Spectral analysis of the Euler-Bernoulli beam model with fully nonconservative feedback matrix: Spectral analysis of the Euler-Bernoulli beam model with fully nonconservative feedback matrix
Spectral analysis of the Euler-Bernoulli beam model with fully nonconservative feedback matrix: Spectral analysis of the Euler-Bernoulli beam model with fully nonconservative feedback matrix
复制标题
具有完全非保守反馈矩阵的欧拉-伯努利梁模型的谱分析：具有完全非保守反馈矩阵的欧拉-伯努利梁模型的谱分析
DOI：
10.1002/mma.4922
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发表时间：
2018
影响因子：
2.9
通讯作者：
Kindrat, Laszlo P.
中科院分区：
文献类型：
--
作者：
Shubov, Marianna A.;Kindrat, Laszlo P.
研究了具有反馈控制型完全非保守边界条件的Euler‐Bernoulli梁模型。输出矢量（右端的剪力和力矩）通过一个2 × 2矩阵（边界控制矩阵）与观测矢量（右端的速度及其空间导数）相连，该矩阵的所有元素都是非零的真实的数。对于边界参数的任意组合，模型的动力学生成元是状态希尔伯特空间中的非自伴矩阵微分算子。一组4个自伴算子，由相同的微分表达式定义为 在不同的领域，它被介绍。证明了这些算子中的每一个，以及，是悬臂梁模型的相同自伴动力学生成器的有限秩扰动。它还表明，非自伴算子，，共享一些特定于其自伴对应物的谱性质，如（1）本征函数的边界不等式，（2）本征值的几何重数，和（3）真实的本征值的存在性。这些结果对于我们下一篇研究多参数梁模型的谱渐近性和稳定性的论文是很重要的。
The Euler‐Bernoulli beam model with fully nonconservative boundary conditions of feedback control type is investigated. The output vector (the shear and the moment at the right end) is connected to the observation vector (the velocity and its spatial derivative on the right end) by a 2 × 2 matrix (the boundary control matrix), all entries of which are nonzero real numbers. For any combination of the boundary parameters, the dynamics generator, , of the model is a non–self‐adjoint matrix differential operator in the state Hilbert space. A set of 4 self‐adjoint operators, defined by the same differential expression as on different domains, is introduced. It is proven that each of these operators, as well as , is a finite‐rank perturbation of the same self‐adjoint dynamics generator of a cantilever beam model. It is also shown that the non–self‐adjoint operator, , shares a number of spectral properties specific to its self‐adjoint counterparts, such as (1) boundary inequalities for the eigenfunctions, (2) the geometric multiplicities of the eigenvalues, and (3) the existence of real eigenvalues. These results are important for our next paper on the spectral asymptotics and stability for the multiparameter beam model.