Symmetry conditions for spacetime observability of wave equations on the torus

TL;DR

This work shows that observability of the 1D wave equation on the torus from spacetime observation sets is controlled not only by the geometric control condition (GCC) but also by a new symmetry requirement (Observable Symmetry Condition, OSC). A conservation law tied to characteristic transports is established, revealing why GCC alone can fail and how OSC remedies this deficiency. The authors prove that observability holds if and only if both GCC and OSC are satisfied, and that unique continuation requires OSC together with a weak form of GCC. The results refine the geometric criteria for exact controllability in spacetime frameworks and highlight the role of symmetry in wave propagation on manifolds with periodic directions.

Abstract

We study observability for the one-dimensional wave equation on the torus from spacetime measurable observation sets. While the Geometric Control Condition (GCC) provides a sufficient criterion in many classical settings, it is no longer sufficient in this framework. We construct explicit counterexamples showing the failure of observability despite the validity of GCC. This leads to the introduction of an additional symmetry condition on the observation set, referred to as the Observable Symmetry Condition (OSC). We prove that observability holds if and only if both GCC and OSC are satisfied. We also show that unique continuation holds if and only if both OSC and a weak form of GCC are satisfied.

Figures (5)

• Figure 1: Observability fails on $G$, though $G$ satisfies ( GCC).
• Figure 2:
• Figure 3: $G$ is a subset of the blue and red part.
• Figure :
• Figure :

Theorems & Definitions (44)

• Definition 1.1
• Proposition 1.2: Conservation law
• Theorem 1.3
• Theorem 1.4
• Remark 2.1: A variant
• Lemma 3.1
• proof : Proof of Proposition \ref{['prop-sym-conser']}
• Proposition 3.2: Controllability implies OSC
• proof
• Proposition 3.3: UCP implies OSC