| Contents |
| We consider a parabolic optimal control problem with a directional sparsity
functional, where the control variable is searched for in the space
of vector measures \(\mathcal{M}(\Omega_c, L^2(I))\).
\begin{equation*}
\text{Minimize }\;
\frac12 {\|y - y_d\|}_{L^2(I \times \Omega_o)}^2 + \alpha {\|u\|}_{\mathcal{M}(\Omega_c, L^2(I))},
\end{equation*}
\begin{equation*}
\text{subject to }\; \partial_t y + A y = u
\quad\text{in } I\times\Omega.
\end{equation*}
The optimal solutions of this problem are localized in space, where the spatial support
is independent of time. We establish an appropriate function space setting for the
problem and derive structural properties of the minimizer from the
optimality conditions. A typical solution given by a sum of point sources with time
dependent intensities, \(u = \sum_i u_i(t) \delta_{x_i}\).
We motivate this problem formulation by discussing an application to a
deconvolution problem. Furthermore we discuss a suitable finite element discretization
and an efficient solution method within a path-following Newton framework.
We provide an a priori error analysis for the discretization of the optimal control problem. |
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