eect.2020052

[9]

Chueh-Hsin Chang, Chiun-Chuan Chen, Chih-Chiang Huang. Traveling wave solutions of a free boundary problem with latent heat effect. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021028

[10]

Franck Davhys Reval Langa, Morgan Pierre. A doubly splitting scheme for the Caginalp system with singular potentials and dynamic boundary conditions. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 653-676. doi: 10.3934/dcdss.2020353

[11]

Dong-Ho Tsai, Chia-Hsing Nien. On space-time periodic solutions of the one-dimensional heat equation. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3997-4017. doi: 10.3934/dcds.2020037

[12]

Haruki Umakoshi. A semilinear heat equation with initial data in negative Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2021, 14 (2) : 745-767. doi: 10.3934/dcdss.2020365

[13]

Feifei Cheng, Ji Li. Geometric singular perturbation analysis of Degasperis-Procesi equation with distributed delay. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 967-985. doi: 10.3934/dcds.2020305

[14]

Mokhtar Bouloudene, Manar A. Alqudah, Fahd Jarad, Yassine Adjabi, Thabet Abdeljawad. Nonlinear singular $ p $ -Laplacian boundary value problems in the frame of conformable derivative. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020442

[15]

Manuel del Pino, Monica Musso, Juncheng Wei, Yifu Zhou. Type Ⅱ finite time blow-up for the energy critical heat equation in $ \mathbb{R}^4 $. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3327-3355. doi: 10.3934/dcds.2020052

[16]

Duy Phan. Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set. Evolution Equations & Control Theory, 2021, 10 (1) : 199-227. doi: 10.3934/eect.2020062

[17]

Maho Endo, Yuki Kaneko, Yoshio Yamada. Free boundary problem for a reaction-diffusion equation with positive bistable nonlinearity. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3375-3394. doi: 10.3934/dcds.2020033

[18]

Pavel Eichler, Radek Fučík, Robert Straka. Computational study of immersed boundary - lattice Boltzmann method for fluid-structure interaction. Discrete & Continuous Dynamical Systems - S, 2021, 14 (3) : 819-833. doi: 10.3934/dcdss.2020349

[19]

Zhiting Ma. Navier-Stokes limit of globally hyperbolic moment equations. Kinetic & Related Models, 2021, 14 (1) : 175-197. doi: 10.3934/krm.2021001

[20]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

2019 Impact Factor: 0.857

Metrics

  • PDF downloads (98)
  • HTML views (555)
  • Cited by (0)

Other articles
by authors

[Back to Top]