## Numerical Approximation of Boundary Control for the Wave Equation - with Application to an Inverse Problem

Publication: Research › Ph.D. thesis – Annual report year: 2009

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 so-called controllability operator.
In this project, we are concerned with the numerical approximation of HUM control
for the one-dimensional wave equation. We study two semi-discretizations of
the wave equation: a linear finite element method (L-FEM) and a discontinuous
Galerkin-FEM (DG-FEM).
The controllability operator is discretized with both L-FEM and DG-FEM
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.
DG-FEM 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 L-FEM. Increasing the order of DG-FEM 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 L-FEM controls and DG-FEM controls. The reconstruction formula is seen
to be quite sensitive to control inaccuracies which indeed favors DG-FEM over
L-FEM.

Original language | English |
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Place of publication | Kgs. Lyngby, Denmark |
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Publisher | Technical University of Denmark (DTU) |

Number of pages | 169 |

State | Published - Sep 2009 |

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