Observability of the 1-D wave equation with mixed boundary conditions in a non-cylindrical domain
The homogeneous, one space dimension, wave equation with mixed boundary conditions, i.e. Neumann type condition at \(x=0\) and Dirichlet condition at moving point \(x=lt\) is considered. Under some assumptions, one proves the existence and uniqueness of a solution and one obtains the exact solution developed in a series. Further, energy estimates and observability at the endpoints \(x=0\) and \(x=lt\) are investigated. Results on existence and uniqueness of solution as well as energy estimates and observability at endpoints are shown also for the reversed mixed boundary conditions, that is Dirichlet condition at \(x=0\) and Neumann type condition at the moving point \(x=lt\).
scientific article; zbMATH DE number 6879902