In this paper, we are concerned with a wave equation on a time-dependent domain with a Dirichlet boundary condition at the endpoint \(x=0\) and a boundary feedback at the moving endpoint \(x=kt\). We discuss the stabilization and exact boundary observability of the 1-dimensional wave equation with moving boundary feedback. By using generalized Fourier series and Parseval’s equality in weighted \(L^{2}-\)spaces, we derive a precise polynomial asymptotic stability for the energy function of the solution. Moreover, the exact boundary observabilities of the solution are established in minimal time. The observability constants are explicitly given at each endpoint.

在本文中，我们关注时间相关域上的波动方程，在端点\（x = 0 \）上具有Dirichlet边界条件，在运动端点\（x = kt \）上具有边界反馈。我们讨论带有运动边界反馈的一维波动方程的稳定性和精确的边界可观测性。通过在加权\（L ^ {2}-\）空间中使用广义傅立叶级数和Parseval等式，我们得出了该解的能量函数的精确多项式渐近稳定性。此外，可以在最短的时间内建立解决方案的精确边界可观察性。可观察性常数在每个端点处明确给出。