The Spectral Analysis and Exponential Stability of a 1-d 2 × 2 Hyperbolic System with Proportional Feedback Control
The Spectral Analysis and Exponential Stability of a 1-d 2 × 2 Hyperbolic System with Proportional Feedback Control
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DOI：
10.1007/s12555-021-0507-0
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发表时间：
2022-07
期刊：
影响因子：
--
通讯作者：
Dong‐xia Zhao;Dong Fan;Yaping Guo
中科院分区：
文献类型：
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作者：
Dong‐xia Zhao;Dong Fan;Yaping Guo
研究了一类1-d2 × 2双曲型系统在比例反馈控制下的适定性和稳定性.利用半群方法和范数等价定理得到了控制参数的显式耗散条件。给出了系统的渐近谱分析。当特征值模趋于+∞时，特征值的真实的部分趋于某个负常数.此外，我们还发现存在一组广义本征函数，它们构成了Hilbert状态空间的Riesz基。因此，谱决定的增长条件成立，从而建立指数稳定性。
In this paper, the well-posedness and stability of a 1-d 2 × 2 hyperbolic system under proportional feedback control is addressed. The semigroup approach and norm equivalence theorem is adopted to obtain the explicit dissipative condition for control parameters. The asymptotic spectral analysis of the system is presented. It is shown that the real part of the eigenvalues goes to some negative constant as the modulus of eigenvalues goes to +∞. Moreover, we find that there is a group of generalized eigenfunctions which forms a Riesz basis of the Hilbert state space. Hence, the spectrum determined growth condition holds, and then the exponential stability is established.