On the lack of observability for wave equations: a Gaussian beam approach

By F. Macià, Enrique Zuazua. Asymptotic Analysis, 2002

47 citations recorded by the source corpus.

This paper is devoted to study the property of observability for wave equations guaranteeing that the total energy of solutions may be estimated by means of the energy concentrated on a subset of the domain or of the boundary. We prove that this property fails in three different situations. First, we consider the wave equation with piecewise smooth coefficients when the observation is made in the exterior boundary. We also present a wave equation with highly oscillating Hölder continuous coefficients for which observability fails from any open set that does not contain the origin. Finally, lack of observability is proved for the constant coefficient wave equation when the observation is made from an interior hypersurface. All the counterexamples presented here are constructed using highly localized solutions known as Gaussian beams.

Subfields: Computational Theory and Mathematics, Computer Networks and Communications, Control and Systems Engineering

View Paper: https://www.semanticscholar.org/paper/868eb3f91e1fa912c0bc6815c99731cc7c7fe492

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Citations in the corpus

Controllability and observability of a heat equation with hyperbolic memory kernel (2009, methodology): “First, by [17,22], we choose M(t) ∈ C2([−T , T ];Cn×n) appearing in (5.”
Controllability and Observability of Partial Differential Equations: Some Results and Open Problems (2007, methodology): “But for variable coefficients the dynamics is more complex and can lead to some unexpected behaviour ([140]).”
Propagation, Observation, and Control of Waves Approximated by Finite Difference Methods (2005, background): “When the coefficient a = a(x) varies in space, the dynamics of this system may be quite complex and can lead to some unexpected behavior [74].”
The Wave Equation: Control and Numerics (2012, background): “x/ varies in space, the dynamics of this system may be quite complex and can lead to some unexpected behavior [74].”
On the observability of time-discrete conservative linear systems (2008, background): “The symbol of the operator in (4.19), that can be obtained by taking the Fourier transform of the dierential operator in space-time is of the form (see for instance [ 16 ])”