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doi: 10.3934/mcrf.2021032
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Existence and cost of boundary controls for a degenerate/singular parabolic equation

1. 

Chair of Computational Mathematics, Fundación Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain, Facultad de Ingeniería, Universidad de Deusto, Avda. de las Universidades 24, 48007 Bilbao, Basque Country, Spain

2. 

Instituto de Matemáticas, Universidad Nacional Autónoma de México, Circuito Exterior, C.U., C.P. 04510 CDMX, Mexico

3. 

Institut de mathématiques de Toulouse, UMR5219; Université de Toulouse, CNRS, UPS IMT, F-31062 Toulouse Cedex 9, France

* Corresponding author: Umberto Biccari

Received  January 2020 Revised  December 2020 Early access June 2021

Fund Project: This project has received funding from the European Research Council (ERC) under the European Unions Horizon 2020 research and innovation programme (grant agreement NO. 694126-DyCon). The work of U. B. was partially supported by the Grant MTM2017-92996-C2-1-R COSNET of MINECO (Spain), by the Elkartek grant KK-2020/00091 CONVADP and by the Air Force Office of Scientific Research (AFOSR) under Award NO. FA9550-18-1-0242. The work of V. H.-S. was supported by the programme "Estancias posdoctorales por México" of CONACyT, Mexico

In this paper, we consider the following degenerate/singular parabolic equation
$ \begin{align*} u_t -(x^\alpha u_{x})_x - \frac{\mu}{x^{2-\alpha}} u = 0, \qquad x\in (0,1), \ t \in (0,T), \end{align*} $
where
$ 0\leq \alpha <1 $
and
$ \mu\leq (1-\alpha)^2/4 $
are two real parameters. We prove the boundary null controllability by means of a
$ H^1(0,T) $
control acting either at
$ x = 1 $
or at the point of degeneracy and singularity
$ x = 0 $
. Besides we give sharp estimates of the cost of controllability in both cases in terms of the parameters
$ \alpha $
and
$ \mu $
. The proofs are based on the classical moment method by Fattorini and Russell and on recent results on biorthogonal sequences.
Citation: U. Biccari, V. Hernández-Santamaría, J. Vancostenoble. Existence and cost of boundary controls for a degenerate/singular parabolic equation. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021032
References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

[2]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[3]

G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 601-616.   Google Scholar

[4]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[5]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.  Google Scholar

[6]

A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Systems and Control: Foundations and Appliactions, Birkhauser-Boston, 1992.  Google Scholar

[7]

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

[8]

U. Biccari, Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential, Math. Control Relat. Fields, 9 (2019), 191-219.  doi: 10.3934/mcrf.2019011.  Google Scholar

[9]

P. CannarsaG. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616.  doi: 10.1007/s00028-008-0353-34.  Google Scholar

[10]

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

[11]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016), ix+209 pp. doi: 10.1090/memo/1133.  Google Scholar

[12]

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

[13]

P. CannarsaP. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 1441-1472.  doi: 10.3934/dcdss.2020082.  Google Scholar

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM: Control Optim. Calc. Var., 26 (2020), 50pp. doi: 10.1051/cocv/2018007.  Google Scholar

[15]

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

[16]

J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257.   Google Scholar

[17]

G. Da Prato, An Introduction to Infinite Dimensional Analysis, Scuola Normale Superiore, Pisa, 2001.  Google Scholar

[18]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210.  Google Scholar

[19]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.  Google Scholar

[20]

H. O. Fattorini and D. L. Russel, 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

[21]

H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[22]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst., 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5.  Google Scholar

[23]

O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035.  Google Scholar

[24]

F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problem: The linear case, SIAM J. Control Optim., 37 (1999), 1195-1221.  doi: 10.1137/S0363012996312763.  Google Scholar

[25]

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

[26]

E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, J. Math. Anal. Appl., 110 (1985), 519-527.  doi: 10.1016/0022-247X(85)90313-0.  Google Scholar

[27]

A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 25pp.  Google Scholar

[28]

S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U.  Google Scholar

[29]

I. Lasiecka and T. I. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120.   Google Scholar

[30]

P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676.  doi: 10.1137/140951746.  Google Scholar

[31]

P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352.  doi: 10.1016/j.jde.2015.06.031.  Google Scholar

[32]

L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226.  doi: 10.1016/j.jde.2004.05.007.  Google Scholar

[33]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.  Google Scholar

[34]

N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York, 1972.  Google Scholar

[35]

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

[36]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6.  Google Scholar

[37]

P. Martinez and J. Vancostenoble, The cost of boundary controllability for a parabolic equation with inverse square potential, Evol. Equ. Control Theory, 8 (2019), 397-422.  doi: 10.3934/eect.2019020.  Google Scholar

[38]

C. K. Qu and R. Wong, "Best possible" upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859.  doi: 10.1090/S0002-9947-99-02165-0.  Google Scholar

[39]

T. I. Seidman, How violent are fast controls?, Math. Control Signals Systems, 1 (1988), 89-95.  doi: 10.1007/BF02551238.  Google Scholar

[40]

T. I. Seidman and J. Yong, How violent are fast controls? Ⅱ, Math. Control Signals Systems, 9 (1996), 327-340.  doi: 10.1007/BF01211854.  Google Scholar

[41]

T. I. SeidmanS. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254.  doi: 10.1007/BF02511154.  Google Scholar

[42]

G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019.  Google Scholar

[43]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discr. Cont. Dyn. Syst., 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.  Google Scholar

[44]

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

[45]

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

[46]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, The Macmillan Company, New York, 1944  Google Scholar

show all references

References:
[1]

F. Alabau-BoussouiraP. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204.  doi: 10.1007/s00028-006-0222-6.  Google Scholar

[2]

F. Ammar KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590.  doi: 10.1016/j.matpur.2011.06.005.  Google Scholar

[3]

G. Avalos and I. Lasiecka, Optimal blowup rates for the minimal energy null control of the strongly damped abstract wave equation, Ann. Sc. Norm. Super. Pisa Cl. Sci., 2 (2003), 601-616.   Google Scholar

[4]

G. Avalos and I. Lasiecka, The null controllability of thermoelastic plates and singularity of the associated minimal energy function, J. Math. Anal. Appl., 294 (2004), 34-61.  doi: 10.1016/j.jmaa.2004.01.035.  Google Scholar

[5]

A. BenabdallahF. BoyerM. González-Burgos and G. Olive, Sharp estimates of the one-dimensional boundary control cost for parabolic systems and applications to the $N$-dimensional boundary null controllability in cylindrical domains, SIAM J. Control Optim., 52 (2014), 2970-3001.  doi: 10.1137/130929680.  Google Scholar

[6]

A. Bensoussan, G. Da Prato, M.C. Delfour and S.K. Mitter, Representation and Control of Infinite Dimensional Systems, Vol. I, Systems and Control: Foundations and Appliactions, Birkhauser-Boston, 1992.  Google Scholar

[7]

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

[8]

U. Biccari, Boundary controllability for a one-dimensional heat equation with a singular inverse-square potential, Math. Control Relat. Fields, 9 (2019), 191-219.  doi: 10.3934/mcrf.2019011.  Google Scholar

[9]

P. CannarsaG. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616.  doi: 10.1007/s00028-008-0353-34.  Google Scholar

[10]

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

[11]

P. Cannarsa, P. Martinez and J. Vancostenoble, Global carleman estimates for degenerate parabolic operators with applications, Memoirs of the American Mathematical Society, 239 (2016), ix+209 pp. doi: 10.1090/memo/1133.  Google Scholar

[12]

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

[13]

P. CannarsaP. Martinez and J. Vancostenoble, Precise estimates for biorthogonal families under asymptotic gap conditions, Discrete Contin. Dyn. Syst. Ser. S., 13 (2020), 1441-1472.  doi: 10.3934/dcdss.2020082.  Google Scholar

[14]

P. Cannarsa, P. Martinez and J. Vancostenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM: Control Optim. Calc. Var., 26 (2020), 50pp. doi: 10.1051/cocv/2018007.  Google Scholar

[15]

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

[16]

J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257.   Google Scholar

[17]

G. Da Prato, An Introduction to Infinite Dimensional Analysis, Scuola Normale Superiore, Pisa, 2001.  Google Scholar

[18]

G. Da Prato and J. Zabczyk, Second Order Partial Differential Equations in Hilbert Spaces, London Mathematical Society Lecture Note Series, 293. Cambridge University Press, Cambridge, 2002. doi: 10.1017/CBO9780511543210.  Google Scholar

[19]

S. Ervedoza, Control and stabilization properties for a singular heat equation with an inverse-square potential, Comm. Partial Differential Equations, 33 (2008), 1996-2019.  doi: 10.1080/03605300802402633.  Google Scholar

[20]

H. O. Fattorini and D. L. Russel, 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

[21]

H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 32 (1974), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[22]

M. Fotouhi and L. Salimi, Null controllability of degenerate/singular parabolic equations, J. Dyn. Control Syst., 18 (2012), 573-602.  doi: 10.1007/s10883-012-9160-5.  Google Scholar

[23]

O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868.  doi: 10.1016/j.jfa.2009.06.035.  Google Scholar

[24]

F. Gozzi and P. Loreti, Regularity of the minimum time function and minimum energy problem: The linear case, SIAM J. Control Optim., 37 (1999), 1195-1221.  doi: 10.1137/S0363012996312763.  Google Scholar

[25]

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

[26]

E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, J. Math. Anal. Appl., 110 (1985), 519-527.  doi: 10.1016/0022-247X(85)90313-0.  Google Scholar

[27]

A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 25pp.  Google Scholar

[28]

S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, J. Math. Anal. Appl., 158 (1991), 487-508.  doi: 10.1016/0022-247X(91)90252-U.  Google Scholar

[29]

I. Lasiecka and T. I. Seidman, Blowup estimates for observability of a thermoelastic system, Asymptot. Anal., 50 (2006), 93-120.   Google Scholar

[30]

P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676.  doi: 10.1137/140951746.  Google Scholar

[31]

P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352.  doi: 10.1016/j.jde.2015.06.031.  Google Scholar

[32]

L. Miller, Geometric bounds on the growth rate of null-controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226.  doi: 10.1016/j.jde.2004.05.007.  Google Scholar

[33]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer Monographs in Mathematics. Springer-Verlag, New York, 2005.  Google Scholar

[34]

N. N. Lebedev, Special Functions and Their Applications, Dover Publications, Inc., New York, 1972.  Google Scholar

[35]

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

[36]

P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq., 6 (2006), 325-362.  doi: 10.1007/s00028-006-0214-6.  Google Scholar

[37]

P. Martinez and J. Vancostenoble, The cost of boundary controllability for a parabolic equation with inverse square potential, Evol. Equ. Control Theory, 8 (2019), 397-422.  doi: 10.3934/eect.2019020.  Google Scholar

[38]

C. K. Qu and R. Wong, "Best possible" upper and lower bounds for the zeros of the Bessel function $J_\nu(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859.  doi: 10.1090/S0002-9947-99-02165-0.  Google Scholar

[39]

T. I. Seidman, How violent are fast controls?, Math. Control Signals Systems, 1 (1988), 89-95.  doi: 10.1007/BF02551238.  Google Scholar

[40]

T. I. Seidman and J. Yong, How violent are fast controls? Ⅱ, Math. Control Signals Systems, 9 (1996), 327-340.  doi: 10.1007/BF01211854.  Google Scholar

[41]

T. I. SeidmanS. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254.  doi: 10.1007/BF02511154.  Google Scholar

[42]

G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrödinger and heat equations, J. Differential Equations, 243 (2007), 70-100.  doi: 10.1016/j.jde.2007.06.019.  Google Scholar

[43]

J. Vancostenoble, Improved Hardy-Poincaré inequalities and sharp Carleman estimates for degenerate/singular parabolic problems, Discr. Cont. Dyn. Syst., 4 (2011), 761-790.  doi: 10.3934/dcdss.2011.4.761.  Google Scholar

[44]

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

[45]

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

[46]

G. N. Watson, A Treatise on the Theory of Bessel Functions, Cambridge University Press, Cambridge, England, The Macmillan Company, New York, 1944  Google Scholar

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