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.
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Dietmar Hömberg; Fredi Tröltzsch. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. Springer, IFIP Advances in Information and Communication Technology, AICT-391, pp.276-285, 2013, System Modeling and Optimization. 〈10.1007/978-3-642-36062-6_28〉
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Salah-Eddine Rebiai. Exponential Stability of the System of Transmission of the Wave Equation with a Delay Term in the Boundary Feedback. Dietmar Hömberg; Fredi Tröltzsch. 25th System Modeling and Optimization (CSMO), Sep 2011, Berlin, Germany. Springer, IFIP Advances in Information and Communication Technology, AICT-391, pp.276-285, 2013, System Modeling and Optimization. 〈10.1007/978-3-642-36062-6_28〉. 〈hal-01347547〉

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