Optimal Control with Lp(Ω), $p\in [0, 1)$, Control Cost

$L^p$ optimal control with $p\in [0,1)$ is investigated. The difficulty of natural lack of convexity and thus of weak lower semicontinuity is addressed by introducing appropriately chosen regularization terms. Existence results and necessary optimality conditions are obtained, and convergence of a monotone scheme is proved. Special attention is given to the particular case of optimal control problems with quadratic tracking and regularized $L^0$ control costs are given. A maximum principle is derived and existence of controls, in some cases relaxed controls, is proved, and an estimate on the consequences of relaxation are estimated.