Boundary observability and null-controllability for non-autonomous degenerate hyperbolic equations via energy estimates

Motivated by the broad applications of wave propagation in non-uniform and time-varying environments, such as in acoustics, elasticity, and seismology, we investigate the controllability of degenerate wave equations with time-dependent wave speeds. In this work, we examine non-autonomous degenerate wave equations in a one-dimensional spatial domain, addressing both divergence and non-divergence forms. A control function is applied at the non-degeneracy boundary point, while Dirichlet or Neumann conditions are imposed at the degeneracy one. Using the generalized energy conservation law, we first establish boundary observability for the homogeneous problem, a key step in our analysis. Building on this, we prove the null-controllability of the non-autonomous degenerate wave systems. To achieve this, we construct solutions using the transposition method, which accommodates low regularity requirements, enabling us to handle weaker smoothness assumptions while maintaining rigorous control over the system’s behavior. We conclude by presenting some insightful observations and potential avenues for future work, which could further advance the understanding of this problem.