doi: 10.3934/mcrf.2021059
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Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping

1. 

Université Polytechnique Hauts-de-France, CÉRAMATHS/DEMAV, Campus Mont Houy, Valenciennes-France

2. 

Université Aix-Marseilles, Laboratoire I2M, Marseille, France

3. 

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

* Corresponding author: Mohammad Akil

Received  June 2021 Revised  October 2021 Early access December 2021

In this paper, we investigate the energy decay of hyperbolic systems of wave-wave, wave-Euler-Bernoulli beam and beam-beam types. The two equations are coupled through boundary connection with only one localized non-smooth fractional Kelvin-Voigt damping. First, we reformulate each system into an augmented model and using a general criteria of Arendt-Batty, we prove that our models are strongly stable. Next, by using frequency domain approach, combined with multiplier technique and some interpolation inequalities, we establish different types of polynomial energy decay rate which depends on the order of the fractional derivative and the type of the damped equation in the system.

Citation: Mohammad Akil, Ibtissam Issa, Ali Wehbe. Energy decay of some boundary coupled systems involving wave\ Euler-Bernoulli beam with one locally singular fractional Kelvin-Voigt damping. Mathematical Control and Related Fields, doi: 10.3934/mcrf.2021059
References:
[1]

Z. AchouriN. E. Amroun and A. Benaissa, The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci., 40 (2017), 3837-3854.  doi: 10.1002/mma.4267.

[2]

R. A. Adams, Sobolev Spaces / Robert A. Adams, Academic Press New York, 1975.

[3]

M. Akil, H. Badawi, S. Nicaise and A. Wehbe, On the stability of Bresse system with one discontinuous local internal kelvin-voigt damping on the axial force, Z. Angew. Math. Phys., 72 (2021), Paper No. 126, 27 pp. doi: 10.1007/s00033-021-01558-y.

[4]

M. AkilH. Badawi and A. Wehbe, Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay, Commun. Pure Appl. Anal., 20 (2021), 2991-3028.  doi: 10.3934/cpaa.2021092.

[5]

M. AkilY. ChitourM. Ghader and A. Wehbe, Stability and exact controllability of a timoshenko system with only one fractional damping on the boundary, Asymptotic Analysis, 119 (2020), 221-280.  doi: 10.3233/ASY-191574.

[6]

M. Akil and A. Wehbe, Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions, Math. Control Relat. Fields, 9 (2019), 97-116.  doi: 10.3934/mcrf.2019005.

[7]

H. AllouniM. Kesri and A. Benaissa, On the asymptotic behaviour of two coupled strings through a fractional joint damper, Rend. Circ. Mat. Palermo (2), 69 (2020), 613-640.  doi: 10.1007/s12215-019-00423-2.

[8]

M. Alves, J. M. Rivera, M. 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.

[9]

K. AmmariH. Fathi and L. Robbiano, Fractional-feedback stabilization for a class of evolution systems, J. Differential Equations, 268 (2020), 5751-5791.  doi: 10.1016/j.jde.2019.11.022.

[10]

K. AmmariM. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system, Netw. Heterog. Media, 4 (2009), 19-34.  doi: 10.3934/nhm.2009.4.19.

[11]

K. AmmariZ. Liu and F. Shel, Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100 (2020), 364-382.  doi: 10.1007/s00233-019-10064-7.

[12]

K. Ammari and M. Mehrenberger, Study of the nodal feedback stabilization of a string-beams network, J. Appl. Math. Comput., 36 (2011), 441-458.  doi: 10.1007/s12190-010-0412-9.

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K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.

[14]

K. Ammari and G. Vodev, Boundary stabilization of the transmission problem for the Bernoulli-Euler plate equation, Cubo, 11 (2009), 39-49. 

[15]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[16]

R. L. Bagley and P. J. Torvik, Fractional calculus - a different approach to the analysis of viscoelastically damped structures, AIAA Journal, 21 (1983), 741-748.  doi: 10.2514/3.8142.

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R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27 (1983), 201-210.  doi: 10.1122/1.549724.

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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.

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J. Bartolomeo and R. Triggiani, Uniform energy decay rates for Euler-Bernoulli equations with feedback operators in the Dirichlet/Neumann boundary conditions, SIAM J. Math. Anal., 22 (1991), 46-71.  doi: 10.1137/0522004.

[20]

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

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S. K. Biswas and N. U. Ahmed, Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations, Math. Control. Signals Syst., 2 (1989), 1-18.  doi: 10.1007/BF02551358.

[22]

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.

[23]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal, 13 (1967), 529-539. 

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M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73-85. 

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G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim., 25, (1987), 526–546. doi: 10.1137/0325029.

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G. ChenS. A. FullingF. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math., 51 (1991), 266-301.  doi: 10.1137/0151015.

[27]

S. ChenK. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local kelvin–voigt damping, SIAM J. Appl. Math., 59 (1999), 651-668.  doi: 10.1137/S0036139996292015.

[28]

R. Denk and F. Kammerlander, Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.  doi: 10.1093/imamat/hxy002.

[29]

X. Fu and Q. Lu, Stabilization of the weakly coupled wave-plate system with one internal damping, 2017.

[30]

B.-Z. Guo and H.-J. Ren, Stability and regularity transmission for coupled beam and wave equations through boundary weak connections, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 73, 29 pp. doi: 10.1051/cocv/2019056.

[31]

Y.-P. Guo, J.-M. Wang and D.-X. Zhao, Energy decay estimates for a two-dimensional coupled wave-plate system with localized frictional damping, ZAMM Z. Angew. Math. Mech., 100 (2020), e201900030, 14 pp. doi: 10.1002/zamm.201900030.

[32]

Z.-J. Han and Z. Liu, Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 51, 14 pp. doi: 10.1051/cocv/2018060.

[33]

F. Hassine, Energy decay estimates of elastic transmission wave/beam systems with a local kelvin-voigt damping, International Journal of Control, (2015), 1–29.

[34]

F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1757-1774.  doi: 10.3934/dcdsb.2016021.

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F. L. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.  doi: 10.1137/0326041.

[36]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[37]

G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior, J. Math. Anal. Appl., 229 (1999), 452-479.  doi: 10.1006/jmaa.1998.6170.

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J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.  doi: 10.1137/0325078.

[39]

J. E. Lagnese, Uniform boundary stabilization of homogeneous isotropic plates, Part of the Lecture Notes in Control and Information Sciences, (1987), 204–215. doi: 10.1007/BFb0041992.

[40]

I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations, 79 (1989), 340-381.  doi: 10.1016/0022-0396(89)90107-1.

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I. Lasiecka, Asymptotic behavior of solutions to plate equations with nonlinear dissipation occurring through shear forces and bending moments, Appl. Math. Optim., 21 (1990), 167-189.  doi: 10.1007/BF01445162.

[42]

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.

[43]

Y.-F. LiZ.-J. Han and G.-Q. Xu, Explicit decay rate for coupled string-beam system with localized frictional damping, Appl. Math. Lett., 78 (2018), 51-58.  doi: 10.1016/j.aml.2017.11.003.

[44]

K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.

[45]

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.

[46]

Z. Liu and Q. Zhang, Stability of a string with local Kelvin–Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.

[47]

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.

[48]

M. Mainardi and E. Bonetti, The application of real-order derivatives in linear viscoelasticity, In H. Giesekus and M. F. Hibberd, editors, Progress and Trends in Rheology II, pages 64–67, Heidelberg, 1988. Steinkopff. doi: 10.1007/978-3-642-49337-9_11.

[49]

D. Matignon and C. Prieur, Asymptotic stability of Webster-Lokshin equation, Math. Control Relat. Fields, 4 (2014), 481-500.  doi: 10.3934/mcrf.2014.4.481.

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B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inform., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056.

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B. Mbodje and G. Montseny, Boundary fractional derivative control of the wave equation, IEEE Trans. Automat. Control, 40 (1995), 378-382.  doi: 10.1109/9.341815.

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L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 20 (1966), 733-737. 

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M. L. Santos and J. E. Muñoz Rivera, Analytic property of a coupled system of wave-plate type with thermal effect, Differential and Integral Equations, 24 (2011), 965-972. 

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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), Paper No. 23, 46 pp. doi: 10.1007/s10440-021-00384-8.

[61]

Q. Zhang, Exponential stability of an elastic string with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 61 (2010), 1009-1015.  doi: 10.1007/s00033-010-0064-5.

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show all references

References:
[1]

Z. AchouriN. E. Amroun and A. Benaissa, The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci., 40 (2017), 3837-3854.  doi: 10.1002/mma.4267.

[2]

R. A. Adams, Sobolev Spaces / Robert A. Adams, Academic Press New York, 1975.

[3]

M. Akil, H. Badawi, S. Nicaise and A. Wehbe, On the stability of Bresse system with one discontinuous local internal kelvin-voigt damping on the axial force, Z. Angew. Math. Phys., 72 (2021), Paper No. 126, 27 pp. doi: 10.1007/s00033-021-01558-y.

[4]

M. AkilH. Badawi and A. Wehbe, Stability results of a singular local interaction elastic/viscoelastic coupled wave equations with time delay, Commun. Pure Appl. Anal., 20 (2021), 2991-3028.  doi: 10.3934/cpaa.2021092.

[5]

M. AkilY. ChitourM. Ghader and A. Wehbe, Stability and exact controllability of a timoshenko system with only one fractional damping on the boundary, Asymptotic Analysis, 119 (2020), 221-280.  doi: 10.3233/ASY-191574.

[6]

M. Akil and A. Wehbe, Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions, Math. Control Relat. Fields, 9 (2019), 97-116.  doi: 10.3934/mcrf.2019005.

[7]

H. AllouniM. Kesri and A. Benaissa, On the asymptotic behaviour of two coupled strings through a fractional joint damper, Rend. Circ. Mat. Palermo (2), 69 (2020), 613-640.  doi: 10.1007/s12215-019-00423-2.

[8]

M. Alves, J. M. Rivera, M. 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.

[9]

K. AmmariH. Fathi and L. Robbiano, Fractional-feedback stabilization for a class of evolution systems, J. Differential Equations, 268 (2020), 5751-5791.  doi: 10.1016/j.jde.2019.11.022.

[10]

K. AmmariM. Jellouli and M. Mehrenberger, Feedback stabilization of a coupled string-beam system, Netw. Heterog. Media, 4 (2009), 19-34.  doi: 10.3934/nhm.2009.4.19.

[11]

K. AmmariZ. Liu and F. Shel, Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100 (2020), 364-382.  doi: 10.1007/s00233-019-10064-7.

[12]

K. Ammari and M. Mehrenberger, Study of the nodal feedback stabilization of a string-beams network, J. Appl. Math. Comput., 36 (2011), 441-458.  doi: 10.1007/s12190-010-0412-9.

[13]

K. Ammari and S. Nicaise, Stabilization of a transmission wave/plate equation, J. Differential Equations, 249 (2010), 707-727.  doi: 10.1016/j.jde.2010.03.007.

[14]

K. Ammari and G. Vodev, Boundary stabilization of the transmission problem for the Bernoulli-Euler plate equation, Cubo, 11 (2009), 39-49. 

[15]

W. Arendt and C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Amer. Math. Soc., 306 (1988), 837-852.  doi: 10.1090/S0002-9947-1988-0933321-3.

[16]

R. L. Bagley and P. J. Torvik, Fractional calculus - a different approach to the analysis of viscoelastically damped structures, AIAA Journal, 21 (1983), 741-748.  doi: 10.2514/3.8142.

[17]

R. L. Bagley and P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of Rheology, 27 (1983), 201-210.  doi: 10.1122/1.549724.

[18]

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.

[19]

J. Bartolomeo and R. Triggiani, Uniform energy decay rates for Euler-Bernoulli equations with feedback operators in the Dirichlet/Neumann boundary conditions, SIAM J. Math. Anal., 22 (1991), 46-71.  doi: 10.1137/0522004.

[20]

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

[21]

S. K. Biswas and N. U. Ahmed, Optimal control of large space structures governed by a coupled system of ordinary and partial differential equations, Math. Control. Signals Syst., 2 (1989), 1-18.  doi: 10.1007/BF02551358.

[22]

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.

[23]

M. Caputo, Linear models of dissipation whose Q is almost frequency independent-II, Geophysical Journal, 13 (1967), 529-539. 

[24]

M. Caputo and M. Fabrizio, A new definition of fractional derivative without singular kernel, Prog. Fract. Differ. Appl., 1 (2015), 73-85. 

[25]

G. Chen, M. C. Delfour, A. M. Krall and G. Payre, Modeling, stabilization and control of serially connected beams, SIAM J. Control Optim., 25, (1987), 526–546. doi: 10.1137/0325029.

[26]

G. ChenS. A. FullingF. J. Narcowich and S. Sun, Exponential decay of energy of evolution equations with locally distributed damping, SIAM J. Appl. Math., 51 (1991), 266-301.  doi: 10.1137/0151015.

[27]

S. ChenK. Liu and Z. Liu, Spectrum and stability for elastic systems with global or local kelvin–voigt damping, SIAM J. Appl. Math., 59 (1999), 651-668.  doi: 10.1137/S0036139996292015.

[28]

R. Denk and F. Kammerlander, Exponential stability for a coupled system of damped-undamped plate equations, IMA J. Appl. Math., 83 (2018), 302-322.  doi: 10.1093/imamat/hxy002.

[29]

X. Fu and Q. Lu, Stabilization of the weakly coupled wave-plate system with one internal damping, 2017.

[30]

B.-Z. Guo and H.-J. Ren, Stability and regularity transmission for coupled beam and wave equations through boundary weak connections, ESAIM Control Optim. Calc. Var., 26 (2020), Paper No. 73, 29 pp. doi: 10.1051/cocv/2019056.

[31]

Y.-P. Guo, J.-M. Wang and D.-X. Zhao, Energy decay estimates for a two-dimensional coupled wave-plate system with localized frictional damping, ZAMM Z. Angew. Math. Mech., 100 (2020), e201900030, 14 pp. doi: 10.1002/zamm.201900030.

[32]

Z.-J. Han and Z. Liu, Regularity and stability of coupled plate equations with indirect structural or Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 25 (2019), Paper No. 51, 14 pp. doi: 10.1051/cocv/2018060.

[33]

F. Hassine, Energy decay estimates of elastic transmission wave/beam systems with a local kelvin-voigt damping, International Journal of Control, (2015), 1–29.

[34]

F. Hassine, Asymptotic behavior of the transmission Euler-Bernoulli plate and wave equation with a localized Kelvin-Voigt damping, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 1757-1774.  doi: 10.3934/dcdsb.2016021.

[35]

F. L. Huang, On the mathematical model for linear elastic systems with analytic damping, SIAM J. Control Optim., 26 (1988), 714-724.  doi: 10.1137/0326041.

[36]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. 

[37]

G. Ji and I. Lasiecka, Nonlinear boundary feedback stabilization for a semilinear Kirchhoff plate with dissipation acting only via moments-limiting behavior, J. Math. Anal. Appl., 229 (1999), 452-479.  doi: 10.1006/jmaa.1998.6170.

[38]

J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429.  doi: 10.1137/0325078.

[39]

J. E. Lagnese, Uniform boundary stabilization of homogeneous isotropic plates, Part of the Lecture Notes in Control and Information Sciences, (1987), 204–215. doi: 10.1007/BFb0041992.

[40]

I. Lasiecka, Stabilization of wave and plate-like equations with nonlinear dissipation on the boundary, J. Differential Equations, 79 (1989), 340-381.  doi: 10.1016/0022-0396(89)90107-1.

[41]

I. Lasiecka, Asymptotic behavior of solutions to plate equations with nonlinear dissipation occurring through shear forces and bending moments, Appl. Math. Optim., 21 (1990), 167-189.  doi: 10.1007/BF01445162.

[42]

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.

[43]

Y.-F. LiZ.-J. Han and G.-Q. Xu, Explicit decay rate for coupled string-beam system with localized frictional damping, Appl. Math. Lett., 78 (2018), 51-58.  doi: 10.1016/j.aml.2017.11.003.

[44]

K. Liu and Z. Liu, Exponential decay of energy of the Euler–Bernoulli beam with locally distributed Kelvin–Voigt damping, SIAM J. Control Optim., 36 (1998), 1086-1098.  doi: 10.1137/S0363012996310703.

[45]

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.

[46]

Z. Liu and Q. Zhang, Stability of a string with local Kelvin–Voigt damping and nonsmooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.

[47]

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.

[48]

M. Mainardi and E. Bonetti, The application of real-order derivatives in linear viscoelasticity, In H. Giesekus and M. F. Hibberd, editors, Progress and Trends in Rheology II, pages 64–67, Heidelberg, 1988. Steinkopff. doi: 10.1007/978-3-642-49337-9_11.

[49]

D. Matignon and C. Prieur, Asymptotic stability of Webster-Lokshin equation, Math. Control Relat. Fields, 4 (2014), 481-500.  doi: 10.3934/mcrf.2014.4.481.

[50]

B. Mbodje, Wave energy decay under fractional derivative controls, IMA J. Math. Control Inform., 23 (2006), 237-257.  doi: 10.1093/imamci/dni056.

[51]

B. Mbodje and G. Montseny, Boundary fractional derivative control of the wave equation, IEEE Trans. Automat. Control, 40 (1995), 378-382.  doi: 10.1109/9.341815.

[52]

L. Nirenberg, An extended interpolation inequality, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 20 (1966), 733-737. 

[53]

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Figure 1.  (EBB)-W$ _{FKV} $ Model
Figure 2.  W-(EBB)$ _{FKV} $ Model
Figure 3.  W-W$ _{FKV} $ Model
Figure 4.  (EBB) $ _{FKV} $ Model
Figure 5.  (EBB)-(EBB)$ _{FKV} $
Table 1.  Decay Results
Model Decay Rate $ \alpha\to 1 $
(EBB)-W$ _{FKV} $ $ t^{\frac{-4}{2-\alpha}} $ $ t^{-4} $
W-W$ _{FKV} $ $ t^{\frac{-4}{2-\alpha}} $ $ t^{-4} $
W-(EBB)$ _{FKV} $ $ t^{\frac{-2}{3-\alpha}} $ $ t^{-1} $
(EBB)$ _{FKV} $ $ t^{\frac{-2}{1-\alpha}} $ Exponential
(EBB)-(EBB)$ _{FKV} $ $ t^{\frac{-2}{3-\alpha}} $ $ t^{-1} $
Model Decay Rate $ \alpha\to 1 $
(EBB)-W$ _{FKV} $ $ t^{\frac{-4}{2-\alpha}} $ $ t^{-4} $
W-W$ _{FKV} $ $ t^{\frac{-4}{2-\alpha}} $ $ t^{-4} $
W-(EBB)$ _{FKV} $ $ t^{\frac{-2}{3-\alpha}} $ $ t^{-1} $
(EBB)$ _{FKV} $ $ t^{\frac{-2}{1-\alpha}} $ Exponential
(EBB)-(EBB)$ _{FKV} $ $ t^{\frac{-2}{3-\alpha}} $ $ t^{-1} $
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