Nodal profile control for networks of geometrically exact beams
Nodal profile control for networks of geometrically exact beams
复制标题
DOI：
10.1016/j.matpur.2021.07.007
复制
发表时间：
2021-03
期刊：
影响因子：
--
通讯作者：
G. Leugering;C. Rodriguez;Yue Wang
中科院分区：
文献类型：
--
作者：
G. Leugering;C. Rodriguez;Yue Wang
在这项工作中，我们考虑网络的所谓几何精确的梁，即，剪切梁，可能会经历大的运动。相应的数学模型，通常写在位移和旋转表示在一个固定的基础（几何精确梁模型，或GEB），有一个准线性的控制系统。然而，该模型也可以用附着在梁上的移动基中表示的内在变量来写（内在GEB模型，或IGEB），并且虽然方程的数量因此加倍，但后一模型具有一阶、双曲和仅半线性的优点。首先，对于任何网络，我们证明了IGEB模型在时间上的半全局经典解的存在性和唯一性（即，对于任意大的时间间隔，只要数据足够小）。然后，对于包含循环的特定网络，我们解决了IGEB模型节点轮廓的局部精确可控性问题-我们通过在简单节点上应用控制来引导解决方案以满足多个节点之一的给定轮廓-通过使用Zhuang，Leugering和Li（2018）的构造性方法[52]。之后，对于任何网络，我们证明了IGEB网络的唯一经典解的存在性意味着相应的GEB网络也是如此，通过使用这两个模型通过非线性变换相关联。特别是，这使我们能够给出相应的存在性，唯一性和可控性的GEB网络的结果。
In this work, we consider networks of so-called geometrically exact beams, namely, shearable beams that may undergo large motions. The corresponding mathematical model, commonly written in terms of displacements and rotations expressed in a fixed basis (Geometrically Exact Beam model, or GEB), has a quasilinear governing system. However, the model may also be written in terms of intrinsic variables expressed in a moving basis attached to the beam (Intrinsic GEB model, or IGEB) and while the number of equations is then doubled, the latter model has the advantage of being of first-order, hyperbolic and only semilinear. First, for any network, we show the existence and uniqueness of semi-global in time classical solutions to the IGEB model (i.e., for arbitrarily large time intervals, provided that the data are small enough). Then, for a specific network containing a cycle, we address the problem of local exact controllability of nodal profiles for the IGEB model – we steer the solution to satisfy given profiles at one of the multiple nodes by means of controls applied at the simple nodes – by using the constructive method of Zhuang, Leugering and Li (2018) [52]. Afterwards, for any network, we show that the existence of a unique classical solution to the IGEB network implies the same for the corresponding GEB network, by using that these two models are related by a nonlinear transformation. In particular, this allows us to give corresponding existence, uniqueness and controllability results for the GEB network.