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

Xue-Lian Jin,Yan Li,Fu Zheng

doi:10.1155/2019/5716729

Xue-Lian Jin, Yan Li + Show 1 more

Open Access

https://doi.org/10.1155/2019/5716729

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Abstract

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.

Highlights

This paper is devoted to analyzing 1-d heat-wave coupled system with a dynamical boundary feedback
Y (0, x) = y0 (x), η (0) = η0, z (0, x) = z0 (x), zt (0, x) = z1 (x), where β is a positive constant, η denotes the dynamical control, and -βη(t) is the negative proportional feedback. This system consists of a wave equation, arising on the interval (0, 1) with state (z, zt) and a heat equation that holds on the interval (−1, 0) with state y
The spectrum and the asymptotic stability of the heat-wave system with dynamical boundary control are investigated through the theory of operator semigroup

Summary

Introduction

This paper is devoted to analyzing 1-d heat-wave coupled system with a dynamical boundary feedback (see [1,2,3]). In 2016, by the frequency domain approach together with multipliers, Han and Zuazua discussed the large time decay rate of a transmission problem coupled heat and wave equations on a planar network in [12]. Under some conditions, the exponential stability of the wave equation with similar dynamic boundary controller was obtained in [18]. In the same time, when the heat equation was present (without dynamical boundary feedback), the system was analyzed in [1] and the asymptotical (but not exponential) stability was shown again. We combine these two results and show that the similar results hold in this note.

Well-Posedness and Asymptotic Stability

Spectral Expansion of the System Operator A

Conclusion