doi: 10.3934/eect.2021055
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Boundary controllability for a coupled system of degenerate/singular parabolic equations

1. 

Hassan First University of Settat, Faculté des Sciences et Techniques, MISI Laboratory, B.P. 577, Settat 26000, Morocco

2. 

Moulay Ismail University of Meknes, FST Errachidia, MAIS Laboratory, MAMCS Group, P.O. Box 509, Boutalamine 52000, Errachidia, Morocco

* Corresponding author: Amine Sbai

Received  April 2021 Early access October 2021

In this paper we study the boundary controllability for a system of two coupled degenerate/singular parabolic equations with a control acting on only one equation. We analyze both approximate and null boundary controllability properties. Besides, we provide an estimate on the null-control cost. The proofs are based on a detailed spectral analysis and the use of the moment method by Fattorini and Russell together with some results on biorthogonal families.

Citation: Brahim Allal, Abdelkarim Hajjaj, Jawad Salhi, Amine Sbai. Boundary controllability for a coupled system of degenerate/singular parabolic equations. Evolution Equations & Control Theory, doi: 10.3934/eect.2021055
References:
[1]

B. Allal, G. Fragnelli and J. Salhi, Controllability for degenerate/singular parabolic systems involving memory terms, submitted. Google Scholar

[2]

B. AllalA. HajjajL. Maniar and J. Salhi, Null controllability for singular cascade systems of $n$-coupled degenerate parabolic equations by one control force, Evol. Equ. Control Theory, 10 (2021), 545-573.  doi: 10.3934/eect.2020080.  Google Scholar

[3]

B. Allal, J. Salhi and A. Sbai, Boundary controllability for a coupled system of parabolic equations with singular potentials, in revision, 2021. Google Scholar

[4]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.  Google Scholar

[5]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151.  doi: 10.1016/j.jfa.2014.07.024.  Google Scholar

[6]

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

[7]

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

[8]

A. BenabdallahF. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. H. Lebesgue, 3 (2020), 717-793.  doi: 10.5802/ahl.45.  Google Scholar

[9]

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

[10]

U. Biccari, V. Hernández-Santamaría and J. Vancostenoble, Existence and Cost of Boundary Controls for a Degenerate/Singular Parabolic Equation, Mathematical Control & Related Fields, 2021. Google Scholar

[11]

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

[12]

P. CannarsaP. Martinez and J. Vancotsenoble, 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. Cannarsa, P. Martinez and J. Vancotsenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 2, 50 pp. doi: 10.1051/cocv/2018007.  Google Scholar

[14]

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

[15]

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

[16]

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[18]

M. Duprez, Controllability of a $2\times2$ parabolic system by one force with space-dependent coupling term of order one, ESAIM Control Optim. Calc. Var., 23 (2017), 1473-1498.  doi: 10.1051/cocv/2016061.  Google Scholar

[19]

Á. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math., 133 (2001), 65-83.  doi: 10.1016/S0377-0427(00)00635-X.  Google Scholar

[20]

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

[21]

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, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[22]

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

[23]

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

[24]

G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates, J. Differential Equations, 260 (2016), 1314-1371.  doi: 10.1016/j.jde.2015.09.019.  Google Scholar

[25]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[26]

M. González-Burgos and G. R. Sousa-Neto, Boundary controllability of a one-dimensional phase-field system with one control force, J. Diffrential Equations, 269 (2020), 4286-4331.  doi: 10.1016/j.jde.2020.03.036.  Google Scholar

[27]

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

[28]

A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), Paper No. 292, 25 pp.  Google Scholar

[29]

C. Heil, A Basis Theory Primer, Expanded Edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-4687-5.  Google Scholar

[30]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005.  Google Scholar

[31]

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

[32]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 3. (French) Travaux et Recherches Mathématiques, No. 20. Dunod, Paris, 1970.  Google Scholar

[33]

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

[34]

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

[35]

J. Salhi, Null controllability for a singular coupled system of degenerate parabolic equations in nondivergence form, Electron. J. Qual. Theory Differ. Equ., (2018), Paper No. 31, 28 pp. doi: 10.14232/ejqtde.2018.1.31.  Google Scholar

[36]

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

[37]

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

[38] G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, 1944.   Google Scholar
[39]

J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser, Boston, 1995. doi: 10.1007/978-0-8176-4733-9.  Google Scholar

show all references

References:
[1]

B. Allal, G. Fragnelli and J. Salhi, Controllability for degenerate/singular parabolic systems involving memory terms, submitted. Google Scholar

[2]

B. AllalA. HajjajL. Maniar and J. Salhi, Null controllability for singular cascade systems of $n$-coupled degenerate parabolic equations by one control force, Evol. Equ. Control Theory, 10 (2021), 545-573.  doi: 10.3934/eect.2020080.  Google Scholar

[3]

B. Allal, J. Salhi and A. Sbai, Boundary controllability for a coupled system of parabolic equations with singular potentials, in revision, 2021. Google Scholar

[4]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, New phenomena for the null controllability of parabolic systems: Minimal time and geometrical dependence, J. Math. Anal. Appl., 444 (2016), 1071-1113.  doi: 10.1016/j.jmaa.2016.06.058.  Google Scholar

[5]

F. Ammar-KhodjaA. BenabdallahM. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: The effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151.  doi: 10.1016/j.jfa.2014.07.024.  Google Scholar

[6]

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

[7]

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

[8]

A. BenabdallahF. Boyer and M. Morancey, A block moment method to handle spectral condensation phenomenon in parabolic control problems, Ann. H. Lebesgue, 3 (2020), 717-793.  doi: 10.5802/ahl.45.  Google Scholar

[9]

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

[10]

U. Biccari, V. Hernández-Santamaría and J. Vancostenoble, Existence and Cost of Boundary Controls for a Degenerate/Singular Parabolic Equation, Mathematical Control & Related Fields, 2021. Google Scholar

[11]

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

[12]

P. CannarsaP. Martinez and J. Vancotsenoble, 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. Cannarsa, P. Martinez and J. Vancotsenoble, The cost of controlling strongly degenerate parabolic equations, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 2, 50 pp. doi: 10.1051/cocv/2018007.  Google Scholar

[14]

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

[15]

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

[16]

J.-M. Coron, Control and Nonlinearity, Mathematical Surveys and Monographs, 136, American Mathematical Society, Providence, RI, 2007. doi: 10.1090/surv/136.  Google Scholar

[17]

E. B. Davies, Spectral Theory And Differential Operators, Cambridge Studies in Advanced Mathematics, vol. 42, Cambridge University Press, Cambridge, 1995. Google Scholar

[18]

M. Duprez, Controllability of a $2\times2$ parabolic system by one force with space-dependent coupling term of order one, ESAIM Control Optim. Calc. Var., 23 (2017), 1473-1498.  doi: 10.1051/cocv/2016061.  Google Scholar

[19]

Á. Elbert, Some recent results on the zeros of Bessel functions and orthogonal polynomials, J. Comput. Appl. Math., 133 (2001), 65-83.  doi: 10.1016/S0377-0427(00)00635-X.  Google Scholar

[20]

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

[21]

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, Quart. Appl. Math., 32 (1974/75), 45-69.  doi: 10.1090/qam/510972.  Google Scholar

[22]

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

[23]

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

[24]

G. Fragnelli, Interior degenerate/singular parabolic equations in nondivergence form: well-posedness and Carleman estimates, J. Differential Equations, 260 (2016), 1314-1371.  doi: 10.1016/j.jde.2015.09.019.  Google Scholar

[25]

G. Fragnelli and D. Mugnai, Carleman estimates for singular parabolic equations with interior degeneracy and non smooth coefficients, Adv. Nonlinear Anal., 6 (2017), 61-84.  doi: 10.1515/anona-2015-0163.  Google Scholar

[26]

M. González-Burgos and G. R. Sousa-Neto, Boundary controllability of a one-dimensional phase-field system with one control force, J. Diffrential Equations, 269 (2020), 4286-4331.  doi: 10.1016/j.jde.2020.03.036.  Google Scholar

[27]

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

[28]

A. Hajjaj, L. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), Paper No. 292, 25 pp.  Google Scholar

[29]

C. Heil, A Basis Theory Primer, Expanded Edition, Applied and Numerical Harmonic Analysis, Birkhäuser/Springer, New York, 2011. doi: 10.1007/978-0-8176-4687-5.  Google Scholar

[30]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005.  Google Scholar

[31]

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

[32]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, Vol. 3. (French) Travaux et Recherches Mathématiques, No. 20. Dunod, Paris, 1970.  Google Scholar

[33]

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

[34]

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

[35]

J. Salhi, Null controllability for a singular coupled system of degenerate parabolic equations in nondivergence form, Electron. J. Qual. Theory Differ. Equ., (2018), Paper No. 31, 28 pp. doi: 10.14232/ejqtde.2018.1.31.  Google Scholar

[36]

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

[37]

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

[38] G. N. Watson, A Treatise on the Theory of Bessel Functions, second edition, Cambridge University Press, Cambridge, 1944.   Google Scholar
[39]

J. Zabczyk, Mathematical Control Theory: An Introduction, Birkhäuser, Boston, 1995. doi: 10.1007/978-0-8176-4733-9.  Google Scholar

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