doi: 10.3934/naco.2022001
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Exact controllability for a degenerate and singular wave equation with moving boundary

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

*Corresponding author: Jawad Salhi

Received  September 2021 Revised  January 2022 Early access February 2022

This paper is concerned with the exact boundary controllability for a degenerate and singular wave equation in a bounded interval with a moving endpoint. By the multiplier method and using an adapted Hardy-poincaré inequality, we prove direct and inverse inequalities for the solutions of the associated adjoint equation. As a consequence, by the Hilbert Uniqueness Method, we deduce the controllability result of the considered system when the control acts on the moving boundary. Furthermore, improved estimates of the speed of the moving endpoint and the controllability time are obtained.

Citation: Alhabib Moumni, Jawad Salhi. Exact controllability for a degenerate and singular wave equation with moving boundary. Numerical Algebra, Control and Optimization, doi: 10.3934/naco.2022001
References:
[1]

F. Alabau-BoussouiraP Cannarsa and G. Leugering, Control and stabilization of degenerate wave equation, SIAM J. Control Optim., 55 (2017), 2052-2087.  doi: 10.1137/15M1020538.

[2]

B. Allal, A. Moumni and J. Salhi, Boundary controllability for a degenerate and singular wave equation, preprint, arXiv: 2108.04159v2.

[3]

F. D. ArarunaG. O. Antunes and L. A. Medeiros, Exact controllability for the semilinear string equation in the non cylindrical domains, Control cybernet, 33 (2004), 237-257. 

[4]

J. Bai and S. Chai, Exact controllability for some degenerate wave equations, Math. Methods Appl. Sci., 43 (2020), 7292-7302.  doi: 10.1002/mma.6464.

[5]

J. Bai and S. Chai, Exact controllability for a one-dimensional degenerate wave equation in domains with moving boundary, Applied Mathematics Letters, 119 (2021), 1-8.  doi: 10.1016/j.aml.2021.107235.

[6]

C. Bardos and G. Chen, Control and stabilization for the wave equation, part Ⅲ: domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.

[7]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

[8]

U. Biccari, V. Hernández-Santamaría and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equationy, Mathematical Control & Related Fields, 2021. doi: 10.3934/mcrf.2021032.

[9]

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.

[10]

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.

[11]

L. Cui, Exact controllability of wave equations with locally distributed control in non-cylindrical domain, J. Math. Anal. Appl., 482 (2020), 1-17.  doi: 10.1016/j.jmaa.2019.123532.

[12]

L. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.

[13] E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511623721.
[14]

V. V. DodonovA. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.

[15]

L. V. Fardigola, Transformation operators in control problems for a degenerate wave equation with variable coefficients, Ukrainian Math. J., 70 (2019), 1300-1318.  doi: 10.1007/s11253-018-1570-4.

[16]

G. Fragnelli and D. Mugnai, Control of Degenerate and Singular Parabolic Equation, Springer International Publishing, 2021. doi: 10.7494/opmath.2019.39.2.207.

[17]

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.

[18]

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.

[19]

B. H. Haak and D. T. Hoang, Exact observability of a $1$-dimensional wave equation on a noncylindrical domain, SIAM J. Control Optim., 57 (2019), 570-589.  doi: 10.1137/17M112960X.

[20]

P. I. KogutO. P. Kupenko and G. Leugering, On boundary exact controllability of one-dimensional wave equations with weak and strong interior degeneration, Math. Methods Appl. Sci., 45 (2022), 770-792.  doi: 10.1002/mma.7811.

[21]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Wiley, Masson, Paris, 1995.

[22]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. 

[23]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, New York, 2005.

[24]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.

[25]

J.-L. Lions, Controlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués, 1, Masson, Paris, 1988.

[26]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[27]

M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Univ., 9 (1996), 435-457. 

[28]

Y. Mokhtari, Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains, Evolution Equations & Control Theory, 2021. doi: 10.3934/eect.2021004.

[29]

G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691. 

[30]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[31]

A. Sengouga, Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints, Mathematical Control & Related Fields, 9 (2020), 1-25.  doi: 10.3934/eect.2020014.

[32]

A. Shao, On Carleman and observability estimates for wave equations on time-dependent domains, Proc. Lond. Math. Soc., 119 (2019), 998-1064.  doi: 10.1112/plms.12253.

[33]

H. SunH. Li and L. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 1-7. 

[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.

[35]

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.

[36]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.

[37]

M. Zhang and H. Gao, Null controllability of some degenerate wave equations, J. Syst. Sci. Complex, 29 (2017), 1-15.  doi: 10.1007/s11424-016-5281-3.

[38]

M. Zhang and H. Gao, Interior controllability of semi-linear degenerate wave equations, J. Math. Anal. Appl., 457 (2018), 10-22.  doi: 10.1016/j.jmaa.2017.07.057.

[39]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.

show all references

References:
[1]

F. Alabau-BoussouiraP Cannarsa and G. Leugering, Control and stabilization of degenerate wave equation, SIAM J. Control Optim., 55 (2017), 2052-2087.  doi: 10.1137/15M1020538.

[2]

B. Allal, A. Moumni and J. Salhi, Boundary controllability for a degenerate and singular wave equation, preprint, arXiv: 2108.04159v2.

[3]

F. D. ArarunaG. O. Antunes and L. A. Medeiros, Exact controllability for the semilinear string equation in the non cylindrical domains, Control cybernet, 33 (2004), 237-257. 

[4]

J. Bai and S. Chai, Exact controllability for some degenerate wave equations, Math. Methods Appl. Sci., 43 (2020), 7292-7302.  doi: 10.1002/mma.6464.

[5]

J. Bai and S. Chai, Exact controllability for a one-dimensional degenerate wave equation in domains with moving boundary, Applied Mathematics Letters, 119 (2021), 1-8.  doi: 10.1016/j.aml.2021.107235.

[6]

C. Bardos and G. Chen, Control and stabilization for the wave equation, part Ⅲ: domain with moving boundary, SIAM J. Control Optim., 19 (1981), 123-138.  doi: 10.1137/0319010.

[7]

C. BardosG. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065.  doi: 10.1137/0330055.

[8]

U. Biccari, V. Hernández-Santamaría and J. Vancostenoble, Existence and cost of boundary controls for a degenerate/singular parabolic equationy, Mathematical Control & Related Fields, 2021. doi: 10.3934/mcrf.2021032.

[9]

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.

[10]

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.

[11]

L. Cui, Exact controllability of wave equations with locally distributed control in non-cylindrical domain, J. Math. Anal. Appl., 482 (2020), 1-17.  doi: 10.1016/j.jmaa.2019.123532.

[12]

L. CuiX. Liu and H. Gao, Exact controllability for a one-dimensional wave equation in non-cylindrical domains, J. Math. Anal. Appl., 402 (2013), 612-625.  doi: 10.1016/j.jmaa.2013.01.062.

[13] E. B. Davies, Spectral Theory and Differential Operators, Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, 1995.  doi: 10.1017/CBO9780511623721.
[14]

V. V. DodonovA. B. Klimov and D. E. Nikonov, Quantum phenomena in resonators with moving walls, J. Math. Phys., 34 (1993), 2742-2756.  doi: 10.1063/1.530093.

[15]

L. V. Fardigola, Transformation operators in control problems for a degenerate wave equation with variable coefficients, Ukrainian Math. J., 70 (2019), 1300-1318.  doi: 10.1007/s11253-018-1570-4.

[16]

G. Fragnelli and D. Mugnai, Control of Degenerate and Singular Parabolic Equation, Springer International Publishing, 2021. doi: 10.7494/opmath.2019.39.2.207.

[17]

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.

[18]

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.

[19]

B. H. Haak and D. T. Hoang, Exact observability of a $1$-dimensional wave equation on a noncylindrical domain, SIAM J. Control Optim., 57 (2019), 570-589.  doi: 10.1137/17M112960X.

[20]

P. I. KogutO. P. Kupenko and G. Leugering, On boundary exact controllability of one-dimensional wave equations with weak and strong interior degeneration, Math. Methods Appl. Sci., 45 (2022), 770-792.  doi: 10.1002/mma.7811.

[21]

V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, Wiley, Masson, Paris, 1995.

[22]

V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. 

[23]

V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, New York, 2005.

[24]

E. Knobloch and R. Krechetnikov, Problems on time-varying domains: Formulation, dynamics, and challenges, Acta Appl. Math., 137 (2015), 123-157.  doi: 10.1007/s10440-014-9993-x.

[25]

J.-L. Lions, Controlabilité Exacte, Perturbation et Stabilisation de Systèmes Distribués, 1, Masson, Paris, 1988.

[26]

J.-L. Lions, Exact controllability, stabilization and perturbations for distributed systems, SIAM Rev., 30 (1988), 1-68.  doi: 10.1137/1030001.

[27]

M. Milla Miranda, Exact controllability for the wave equation in domains with variable boundary, Rev. Mat. Univ., 9 (1996), 435-457. 

[28]

Y. Mokhtari, Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains, Evolution Equations & Control Theory, 2021. doi: 10.3934/eect.2021004.

[29]

G. T. Moore, Quantum theory of electromagnetic field in a variable-length one-dimensional cavity, J. Math. Phys., 11 (1970), 2679-2691. 

[30]

D. L. Russell, Controllability and stabilizability theory for linear partial differential equations: Recent progress and open questions, SIAM Rev., 20 (1978), 639-739.  doi: 10.1137/1020095.

[31]

A. Sengouga, Exact boundary observability and controllability of the wave equation in an interval with two moving endpoints, Mathematical Control & Related Fields, 9 (2020), 1-25.  doi: 10.3934/eect.2020014.

[32]

A. Shao, On Carleman and observability estimates for wave equations on time-dependent domains, Proc. Lond. Math. Soc., 119 (2019), 998-1064.  doi: 10.1112/plms.12253.

[33]

H. SunH. Li and L. Lu, Exact controllability for a string equation in domains with moving boundary in one dimension, Electron. J. Diff. Equations, 2015 (2015), 1-7. 

[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.

[35]

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.

[36]

P. K. C. Wang, Stabilization and control of distributed systems with time-dependent spatial domains, J. Optim. Theory Appl., 65 (1990), 331-362.  doi: 10.1007/BF01102351.

[37]

M. Zhang and H. Gao, Null controllability of some degenerate wave equations, J. Syst. Sci. Complex, 29 (2017), 1-15.  doi: 10.1007/s11424-016-5281-3.

[38]

M. Zhang and H. Gao, Interior controllability of semi-linear degenerate wave equations, J. Math. Anal. Appl., 457 (2018), 10-22.  doi: 10.1016/j.jmaa.2017.07.057.

[39]

E. Zuazua, Exact controllability for semilinear wave equations in one space dimension, Inst. H. Poincaré Anal. Non Linéaire, 10 (1993), 109-129.  doi: 10.1016/S0294-1449(16)30221-9.

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