384 research outputs found
Geometric control condition for the wave equation with a time-dependent observation domain
We characterize the observability property (and, by duality, the
controllability and the stabilization) of the wave equation on a Riemannian
manifold with or without boundary, where the observation (or control)
domain is time-varying. We provide a condition ensuring observability, in terms
of propagating bicharacteristics. This condition extends the well-known
geometric control condition established for fixed observation domains. As one
of the consequences, we prove that it is always possible to find a
time-dependent observation domain of arbitrarily small measure for which the
observability property holds. From a practical point of view, this means that
it is possible to reconstruct the solutions of the wave equation with only few
sensors (in the Lebesgue measure sense), at the price of moving the sensors in
the domain in an adequate way.We provide several illustrating examples, in
which the observationdomain is the rigid displacement in of a fixed
domain, withspeed showing that the observability property depends both on
and on the wave speed. Despite the apparent simplicity of some of
ourexamples, the observability property can depend on nontrivial
arithmeticconsiderations
Optimal shape and location of sensors for parabolic equations with random initial data
In this article, we consider parabolic equations on a bounded open connected
subset of . We model and investigate the problem of optimal
shape and location of the observation domain having a prescribed measure. This
problem is motivated by the question of knowing how to shape and place sensors
in some domain in order to maximize the quality of the observation: for
instance, what is the optimal location and shape of a thermometer? We show that
it is relevant to consider a spectral optimal design problem corresponding to
an average of the classical observability inequality over random initial data,
where the unknown ranges over the set of all possible measurable subsets of
of fixed measure. We prove that, under appropriate sufficient spectral
assumptions, this optimal design problem has a unique solution, depending only
on a finite number of modes, and that the optimal domain is semi-analytic and
thus has a finite number of connected components. This result is in strong
contrast with hyperbolic conservative equations (wave and Schr\"odinger)
studied in [56] for which relaxation does occur. We also provide examples of
applications to anomalous diffusion or to the Stokes equations. In the case
where the underlying operator is any positive (possible fractional) power of
the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the
complexity of the optimal domain may strongly depend on both the geometry of
the domain and on the positive power. The results are illustrated with several
numerical simulations
Spectral shape optimization for the Neumann traces of the Dirichlet-Laplacian eigenfunctions
We consider a spectral optimal design problem involving the Neumann traces of
the Dirichlet-Laplacian eigenfunctions on a smooth bounded open subset
of . The cost functional measures the amount of energy that Dirichlet
eigenfunctions concentrate on the boundary and that can be recovered with a
bounded density function. We first prove that, assuming a constraint on
densities, the so-called {\it Rellich functions} maximize this
functional.Motivated by several issues in shape optimization or observation
theory where it is relevant to deal with bounded densities, and noticing that
the -norm of {\it Rellich functions} may be large, depending on the
shape of , we analyze the effect of adding pointwise constraints when
maximizing the same functional. We investigate the optimality of {\it
bang-bang} functions and {\it Rellich densities} for this problem. We also deal
with similar issues for a close problem, where the cost functional is replaced
by a spectral approximation.Finally, this study is completed by the
investigation of particular geometries and is illustrated by several numerical
simulations
Thermoacoustic tomography with an arbitrary elliptic operator
Thermoacoustic tomography is a term for the inverse problem of determining of
one of initial conditions of a hyperbolic equation from boundary measurements.
In the past publications both stability estimates and convergent numerical
methods for this problem were obtained only under some restrictive conditions
imposed on the principal part of the elliptic operator. In this paper
logarithmic stability estimates are obatined for an arbitrary variable
principal part of that operator. Convergence of the Quasi-Reversibility Method
to the exact solution is also established for this case. Both complete and
incomplete data collection cases are considered.Comment: 16 page
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