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Regional and partial observability and control of waves

Belhassen Dehman, Sylvain Ervedoza an Enrique Zuazua

Abstract

We establish sharp regional observability results for solutions of the wave equation in a bounded domain of $Ω\subset \mathbb{R}^n$, in case where the geometric control condition is not satisfied. Assuming that the waves are observed on a non-empty open subset $ω\subset Ω$ and that the initial data are supported in another open subset $\mathscr{O} \subsetΩ$, we derive estimates for the energy of initial data localized in $\mathscr{O}$, in terms of the energy measured on the observation set $(0,T) \times ω$. This holds under a suitable geometric condition relating the time horizon T and the subdomains $ω$ and $\mathscr{O}$. By duality, we obtain new controllability results for the wave equation, ensuring that the projection of the solution onto $\mathscr{O}$ 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 $(0,T) \times ω$. Applications to wave control are discussed accordingly.

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 of , in case where the geometric control condition is not satisfied. Assuming that the waves are observed on a non-empty open subset and that the initial data are supported in another open subset , we derive estimates for the energy of initial data localized in , in terms of the energy measured on the observation set . This holds under a suitable geometric condition relating the time horizon T and the subdomains and . By duality, we obtain new controllability results for the wave equation, ensuring that the projection of the solution onto can be controlled by means of controls supported in , 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 . Applications to wave control are discussed accordingly.
Paper Structure (32 sections, 14 theorems, 97 equations, 2 figures)

This paper contains 32 sections, 14 theorems, 97 equations, 2 figures.

Table of Contents

  1. Introduction and first results
  2. Problem formulation
  3. Main results
  4. Examples
  5. Some relevant consequences
  6. Methodology of proof
  7. Outline.
  8. Proofs of the main results
  9. Proof of Theorem \ref{['Theo-Obs-1']}
  10. Proof of Theorem \ref{['Theo-Obs-2']}
  11. Some geometric facts, operators and wave fronts
  12. Local geodesic coordinates
  13. Generalized bicharacteristic rays
  14. Sets of interest
  15. Pseudo-differential operators
  16. ...and 17 more sections

Key Result

Theorem 1.1

Given the domain $\Omega$, the observation subdomain $\omega \subset \Omega$, and the time-horizon $T>0$, let the subdomain $\mathscr{O}$ be such that every generalized bicharacteristic ray (see Definition def:gene-bichar for their precise definition) starting from $\{t = 0\}\times \overline{\mathsc for any initial data $(u_{0},u_{1}) \in L^{2}(\Omega)\times H^{-1}(\Omega)$ with support in $\overl

Figures (2)

  • Figure 1: Illustration of the $1$-d example: $\Omega = (-1,1)$, $\omega = (-1,-3/4) \cup (3/4,1)$, $\mathscr{O} = (-1/4, 1/4)$. The critical time given by Theorem \ref{['Theo-Obs-1']} is $T_{0,crit} = 1$, while the times for unique continuation and the GCC coincide and are equal to $T_{UC} = 3/2$.
  • Figure 4: When $\Omega = {\mathbb{R}}^d$, $\omega$ is the unit ball and $\Delta_A$ is the flat Laplacian, the set $\mathcal{O}(\omega_T)$, when projected in the physical space is supported in the ball of size $T$. In fact, in this case, as the bicharacteristics in this setting correspond to straight lines, $\mathcal{O}(\omega_T) = \{(x, \xi) \in T_{b}^{*}\Omega\backslash 0 \text{ s.t. } \exists t \in [0,T] \text{ satisfying } x + \xi t \in \omega \}.$

Theorems & Definitions (42)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.4
  • Remark 1.5
  • Lemma 2.1
  • Corollary 2.2
  • proof
  • Lemma 2.3
  • Remark 2.4
  • ...and 32 more