Stability results for the approximation of weakly coupled wave equations

Abdallah, Farah; Nicaise, Serge; Valein, Julie; Wehbe, Ali

doi:10.1016/j.crma.2011.12.004

Cited by 3 publications

(2 citation statements)

References 16 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…In addition, the simultaneous boundary stabilization of the same system is also studied in [11]. Let us mention the additional references [3], [13], [2], [15], and [10] for indirect stabilization of coupled equations via one order terms. Next, we recall some of the results related to the stability of two equations coupled through displacements.…”

Section: Introductionmentioning

confidence: 99%

“…Last but not least, we recall some results concerning the exponential or polynomial indirect stability of systems which arise from physical problems. For example, we quote [1,9,20,22,28,34,40,41] for the Bresse system and [2,7,12,14,25,33,36] for the Timoshenko system. The Bresse system is usually considered in the study of elastic structures of the arcs type (see [26]) while the Timoshenko system is usually considered in describing the transverse vibration of a beam and it ignores damping effects of any nature (see [39]).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Optimal indirect stability of a weakly damped elastic abstract system of second order equations coupled by velocities

Abdallah¹,

Ghader²,

Wehbe³

et al. 2019

Communications on Pure &Amp; Applied Analysis

Self Cite

View full text Add to dashboard Cite

In this paper, by means of the Riesz basis approach, we study the stability of a weakly damped system of two second order evolution equations coupled through the velocities (see (1.1)). If the fractional order damping becomes viscous and the waves propagate with equal speeds, we prove exponential stability of the system and, otherwise, we establish an optimal polynomial decay rate. Finally, we provide some illustrative examples.1 γ and that this decay rate is in some sense optimal. Regarding System (1.1) when α = 0, it reduces to System (1.2) with B = A γ and a = 1. In this case, we recover the results of [29].When the coupling acts through displacements, Loreti and Rao studied in [31] the stability of the following abstract system of coupled equations (1.3) u tt + Au + A γ u t + αy = 0, y tt + Ay − αu = 0.They proved that System (1.3) is not exponentially stable and an optimal polynomial energy decay rate of type t −τ (γ) is obtained where

show abstract