B. Dehman, S. Ervedoza, E. Zuazua (2025) Regional and Partial Observability and Control of Waves
Abstract.We establish sharp regional observability results for solutions of the wave equation in a bounded domain Ω ⊂ R n , in arbitrary spatial dimension. Assuming the waves are observed on a non-empty open subset ω ⊂ Ω and that the initial data are supported in another open subset O ⊂ Ω, we derive estimates for the energy of initial data localized in O, in terms of the energy measured on the observation set (0, T) × ω. This holds under a suitable geometric condition relating the time horizon T and the pair of subdomains (ω, O). Roughly speaking, this geometric condition requires that all rays of geometric optics in Ω, emanating from O, must reach the observation region (0, T) × ω. Our result significantly generalizes classical observability results, which recover the total energy of all solutions when the observation set ω satisfies the so-called Geometric Control Condition (GCC)—a particular case corresponding to O = Ω. A notable feature of our approach is that it remains effective in settings where Holmgren’s uniqueness does not guarantee unique continuation. As a consequence of our analysis, unique continuation is nonetheless recovered for wave solutions observed on (0, T) × ω with initial data supported in O. The proof of this previously unnoticed result combines a high-frequency observability estimate-based on the propagation of singularities—with a compactness-uniqueness argument that exploits the unique continuation properties of elliptic operators. By duality, this observability result leads to new controllability results for the wave equation, ensuring that the projection of the solution onto O can be controlled by means of controls supported in ω, with optimal spatial support. We also present several extensions of the main result, including the case of boundary observations, as well as a characterization of the observable fraction of the energy of the initial data from partial measurements on (0, T) × ω. Applications to wave control are discussed accordingly.
