In this paper, we study the indirect boundary stability and exact controllability of a one-dimensional Timoshenko system. In the first part of the paper, we consider the Timoshenko system with only one boundary fractional damping. We first show that the system is strongly stable but not uniformly stable. Hence, we look for a polynomial decay rate for smooth initial data. Using frequency domain arguments combined with the multiplier method, we prove that the energy decay rate depends on coefficients appearing in the PDE and on the order of the fractional damping. Moreover, under the equal speed propagation condition, we obtain the optimal polynomial energy decay rate. In the second part of this paper, we study the indirect boundary exact controllability of the Timoshenko system with mixed Dirichlet–Neumann boundary conditions and boundary control. Using non-harmonic analysis, we first establish a weak observability inequality, which depends on the ratio of the waves propagation speeds. Next, using the HUM method, we prove that the system is exactly controllable in appropriate spaces and that the control time can be small.

中文翻译：

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边界上只有一个分数阻尼的Timoshenko系统的稳定性和精确可控制性

在本文中，我们研究了一维Timoshenko系统的间接边界稳定性和精确可控性。在本文的第一部分中，我们考虑只有一个边界分数阻尼的Timoshenko系统。我们首先显示系统是高度稳定的，但不是一致稳定的。因此，我们寻找平滑初始数据的多项式衰减率。使用频域自变量结合乘数方法，我们证明了能量衰减率取决于PDE中出现的系数以及分数阻尼的阶数。此外，在等速传播条件下，我们获得了最佳的多项式能量衰减率。在本文的第二部分中，我们研究了混合Dirichlet-Neumann边界条件和边界控制的Timoshenko系统的间接边界精确可控制性。使用非谐波分析，我们首先建立一个弱的可观测性不等式，这取决于波的传播速度之比。接下来，使用HUM方法，我们证明了该系统在适当的空间内是完全可控的，并且控制时间可以很小。