Second-Order and Stability Analysis for State-Constrained Elliptic Optimal Control Problems with Sparse Controls

An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the L¹-norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing L²-regularization parameter for the control. The associated convergence rate is estimated.