几何精确梁网的节点轮廓控制
Nodal profile control for networks of geometrically exact beams
摘要：
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，通常用位移和旋转来表示，以固定基（几何精确梁模型，或GEB），具有准线性控制系统。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，对于一个含有环的特定网络，我们利用Zhuang，Leugering和Li [Exact boundary controllability of nodal profile for Saint-Venant systems on a network with loops，在J.Math.PuresAppl.中，2018年]。然后，对任意网络，通过非线性变换证明了IGEB网络存在唯一的经典解，这对相应的GEB网络也是一样的.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。
英文摘要：
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 [Exact boundary controllability of nodal profile for Saint-Venant system on a network with loops, in J. Math. Pures Appl., 2018]. 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.
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