Exact controllability of damped coupled Euler-Bernoulli and Timoshenko beam model
Exact controllability of damped coupled Euler-Bernoulli and Timoshenko beam model
复制标题
阻尼耦合欧拉-伯努利和铁莫申科梁模型的精确可控性
DOI：
10.1093/imamci/dni059
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发表时间：
2006
期刊：
影响因子：
--
通讯作者：
M. Shubov
中科院分区：
文献类型：
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作者：
M. Shubov
本文研究了Euler-Bernoulli梁和Brachenko梁耦合振动的双曲型方程组的零能控性问题。该系统在有限区间上考虑，具有包含阻尼项的物理意义边界条件的双参数族。在两个方程的右侧，控制被引入为可分离的强迫项gi（x）fi（t），i= 1，2。假定力分布函数gi（x），i= 1，2是给定的。为了构造使系统的给定初始状态在特定时间区间[0，T]上为零的控制fi（t），i= 1，2，本文应用了谱分解方法。在本论文中使用的方法，是基于作者和合作者在最近的作品中所获得的结果。在这些工作中，详细的渐近和谱分析的非自伴算子产生的耦合梁的动力学已经进行。已经证明，对于每一组边界参数，上述算子是Riesz谱的，即其广义本征向量形成能量空间中的Riesz基。还导出了双支谱的显式渐近公式。基于这些谱的结果，控制问题已减少到相应的时刻的问题。为了解决这一矩问题，谱的渐近表示和广义特征向量的Riesz基性质已被使用。本文证明了精确能控的充分必要条件，并给出了控制律的显式表达式。文中还讨论了近似能控性的情形。
The zero controllability problem for the system of two coupled hyperbolic equations which governs the vibrations of the coupled Euler–Bernoulli and Timoshenko beam model is studied in the paper. The system is considered on a finite interval with a two-parameter family of physically meaningful boundary conditions containing damping terms. The controls are introduced as separable forcing termsgi(x)fi(t),i= 1, 2, on the right-hand sides of both equations. The force profile functionsgi(x),i= 1, 2, are assumed to be given. To construct the controlsfi(t),i= 1, 2, which bring a given initial state of the system to zero on the specific time interval [0,T], the spectral decomposition method has been applied. The approach, used in the present paper, is based on the results obtained in the recent works by the author and the collaborators. In these works, the detailed asymptotic and spectral analyses of the non-self-adjoint operators generating the dynamics of the coupled beam have been carried out. It has been shown that for each set of the boundary parameters, the aforementioned operator is Riesz spectral, i.e. its generalized eigenvectors form a Riesz basis in the energy space. Explicit asymptotic formulas for the two-branch spectrum have also been derived. Based on these spectral results, the control problem has been reduced to the corresponding moment problem. To solve this moment problem, the asymptotical representation of the spectrum and the Riesz basis property of the generalized eigenvectors have been used. The necessary and/or sufficient conditions for the exact controllability are proven in the paper and the explicit formulas for the control laws are given. The case of the approximate controllability is discussed in the paper as well.