当前位置: X-MOL 学术SIAM J. Control Optim. › 论文详情
Our official English website, www.x-mol.net, welcomes your feedback! (Note: you will need to create a separate account there.)
Boundary Feedback Stabilization for the Intrinsic Geometrically Exact Beam Model
SIAM Journal on Control and Optimization ( IF 2.2 ) Pub Date : 2020-11-24 , DOI: 10.1137/20m1340010
Charlotte Rodriguez , Günter Leugering

SIAM Journal on Control and Optimization, Volume 58, Issue 6, Page 3533-3558, January 2020.
In this work we address the problem of boundary feedback stabilization for a geometrically exact shearable beam, allowing for large deflections and rotations and small strains. The corresponding mathematical model may be written in terms of displacements and rotations (geometrically exact beam), or intrinsic variables (intrinsic geometrically exact beam). A nonlinear transformation relates both models, allowing us to take advantage of the fact that the latter model is a one-dimensional first-order semilinear hyperbolic system, and deduce stability properties for both models. By applying boundary feedback controls at one end of the beam while the other end is clamped, we show that the zero steady state of the intrinsic geometrically exact beam model is locally exponentially stable for the $H^1$ and $H^2$ norms. The proof rests on the construction of a Lyapunov function, where the theory of Bastin and Coron [Stability and Boundary Stabilization of $1$-D Hyperbolic Systems, in Progr. Nonlinear Differential Equations Appl. 88, Birkhäuser/Springer, Cham, 2016] plays a crucial role. The major difficulty in applying this theory stems from the complicated nature of the nonlinearity and lower order term where no smallness arguments apply. Using the relationship between both models, we deduce the existence of a unique solution to the geometrically exact beam model, and properties of this solution as time goes to $+\infty$.


中文翻译:

本征几何精确梁模型的边界反馈稳定

SIAM控制与优化杂志,第58卷,第6期,第3533-3558页,2020年1月。
在这项工作中,我们解决了几何上精确的可剪切梁的边界反馈稳定问题,允许较大的偏转和旋转以及较小的应变。可以根据位移和旋转(几何上精确的射束)或固有变量(本征上几何上的精确射束)来编写相应的数学模型。非线性变换将两个模型联系在一起,这使我们可以利用后一个模型是一维一阶半线性双曲线系统这一事实,并推论这两个模型的稳定性。通过在梁的一端被夹紧而另一端被夹紧的情况下应用边界反馈控制,我们证明了固有几何精确梁模型的零稳态对于$ H ^ 1 $和$ H ^ 2 $范数是局部指数稳定的。证明建立在Lyapunov函数的构造上,其中Bastin和Coron理论[Progr中的$ 1 $ -D双曲系统的稳定性和边界稳定]。非线性微分方程 88,Birkhäuser/ Springer,Cham,2016]起着至关重要的作用。应用该理论的主要困难在于非线性和低阶项的复杂性质,其中没有适用任何小论证。利用这两个模型之间的关系,我们推导出了几何精确束模型的唯一解的存在,并且随着时间的流逝,该解的性质也随之增加。应用该理论的主要困难在于非线性和低阶项的复杂性质,其中没有适用任何小论证。利用这两个模型之间的关系,我们推导出了几何精确束模型的唯一解的存在,并且随着时间的流逝,该解的性质也随之增加。应用该理论的主要困难在于非线性和低阶项的复杂性质,其中没有适用任何小论证。利用这两个模型之间的关系,我们推导出了几何精确束模型的唯一解的存在,并且随着时间的流逝,该解的性质也随之增加。
更新日期:2020-11-25
down
wechat
bug