引用本文:杜燕, 许跟起.具有边界控制的线性Timoshenko型系统的指数稳定性[J].控制理论与应用,2008,25(1):33~39.[点击复制]
DU Yan, XU Gen-qi.Exponential stability of a system of linear Timoshenko type with boundary controls[J].Control Theory and Technology,2008,25(1):33~39.[点击复制]
具有边界控制的线性Timoshenko型系统的指数稳定性
Exponential stability of a system of linear Timoshenko type with boundary controls
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DOI编号  10.7641/j.issn.1000-8152.2008.1.006
  2008,25(1):33-39
中文关键词  线性Timoshenko型系统  边界反馈控制  Riesz基  指数稳定性
英文关键词  linear Timoshenko type system  boundary feedback control  Riesz basis  exponential stability
基金项目  国家自然科学基金资助项目(60474017); 教育部南开–天津大学刘徽应用数学中心资助项目.
作者单位
杜燕, 许跟起 天津大学数学系, 天津300072 
中文摘要
      本文研究多孔弹性材料在实际应用中的镇定问题. 多孔物体的动力学行为由线性Timoshenko型方程描述,这样的系统一般只是渐近稳定但不指数稳定. 假定系统两端都是自由的, 在自由端对系统施加边界速度反馈控制,本文讨论闭环系统的适定性和指数稳定性. 首先, 利用有界线性算子半群理论得到了系统的适定性. 进一步对系统算子的本征值的渐近值估计, 得到算子谱分布在一个带域, 相互分离的, 模充分大的本征值都是简单本征值. 通过引入一个辅助算子, 利用它的谱性质以及有界线性算子的扰动理论, 得到系统的广义本征向量的完整性以及Riesz基性质. 最后利用Riesz基性质和谱分布得到闭环系统的指数稳定性.
英文摘要
      In this paper we consider the stabilization problem of porous elastic solids in real world. The kinetic behavior of porous solids is governed by equations of linear Timoshenko type which is usually asymptotically stable but not exponentially stable. We apply boundary velocity feedback controls to the system with both ends free, then examine the well-posedness and exponential stability of the closed loop system. Firstly, we obtain the well-posedness of the system by the semigroup theory of bounded linear operators. Secondly, we get the asymptotic values of eigenvalues of the system, which are isolated and lie in a strip area under certain condition. Moreover, we introduce an auxiliary operator, and then by means of its spectral properties and theory of perturbations of bounded linear operators to prove that there is a sequence of generalized eigenvector system which forms a Riesz basis for Hilbert state space. Finally, we obtain the exponential stability of the closed loop system by using the Riesz basis property and spectral distribution .