Extension of controllability to L² initial data

Determine whether the null controllability conclusion for the Schrödinger equation on a convex polyhedron, established for initial data in H^s(P) with s>d/2 and control supported in a nonempty open neighborhood of the boundary singular set, remains valid when the initial data belong merely to L²(P).

Background

The paper proves observability and null controllability for the Schrödinger equation on a convex polyhedron P when the initial datum u_0 has Sobolev regularity Hs(P) with s>d/2. This regularity ensures, by Sobolev embedding, that u_0 and the corresponding solution are continuous in the spatial variables.

Spatial continuity is used in the proof to work with almost-periodic functions along periodic billiard tubes and to apply a dimension-reduction argument. The authors explicitly leave unresolved whether the same controllability conclusion can be obtained under the weaker natural energy-space assumption u_0∈L²(P).

References

This then raises the question of whether the same conclusion holds when $u_0$ merely belongs to $L2(P)$.

— Observability and controllability for the Schrödinger equation on polyhedra  (2609.10096 - Qu et al., 9 Sep 2026) in Section 1, Introduction, immediately following Theorem 1.2