Boundary stabilization of a vibrating string with variable length

We study small vibrations of a string with a length ⠂(t) varying in time at a speed less than the speed of propagation of vibrations. We establish lower and upper estimates for the energy of the string when a dash-pot with constant damping factor eta is placed at the moving boundary. The estimates depend explicitly on ⠂(t), eta and a function phi that solves the functional equation phi(t + ⠂(t)) - phi(t - ⠂(t)) = 2. (c) 2023 Elsevier Inc. All rights reserved.

Automated summary

This paper studies small vibrations of a string with a time-varying length ⠂(t) at a speed slower than the speed of vibration propagation. We establish lower and upper bounds for the energy of the string when a dash-pot with a constant damping factor eta is placed at the moving boundary. The estimates explicitly depend on ⠂(t), eta, and a function phi that satisfies the functional equation phi(t + ⠂(t)) - phi(t - ⠂(t)) = 2. (c) 2023 Elsevier Inc. All rights reserved.