American Institute of Mathematical Sciences

Advanced
March  2020, 40(3): 1493-1515. doi: 10.3934/dcds.2020084

Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay

Xin-Guang Yang 1, , Jing Zhang 2,, and  Shu Wang 3,

1. 

College of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, China
2. 

Department of Mathematics and Economics, Virginia State University, Petersburg, VA 23806, USA
3. 

College of Applied Science, Beijing University of Technology, Beijing 100124, China

* Corresponding author: Jing Zhang

Received  February 2019 Revised  October 2019 Published  December 2019

Fund Project: Research partly supported by NSFC (No. 11831003, No. 11771031, No. 11531010 and No. 11726625), the Fund of Young Backbone Teacher in Henan Province (No. 2018GGJS039).

This paper is concerned with the stability and dynamics of a weak viscoelastic system with nonlinear time-varying delay. By imposing appropriate assumptions on the memory and sub-linear delay operator, we prove the global well-posedness and stability which generates a gradient system. The gradient system possesses finite fractal dimensional global and exponential attractors with unstable manifold structure. Moreover, the effect and balance between damping and time-varying delay are also presented.

Keywords: Viscoelastic system, time varying delay, quasi-stability, unstable manifold, exponential attractor.
Mathematics Subject Classification: 35B40, 35B41, 35Q30, 76D03, 76D05.
Citation: Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete and Continuous Dynamical Systems, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084
References:
[1]

M. Aassila, M. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602.  doi: 10.1137/S0363012998344981.

[2]

C. Abdallah, P. Dorato and R. Byrne, Delayed positive feedback can stabilize oscillatory system, American Control Conference, San Francisco, 1993, 3106–3107. doi: 10.23919/ACC.1993.4793475.

[3]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[4]

J. Appleby, M. Fabrizio, B. Lazzari and D. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.  doi: 10.1142/S0218202506001674.

[5]

M. M. Cavalcanti, D. V. N. Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations, 2002 (2002), 1-14. 

[6]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., (195) (2008). doi: 10.1090/memo/0912.

[7]

I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[8]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[10]

Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6.

[11]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.

[12]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264.  doi: 10.1080/0003681021000035588.

[13]

B. Feng, General decay for a viscoelastic wave equation with density and time delay term in $\mathbb{R}^n$, Taiwanese J. Math., 22 (2018), 205-223.  doi: 10.11650/tjm/8105.

[14]

E. Fridman, Introduction to Time-Delay Systems. Analysis and Control, Systems & Control: Foundations & Applications, Birkhäser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.

[15]

C. Giorgi, J. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[16]

A. Guesmia and S. A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.  doi: 10.1016/j.nonrwa.2011.08.004.

[17]

Y. Guo, M. A. Rammaha and S. Sakuntasathien, Energy decay of a viscoelastic wave equation with supercritical nonlinearities, Z. Angew. Math. Phys., 69 (2018), 28pp. doi: 10.1007/s00033-018-0961-6.

[18]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.

[19]

W. Liu, General decay rate estimate for the energy of a weak viscoealstic equation with internal time-varying delay term, Taiwanese J. Math., 17 (2013), 2101-2115.  doi: 10.11650/tjm.17.2013.2968.

[20]

G. Liu and L. Diao, Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta. Appl. Math., 155 (2018), 9-19.  doi: 10.1007/s10440-017-0142-1.

[21]

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

[22]

S. A. Messaoudi, General decay of solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.

[23]

S. A. Messaoudi, General decay of solutions of a weak viscoelastic equation, Arab. J. Sci. Eng., 36 (2011), 1569-1579.  doi: 10.1007/s13369-011-0132-y.

[24]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[25]

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 Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[26]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958. 

[27]

S. Nicaise, C. Pignotti and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[28]

S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[29]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[30]

C. Pignotti, Stability results for second-order evolution equations with memory and switching time-delay, J. Dynam. Differential Equations, 29 (2017), 1309-1324.  doi: 10.1007/s10884-016-9545-3.

[31]

Y. Qin, J. Ren and T. Wei, Global existence, asymptotic behavior, and uniform attractor for a nonautonomous equation, Math. Methods Appl. Sci., 36 (2013), 2540-2553.  doi: 10.1002/mma.2774.

[32]

B. Said-Houari, Asymptotic behaviors of solutions for viscoelastic wave equation with space-time dependent damping term, J. Math. Anal. Appl., 387 (2012), 1088-1105.  doi: 10.1016/j.jmaa.2011.10.017.

[33]

I. H. Suh and Z. Bien, Use of time delay actions in the controller design, IEEE Trans. Automatic Control, 25 (1980), 600-603.  doi: 10.1109/TAC.1980.1102347.

[34]

C. Q. Xu, S. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.

show all references

References:
[1]

M. Aassila, M. M. Cavalcanti and J. A. Soriano, Asymptotic stability and energy decay rates for solutions of the wave equation with memory in a star-shaped domain, SIAM J. Control Optim., 38 (2000), 1581-1602.  doi: 10.1137/S0363012998344981.

[2]

C. Abdallah, P. Dorato and R. Byrne, Delayed positive feedback can stabilize oscillatory system, American Control Conference, San Francisco, 1993, 3106–3107. doi: 10.23919/ACC.1993.4793475.

[3]

F. Alabau-Boussouira, P. Cannarsa and D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342-1372.  doi: 10.1016/j.jfa.2007.09.012.

[4]

J. Appleby, M. Fabrizio, B. Lazzari and D. Reynolds, On exponential asymptotic stability in linear viscoelasticity, Math. Models Methods Appl. Sci., 16 (2006), 1677-1694.  doi: 10.1142/S0218202506001674.

[5]

M. M. Cavalcanti, D. V. N. Cavalcanti and J. A. Soriano, Exponential decay for the solution of semilinear viscoelastic wave equations with localized damping, Electron. J. Differential Equations, 2002 (2002), 1-14. 

[6]

I. D. Chueshov and I. Lasiecka, Long-time behavior of second order evolution equations with nonlinear damping, Mem. Amer. Math. Soc., (195) (2008). doi: 10.1090/memo/0912.

[7]

I. D. Chueshov and I. Lasiecka, Von Karman Evolution Equations. Well-Posedness and Long Time Dynamics, Springer Monographs in Mathematics, Springer, New York, 2010. doi: 10.1007/978-0-387-87712-9.

[8]

M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity, Commun. Pure Appl. Anal., 4 (2005), 705-720.  doi: 10.3934/cpaa.2005.4.705.

[9]

C. M. Dafermos, Asymptotic stability in viscoelasticity, Arch. Rational Mech. Anal., 37 (1970), 297-308.  doi: 10.1007/BF00251609.

[10]

Q. Dai and Z. Yang, Global existence and exponential decay of the solution for a viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 65 (2014), 885-903.  doi: 10.1007/s00033-013-0365-6.

[11]

R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.  doi: 10.1137/0324007.

[12]

M. Fabrizio and S. Polidoro, Asymptotic decay for some differential systems with fading memory, Appl. Anal., 81 (2002), 1245-1264.  doi: 10.1080/0003681021000035588.

[13]

B. Feng, General decay for a viscoelastic wave equation with density and time delay term in $\mathbb{R}^n$, Taiwanese J. Math., 22 (2018), 205-223.  doi: 10.11650/tjm/8105.

[14]

E. Fridman, Introduction to Time-Delay Systems. Analysis and Control, Systems & Control: Foundations & Applications, Birkhäser/Springer, Cham, 2014. doi: 10.1007/978-3-319-09393-2.

[15]

C. Giorgi, J. Muñoz Rivera and V. Pata, Global attractors for a semilinear hyperbolic equation in viscoelasticity, J. Math. Anal. Appl., 260 (2001), 83-99.  doi: 10.1006/jmaa.2001.7437.

[16]

A. Guesmia and S. A. Messaoudi, A general decay result for a viscoelastic equation in the presence of past and finite history memories, Nonlinear Anal. Real World Appl., 13 (2012), 476-485.  doi: 10.1016/j.nonrwa.2011.08.004.

[17]

Y. Guo, M. A. Rammaha and S. Sakuntasathien, Energy decay of a viscoelastic wave equation with supercritical nonlinearities, Z. Angew. Math. Phys., 69 (2018), 28pp. doi: 10.1007/s00033-018-0961-6.

[18]

M. Kirane and B. Said-Houari, Existence and asymptotic stability of viscoelastic wave equation with a delay, Z. Angew. Math. Phys., 62 (2011), 1065-1082.  doi: 10.1007/s00033-011-0145-0.

[19]

W. Liu, General decay rate estimate for the energy of a weak viscoealstic equation with internal time-varying delay term, Taiwanese J. Math., 17 (2013), 2101-2115.  doi: 10.11650/tjm.17.2013.2968.

[20]

G. Liu and L. Diao, Energy decay of the solution for a weak viscoelastic equation with a time-varying delay, Acta. Appl. Math., 155 (2018), 9-19.  doi: 10.1007/s10440-017-0142-1.

[21]

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

[22]

S. A. Messaoudi, General decay of solution energy in a viscoelastic equation with a nonlinear source, Nonlinear Anal., 69 (2008), 2589-2598.  doi: 10.1016/j.na.2007.08.035.

[23]

S. A. Messaoudi, General decay of solutions of a weak viscoelastic equation, Arab. J. Sci. Eng., 36 (2011), 1569-1579.  doi: 10.1007/s13369-011-0132-y.

[24]

M. I. Mustafa, Optimal decay rates for the viscoelastic wave equation, Math. Methods Appl. Sci., 41 (2018), 192-204.  doi: 10.1002/mma.4604.

[25]

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 Optim., 45 (2006), 1561-1585.  doi: 10.1137/060648891.

[26]

S. Nicaise and C. Pignotti, Stabilization of the wave equation with boundary or internal distributed delay, Differential Integral Equations, 21 (2008), 935-958. 

[27]

S. Nicaise, C. Pignotti and E. Fridman, Stability of the heat and of the wave equations with boundary time-varying delays, Discrete Contin. Dyn. Syst. Ser. S, 2 (2009), 559-581.  doi: 10.3934/dcdss.2009.2.559.

[28]

S. Nicaise, C. Pignotti and J. Valein, Exponential stability of the wave equation with boundary time-varying delay, Discrete Contin. Dyn. Syst. Ser. S, 4 (2011), 693-722.  doi: 10.3934/dcdss.2011.4.693.

[29]

V. Pata, Stability and exponential stability in linear viscoelasticity, Milan J. Math., 77 (2009), 333-360.  doi: 10.1007/s00032-009-0098-3.

[30]

C. Pignotti, Stability results for second-order evolution equations with memory and switching time-delay, J. Dynam. Differential Equations, 29 (2017), 1309-1324.  doi: 10.1007/s10884-016-9545-3.

[31]

Y. Qin, J. Ren and T. Wei, Global existence, asymptotic behavior, and uniform attractor for a nonautonomous equation, Math. Methods Appl. Sci., 36 (2013), 2540-2553.  doi: 10.1002/mma.2774.

[32]

B. Said-Houari, Asymptotic behaviors of solutions for viscoelastic wave equation with space-time dependent damping term, J. Math. Anal. Appl., 387 (2012), 1088-1105.  doi: 10.1016/j.jmaa.2011.10.017.

[33]

I. H. Suh and Z. Bien, Use of time delay actions in the controller design, IEEE Trans. Automatic Control, 25 (1980), 600-603.  doi: 10.1109/TAC.1980.1102347.

[34]

C. Q. Xu, S. P. Yung and L. K. Li, Stabilization of the wave system with input delay in the boundary control, ESAIM Control Optim. Calc. Var., 12 (2006), 770-785.  doi: 10.1051/cocv:2006021.

[1]

Moncef Aouadi, Alain Miranville. Quasi-stability and global attractor in nonlinear thermoelastic diffusion plate with memory. Evolution Equations and Control Theory, 2015, 4 (3) : 241-263. doi: 10.3934/eect.2015.4.241
[2]

Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[3]

Jason S. Howell, Irena Lasiecka, Justin T. Webster. Quasi-stability and exponential attractors for a non-gradient system---applications to piston-theoretic plates with internal damping. Evolution Equations and Control Theory, 2016, 5 (4) : 567-603. doi: 10.3934/eect.2016020
[4]

Luci H. Fatori, Marcio A. Jorge Silva, Vando Narciso. Quasi-stability property and attractors for a semilinear Timoshenko system. Discrete and Continuous Dynamical Systems, 2016, 36 (11) : 6117-6132. doi: 10.3934/dcds.2016067
[5]

Baowei Feng. On a semilinear Timoshenko-Coleman-Gurtin system: Quasi-stability and attractors. Discrete and Continuous Dynamical Systems, 2017, 37 (9) : 4729-4751. doi: 10.3934/dcds.2017203
[6]

Xin-Guang Yang. An Erratum on "Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay" (Discrete Continuous Dynamic Systems, 40(3), 2020, 1493-1515). Discrete and Continuous Dynamical Systems, 2022, 42 (3) : 1493-1494. doi: 10.3934/dcds.2021161
[7]

Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure and Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667
[8]

István Györi, Ferenc Hartung. Exponential stability of a state-dependent delay system. Discrete and Continuous Dynamical Systems, 2007, 18 (4) : 773-791. doi: 10.3934/dcds.2007.18.773
[9]

Tibor Krisztin. A local unstable manifold for differential equations with state-dependent delay. Discrete and Continuous Dynamical Systems, 2003, 9 (4) : 993-1028. doi: 10.3934/dcds.2003.9.993
[10]

Yujun Zhu. Topological quasi-stability of partially hyperbolic diffeomorphisms under random perturbations. Discrete and Continuous Dynamical Systems, 2014, 34 (2) : 869-882. doi: 10.3934/dcds.2014.34.869
[11]

Yangzi Hu, Fuke Wu. The improved results on the stochastic Kolmogorov system with time-varying delay. Discrete and Continuous Dynamical Systems - B, 2015, 20 (5) : 1481-1497. doi: 10.3934/dcdsb.2015.20.1481
[12]

Carlos Nonato, Manoel Jeremias dos Santos, Carlos Raposo. Dynamics of Timoshenko system with time-varying weight and time-varying delay. Discrete and Continuous Dynamical Systems - B, 2022, 27 (1) : 523-553. doi: 10.3934/dcdsb.2021053
[13]

Baowei Feng, Carlos Alberto Raposo, Carlos Alberto Nonato, Abdelaziz Soufyane. Analysis of exponential stabilization for Rao-Nakra sandwich beam with time-varying weight and time-varying delay: Multiplier method versus observability. Mathematical Control and Related Fields, 2022  doi: 10.3934/mcrf.2022011
[14]

Wenjun Liu, Biqing Zhu, Gang Li, Danhua Wang. General decay for a viscoelastic Kirchhoff equation with Balakrishnan-Taylor damping, dynamic boundary conditions and a time-varying delay term. Evolution Equations and Control Theory, 2017, 6 (2) : 239-260. doi: 10.3934/eect.2017013
[15]

Xiang Xie, Honglei Xu, Xinming Cheng, Yilun Yu. Improved results on exponential stability of discrete-time switched delay systems. Discrete and Continuous Dynamical Systems - B, 2017, 22 (1) : 199-208. doi: 10.3934/dcdsb.2017010
[16]

Aowen Kong, Carlos Nonato, Wenjun Liu, Manoel Jeremias dos Santos, Carlos Raposo. Equivalence between exponential stabilization and observability inequality for magnetic effected piezoelectric beams with time-varying delay and time-dependent weights. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 2959-2978. doi: 10.3934/dcdsb.2021168
[17]

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
[18]

Radosław Czaja. Pullback attractors via quasi-stability for non-autonomous lattice dynamical systems. Discrete and Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021276
[19]

Ling Zhang, Xiaoqi Sun. Stability analysis of time-varying delay neural network for convex quadratic programming with equality constraints and inequality constraints. Discrete and Continuous Dynamical Systems - B, 2022  doi: 10.3934/dcdsb.2022035
[20]

Quan Hai, Shutang Liu. Mean-square delay-distribution-dependent exponential synchronization of chaotic neural networks with mixed random time-varying delays and restricted disturbances. Discrete and Continuous Dynamical Systems - B, 2021, 26 (6) : 3097-3118. doi: 10.3934/dcdsb.2020221

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (395)
  • HTML views (150)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy