This note establishes the exponential input-to-state stability (EISS) property for a clamped-free damped string with respect to distributed and boundary disturbances. While efficient methods for establishing ISS properties for distributed parameter systems with respect to distributed disturbances have been developed during the last decades, establishing ISS properties with respect to boundary disturbances remains challenging. One of the well-known methods for well-posedness analysis of systems with boundary inputs is the use of a lifting operator for transferring the boundary disturbance to a distributed one. However, the resulting distributed disturbance involves time derivatives of the boundary perturbation. Thus, the subsequent ISS estimate depends on its amplitude, and may not be expressed in the strict form of ISS properties. To solve this problem, we show for a clamped-free damped string equation that the projection of the original system trajectories in an adequate Riesz basis can be used to establish the desired EISS property.

中文翻译：

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存在分布和边界扰动时无钳位阻尼弦的输入状态稳定性


本文建立了无钳位阻尼弦相对于分布和边界扰动的指数输入状态稳定性 (EISS) 属性。虽然在过去几十年中已经开发出建立分布式参数系统关于分布式扰动的 ISS 属性的有效方法，但建立关于边界扰动的 ISS 属性仍然具有挑战性。对具有边界输入的系统进行适定性分析的一种众所周知的方法是使用提升算子将边界扰动转移到分布式扰动。然而，由此产生的分布式扰动涉及边界扰动的时间导数。因此，后续的 ISS 估计取决于其幅度，并且可能不会以 ISS 属性的严格形式表示。为了解决这个问题，我们证明了对于无钳位阻尼弦方程，原始系统轨迹在适当的 Riesz 基上的投影可用于建立所需的 EISS 属性。