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Journal Articles Mathematical News / Mathematische Nachrichten Year : 2018

Indirect stability of the wave equation with a dynamic boundary control

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Abstract

In this paper, we consider a damped wave equation with a dynamic boundary control. First, combining a general criteria of Arendt and Batty with Holmgren's theorem we show the strong stability of our system. Next, we show that our system is not uniformly stable in general, since it is the case for the unit disk. Hence, we look for a polynomial decay rate for smooth initial data for our system by applying a frequency domain approach. In a first step, by giving some sufficient conditions on the boundary of our domain and by using the exponential decay of the wave equation with a standard damping, we prove a polynomial decay in 1/t1/4 of the energy. In a second step, under appropriated conditions on the boundary, called the multiplier control conditions, we establish a polynomial decay in 1/t of the energy. Later, we show in a particular case that such a polynomial decay is available even if the previous conditions are not satisfied. For this aim, we consider our system on the unit square of the plane. Using a method based on a Fourier analysis and a specific analysis of the obtained 1-d problems combining Ingham's inequality and an interpolation method, we establish a polynomial decay in 1/t of the energy for sufficiently smooth initial data. Finally, in the case of the unit disk, using the real part of the asymptotic expansion of eigenvalues of the damped system, we prove that the obtained decay is optimal in the domain of the operator.
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Dates and versions

hal-03227562 , version 1 (17-05-2021)

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Denis Mercier, Serge Nicaise, Mohamad Ali Sammoury, A. Wehbe. Indirect stability of the wave equation with a dynamic boundary control. Mathematical News / Mathematische Nachrichten, 2018, 291 (7), pp.1114-1146. ⟨10.1002/mana.201700021⟩. ⟨hal-03227562⟩
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