Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control

Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control

In this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1-d heat-wave system coupled at the interface and the long-time behavior of the system is discussed. The system is formulated into an abstract Cauchy problem on the energy space....

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Main Authors: Xue-Lian Jin, Yan Li, Fu Zheng
Format: Article
Language:English
Published: Hindawi Limited 2019-01-01
Series:Mathematical Problems in Engineering
Online Access:http://dx.doi.org/10.1155/2019/5716729
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spelling doaj-827504fd81c74ac7881cf77a55e790eb2020-11-24T21:21:29ZengHindawi LimitedMathematical Problems in Engineering1024-123X1563-51472019-01-01201910.1155/2019/57167295716729Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary ControlXue-Lian Jin0Yan Li1Fu Zheng2Science College, Liaoning University of Technology, Jinzhou, 121001, ChinaCollege of Mathematics and Physics, Bohai University, Jinzhou, 121013, ChinaCollege of Mathematics and Physics, Bohai University, Jinzhou, 121013, ChinaIn this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1-d heat-wave system coupled at the interface and the long-time behavior of the system is discussed. The system is formulated into an abstract Cauchy problem on the energy space. The energy of the system does not increase because the semigroup generated by the system operator is contracted. In the meanwhile, the asymptotic stability of the system is derived in light of the spectral configuration of the system operator. Furthermore, the spectral expansions of the system operator are precisely investigated and the asymptotical stability is not exponential and is shown in view of the spectral expansions.http://dx.doi.org/10.1155/2019/5716729
collection DOAJ
language English
format Article
sources DOAJ
author Xue-Lian Jin
Yan Li
Fu Zheng
spellingShingle Xue-Lian Jin
Yan Li
Fu Zheng
Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control
Mathematical Problems in Engineering
author_facet Xue-Lian Jin
Yan Li
Fu Zheng
author_sort Xue-Lian Jin
title Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control
title_short Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control
title_full Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control
title_fullStr Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control
title_full_unstemmed Spectrum and Stability of a 1-d Heat-Wave Coupled System with Dynamical Boundary Control
title_sort spectrum and stability of a 1-d heat-wave coupled system with dynamical boundary control
publisher Hindawi Limited
series Mathematical Problems in Engineering
issn 1024-123X
1563-5147
publishDate 2019-01-01
description In this paper, the negative proportional dynamic feedback is designed in the right boundary of the wave component of the 1-d heat-wave system coupled at the interface and the long-time behavior of the system is discussed. The system is formulated into an abstract Cauchy problem on the energy space. The energy of the system does not increase because the semigroup generated by the system operator is contracted. In the meanwhile, the asymptotic stability of the system is derived in light of the spectral configuration of the system operator. Furthermore, the spectral expansions of the system operator are precisely investigated and the asymptotical stability is not exponential and is shown in view of the spectral expansions.
url http://dx.doi.org/10.1155/2019/5716729
work_keys_str_mv AT xuelianjin spectrumandstabilityofa1dheatwavecoupledsystemwithdynamicalboundarycontrol
AT yanli spectrumandstabilityofa1dheatwavecoupledsystemwithdynamicalboundarycontrol
AT fuzheng spectrumandstabilityofa1dheatwavecoupledsystemwithdynamicalboundarycontrol
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