September  2021, 20(9): 2991-3028. doi: 10.3934/cpaa.2021092

Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay

1. 

Université Savoie Mont Blanc, LAMA, Chambéry, France

2. 

Université Polytechnique Hauts-de-France, LAMAV, Valenciennes, France

3. 

Lebanese University, Faculty of sciences, Khawarizmi Laboratory of Mathematics and Applications-KALMA, Hadath-Beirut, Lebanon

* Corresponding author

Received  February 2021 Revised  April 2021 Published  September 2021 Early access  June 2021

The purpose of this paper is to investigate the stabilization of a locally coupled wave equations with non smooth localized viscoelastic damping of Kelvin-Voigt type and localized time delay. Using a general criteria of Arendt-Batty, we show the strong stability of our system in the absence of the compactness of the resolvent. Finally, using frequency domain approach combined with the multiplier method, we prove a polynomial energy decay rate of order $ t^{-1} $.

Citation: Mohammad Akil, Haidar Badawi, Ali Wehbe. Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay. Communications on Pure and Applied Analysis, 2021, 20 (9) : 2991-3028. doi: 10.3934/cpaa.2021092
References:
[1]

F. AbdallahM. Ghader and A. Wehbe, Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities, Commun. Pure & Appl. Anal., 18 (2019), 2789-2818.  doi: 10.3934/cpaa.2019125.

[2]

F. AbdallahM. Ghader and A. Wehbe, Stability results of a distributed problem involving bresse system with history and/or cattaneo law under fully dirichlet or mixed boundary conditions, Math. Methods Appl. Sci., 41 (2018), 1876-1907.  doi: 10.1002/mma.4717.

[3]

M. AkilY. ChitourM. Ghader and A. Wehbe, Stability and exact controllability of a timoshenko system with only one fractional damping on the boundary, Asympt. Anal., 10 (2019), 1-60.  doi: 10.3233/asy-191574.

[4]

M. Akil, M. Ghader and A. Wehbe, The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization, SeMA Journal, Nov 2020.

[5]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328 (1999), 1015-1020.  doi: 10.1016/S0764-4442(99)80316-4.

[6]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.

[7]

M. AlvesJ. M. RiveraM. Sepúlveda and O. V. Villagrán, The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin–Voigt Dissipation, SIAM J. Appl. Math., 74 (2014), 345-365.  doi: 10.1137/130923233.

[8]

M. AlvesJ. M. RiveraM. SepúlvedaO. V. Villagrán and M. Z. Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Mathematische Nachrichten, 287 (2013), 483-497.  doi: 10.1002/mana.201200319.

[9]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar., 123 (2009), 1-10.  doi: 10.1007/s10474-009-8011-7.

[10]

K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems & Control Letters, 59 (2010), 623–628. doi: 10.1016/j.sysconle.2010.07.007.

[11]

K. AmmariS. Nicaise and C. Pignotti, Stability of an abstract-wave equation with delay and a Kelvin–Voigt damping, Asymptotic Analysis, 95 (2015), 21-38.  doi: 10.3233/ASY-151317.

[12]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.2307/2000826.

[13]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[14]

E. M. A. BenhassiK. AmmariS. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay, J. Evol. Equ., 9 (2009), 103-121.  doi: 10.1007/s00028-009-0004-z.

[15]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[16]

Y. Cui and Z. Wang, Asymptotic stability of wave equations coupled by velocities, Mathematical Control and Related Fields, 6 (2016), 429-446.  doi: 10.3934/mcrf.2016010.

[17]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697-713.  doi: 10.1137/0326040.

[18]

R. Datko, Two questions concerning the boundary control of certain elastic systems, Journal of Differential Equations, 92 (1991), 27-44.  doi: 10.1016/0022-0396(91)90062-E.

[19]

R. Datko, Two examples of ill-posedness with respect to time delays revisited, IEEE Transactions on Automatic Control, 42 (1997), 511-515.  doi: 10.1109/9.566660.

[20]

R. Datko, J. Lagnese and M. Polis, An example of the effect of time delays in boundary feedback stabilization of wave equations, In 1985 24th IEEE Conference on Decision and Control. IEEE, Dec. 1985. doi: 10.1137/0324007.

[21]

H. DemchenkoA. Anikushyn and M. Pokojovy, On a Kelvin-Voigt viscoelastic wave equation with strong delay, SIAM J. Math. Anal., 51 (2019), 4382-4412.  doi: 10.1137/18M1219308.

[22]

M. DreherR. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379.  doi: 10.1016/j.aml.2009.03.010.

[23]

U. ErnstK. Pawelzik and T. Geisel, Delay-induced multistable synchronization of biological oscillators, Phys. Rev. E, 57 (1998), 2150-2162.  doi: 10.1103/PhysRevE.57.2150.

[24]

M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations, IMA Journal of Mathematical Control and Information, 27 (2010), 189–203. doi: 10.1093/imamci/dnq007.

[25]

B. Z. GUO and C. Z. XU, Boundary Output Feedback Stabilization of A One-Dimensional Wave Equation System With Time Delay. IFAC Proceedings Volumes, 41 (2008), 8755–8760.

[26]

F. Hassine, Stability of elastic transmission systems with a local Kelvin–Voigt damping, European Journal of Control, 23 (2015), 84-93.  doi: 10.1016/j.ejcon.2015.03.001.

[27]

F. Hassine, Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin–Voigt damping, International Journal of Control, 89 (2016), 1933-1950.  doi: 10.1080/00207179.2015.1135509.

[28]

F. Hassine and N. Souayeh, Stability for coupled waves with locally disturbed kelvin–voigt damping, Semigroup Forum, 102 (2021), 134-159.  doi: 10.1007/s00233-020-10142-1.

[29]

A. HayekS. NicaiseZ. Salloum and A. Wehbe, A transmission problem of a system of weakly coupled wave equations with kelvin–voigt dampings and non-smooth coefficient at the interface, SeMA Journal, 77 (2020), 305-338.  doi: 10.1007/s40324-020-00218-x.

[30]

F. Huang, On the Mathematical Model for Linear Elastic Systems with Analytic Damping, SIAM Journal on Control and Optimization, 26 (1988), 714–724. doi: 10.1137/0326041.

[31]

T. Kato, Perturbation Theory for Linear Operators. Springer Berlin Heidelberg, 1995.

[32]

V. Kolmanoviskii and A. Mishkis, Introduction of the Theory and Applications of Functional and Differential Equations, Dordrecht, 199. doi: 10.1007/978-94-017-1965-0.

[33]

J. L. Lions. Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées. Masson, Paris, 1988.

[34]

K. LiuS. Chen and Z. Liu, Spectrum and Stability for Elastic Systems with Global or Local Kelvin–Voigt Damping, SIAM J. Appl. Math., 59 (1998), 651-668.  doi: 10.1137/S0036139996292015.

[35]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[36]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881.  doi: 10.1016/j.jmaa.2007.02.021.

[37]

Z. Liu and Q. Zhang, Stability of a String with Local Kelvin–Voigt Damping and Nonsmooth Coefficient at Interface. SIAM Journal on Control and Optimization, 54 (2016), 1859–1871. doi: 10.1137/15M1049385.

[38]

Z. Liu and S. Zheng, Semigroups associated with dissipative systems, volume 398 of Chapman & Hall/CRC Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 1999.

[39]

D. Mercier and V. Règnier, Decay rate of the timoshenko system with one boundary damping, Evol. Equ. Control Theor., 8 (2019), 423-445.  doi: 10.3934/eect.2019021.

[40]

S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, J. Math. Phys., 57 (2016), 111501. doi: 10.1063/1.4966551.

[41]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of bresse systems, Electron. J. Differ. Equ., 2014 (2014), 19pp.

[42]

S. Nicaise and C. Pignotti, Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks, SIAM J. Control and Optimization, 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[43]

S. Nicaise and C. Pignotti., Stability of the wave equation with localized Kelvin–Voigt damping and boundary delay feedback, Discrete and Continuous Dynamical Systems-Series S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.

[44]

H. P. Oquendo, Frictional versus Kelvin–Voigt damping in a transmission problem, Mathematical Methods in the Applied Sciences, 40 (2017), 7026-7032.  doi: 10.1002/mma.4510.

[45]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, in Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[46]

C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Systems & Control Letters, 61 (2012), 92–97. doi: 10.1016/j.sysconle.2011.09.016.

[47]

J. E. M. RiveraO. V. Villagran and M. Sepulveda, Stability to localized viscoelastic transmission problem, Commun. Partial Differ. Equ., 43 (2018), 821-838.  doi: 10.1080/03605302.2018.1475490.

[48]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.

[49]

J. M. WangB. Z. Guo and M. Krstic, Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time, SIAM Journal on Control and Optimization, 49 (2011), 517-554.  doi: 10.1137/100796261.

[50]

A. Wehbe, I. Issa and M. Akil, Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients, Acta Appl. Math., 171 (2021), 46pp. doi: 10.1007/s10440-021-00384-8.

[51]

Y. Xie and G. Xu, Exponential stability of 1-d wave equation with the boundary time delay based on the interior control, Discrete & Continuous Dynamical Systems-S, 10 (2017), 557–579. doi: 10.3934/dcdss.2017028.

[52]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 770–785. doi: 10.1051/cocv:2006021.

[53]

X. Zhang and E. Zuazua, Polynomial decay and control of a $1-d$ hyperbolic-parabolic coupled system, J. Differ. Equ., 204 (2004), 380-438.  doi: 10.1016/j.jde.2004.02.004.

show all references

References:
[1]

F. AbdallahM. Ghader and A. Wehbe, Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities, Commun. Pure & Appl. Anal., 18 (2019), 2789-2818.  doi: 10.3934/cpaa.2019125.

[2]

F. AbdallahM. Ghader and A. Wehbe, Stability results of a distributed problem involving bresse system with history and/or cattaneo law under fully dirichlet or mixed boundary conditions, Math. Methods Appl. Sci., 41 (2018), 1876-1907.  doi: 10.1002/mma.4717.

[3]

M. AkilY. ChitourM. Ghader and A. Wehbe, Stability and exact controllability of a timoshenko system with only one fractional damping on the boundary, Asympt. Anal., 10 (2019), 1-60.  doi: 10.3233/asy-191574.

[4]

M. Akil, M. Ghader and A. Wehbe, The influence of the coefficients of a system of wave equations coupled by velocities on its stabilization, SeMA Journal, Nov 2020.

[5]

F. Alabau, Stabilisation frontière indirecte de systèmes faiblement couplés, Comptes Rendus de l'Académie des Sciences-Series I-Mathematics, 328 (1999), 1015-1020.  doi: 10.1016/S0764-4442(99)80316-4.

[6]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.

[7]

M. AlvesJ. M. RiveraM. Sepúlveda and O. V. Villagrán, The Lack of Exponential Stability in Certain Transmission Problems with Localized Kelvin–Voigt Dissipation, SIAM J. Appl. Math., 74 (2014), 345-365.  doi: 10.1137/130923233.

[8]

M. AlvesJ. M. RiveraM. SepúlvedaO. V. Villagrán and M. Z. Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Mathematische Nachrichten, 287 (2013), 483-497.  doi: 10.1002/mana.201200319.

[9]

K. Ammari and M. Mehrenberger, Stabilization of coupled systems, Acta Math. Hungar., 123 (2009), 1-10.  doi: 10.1007/s10474-009-8011-7.

[10]

K. Ammari, S. Nicaise and C. Pignotti, Feedback boundary stabilization of wave equations with interior delay, Systems & Control Letters, 59 (2010), 623–628. doi: 10.1016/j.sysconle.2010.07.007.

[11]

K. AmmariS. Nicaise and C. Pignotti, Stability of an abstract-wave equation with delay and a Kelvin–Voigt damping, Asymptotic Analysis, 95 (2015), 21-38.  doi: 10.3233/ASY-151317.

[12]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.2307/2000826.

[13]

C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780.  doi: 10.1007/s00028-008-0424-1.

[14]

E. M. A. BenhassiK. AmmariS. Boulite and L. Maniar, Feedback stabilization of a class of evolution equations with delay, J. Evol. Equ., 9 (2009), 103-121.  doi: 10.1007/s00028-009-0004-z.

[15]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[16]

Y. Cui and Z. Wang, Asymptotic stability of wave equations coupled by velocities, Mathematical Control and Related Fields, 6 (2016), 429-446.  doi: 10.3934/mcrf.2016010.

[17]

R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 26 (1988), 697-713.  doi: 10.1137/0326040.

[18]

R. Datko, Two questions concerning the boundary control of certain elastic systems, Journal of Differential Equations, 92 (1991), 27-44.  doi: 10.1016/0022-0396(91)90062-E.

[19]

R. Datko, Two examples of ill-posedness with respect to time delays revisited, IEEE Transactions on Automatic Control, 42 (1997), 511-515.  doi: 10.1109/9.566660.

[20]

R. Datko, J. Lagnese and M. Polis, An example of the effect of time delays in boundary feedback stabilization of wave equations, In 1985 24th IEEE Conference on Decision and Control. IEEE, Dec. 1985. doi: 10.1137/0324007.

[21]

H. DemchenkoA. Anikushyn and M. Pokojovy, On a Kelvin-Voigt viscoelastic wave equation with strong delay, SIAM J. Math. Anal., 51 (2019), 4382-4412.  doi: 10.1137/18M1219308.

[22]

M. DreherR. Quintanilla and R. Racke, Ill-posed problems in thermomechanics, Applied Mathematics Letters, 22 (2009), 1374-1379.  doi: 10.1016/j.aml.2009.03.010.

[23]

U. ErnstK. Pawelzik and T. Geisel, Delay-induced multistable synchronization of biological oscillators, Phys. Rev. E, 57 (1998), 2150-2162.  doi: 10.1103/PhysRevE.57.2150.

[24]

M. Gugat, Boundary feedback stabilization by time delay for one-dimensional wave equations, IMA Journal of Mathematical Control and Information, 27 (2010), 189–203. doi: 10.1093/imamci/dnq007.

[25]

B. Z. GUO and C. Z. XU, Boundary Output Feedback Stabilization of A One-Dimensional Wave Equation System With Time Delay. IFAC Proceedings Volumes, 41 (2008), 8755–8760.

[26]

F. Hassine, Stability of elastic transmission systems with a local Kelvin–Voigt damping, European Journal of Control, 23 (2015), 84-93.  doi: 10.1016/j.ejcon.2015.03.001.

[27]

F. Hassine, Energy decay estimates of elastic transmission wave/beam systems with a local Kelvin–Voigt damping, International Journal of Control, 89 (2016), 1933-1950.  doi: 10.1080/00207179.2015.1135509.

[28]

F. Hassine and N. Souayeh, Stability for coupled waves with locally disturbed kelvin–voigt damping, Semigroup Forum, 102 (2021), 134-159.  doi: 10.1007/s00233-020-10142-1.

[29]

A. HayekS. NicaiseZ. Salloum and A. Wehbe, A transmission problem of a system of weakly coupled wave equations with kelvin–voigt dampings and non-smooth coefficient at the interface, SeMA Journal, 77 (2020), 305-338.  doi: 10.1007/s40324-020-00218-x.

[30]

F. Huang, On the Mathematical Model for Linear Elastic Systems with Analytic Damping, SIAM Journal on Control and Optimization, 26 (1988), 714–724. doi: 10.1137/0326041.

[31]

T. Kato, Perturbation Theory for Linear Operators. Springer Berlin Heidelberg, 1995.

[32]

V. Kolmanoviskii and A. Mishkis, Introduction of the Theory and Applications of Functional and Differential Equations, Dordrecht, 199. doi: 10.1007/978-94-017-1965-0.

[33]

J. L. Lions. Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués. Tome 1, volume 8 of Recherches en Mathématiques Appliquées. Masson, Paris, 1988.

[34]

K. LiuS. Chen and Z. Liu, Spectrum and Stability for Elastic Systems with Global or Local Kelvin–Voigt Damping, SIAM J. Appl. Math., 59 (1998), 651-668.  doi: 10.1137/S0036139996292015.

[35]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[36]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881.  doi: 10.1016/j.jmaa.2007.02.021.

[37]

Z. Liu and Q. Zhang, Stability of a String with Local Kelvin–Voigt Damping and Nonsmooth Coefficient at Interface. SIAM Journal on Control and Optimization, 54 (2016), 1859–1871. doi: 10.1137/15M1049385.

[38]

Z. Liu and S. Zheng, Semigroups associated with dissipative systems, volume 398 of Chapman & Hall/CRC Research Notes in Mathematics. Chapman & Hall/CRC, Boca Raton, FL, 1999.

[39]

D. Mercier and V. Règnier, Decay rate of the timoshenko system with one boundary damping, Evol. Equ. Control Theor., 8 (2019), 423-445.  doi: 10.3934/eect.2019021.

[40]

S. A. Messaoudi, A. Fareh and N. Doudi, Well posedness and exponential stability in a wave equation with a strong damping and a strong delay, J. Math. Phys., 57 (2016), 111501. doi: 10.1063/1.4966551.

[41]

N. Najdi and A. Wehbe, Weakly locally thermal stabilization of bresse systems, Electron. J. Differ. Equ., 2014 (2014), 19pp.

[42]

S. Nicaise and C. Pignotti, Stability and Instability Results of the Wave Equation with a Delay Term in the Boundary or Internal Feedbacks, SIAM J. Control and Optimization, 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[43]

S. Nicaise and C. Pignotti., Stability of the wave equation with localized Kelvin–Voigt damping and boundary delay feedback, Discrete and Continuous Dynamical Systems-Series S, 9 (2016), 791-813.  doi: 10.3934/dcdss.2016029.

[44]

H. P. Oquendo, Frictional versus Kelvin–Voigt damping in a transmission problem, Mathematical Methods in the Applied Sciences, 40 (2017), 7026-7032.  doi: 10.1002/mma.4510.

[45]

A. Pazy, Semigroups of linear operators and applications to partial differential equations, in Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[46]

C. Pignotti, A note on stabilization of locally damped wave equations with time delay, Systems & Control Letters, 61 (2012), 92–97. doi: 10.1016/j.sysconle.2011.09.016.

[47]

J. E. M. RiveraO. V. Villagran and M. Sepulveda, Stability to localized viscoelastic transmission problem, Commun. Partial Differ. Equ., 43 (2018), 821-838.  doi: 10.1080/03605302.2018.1475490.

[48]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, J. Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.

[49]

J. M. WangB. Z. Guo and M. Krstic, Wave Equation Stabilization by Delays Equal to Even Multiples of the Wave Propagation Time, SIAM Journal on Control and Optimization, 49 (2011), 517-554.  doi: 10.1137/100796261.

[50]

A. Wehbe, I. Issa and M. Akil, Stability results of an elastic/viscoelastic transmission problem of locally coupled waves with non smooth coefficients, Acta Appl. Math., 171 (2021), 46pp. doi: 10.1007/s10440-021-00384-8.

[51]

Y. Xie and G. Xu, Exponential stability of 1-d wave equation with the boundary time delay based on the interior control, Discrete & Continuous Dynamical Systems-S, 10 (2017), 557–579. doi: 10.3934/dcdss.2017028.

[52]

G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 770–785. doi: 10.1051/cocv:2006021.

[53]

X. Zhang and E. Zuazua, Polynomial decay and control of a $1-d$ hyperbolic-parabolic coupled system, J. Differ. Equ., 204 (2004), 380-438.  doi: 10.1016/j.jde.2004.02.004.

Figure 1.  Local Kelvin-Voigt damping and local time delay feedback
Figure 2.  Geometric description of the functions $ \chi_1 $ and $ \chi_2 $
Figure 3.  Geometric description of the functions $ \theta_1 $, $ \theta_2 $ and $ \theta_3 $
Figure 4.  Geometric description of the functions $ \theta_4 $ and $ \theta_5 $
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