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doi: 10.3934/dcdss.2022071
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Controllability for degenerate/singular parabolic systems involving memory terms

1. 

Department of Mathematics, Faculty of Applied Sciences, Ibn Zohr University, Aït-Melloul 86153, Morocco

2. 

Department of Ecology and Biology, Tuscia University, Largo dell'Università, 01100 Viterbo, Italy

3. 

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

* Corresponding author: Genni Fragnelli

To the memory of Rosa Maria Mininni

Received  September 2021 Revised  February 2022 Early access March 2022

Fund Project: $ ^* $ G.F. is a member of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM) and a member of UMI "Modellistica Socio-Epidemiologica (MSE)". She is supported by the FFABR Fondo per il finanziamento delle attività base di ricerca 2017, by the INdAM - GNAMPA Project 2020 Problemi inversi e di controllo per equazioni di evoluzione e loro applicazioni.
** J.S. thanks the University of Bari Aldo Moro, where he was Visiting Professor when this work was written

In this paper we deal with the null controllability for degenerate/singular parabolic systems with memory terms. To this aim, we first prove the null controllability property for some auxiliary nonhomogeneous degenerate/singular problems via new Carleman estimates for their corresponding adjoint systems. Then, under a condition on the kernels, using the Kakutani's fixed point theorem, we deduce null controllability results for the initial problems with memory.

Citation: Brahim Allal, Genni Fragnelli, Jawad Salhi*. Controllability for degenerate/singular parabolic systems involving memory terms. Discrete and Continuous Dynamical Systems - S, doi: 10.3934/dcdss.2022071
References:
[1]

E. M. Ait Ben HassiF. Ammar KhodjaA. Hajjaj and L. Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory, 2 (2013), 441-459.  doi: 10.3934/eect.2013.2.441.

[2]

B. Allal and G. Fragnelli, Null controllability of degenerate parabolic equation with memory, Math. Methods Appl. Sci., 44 (2021), 9163-9190.  doi: 10.1002/mma.7342.

[3]

B. Allal, G. Fragnelli and J. Salhi, Null controllability for a singular heat equation with a memory term, Electron. J. Qual. Theory Differ. Equ., (2021), Paper No. 14, 24 pp. doi: 10.14232/ejqtde.2021.1.1.

[4]

B. Allal, G. Fragnelli and J. Salhi, On the null controllability of coupled systems of degenerate parabolic integro-differential equations, submitted.

[5]

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.

[6]

B. AllalA. HajjajL. Maniar and J. Salhi, Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component, Math. Control Relat. Fields, 10 (2020), 643-667.  doi: 10.3934/mcrf.2020014.

[7]

G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials With Memory: Theory and Applications, Theory and applications. Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.

[8]

F. Ammar-KhodjaA. Benabdallah and C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 320 (2006), 928-943.  doi: 10.1016/j.jmaa.2005.07.060.

[9]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. Gonzalez-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.

[10]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. Gonzalez-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.

[11]

F. Ammar-KhodjaA. BenabdallahC. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var., 11 (2005), 426-448.  doi: 10.1051/cocv:2005013.

[12]

J.-P. Aubin, L'Analyse non Linéaire et ses Motivations Économiques, Collection Mathématiques Appliquées pour la Maîtrise Masson, Paris, 1984.

[13]

F. Bloom, Ill-Posed Problems for Integro-Differential Equations in Mechanics and Electromagnetic Theory, SIAM Studies in Applied Mathematics, 3. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.

[14]

P. Cannarsa and L. de Teresa, Controllability of $1$-D coupled degenerate parabolic equations, addendum and corrigendum, Electron. J. Differential Equations, (2009), No. 73, 21 pp.

[15]

P. CannarsaD. Rocchetti and J. Vancostenoble, Generation of analytic semi-groups in $L^2$ for a class of second order degenerate elliptic operators, Control Cybernet., 37 (2008), 831-878. 

[16]

F. W. Chaves-SilvaL. Rosier and E. Zuazua., Null controllability of a system of viscoelasticity with a moving control, J. Math. Pures Appl., 101 (2014), 198-222.  doi: 10.1016/j.matpur.2013.05.009.

[17]

F. W. Chaves-SilvaX. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM Journal on Control and Optimization, 55 (2017), 2437-2459.  doi: 10.1137/151004239.

[18]

E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721.

[19]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to null controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[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. Fragnelli and D. Mugnai, Control of Degenerate and Singular Parabolic Equation, BCAM SpringerBrief, 2021.

[24]

G. Fragnelli and D. Mugnai, Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case, Discrete Contin. Dyn. Syst.-S, 13 (2020), 1495-1511.  doi: 10.3934/dcdss.2020084.

[25]

G. Fragnelli and D. Mugnai, Controllability of degenerate and singular parabolic problems: The double strong case with Neumann boundary conditions, Opuscula Math., 39 (2019), 207-225.  doi: 10.7494/OpMath.2019.39.2.207.

[26]

G. Fragnelli, D. Mugnai, Controllability of strongly degenerate parabolic problems with strongly singular potentials, Electron. J. Qual. Theory Differ. Equ., (2018), 11 pp. doi: 10.14232/ejqtde.2018.1.50.

[27]

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.

[28]

X. FuJ. Yong and X. Zhang, Controllability and observability of the heat equations with hyperbolic memory kernel, J. Differatial Equations, 247 (2009), 2395-2439.  doi: 10.1016/j.jde.2009.07.026.

[29]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[30]

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

[31]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013.

[32]

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

[33]

A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems Control Lett., 61 (2012), 999-1002.  doi: 10.1016/j.sysconle.2012.07.002.

[34]

V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations. Stability and Control: Theory, Methods and Applications, 1. Gordon and Breach Science Publishers, Lausanne, 1995.

[35]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der Mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.

[36]

J. L. Lions, Contrôle des Systèmes Distribués Singuliers, Méthodes Mathématiques de l'Informatique, 13. Gauthier-Villars, Montrouge, 1983.

[37]

Q. LüX. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl., 108 (2017), 500-531.  doi: 10.1016/j.matpur.2017.05.001.

[38]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[39]

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

[40]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[41]

Q. Tao and H. Gao, On the null controllability of heat equation with memory, J. Math. Anal. Appl., 440 (2016), 1-13.  doi: 10.1016/j.jmaa.2016.03.036.

[42]

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.

[43]

G. Wang, Y. Zhang and E. Zuazua, Reachable subspaces, control regions and heat equations with memory, preprint, arXiv: 2101.10615v1.

[44]

X. Zhou and H. Gao, Interior approximate and null controllability of the heat equation with memory, Comput. Math. Appl., 67 (2014), 602-613.  doi: 10.1016/j.camwa.2013.12.005.

show all references

References:
[1]

E. M. Ait Ben HassiF. Ammar KhodjaA. Hajjaj and L. Maniar, Carleman estimates and null controllability of coupled degenerate systems, Evol. Equ. Control Theory, 2 (2013), 441-459.  doi: 10.3934/eect.2013.2.441.

[2]

B. Allal and G. Fragnelli, Null controllability of degenerate parabolic equation with memory, Math. Methods Appl. Sci., 44 (2021), 9163-9190.  doi: 10.1002/mma.7342.

[3]

B. Allal, G. Fragnelli and J. Salhi, Null controllability for a singular heat equation with a memory term, Electron. J. Qual. Theory Differ. Equ., (2021), Paper No. 14, 24 pp. doi: 10.14232/ejqtde.2021.1.1.

[4]

B. Allal, G. Fragnelli and J. Salhi, On the null controllability of coupled systems of degenerate parabolic integro-differential equations, submitted.

[5]

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.

[6]

B. AllalA. HajjajL. Maniar and J. Salhi, Lipschitz stability for some coupled degenerate parabolic systems with locally distributed observations of one component, Math. Control Relat. Fields, 10 (2020), 643-667.  doi: 10.3934/mcrf.2020014.

[7]

G. Amendola, M. Fabrizio and J. M. Golden, Thermodynamics of Materials With Memory: Theory and Applications, Theory and applications. Springer, New York, 2012. doi: 10.1007/978-1-4614-1692-0.

[8]

F. Ammar-KhodjaA. Benabdallah and C. Dupaix, Null-controllability of some reaction-diffusion systems with one control force, J. Math. Anal. Appl., 320 (2006), 928-943.  doi: 10.1016/j.jmaa.2005.07.060.

[9]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. Gonzalez-Burgos, A generalization of the Kalman rank condition for time-dependent coupled linear parabolic systems, Differ. Equ. Appl., 1 (2009), 427-457.  doi: 10.7153/dea-01-24.

[10]

F. Ammar KhodjaA. BenabdallahC. Dupaix and M. Gonzalez-Burgos, A Kalman rank condition for the localized distributed controllability of a class of linear parabolic systems, J. Evol. Equ., 9 (2009), 267-291.  doi: 10.1007/s00028-009-0008-8.

[11]

F. Ammar-KhodjaA. BenabdallahC. Dupaix and I. Kostin, Null-controllability of some systems of parabolic type by one control force, ESAIM Control Optim. Calc. Var., 11 (2005), 426-448.  doi: 10.1051/cocv:2005013.

[12]

J.-P. Aubin, L'Analyse non Linéaire et ses Motivations Économiques, Collection Mathématiques Appliquées pour la Maîtrise Masson, Paris, 1984.

[13]

F. Bloom, Ill-Posed Problems for Integro-Differential Equations in Mechanics and Electromagnetic Theory, SIAM Studies in Applied Mathematics, 3. Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa., 1981.

[14]

P. Cannarsa and L. de Teresa, Controllability of $1$-D coupled degenerate parabolic equations, addendum and corrigendum, Electron. J. Differential Equations, (2009), No. 73, 21 pp.

[15]

P. CannarsaD. Rocchetti and J. Vancostenoble, Generation of analytic semi-groups in $L^2$ for a class of second order degenerate elliptic operators, Control Cybernet., 37 (2008), 831-878. 

[16]

F. W. Chaves-SilvaL. Rosier and E. Zuazua., Null controllability of a system of viscoelasticity with a moving control, J. Math. Pures Appl., 101 (2014), 198-222.  doi: 10.1016/j.matpur.2013.05.009.

[17]

F. W. Chaves-SilvaX. Zhang and E. Zuazua, Controllability of evolution equations with memory, SIAM Journal on Control and Optimization, 55 (2017), 2437-2459.  doi: 10.1137/151004239.

[18]

E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, 42, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511623721.

[19]

E. Fernández-Cara and S. Guerrero, Global Carleman inequalities for parabolic systems and applications to null controllability, SIAM J. Control Optim., 45 (2006), 1395-1446.  doi: 10.1137/S0363012904439696.

[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. Fragnelli and D. Mugnai, Control of Degenerate and Singular Parabolic Equation, BCAM SpringerBrief, 2021.

[24]

G. Fragnelli and D. Mugnai, Singular parabolic equations with interior degeneracy and non smooth coefficients: The Neumann case, Discrete Contin. Dyn. Syst.-S, 13 (2020), 1495-1511.  doi: 10.3934/dcdss.2020084.

[25]

G. Fragnelli and D. Mugnai, Controllability of degenerate and singular parabolic problems: The double strong case with Neumann boundary conditions, Opuscula Math., 39 (2019), 207-225.  doi: 10.7494/OpMath.2019.39.2.207.

[26]

G. Fragnelli, D. Mugnai, Controllability of strongly degenerate parabolic problems with strongly singular potentials, Electron. J. Qual. Theory Differ. Equ., (2018), 11 pp. doi: 10.14232/ejqtde.2018.1.50.

[27]

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.

[28]

X. FuJ. Yong and X. Zhang, Controllability and observability of the heat equations with hyperbolic memory kernel, J. Differatial Equations, 247 (2009), 2395-2439.  doi: 10.1016/j.jde.2009.07.026.

[29]

A. V. Fursikov and O. Y. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Series, 34. Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1996.

[30]

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

[31]

S. Guerrero and O. Y. Imanuvilov, Remarks on non controllability of the heat equation with memory, ESAIM Control Optim. Calc. Var., 19 (2013), 288-300.  doi: 10.1051/cocv/2012013.

[32]

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

[33]

A. Halanay and L. Pandolfi, Lack of controllability of the heat equation with memory, Systems Control Lett., 61 (2012), 999-1002.  doi: 10.1016/j.sysconle.2012.07.002.

[34]

V. Lakshmikantham and M. Rama Mohana Rao, Theory of Integro-Differential Equations. Stability and Control: Theory, Methods and Applications, 1. Gordon and Breach Science Publishers, Lausanne, 1995.

[35]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Die Grundlehren der Mathematischen Wissenschaften, Band 170 Springer-Verlag, New York-Berlin, 1971.

[36]

J. L. Lions, Contrôle des Systèmes Distribués Singuliers, Méthodes Mathématiques de l'Informatique, 13. Gauthier-Villars, Montrouge, 1983.

[37]

Q. LüX. Zhang and E. Zuazua, Null controllability for wave equations with memory, J. Math. Pures Appl., 108 (2017), 500-531.  doi: 10.1016/j.matpur.2017.05.001.

[38]

J. Prüss, Evolutionary Integral Equations and Applications, Monographs in Mathematics, 87. Birkhäuser Verlag, Basel, 1993. doi: 10.1007/978-3-0348-8570-6.

[39]

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

[40]

J. Simon, Compact sets in the space $L^p(0, T;B)$, Ann. Mat. Pura Appl., 146 (1987), 65-96.  doi: 10.1007/BF01762360.

[41]

Q. Tao and H. Gao, On the null controllability of heat equation with memory, J. Math. Anal. Appl., 440 (2016), 1-13.  doi: 10.1016/j.jmaa.2016.03.036.

[42]

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.

[43]

G. Wang, Y. Zhang and E. Zuazua, Reachable subspaces, control regions and heat equations with memory, preprint, arXiv: 2101.10615v1.

[44]

X. Zhou and H. Gao, Interior approximate and null controllability of the heat equation with memory, Comput. Math. Appl., 67 (2014), 602-613.  doi: 10.1016/j.camwa.2013.12.005.

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