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Article Dans Une Revue SIAM Journal on Control and Optimization Année : 2009

Stabilization of the Wave Equation on 1-d Networks

Enrique Zuazua

Résumé

In this paper we study the stabilization of the wave equation on general 1-d networks. For that, we transfer known observability results in the context of control problems of conservative systems (see [R. Dáger and E. Zuazua, Wave Propagation, Observation, and Control in 1-d Flexible Multi-structures, Math. Appl. 50, Springer-Verlag, Berlin, 2006]) into a weighted observability estimate for dissipative systems. Then we use an interpolation inequality similar to the one proved in [P. Bégout and F. Soria, J. Differential Equations, 240 (2007), pp. 324-356] to obtain the explicit decay estimates of the energy for smooth initial data. The obtained decay rate depends on the geometric and topological properties of the network. We also give some examples of particular networks in which our results apply, yielding different decay rates.
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Dates et versions

hal-03550152 , version 1 (31-01-2022)

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Julie Valein, Enrique Zuazua. Stabilization of the Wave Equation on 1-d Networks. SIAM Journal on Control and Optimization, 2009, 48 (4), pp.2771-2797. ⟨10.1137/080733590⟩. ⟨hal-03550152⟩
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