Abstract We study the stabilization of two vibrating strings with variable physical coefficients joined by a feedback control at a common endpoint and subject to non-symmetrical boundary conditions. We prove that this system is exponentially stable under sufficient conditions on the physical coefficients. For this result, we show that the system has a sequence of the generalized eigenvectors which forms a Riesz basis with parentheses for the state Hilbert space, and as a consequence the spectrum-determined growth condition holds.

中文翻译：

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变系数联合阻尼下两弦的指数稳定性

摘要 我们研究了两个具有可变物理系数的振动弦的稳定性，这些振动弦通过公共端点处的反馈控制连接并受到非对称边界条件的影响。我们证明了该系统在物理系数的充分条件下是指数稳定的。对于这个结果，我们表明系统具有一系列广义特征向量，这些向量形成了带有括号的 Riesz 基，用于状态 Hilbert 空间，因此频谱确定的增长条件成立。