2,133 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
Fully discrete finite element data assimilation method for the heat equation
We consider a finite element discretization for the reconstruction of the
final state of the heat equation, when the initial data is unknown, but
additional data is given in a sub domain in the space time. For the
discretization in space we consider standard continuous affine finite element
approximation, and the time derivative is discretized using a backward
differentiation. We regularize the discrete system by adding a penalty of the
-semi-norm of the initial data, scaled with the mesh-parameter. The
analysis of the method uses techniques developed in E. Burman and L. Oksanen,
Data assimilation for the heat equation using stabilized finite element
methods, arXiv, 2016, combining discrete stability of the numerical method with
sharp Carleman estimates for the physical problem, to derive optimal error
estimates for the approximate solution. For the natural space time energy norm,
away from , the convergence is the same as for the classical problem with
known initial data, but contrary to the classical case, we do not obtain faster
convergence for the -norm at the final time
Mini-Workshop: Recent Developments on Approximation Methods for Controlled Evolution Equations
This mini-workshop brought together mathematicians engaged in partial differential equations, functional analysis, numerical analysis and systems theory in order to address a number of current problems in the approximation of controlled evolution equations
Inverse problems for linear hyperbolic equations using mixed formulations
We introduce in this document a direct method allowing to solve numerically
inverse type problems for linear hyperbolic equations. We first consider the
reconstruction of the full solution of the wave equation posed in - a bounded subset of - from a partial
distributed observation. We employ a least-squares technique and minimize the
-norm of the distance from the observation to any solution. Taking the
hyperbolic equation as the main constraint of the problem, the optimality
conditions are reduced to a mixed formulation involving both the state to
reconstruct and a Lagrange multiplier. Under usual geometric optic conditions,
we show the well-posedness of this mixed formulation (in particular the inf-sup
condition) and then introduce a numerical approximation based on space-time
finite elements discretization. We prove the strong convergence of the
approximation and then discussed several examples for and . The
problem of the reconstruction of both the state and the source term is also
addressed
On Characterization of Inverse Data in the Boundary Control Method
We deal with a dynamical system
\begin{align*} & u_{tt}-\Delta u+qu=0 && {\rm in}\,\,\,\Omega \times (0,T)\\
& u\big|_{t=0}=u_t\big|_{t=0}=0 && {\rm in}\,\,\,\overline \Omega\\ &
\partial_\nu u = f && {\rm in}\,\,\,\partial\Omega \times [0,T]\,,
\end{align*} where is a bounded domain, a real-valued function, the outward normal to , a solution. The input/output correspondence is realized
by a response operator
and its relevant extension by hyperbolicity . Ope\-rator is
determined by , where . The inverse problem is: Given to
recover in . We solve this problem by the boundary control method
and describe the {\it ne\-ces\-sary and sufficient} conditions on ,
which provide its solvability.Comment: 33 pages, 1 figur
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