Stability to weakly dissipative Timoshenko systems

Abstract

In this paper, we consider the Timoshenko systems with frictional dissipation working only on the vertical displacement. We prove that the system is exponentially stable if and only if the wave speeds are the same. On the contrary, we show that the Timoshenko systems is polynomially stable giving the optimal decay rate. Copyright © 2013 John Wiley & Sons, Ltd.

Asymptotic behavior of weakly dissipative Bresse‐Timoshenko system on influence of the second spectrum of frequency

“…We prove that frictional damping acting on vertical displacement of this system is exponentially stable regardless the equality between velocities of wave propagation. This result is new and completely different from ones obtained by Almeida Júnior et al [4] for the Timoshenko system with the same damping acting on vertical displacement. Our approach is strongly inspired in a recent result due to Almeida Júnior and Ramos [2] where the authors justified, from physical point of view, a classical and pioneering result due to Soufyane.…”

“…On the other hand, if there exists only one dissipative mechanism acting only on shear force or acting only on bending moment , the exponential decay depends on a particular relationship between coefficients of the system. This has been the case of several works (see for example [3][4][5]10,[25][26][27][28]33,36,37] and references contained therein) since the work due to Soufyane. [36] In some cases of these references, being 1 ∶= √ ′ ∕ and 2 ∶= √ ∕ the velocities of wave propagation for Bresse-Timoshenko beam model, the condition given by = ∕ ′ (resulting of the equality 1 = 2 ) is necessary and sufficient to achieve the exponential decay.…”

On the decay rates of Timoshenko system with second sound

“…In this regard, we quote, among others, the work of Soufyane and Wehbe , Guesmia and Messaoudi Rivera and Fernández Sare , Rivera and Racke , Messaoudi and Mustafa , and Messaoudi and Said‐Houari . Similar results were obtained by Almeida Júnior et al , when the damping term is in the first equation.…”

Well‐posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks

“…See, for example, the literature. () For laminated Timoshenko beams without time delay, there are just a few published works. Wang et al considered the boundary with one end fixed: and at the other end and established the exponential stability by assuming and k i ≠ r i ,( i =1,2).…”