Spatially sparse optimization problems in fractional order Sobolev spaces

We investigate time-dependent optimization problems in fractional Sobolev spaces with the sparsity promoting Lᵖ-pseudo norm for $0<p<1$ in the objective functional. In order to avoid computing the fractional Laplacian on the time-space cylinder I× Ω, we introduce an auxiliary function w on Ω that is an upper bound for the function u∈ L²(I×Ω). We prove existence and regularity results and derive a necessary optimality condition. This is done by smoothing the Lᵖ-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.