{"title":"Stabilization for Wave Equation with Localized Kelvin–Voigt Damping on Cuboidal Domain: A Degenerate Case","authors":"Zhong-Jie Han, Zhuangyi Liu, Kai Yu","doi":"10.1137/22m153210x","DOIUrl":null,"url":null,"abstract":"SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 441-465, February 2024. <br/> Abstract. In this paper, we study the stabilization issue for a multidimensional wave equation with localized Kelvin–Voigt damping on a cuboidal domain, in which the damping region does not satisfy the geometric control condition (GCC). The variable damping coefficient is assumed to be degenerate near the interface. We prove that the system is polynomially stable with a decay rate depending on the degree of the degeneration [math]. A relationship between the decay order and [math] is identified. In particular, this decay rate is consistent with the optimal one for the corresponding system with constant damping coefficient (i.e., [math]) obtained in [K. Yu and Z.-J. Han, SIAM J. Control Optim., 59 (2021), pp. 1973–1988]. Moreover, it is the first result on the decay rates of the solutions to multidimensional wave equations with localized degenerate Kelvin–Voigt damping when GCC is not satisfied.","PeriodicalId":49531,"journal":{"name":"SIAM Journal on Control and Optimization","volume":null,"pages":null},"PeriodicalIF":2.2000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"SIAM Journal on Control and Optimization","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1137/22m153210x","RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"AUTOMATION & CONTROL SYSTEMS","Score":null,"Total":0}
引用次数: 0
Abstract
SIAM Journal on Control and Optimization, Volume 62, Issue 1, Page 441-465, February 2024. Abstract. In this paper, we study the stabilization issue for a multidimensional wave equation with localized Kelvin–Voigt damping on a cuboidal domain, in which the damping region does not satisfy the geometric control condition (GCC). The variable damping coefficient is assumed to be degenerate near the interface. We prove that the system is polynomially stable with a decay rate depending on the degree of the degeneration [math]. A relationship between the decay order and [math] is identified. In particular, this decay rate is consistent with the optimal one for the corresponding system with constant damping coefficient (i.e., [math]) obtained in [K. Yu and Z.-J. Han, SIAM J. Control Optim., 59 (2021), pp. 1973–1988]. Moreover, it is the first result on the decay rates of the solutions to multidimensional wave equations with localized degenerate Kelvin–Voigt damping when GCC is not satisfied.
期刊介绍：
SIAM Journal on Control and Optimization (SICON) publishes original research articles on the mathematics and applications of control theory and certain parts of optimization theory. Papers considered for publication must be significant at both the mathematical level and the level of applications or potential applications. Papers containing mostly routine mathematics or those with no discernible connection to control and systems theory or optimization will not be considered for publication. From time to time, the journal will also publish authoritative surveys of important subject areas in control theory and optimization whose level of maturity permits a clear and unified exposition.
The broad areas mentioned above are intended to encompass a wide range of mathematical techniques and scientific, engineering, economic, and industrial applications. These include stochastic and deterministic methods in control, estimation, and identification of systems; modeling and realization of complex control systems; the numerical analysis and related computational methodology of control processes and allied issues; and the development of mathematical theories and techniques that give new insights into old problems or provide the basis for further progress in control theory and optimization. Within the field of optimization, the journal focuses on the parts that are relevant to dynamic and control systems. Contributions to numerical methodology are also welcome in accordance with these aims, especially as related to large-scale problems and decomposition as well as to fundamental questions of convergence and approximation.
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