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Abstract
We consider a control problem for the wave equation: Given the initial state,
find a specific boundary condition, called a control, that steers the system to
a desired final state. The Hilbert uniqueness method (HUM) is a mathematical
method for the solution of such control problems. It builds on the duality
between the control system and its adjoint system, and these systems are connected
via a socalled controllability operator.
In this project, we are concerned with the numerical approximation of HUM control
for the onedimensional wave equation. We study two semidiscretizations of
the wave equation: a linear finite element method (LFEM) and a discontinuous
GalerkinFEM (DGFEM).
The controllability operator is discretized with both LFEM and DGFEM
to obtain a HUM matrix. We show that formulating HUM in a sine basis is
beneficial for several reasons: (i) separation of low and high frequency waves,
(ii) close connection to the dispersive relation, (iii) simple and effective filtering.
The dispersive behavior of a discretization is very important for its ability to
solve control problems. We demonstrate that the group velocity is determining
for a scheme’s success in relation to HUM. The vanishing group velocity for
high wavenumbers results in a dramatic decay of the corresponding eigenvalues
of the HUM matrix and thereby also in a huge condition number. We show
that, provided sufficient filtering, the phase velocity decides the accuracy of the
computed controls.
DGFEM shows very suitable for the treatment of control problems. The
good dispersive behavior is an important virtue and a decisive factor in the success
over LFEM. Increasing the order of DGFEM even give results of spectral
accuracy.
The field of control is closely related to other fields of mathematics among these
are inverse problems. As an example, we employ a HUM solution to an inverse
source problem for the wave equation: Given boundary measurements for a
wave problem with a separable source, find the spatial part of the source term.
The reconstruction formula depends on a set of HUM eigenfunction controls; we
suggest a discretization and show its convergence. We compare results obtained
by LFEM controls and DGFEM controls. The reconstruction formula is seen
to be quite sensitive to control inaccuracies which indeed favors DGFEM over
LFEM.
Original language  English 

Place of Publication  Kgs. Lyngby, Denmark 

Publisher  Technical University of Denmark 
Number of pages  169 
Publication status  Published  Sept 2009 
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 1 Finished

Numerisk approksimation af randkontrol problemer
Mariegaard, J. S., Knudsen, K., Hansen, P. C., Pedersen, M., Sørensen, M. P., Hesthaven, J. & Hugger, J.
01/09/2005 → 23/09/2009
Project: PhD