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 & Related Fields, doi: 10.3934/mcrf.2021059
References:
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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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

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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.  Google Scholar

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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.  Google Scholar

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[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.  Google Scholar

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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.   Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[14]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

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

[24]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[29]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[36]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[46]

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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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