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Pré-Publication, Document De Travail Année : 2021

Stabilization of the Wave Equation through Nonlinear Dirichlet Actuation

Résumé

In this paper, we consider the problem of nonlinear (in particular, saturated) stabilization of the high-dimensional wave equation in the Dirichlet boundary condition. The wave dynamics are subject to a dissipative nonlinear velocity feedback and generate a strongly continuous semigroup of contractions on the optimal energy space L 2 (Ω) × H −1 (Ω). It is first proved that any solution to the closed-loop equations converges to zero in the aforementioned topology. Secondly, under the condition that the feedback nonlinearity has linear growth around zero, polynomial energy decay rates are established for solutions with smooth initial data. This constitutes new Dirichlet counterparts to well-known results pertaining to nonlinear stabilization in H 1 (Ω) × L 2 (Ω) of the wave equation in the Neumann boundary conditions.
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Dates et versions

hal-03363940 , version 1 (04-10-2021)
hal-03363940 , version 2 (26-08-2022)

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  • HAL Id : hal-03363940 , version 1

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Nicolas Vanspranghe, Francesco Ferrante, Christophe Prieur. Stabilization of the Wave Equation through Nonlinear Dirichlet Actuation. 2021. ⟨hal-03363940v1⟩
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