  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.
