September  2021, 10(3): 545-573. doi: 10.3934/eect.2020080

Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force

1. 

Faculté des Sciences et Techniques, Université Hassan 1er, Laboratoire MISI, B.P. 577, Settat 26000, Morocco

2. 

Département de Mathématiques, Faculté des Sciences Semlalia, LMDP, UMMISCO (IRD-UPMC), Université Cadi Ayyad, Marrakech 40000, B.P. 2390, Morocco

* Corresponding author: Lahcen Maniar

Received  June 2019 Revised  March 2020 Published  September 2021 Early access  July 2020

In this paper, we consider a class of cascade systems of $ n $-coupled degenerate parabolic equations with singular lower order terms. We assume that both degeneracy and singularity occur in the interior of the space domain and we focus on null controllability problem. To this aim, we prove first Carleman estimates for the associated adjoint problem, then, we infer from it an indirect observability inequality. As a consequence, we deduce null controllability result when a unique distributed control is exerted on the system.

Citation: Brahim Allal, Abdelkarim Hajjaj, Lahcen Maniar, Jawad Salhi. Null controllability for singular cascade systems of $ n $-coupled degenerate parabolic equations by one control force. Evolution Equations and Control Theory, 2021, 10 (3) : 545-573. doi: 10.3934/eect.2020080
References:
[1]

F. Alabau-Boussouira, A hierarchic multi-level energy method for the control of bidiagonal and mixed $n$-coupled cascade systems of PDE's by a reduced number of controls, Adv. Differential Equations, 8 (2013), 1005-1072. 

[2]

K. AtifiI. BoutaayamouH. O. Sidi and J. Salhi, An inverse source problem for singular parabolic equations with interior degeneracy, Abstract and Applied Analysis, 2018 (2018), 1-16.  doi: 10.1155/2018/2067304.

[3]

F. Ammar KhodjaA. BenabdallahM. Gonzalez-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[4]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy Problems, Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[5]

J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constant formula, Proc. Amer. Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.

[6]

U. Biccari, V. Hernandez-Santamaria and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equations, preprint, arXiv: 2001.11403.

[7]

I. BoutaayamouG. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, 2014 (2014), 1-26. 

[8]

I. Boutaayamou and J. Salhi, Null controllability for linear parabolic cascade systems with interior degeneracy, Electron. J. Differential Equations, 2016 (2016), 1-22. 

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.

[10]

P. Cannarsa, G. Floridia, F. Golgeleteyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 22 pp. doi: 10.1088/1361-6420/ab1c69.

[11]

P. CannarsaR. Ferretti and P. Martinez, Null controllability for parabolic operators with interior degeneracy and one-sided control, SIAM J. Control Optim., 57 (2019), 900-924.  doi: 10.1137/18M1198442.

[12]

P. Cannarsa and L. De Teresa, Controllability of $1$-D coupled degenerate parabolic equations, addendum and corrigendum, Electron. J. Differential Equations, 2009 (2009), 1-24. 

[13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998. 
[14]

John B. Conway, A Course in Functional Analysis, 2$^nd$ edition, Springer-Verlag, New York, 1990.

[15]

M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of $m$ equations with $m-1$ controls involving coupling terms of zero or first order, J. Math. Pures Appl., 9 (2016), 905-934.  doi: 10.1016/j.matpur.2016.03.016.

[16]

M. Fadili and L. Maniar, Null controllability of $n$-coupled degenerate parabolic systems with $m$-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.

[17]

E. Fernández-CaraM. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 7 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.

[18]

J. Carmelo Flores and L. de Teresa, Null controllability of one dimensional degenerate parabolic equations with first order terms, Discrete & Continuous Dynamical Systems - B, 22 (2017). doi: 10.3934/dcdsb.2020136.

[19]

G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, ESAIM Control Optim. Calc. Var., 26 (2020), 34 pp. doi: 10.1051/cocv/2019066.

[20]

M. Fotouhi and L. Salimi, Controllability results for a class of one dimensional degenerate/singular parabolic equations, Commun. Pure Appl. Anal., 12 (2013), 1415-1430.  doi: 10.3934/cpaa.2013.12.1415.

[21]

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.

[22]

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.

[23]

G. FragnelliG. R. GoldsteinJ. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$ type, Electron. J. Differential Equations, 2012 (2012), 1-30. 

[24]

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.

[25]

G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016), 84 pp. doi: 10.1090/memo/1146.

[26]

G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal., 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015.

[27]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lect. Notes Ser., 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[28]

M. González-Burgos and L. De Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.

[29]

A. HajjajL. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 1-25. 

[30]

A.Y. Khapalov, Global non-negative controllability of the semilinear parabolic equation governed by bilinear control, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 269-283.  doi: 10.1051/cocv:2002011.

[31]

J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.

[32]

J. Salhi, Null controllability for a singular coupled system of degenerate parabolic equations in nondivergence form, Electron. J. Qual. Theory Differ. Equ., 13 (2018), 1-28.  doi: 10.14232/ejqtde.2018.1.31.

[33]

J. Vancostenoble, Global non-negative approximate controllability of parabolic equations with singular potentials, in Trends in Control Theory and Partial Differential Equations, , Springer INdAM Series, 32, Springer, Cham, 2019,255–276.

[34]

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.

show all references

References:
[1]

F. Alabau-Boussouira, A hierarchic multi-level energy method for the control of bidiagonal and mixed $n$-coupled cascade systems of PDE's by a reduced number of controls, Adv. Differential Equations, 8 (2013), 1005-1072. 

[2]

K. AtifiI. BoutaayamouH. O. Sidi and J. Salhi, An inverse source problem for singular parabolic equations with interior degeneracy, Abstract and Applied Analysis, 2018 (2018), 1-16.  doi: 10.1155/2018/2067304.

[3]

F. Ammar KhodjaA. BenabdallahM. Gonzalez-Burgos and L. de Teresa, Recent results on the controllability of linear coupled parabolic problems: A survey, Math. Control Relat. Fields, 1 (2011), 267-306.  doi: 10.3934/mcrf.2011.1.267.

[4]

W. Arendt, C. J. K. Batty, M. Hieber and F. Neubrander, Vector-valued Laplace transforms and Cauchy Problems, Monographs in Mathematics, 96, Birkhäuser Verlag, Basel, 2001. doi: 10.1007/978-3-0348-5075-9.

[5]

J. M. Ball, Strongly continuous semigroups, weak solutions and the variation of constant formula, Proc. Amer. Math. Soc., 63 (1977), 370-373.  doi: 10.2307/2041821.

[6]

U. Biccari, V. Hernandez-Santamaria and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equations, preprint, arXiv: 2001.11403.

[7]

I. BoutaayamouG. Fragnelli and L. Maniar, Lipschitz stability for linear parabolic systems with interior degeneracy, Electron. J. Differential Equations, 2014 (2014), 1-26. 

[8]

I. Boutaayamou and J. Salhi, Null controllability for linear parabolic cascade systems with interior degeneracy, Electron. J. Differential Equations, 2016 (2016), 1-22. 

[9]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer-Verlag, New York, 2011.

[10]

P. Cannarsa, G. Floridia, F. Golgeleteyen and M. Yamamoto, Inverse coefficient problems for a transport equation by local Carleman estimate, Inverse Problems, 35 (2019), 22 pp. doi: 10.1088/1361-6420/ab1c69.

[11]

P. CannarsaR. Ferretti and P. Martinez, Null controllability for parabolic operators with interior degeneracy and one-sided control, SIAM J. Control Optim., 57 (2019), 900-924.  doi: 10.1137/18M1198442.

[12]

P. Cannarsa and L. De Teresa, Controllability of $1$-D coupled degenerate parabolic equations, addendum and corrigendum, Electron. J. Differential Equations, 2009 (2009), 1-24. 

[13] T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, Clarendon Press, Oxford, 1998. 
[14]

John B. Conway, A Course in Functional Analysis, 2$^nd$ edition, Springer-Verlag, New York, 1990.

[15]

M. Duprez and P. Lissy, Indirect controllability of some linear parabolic systems of $m$ equations with $m-1$ controls involving coupling terms of zero or first order, J. Math. Pures Appl., 9 (2016), 905-934.  doi: 10.1016/j.matpur.2016.03.016.

[16]

M. Fadili and L. Maniar, Null controllability of $n$-coupled degenerate parabolic systems with $m$-controls, J. Evol. Equ., 17 (2017), 1311-1340.  doi: 10.1007/s00028-017-0385-3.

[17]

E. Fernández-CaraM. González-Burgos and L. de Teresa, Boundary controllability of parabolic coupled equations, J. Funct. Anal., 7 (2010), 1720-1758.  doi: 10.1016/j.jfa.2010.06.003.

[18]

J. Carmelo Flores and L. de Teresa, Null controllability of one dimensional degenerate parabolic equations with first order terms, Discrete & Continuous Dynamical Systems - B, 22 (2017). doi: 10.3934/dcdsb.2020136.

[19]

G. Floridia, C. Nitsch and C. Trombetti, Multiplicative controllability for nonlinear degenerate parabolic equations between sign-changing states, ESAIM Control Optim. Calc. Var., 26 (2020), 34 pp. doi: 10.1051/cocv/2019066.

[20]

M. Fotouhi and L. Salimi, Controllability results for a class of one dimensional degenerate/singular parabolic equations, Commun. Pure Appl. Anal., 12 (2013), 1415-1430.  doi: 10.3934/cpaa.2013.12.1415.

[21]

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.

[22]

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.

[23]

G. FragnelliG. R. GoldsteinJ. A. Goldstein and S. Romanelli, Generators with interior degeneracy on spaces of $L^2$ type, Electron. J. Differential Equations, 2012 (2012), 1-30. 

[24]

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.

[25]

G. Fragnelli and D. Mugnai, Carleman estimates, observability inequalities and null controllability for interior degenerate non smooth parabolic equations, Mem. Amer. Math. Soc., 242 (2016), 84 pp. doi: 10.1090/memo/1146.

[26]

G. Fragnelli and D. Mugnai, Carleman estimates and observability inequalities for parabolic equations with interior degeneracy, Adv. Nonlinear Anal., 2 (2013), 339-378.  doi: 10.1515/anona-2013-0015.

[27]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of evolution equations, Lect. Notes Ser., 34, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[28]

M. González-Burgos and L. De Teresa, Controllability results for cascade systems of $m$ coupled parabolic PDEs by one control force, Portugal. Math., 67 (2010), 91-113.  doi: 10.4171/PM/1859.

[29]

A. HajjajL. Maniar and J. Salhi, Carleman estimates and null controllability of degenerate/singular parabolic systems, Electron. J. Differential Equations, 2016 (2016), 1-25. 

[30]

A.Y. Khapalov, Global non-negative controllability of the semilinear parabolic equation governed by bilinear control, ESAIM: Control, Optimisation and Calculus of Variations, 7 (2002), 269-283.  doi: 10.1051/cocv:2002011.

[31]

J. Le Rousseau and G. Lebeau, On carleman estimates for elliptic and parabolic operators. Applications to unique continuation and control of parabolic equations, ESAIM Control Optim. Calc. Var., 18 (2012), 712-747.  doi: 10.1051/cocv/2011168.

[32]

J. Salhi, Null controllability for a singular coupled system of degenerate parabolic equations in nondivergence form, Electron. J. Qual. Theory Differ. Equ., 13 (2018), 1-28.  doi: 10.14232/ejqtde.2018.1.31.

[33]

J. Vancostenoble, Global non-negative approximate controllability of parabolic equations with singular potentials, in Trends in Control Theory and Partial Differential Equations, , Springer INdAM Series, 32, Springer, Cham, 2019,255–276.

[34]

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.

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