scholarly journals Coupled second order evolution equations with fading memory: Optimal energy decay rate

2014 ◽  
Vol 257 (5) ◽  
pp. 1501-1528 ◽  
Author(s):  
Kun-Peng Jin ◽  
Jin Liang ◽  
Ti-Jun Xiao
Keyword(s):  
Decay Rate ◽  
Evolution Equations ◽  
Energy Decay ◽  
Second Order ◽  
Fading Memory ◽  
Optimal Energy ◽  
Energy Decay Rate

Sharp decay rates for a class of nonlinear viscoelastic plate models

2017 ◽  
Vol 20 (02) ◽  
pp. 1750010 ◽  
Author(s):  
E. H. Gomes Tavares ◽  
M. A. Jorge Silva ◽  
T. F. Ma
Keyword(s):  
Decay Rate ◽  
Evolution Equations ◽  
Viscoelastic Plate ◽  
Energy Decay ◽  
Decay Rates ◽  
Memory Kernel ◽  
Energy Decay Rate ◽  
Plate Equations ◽  
Energy Decay Rates ◽  
With Memory

This paper is concerned with uniform stability of the energy corresponding to a class of nonlinear plate equations with memory. It is assumed that the memory kernel [Formula: see text] satisfies the condition [Formula: see text] of Alabau-Boussouira and Cannarsa [A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Acad. Sci. Paris Ser. I 347 (2009) 867–872], where [Formula: see text] is positive, convex, increasing, and satisfies [Formula: see text]. Then, we obtain sharp energy decay rate in the sense that it recovers the decay rate assumed to the memory kernel. To this end we use a recent approach proposed by Lasiecka and Wang [Intrinsic decay rate estimates for semilinear abstract second order equations with memory, in New Prospects in Direct, Inverse and Control Problems for Evolution Equations, Springer Series INDAM, Vol. 10 (Springer, 2014), pp. 271–303].

scholarly journals Optimal energy decay rate of Rayleigh beam equation with only one dynamic boundary control

2017 ◽  
Vol 35 (3) ◽  
pp. 131-171 ◽  
Author(s):  
Denis Mercier ◽  
Serge Nicaise ◽  
Mohamad Sammoury ◽  
Ali Wehbe
Keyword(s):  
Asymptotic Expansion ◽  
Decay Rate ◽  
Boundary Control ◽  
Energy Decay ◽  
Beam Equation ◽  
Control Force ◽  
Rayleigh Beam ◽  
Optimal Energy ◽  
Energy Decay Rate ◽  
Boundary Controls

In \cite{WehbeRayleigh:06}, Wehbe considered the Rayleigh beam equation with two dynamical boundary controls and established the optimal polynomial energy decay rate of type $\dfrac{1}{t}$. The proof exploits in an explicit way the presence of two boundary controls, hence the case of the Rayleigh beam damped by only one dynamical boundary control remained open. In this paper, we fill this gap by considering a clamped Rayleigh beam equation subject to only on dynamical boundary feedback. First, we consider the Rayleigh beam equation subject to only one dynamical boundary control moment. We give the asymptotic expansion of eigenvalues and eigenfunctions of the undamped underlying system and we establish a polynomial energy decay rate of type $\frac{1}{t}$ for smooth initial data via an observability inequality of the corresponding undamped problem combined with the boundedness property of the transfer function of the associated undamped problem. Moreover, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained energy decay rate is optimal. Next, we consider the Rayleigh beam equation subject to only one dynamical boundary control force. We start by giving the asymptotic expansion of the eigenvalues and the eigenfunctions of the damped and undamped systems using an explicit approximation of the characteristic equation determining these eigenvalues. We next show that the system of eigenvectors of the damped problem form a Riesz basis. Finally, we establish the optimal energy decay rate of polynomial type $\frac{1}{\sqrt{t}}$.

scholarly journals Energy decay rate for a quasi-linear wave equation with localized strong dissipation

2011 ◽  
Vol 62 (1) ◽  
pp. 164-172 ◽  
Author(s):  
Daewook Kim ◽  
Yong Han Kang ◽  
Mi Jin Lee ◽  
Il Hyo Jung
Keyword(s):  
Wave Equation ◽  
Decay Rate ◽  
Energy Decay ◽  
Linear Wave ◽  
Linear Wave Equation ◽  
Energy Decay Rate
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