Shape Design for Degenerate Hyperbolic Equation with Degenerate Boundary and Its Application to Observability

Abstract

In this paper, we study the observability and controllability of a class of degenerate hyperbolic equations with a control region intersecting the degenerate set. Unlike the existing results that mainly deal with control regions separated from the degeneracy, we consider the case where the control region reaches the degenerate part. To handle the difficulty caused by the degeneracy, we introduce a shape-design-based approximation method based on shape design by approximating the degenerate equation with a family of uniformly hyperbolic equations. The proof relies on the spectral approximation of the associated operators, precise estimates for weak solutions, and the multiplier method. We first establish observability inequalities for the approximating equations with constants independent of the approximation parameter. Then, by passing to the degenerate limit, we obtain the observability inequality for the original degenerate equation.