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Extension of the strong law of large numbers for capacities
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 |
$\begin{align*} u_t-u_{xx}-\frac{\mu}{x^2}u = 0, \;\;\; (x, t)\in(0, 1)\times(0, T).\end{align*}$ |
$\mu<1/4$ |
$f\in H^1(0, T)$ |
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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