Academic Journal
Indirect stability of the wave equation with a dynamic boundary control
| العنوان: | Indirect stability of the wave equation with a dynamic boundary control |
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| المؤلفون: | Mercier, Denis, Nicaise, Serge, Sammoury, Mohamad, Ali, Wehbe, Ali |
| المساهمون: | Sorbonne Université (SU), Laboratoire de géographie physique : Environnements Quaternaires et Actuels (LGP), Université Paris 1 Panthéon-Sorbonne (UP1)-Université Paris-Est Créteil Val-de-Marne - Paris 12 (UPEC UP12)-Centre National de la Recherche Scientifique (CNRS), Université Polytechnique Hauts-de-France (UPHF), Laboratoire de Mathématiques et leurs Applications de Valenciennes - EA 4015 (LAMAV), Université de Valenciennes et du Hainaut-Cambrésis (UVHC)-Centre National de la Recherche Scientifique (CNRS), Faculté des Sciences, الجامعة اللبنانية بيروت = Lebanese University Beirut = Université libanaise Beyrouth (LU / ULB) |
| المصدر: | ISSN: 0025-584X. |
| بيانات النشر: | HAL CCSD Wiley-VCH Verlag |
| سنة النشر: | 2018 |
| المجموعة: | Université Paris 1 Panthéon-Sorbonne: HAL |
| مصطلحات موضوعية: | Dynamic control, indirect stability, MSC (2010): 35B35, 35L35, [MATH.MATH-OC]Mathematics [math]/Optimization and Control [math.OC] |
| الوصف: | International audience ; 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/(t^1/4) 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. |
| نوع الوثيقة: | article in journal/newspaper |
| اللغة: | English |
| Relation: | hal-01956619; https://hal.science/hal-01956619; https://hal.science/hal-01956619/document; https://hal.science/hal-01956619/file/Nic270%20%282%29.pdf |
| DOI: | 10.1002/mana.201700021 |
| الاتاحة: | https://hal.science/hal-01956619 https://hal.science/hal-01956619/document https://hal.science/hal-01956619/file/Nic270%20%282%29.pdf https://doi.org/10.1002/mana.201700021 |
| Rights: | info:eu-repo/semantics/OpenAccess |
| رقم الانضمام: | edsbas.7FD33D6C |
| قاعدة البيانات: | BASE |
| DOI: | 10.1002/mana.201700021 |
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