We analyze an observer strategy based on partial—that is, in a subdomain—measurements of the solution of a wave equation, in order to compensate for uncertain initial conditions. We prove the exponential convergence of this observer under a nonstandard observability condition, whereas using measurements of the time derivative of the solution would lead to a standard observability condition arising in stabilization and exact controlabillity. Nevertheless, we directly relate our specific observability condition to the classical geometric control condition. Finally, we provide some numerical illustrations of the effectiveness of the approach.

Observer theory has been established for decades [

In this paper we consider an observer strategy originally proposed in [

Let

For this system, we consider that some measurements are available—assumed to be without noise in this paper—in

The aim in observer design is to define a system

In order to assess the efficiency of the observer, we can consider the dynamics followed by the error

In the case of time-derivative measurements (

In the first case of direct measurements of the field (

The above choices of gain operator

We consider in this paper a known source term, but the observers considered here can be extended to also estimate an unknown source term, following the strategy introduced in [

In order to establish that (

Assume that we have the observability condition

Let us consider the space

However, the observability condition (

Assume that the geometric control condition of Bardos et al. [

Since the geometric control condition holds, we have the classical observability condition [

Note that this proof requires a geometric control condition slightly stronger than that directly associated with the measurement domain

For both observer strategies, the potential noise in the measurements—disregarded in this paper—would simply entail an additional source term in the error equations, without amplification of this error term by time or space differentiation. The parameter

In order to illustrate the effectiveness of our Schur Displacement Feedback (SDF) observer approach—and compare it to the classical Direct Velocity Feedback (DVF) observer—we consider the two-dimensional domain shown in Figure

Geometry and two observation domains considered.

Large observation domain

Small observation domain

Initial condition in mesh used.

We show in Figures

Poles for SDF and DVF stabilization (up and down, resp.) with large and small observation domains (left and right, resp.).

SDF—large observation domain

SDF—small observation domain

DVF—large observation domain

DVF—small observation domain

This is further illustrated in Figure

Energies of the undamped and SDF-stabilized solutions.