Abstract
This paper considers a viscoelastic Euler-Bernoulli plate system with variable coefficients and time-varying delay. By the multiplier method, Riemannian geometry method, and under suitable assumptions on viscoelastic term and time delay, the authors obtain the general stability of the solution for the system which depends on the behavior of the relaxation function.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Fabrizio M, Giorgi C, and Pata V, A new approach to equations with memory, Archive for Rational Mechanics and Analysis, 2010, 198: 198–232.
Datko R, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM Journal on Control and Optimization, 1998, 26(3): 697–713.
Datko R, Lagnese J, and Polis M P, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM Journal on Control and Optimization, 1986, 24: 152–156.
Nicaise S and Pignotti C, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM Journal on Control and Optimization, 2006, 45(5): 1561–1585.
Ammari K, Nicaise S, and Pignotti C, Feedback boundary stabilization of wave equations with interior delay, Systems & Control Letters, 2010, 59: 623–628.
Kirane M and Said-Houari B, Existence and asymptotic stability of a viscoelastic wave equation with a delay, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 2011, 62: 1065–1082.
Nicaise S, Pignotti C, and Valein J, Exponential stability of the wave equation with boundary time-varying delay, Discrete and Continuous Dynamical Systems — Series S, 2011, 4: 693–722.
Benaissa A and Louhibi N, Global existence and energy decay of solutions to a nonlinear wave equation with a delay term, Georgian Mathmatical Journal, 2013, 20: 1–24.
Park S H, Decay rate estimates for a weak viscoelastic beam equation with time-varying delay, Applied Mathematics Letters, 2014, 31: 46–51.
Yang Z F, Existence and energy decay of solutions for the Euler-Bernoulli viscoelastic equation with a delay, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 2015, 66: 727–745.
Mustafa M I and Kafini M, Decay rates for memory-type plate system with delay and source term, Mathematical Methods in the Applied Sciences, 2017, 40: 883–895.
Yao P F, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM Journal on Control and Optimization, 1999, 37(5): 1568–1599.
Cao X M, Energy decay of solutions for a variable-coefficient viscoelastic wave equation with a weak nonlinear dissipation, Journal of Mathmatical Physics, 2016, 57, 021509.
Liu K S, Locally distributed control and damping for the conservative systems, SIAM Journal on Control and Optimization, 1997, 35(5): 1574–1590.
Feng S J and Feng D X, Nonlinear internal damping of wave equations with variable coefficients, Acta Mathematica Sinica, English Series, 2004, 20(6): 1057–1072.
Cao X M and Yao P F, General decay rate estimates for viscoelastic wave equation with variable coefficients, Journal of Systems Science & Complexity, 2014, 27(5): 836–852.
Li J and Chai S G, Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback, Nonlinear Analysis, 2015, 112: 105–117.
Li J, Chai S G, and Wu J Q, Energy decay of the wave equation with variable coefficients and a localized half-linear dissipation in an exterior domain, Zeitschrift für Angewandte Mathematik und Physik ZAMP, 2015, 66: 95–112.
Li J and Chai S G, Existence and energy decay rates of solutions to the variable-coefficient Euler- Bernoulli plate with a delay in localized nonlinear internal feedback, Journal of Mathematical Analysis and Applications, 2016, 443: 981–1006.
Zhang W and Zhang Z F, Stabilization of transmission coupled wave and Euler-Bernoulli equations on Riemannian manifolds by nonlinear feedbacks, Journal of Mathematical Analysis and Applications, 2015, 422: 1504–1526.
Yao P F, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, Differential Geometric Methods in the Control of Partial Differential Equations, Contemporary Mathematics, 268, American Mathematical Society, Providence, RI, 2000, 383–406.
Guo Y X and Yao P F, Stabilization of Euler-Bernoulli plate equation with variable coefficients by nonlinear boundary feedback, Journal of Mathematical Analysis and Applications, 2006, 317: 50–70.
Wu H, Shen C L, and Yu Y L, An Introduction to Riemannian Geometry, Peking University Press, Beijing, 1989.
Yao P F, Modeling and Control in Vibrational and Structural Dynamics — A Differential Geometric Approach, Chapman and Hall/CRC Applied Mathematics and Nonlinear Science Series, CRC Press, 2011.
Martinez P, A new method to obtain decay rate estimates for dissipative systems, ESAIM: Control, Optimisation and Calculus of Variations, 1999, 4: 419–444.
Simon J, Compact sets in the space Lp(0, T;B), Annali di Matematica Pura ed Applicata, 1987, 146(4): 65–96.
Author information
Authors and Affiliations
Corresponding author
Additional information
This research was partially supported by the National Natural Science Foundation of China under Grant Nos. 11871315, 61374089, and the Natural Science Foundation of Shanxi Province under Grant No. 2014011005–2.
This paper was recommended for publication by Editor SUN Jian.
Rights and permissions
About this article
Cite this article
Wang, P., Hao, J. Asymptotic Stability of Memory-Type Euler-Bernoulli Plate with Variable Coefficients and Time Delay. J Syst Sci Complex 32, 1375–1392 (2019). https://doi.org/10.1007/s11424-018-7370-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11424-018-7370-y