The existence, uniqueness, strong and exponential stability of a generalized telegraph equation set on one dimensional star shaped networks are established. It is assumed that a dissipative boundary condition is applied at all the external vertices and an improved Kirchhoff law at the common internal vertex is considered. First, using a general criteria of Arendt-Batty (see Arendt and Batty in Trans. Am. Math. Soc. 306(2):837–852, 1988 ), combined with a new uniqueness result, we prove that our system is strongly stable. Next, using a frequency domain approach, combined with a multiplier technique and the construction of a new multiplier satisfying some ordinary differential inequalities, we show that the energy of the system decays exponentially to zero.

中文翻译：

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星型网络上广义电报方程解的存在性，唯一性和稳定性

建立了一维星形网络上广义电报方程组的存在性，唯一性，强指数稳定性。假定在所有外部顶点上都应用了耗散边界条件，并考虑了在公共内部顶点处的改进的Kirchhoff定律。首先，使用Arendt-Batty的一般标准（请参见Trans。Am。Math。Soc。306（2）：837–852中的Arendt和Batty， 1988年 ），结合新的唯一性结果，我们证明我们的系统非常稳定。接下来，使用频域方法，结合乘数技术并构造满足某些普通微分不等式的新乘数，我们证明系统的能量呈指数衰减至零。