Citation:
Rebiai S-E, Sidi-Ali FZ. Uniform exponential stability of the transmission wave equation with a delay term in the boundary feedback. IMA Journal of Mathematical Control and InformationIMA Journal of Mathematical Control and Information. 2016;33 :1 - 20.
Date Published:
2016Abstract:
We consider a system of transmission of the wave equation with Neumann feedback control that contains a delay term and acts on the exterior boundary. First, we prove under some assumptions that the closed-loop system generates a C 0 -semigroup of contractions on an appropriate Hilbert space. Then, under further assumptions, we show that the closed-loop system is exponentially stable. To establish this result, we introduce a suitable energy function and use multiplier method together with an estimate taken from Lasiecka & Triggiani (1992, Appl. Math. Optim., 25, 189–244.) (Lemma 7.2) and compactness-uniqueness argument.