Uniform stabilization of the wave equation with dirichlet-feedback control without geometrical conditions

In this paper we eliminate altogether geometrical conditions that were assumed (even) with control action on the entire boundary in prior literature: (i) strict convexity of our paper [LT4] on uniform stabilization of the wave equation in the (optimal) state spaceL2(Ω)×H−1(Ω) withL2(Σ) Dirichlet feedback control, as well as (ii) “star-shaped” conditions in papers [C1], [La1], and [Tr1] on uniform stabilization and [Lio1] and [LT5] on exact controllability in the energy spaceH1(Ω)×L2(Ω) of the wave equation withL2(Σ)-Neumann feedback control. Key to the present improvements is a pseudodifferential analysis which permits us to express certain boundary traces of the solution in terms of other traces modulo lower-order interior terms. See Lemma 3.1 for the Dirichlet case and Lemma 7.2 for the Neumann case.

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