Regional and partial observability and control of waves

We establish sharp regional observability results for solutions of the wave equation in a bounded domain <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>Ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:msup> <mml:mi>ℝ</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:mrow> </mml:math> , in arbitrary spatial dimension. Assuming the waves are observed on a non-empty open subset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>ω</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> and that the initial data are supported in another open subset <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒪</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> , we derive estimates for the energy of initial data localized in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒪</mml:mi> </mml:math> , in terms of the energy measured on the observation set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> . This holds under a suitable geometric condition relating the time horizon <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>T</mml:mi> </mml:math> and the pair of subdomains <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>ω</mml:mi> <mml:mo>,</mml:mo> <mml:mi>𝒪</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> . Roughly speaking, this geometric condition requires that all rays of geometric optics in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>Ω</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , emanating from <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mover> <mml:mi>𝒪</mml:mi> <mml:mo>¯</mml:mo> </mml:mover> </mml:math> , must reach the observation region <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> . Our result generalizes classical observability results, which recover the total energy of all solutions when the observation set <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> satisfies the so-called Geometric Control Condition (GCC) — a particular case corresponding to <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>𝒪</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Ω</mml:mi> </mml:mrow> </mml:math> . 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> with initial data supported in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒪</mml:mi> </mml:math> . The proof of our 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 controllability results for the wave equation, ensuring that the projection of the solution onto <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>𝒪</mml:mi> </mml:math> can be controlled by means of controls supported in <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>ω</mml:mi> </mml:math> , 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 <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mn>0</mml:mn> <mml:mo>,</mml:mo> <mml:mi>T</mml:mi> <mml:mo>)</mml:mo> <mml:mo>×</mml:mo> <mml:mi>ω</mml:mi> </mml:mrow> </mml:math> . Applications to wave control are discussed accordingly.