In this paper the finite time stabilization of an infinite network of vibrating strings is studied. We consider a network with $N\in {\mathbb{N}}^{⁎},N\ge 2$ finite edges and one infinite edge. This network is subject to homogeneous Dirichlet conditions at the endpoints of the finite edges. At the internal vertex 0, we put the continuity condition and a balanced damping condition of the form ${\sum }_{j=0}^N({\partial }_x{u}_j(0,t)-{\partial }_t{u}_j(0,t))=0$. By a spectral analysis technique, we prove that when the finite edges lengths are rationally proportional, the energy of the system becomes constant within a finite time $T_e$. We also prove that for an equilateral network we have $T_e=2\ell$, where ℓ is the common edge length. This is a finite time stabilization result of the system to a state with constant energy. This finite time limit energy is given in terms of the initial data and computed for a generic network. The main idea is to calculate the resolvent and to prove that it is of finite exponential type on some subspace of the energy space.

本文研究了振动弦的无限网络的有限时间稳定。我们考虑与$ñ\in {\mathbb{ñ}}^{⁎}，ñ\ge \mathrm{2个}$有限边和一个无限边。该网络在有限边沿的端点处均质Dirichlet条件。在内部顶点0处，我们将连续条件和平衡阻尼条件的形式设为${\sum }_{Ĵ=0}^{ñ}（{\partial }_X{ü}_{Ĵ}（0，Ť）-{\partial }_{Ť}{ü}_{Ĵ}（0，Ť））=0$。通过频谱分析技术，我们证明了当有限的边长合理地成比例时，系统的能量在有限的时间内变得恒定${Ť}_{Ë}$。我们还证明，对于平等网络，我们拥有${Ť}_{Ë}=\mathrm{2个}\ell$，其中ℓ是公共边的长度。这是系统达到恒定能量状态的有限时间稳定结果。此有限时限能量是根据初始数据给出的，并针对通用网络进行了计算。主要思想是计算分解体，并证明其在能量空间的某些子空间上是有限指数类型的。