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On the sensitivity of the transmission of boundary dissipation for strongly coupled and indirectly damped systems of wave equations

  • Bopeng RaoEmail author
Article
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Abstract

We consider the stability of a system of two strongly coupled wave equations by means of only one boundary feedback. We show that the stability of the system depends in a very complex way on all of the involved factors such as the type of coupling, the hidden regularity and the accordance of boundary conditions. We first show that the system is uniformly exponentially stable if the undamped equation has Dirichlet boundary condition, while it is only polynomially stable if the undamped equation is subject to Neumann boundary condition. Next, by a spectral approach, we show that this sensitivity of stability with respect to the boundary conditions on the undamped equation is intrinsically linked with the transmission of the vibration as well as the dissipation between the equations.

Keywords

Wave equation Indirect damping Transmission of boundary dissipation Exponential stability Polynomial energy decay rate 

Mathematics Subject Classification

93D15 93D20 93C20 35L53 35B40 

Notes

Acknowledgements

The author would like to thank the referees for their very valuable comments.

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© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Institut de Recherche Mathématique AvancéeUniversité de StrasbourgStrasbourgFrance

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