In this work, we are interested in tree-shaped networks of freely vibrating beams which are geometrically exact (GEB) -- in the sense that large motions (deflections, rotations) are accounted for in addition to shearing -- and linked by rigid joints. For the intrinsic GEB formulation, namely that in terms of velocities and internal forces/moments, we derive transmission conditions and show that the network is locally in time well-posed in the classical sense. Applying velocity feedback controls at the external nodes of a star-shaped network, we show by means of a quadratic Lyapunov functional and the theory developed by Bastin \& Coron in \cite{BC2016} that the zero steady state of this network is exponentially stable for the $H^1$ and $H^2$ norms. The major obstacles to overcome in the intrinsic formulation of the GEB network, are that the governing equations are semilinar, containing a quadratic nonlinearity, and that linear lower order terms cannot be neglected.
Nodal profile control for networks of geometrically exact beams
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。
Theorical and Numerical Analysis of the Rapid Pointwise Stabilization of Coupled String-Beam Systems
Institut Supérieur d’Informatique et de Mathématiques de Monastir（莫纳斯特尔信息与数学高等学院）；Université de Monastir（莫纳斯特尔大学）；Institut de Rechrche Mathématique Avancée, Université de Strasbourg（斯特拉斯堡大学高级数学研究所）
