• Previous Article
    Recovering time-dependent diffusion coefficients in a nonautonomous parabolic equation from energy measurements
  • DCDS-S Home
  • This Issue
  • Next Article
    $ L^p $-strong solution for the stationary exterior Stokes equations with Navier boundary condition
June  2022, 15(6): 1421-1438. doi: 10.3934/dcdss.2022073

Polynomial stability in viscoelastic network of strings

1. 

UR Analysis and Control of PDEs, UR13ES64, ISCAE, University of Manouba, Tunisia

2. 

UR Analysis and Control of PDEs, UR13ES64, Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El Manar, Tunisia

* Corresponding author: Rania Yahia

Received  August 2021 Revised  February 2022 Published  June 2022 Early access  March 2022

In this paper we consider star-shaped viscoelastic networks, and study the large-time behaviour of these networks by proving polynomial decay rates. The energy decay rate depends on the irrationality properties of the lengths of the rods.

Citation: Karim El Mufti, Rania Yahia. Polynomial stability in viscoelastic network of strings. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1421-1438. doi: 10.3934/dcdss.2022073
References:
[1]

M. AlvesJ. Muñoz RiveraM. SepùlvedaO. Vera Villagrán and M. Z. Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), 483-497.  doi: 10.1002/mana.201200319.

[2]

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.

[3]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[4]

C. J. K. BattyR. Chill and Y. Tomilov, Fine scales of decay of operator semigroups, J. Eur. Math. Soc. (JEMS), 18 (2016), 853-929.  doi: 10.4171/JEMS/605.

[5]

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

[6]

A. BchatniaK. El Mufti and R. Yahia, Stability of an infinite star-shaped network of strings by a Kelvin-Voigt damping, Math. Methods Appl. Sci., 45 (2022), 2024-2041.  doi: 10.1002/mma.7903.

[7]

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.

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[9]

R. Dàger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques & Applications (Berlin), 50. Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[10]

Z.-J. Han and E. Zuazua, Decay rates for elastic-thermoelastic star-shaped networks, Netw. Heterog. Media, 12 (2017), 461-488.  doi: 10.3934/nhm.2017020.

[11]

F. Hassine, Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface, J. Differential Equations, 297 (2021), 1-24.  doi: 10.1016/j.jde.2021.06.017.

[12]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.

[13]

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.

[14]

K. LiuZ. Liu and Q. Zhang, Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.

[15]

Z. Liu and R. Quintanilla, Energy decay rate of a mixed type Ⅱ and type Ⅲ thermolastic system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433-1444.  doi: 10.3934/dcdsb.2010.14.1433.

[16]

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.

[17]

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.

[18]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, 398. Chapman & Hall/CRC, Boca Raton, FL, 1999.

[19]

B. Muckenhoupt, Hardy's inequality with weights, Stud. Math., 44 (1972), 31-38.  doi: 10.4064/sm-44-1-31-38.

[20]

M. Renardy, On localized Kelvin-Voigt damping, Z. Angew. Math. Mech., 84 (2004), 280-283.  doi: 10.1002/zamm.200310100.

[21]

W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201.  doi: 10.1007/BF02392334.

[22]

V. D. Stepanov, Weighted Hardy inequality, Siberian Math. J, 28 (1987), 515-517. 

[23]

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.

show all references

References:
[1]

M. AlvesJ. Muñoz RiveraM. SepùlvedaO. Vera Villagrán and M. Z. Garay, The asymptotic behavior of the linear transmission problem in viscoelasticity, Math. Nachr., 287 (2014), 483-497.  doi: 10.1002/mana.201200319.

[2]

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.

[3]

A. BátkaiK.-J. EngelJ. Prüss and R. Schnaubelt, Polynomial stability of operator semigroups, Math. Nachr., 279 (2006), 1425-1440.  doi: 10.1002/mana.200410429.

[4]

C. J. K. BattyR. Chill and Y. Tomilov, Fine scales of decay of operator semigroups, J. Eur. Math. Soc. (JEMS), 18 (2016), 853-929.  doi: 10.4171/JEMS/605.

[5]

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

[6]

A. BchatniaK. El Mufti and R. Yahia, Stability of an infinite star-shaped network of strings by a Kelvin-Voigt damping, Math. Methods Appl. Sci., 45 (2022), 2024-2041.  doi: 10.1002/mma.7903.

[7]

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.

[8]

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011.

[9]

R. Dàger and E. Zuazua, Wave Propagation, Observation and Control in 1-d Flexible Multi-Structures, Mathématiques & Applications (Berlin), 50. Springer-Verlag, Berlin, 2006. doi: 10.1007/3-540-37726-3.

[10]

Z.-J. Han and E. Zuazua, Decay rates for elastic-thermoelastic star-shaped networks, Netw. Heterog. Media, 12 (2017), 461-488.  doi: 10.3934/nhm.2017020.

[11]

F. Hassine, Stability of a star-shaped network with local Kelvin-Voigt damping and non-smooth coefficient at interface, J. Differential Equations, 297 (2021), 1-24.  doi: 10.1016/j.jde.2021.06.017.

[12]

K. Liu and Z. Liu, Exponential decay of energy of vibrating strings with local viscoelasticity, Z. Angew. Math. Phys., 53 (2002), 265-280.  doi: 10.1007/s00033-002-8155-6.

[13]

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.

[14]

K. LiuZ. Liu and Q. Zhang, Eventual differentiability of a string with local Kelvin-Voigt damping, ESAIM Control Optim. Calc. Var., 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.

[15]

Z. Liu and R. Quintanilla, Energy decay rate of a mixed type Ⅱ and type Ⅲ thermolastic system, Discrete Contin. Dyn. Syst. Ser. B, 14 (2010), 1433-1444.  doi: 10.3934/dcdsb.2010.14.1433.

[16]

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.

[17]

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.

[18]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics, 398. Chapman & Hall/CRC, Boca Raton, FL, 1999.

[19]

B. Muckenhoupt, Hardy's inequality with weights, Stud. Math., 44 (1972), 31-38.  doi: 10.4064/sm-44-1-31-38.

[20]

M. Renardy, On localized Kelvin-Voigt damping, Z. Angew. Math. Mech., 84 (2004), 280-283.  doi: 10.1002/zamm.200310100.

[21]

W. M. Schmidt, Simultaneous approximation to algebraic numbers by rationals, Acta Math., 125 (1970), 189-201.  doi: 10.1007/BF02392334.

[22]

V. D. Stepanov, Weighted Hardy inequality, Siberian Math. J, 28 (1987), 515-517. 

[23]

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.

Figure 1.  Star-shaped viscoelastic network and the prescribed conditions
[1]

Rachid Assel, Mohamed Ghazel. Energy decay for the damped wave equation on an unbounded network. Evolution Equations and Control Theory, 2018, 7 (3) : 335-351. doi: 10.3934/eect.2018017

[2]

Bopeng Rao, Xu Zhang. Frequency domain approach to decay rates for a coupled hyperbolic-parabolic system. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2789-2809. doi: 10.3934/cpaa.2021119

[3]

George Avalos, Roberto Triggiani. Rational decay rates for a PDE heat--structure interaction: A frequency domain approach. Evolution Equations and Control Theory, 2013, 2 (2) : 233-253. doi: 10.3934/eect.2013.2.233

[4]

Fatiha Alabau-Boussouira, Vincent Perrollaz, Lionel Rosier. Finite-time stabilization of a network of strings. Mathematical Control and Related Fields, 2015, 5 (4) : 721-742. doi: 10.3934/mcrf.2015.5.721

[5]

Marta Lewicka, Piotr B. Mucha. A local existence result for a system of viscoelasticity with physical viscosity. Evolution Equations and Control Theory, 2013, 2 (2) : 337-353. doi: 10.3934/eect.2013.2.337

[6]

Aihua Li. An algebraic approach to building interpolating polynomial. Conference Publications, 2005, 2005 (Special) : 597-604. doi: 10.3934/proc.2005.2005.597

[7]

L.R. Ritter, Akif Ibragimov, Jay R. Walton, Catherine J. McNeal. Stability analysis using an energy estimate approach of a reaction-diffusion model of atherogenesis. Conference Publications, 2009, 2009 (Special) : 630-639. doi: 10.3934/proc.2009.2009.630

[8]

Michiko Yuri. Polynomial decay of correlations for intermittent sofic systems. Discrete and Continuous Dynamical Systems, 2008, 22 (1&2) : 445-464. doi: 10.3934/dcds.2008.22.445

[9]

Zhong-Jie Han, Gen-Qi Xu. Spectrum and dynamical behavior of a kind of planar network of non-uniform strings with non-collocated feedbacks. Networks and Heterogeneous Media, 2010, 5 (2) : 315-334. doi: 10.3934/nhm.2010.5.315

[10]

Walid Boughamda. Boundary stabilization for a star-shaped network of variable coefficients strings linked by a point mass. Discrete and Continuous Dynamical Systems - S, 2022, 15 (5) : 1103-1125. doi: 10.3934/dcdss.2021139

[11]

Vittorino Pata. Exponential stability in linear viscoelasticity with almost flat memory kernels. Communications on Pure and Applied Analysis, 2010, 9 (3) : 721-730. doi: 10.3934/cpaa.2010.9.721

[12]

Giovambattista Amendola, Mauro Fabrizio, John Murrough Golden. Minimum free energy in the frequency domain for a heat conductor with memory. Discrete and Continuous Dynamical Systems - B, 2010, 14 (3) : 793-816. doi: 10.3934/dcdsb.2010.14.793

[13]

Priyanjana M. N. Dharmawardane. Decay property of regularity-loss type for quasi-linear hyperbolic systems of viscoelasticity. Conference Publications, 2013, 2013 (special) : 197-206. doi: 10.3934/proc.2013.2013.197

[14]

Gustavo Alberto Perla Menzala, Julian Moises Sejje Suárez. A thermo piezoelectric model: Exponential decay of the total energy. Discrete and Continuous Dynamical Systems, 2013, 33 (11&12) : 5273-5292. doi: 10.3934/dcds.2013.33.5273

[15]

Mohammed Aassila. On energy decay rate for linear damped systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 851-864. doi: 10.3934/dcds.2002.8.851

[16]

Bopeng Rao. Optimal energy decay rate in a damped Rayleigh beam. Discrete and Continuous Dynamical Systems, 1998, 4 (4) : 721-734. doi: 10.3934/dcds.1998.4.721

[17]

Alberto Ferrero, Filippo Gazzola, Hans-Christoph Grunau. Decay and local eventual positivity for biharmonic parabolic equations. Discrete and Continuous Dynamical Systems, 2008, 21 (4) : 1129-1157. doi: 10.3934/dcds.2008.21.1129

[18]

Stefano Fasani, Sergio Rinaldi. Local stabilization and network synchronization: The case of stationary regimes. Mathematical Biosciences & Engineering, 2010, 7 (3) : 623-639. doi: 10.3934/mbe.2010.7.623

[19]

Jianfeng Feng, Mariya Shcherbina, Brunello Tirozzi. Stability of the dynamics of an asymmetric neural network. Communications on Pure and Applied Analysis, 2009, 8 (2) : 655-671. doi: 10.3934/cpaa.2009.8.655

[20]

Guillaume Cantin, Alexandre Thorel. On a generalized diffusion problem: A complex network approach. Discrete and Continuous Dynamical Systems - B, 2022, 27 (4) : 2345-2365. doi: 10.3934/dcdsb.2021135

2021 Impact Factor: 1.865

Metrics

  • PDF downloads (94)
  • HTML views (64)
  • Cited by (0)

Other articles
by authors

[Back to Top]