Exponential Stability of the System of Transmission of the Wave Equation with a Delay Term in the Boundary Feedback

Abstract : We consider a system of transmission of the wave equation with Neumann feedback control that contains a delay term and that acts on the exterior boundary. First, we prove under some assumptions that the closed-loop system generates a C₀ −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 [3] (Lemma 7.2) and compactness-uniqueness arguments.