  We investigate time-dependent optimization problems in fractional Sobolev
spaces with the sparsity promoting $L^p$-pseudo norm for $0<p<1$ in the
objective functional. In order to avoid computing the fractional Laplacian on
the time-space cylinder $I\times \Omega$, we introduce an auxiliary function
$w$ on $\Omega$ that is an upper bound for the function $u\in
L^2(I\times\Omega)$. We prove existence and regularity results and derive a
necessary optimality condition. This is done by smoothing the $L^p$-pseudo norm
and by penalizing the inequality constraint regarding $u$ and $w$. The problem
is solved numerically with an iterative scheme whose weak limit points satisfy
a weaker form of the necessary optimality condition.
