Spatially sparse optimization problems in fractional order Sobolev spaces

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 $I\times \mathrm{\Omega }$, we introduce an auxiliary function $w$ on $\mathrm{\Omega }$ that is an upper bound for the function $u\in L^2(I\times \mathrm{\Omega })$. 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.