| Abstract: |
| In this talk, I will discuss observability for wave equations, with particular emphasis on symmetry. I will focus on spacetime measurable observation sets on the one-dimensional torus, where a complete characterization is available. will first describe a counterexample showing that the Geometric Control Condition (GCC), while sufficient in many settings, does not guarantee observability when observation is restricted to certain spacetime measurable sets. Inspired by this counterexample, we introduce a symmetry condition, referred to as the Observable Symmetry Condition (OSC), which captures an additional structural constraint underlying observability. We show that OSC, together with GCC, provides a necessary and sufficient characterization of observability on spacetime measurable sets. Finally, motivated by OSC, we introduce a weak GCC and show that, when combined with OSC, it yields a necessary and sufficient condition for unique continuation. |
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