一类抽象线性二阶发展系统的一致同步
Uniform synchronization of an abstract linear second order evolution system
摘要：
虽然有限水平上波动方程同步的数学研究已经比较成熟，但长时间水平上波动方程同步的研究还很少。研究了希尔伯特空间中一类抽象线性二阶发展系统在无穷水平上的一致同步。首先，利用压缩半群的一致稳定性的经典紧扰动理论，我们将建立所考虑系统的一致同步所必需的阻尼数的下界。然后，在最小阻尼数下，阐明了系统的代数结构以及耦合矩阵相容条件的必要性。然后利用紧扰动法建立了系统的一致同步，并给出了渐近轨道的动力学性质。给出了边界反馈或（和）局部分布反馈波动方程组和分布反馈Kirchhoff板系统的各种应用。文章最后提出了一些有待解决的问题，为今后的发展提供参考。研究的基础是半群的同步理论和紧扰动。
英文摘要：
Although the mathematical study on the synchronization of wave equations at finite horizon has been well developed, there was few results on the synchronization of wave equations for long-time horizon. The aim of the paper is to investigate the uniform synchronization at the infinite horizon for one abstract linear second order evolution system in a Hilbert space. First, using the classical compact perturbation theory on the uniform stability of semigroups of contractions, we will establish a lower bound on the number of damping, necessary for the uniform synchronization of the considered system. Then, under the minimum number of damping, we clarify the algebraic structure of the system as well as the necessity of the conditions of compatibility on the coupling matrices. We then establish the uniform synchronization by the compact perturbation method and then give the dynamics of the asymptotic orbit. Various applications are given for the system of wave equations with boundary feedback or (and) locally distributed feedback, and for the system of Kirchhoff plate with distributed feedback. Some open questions are raised at the end of the paper for future development. The study is based on the synchronization theory and the compact perturbation of semigroups.
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