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

Eduardo Casas* and Fredi Tröltzsch

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 $L^1-$norm of the control as part of the objective functional that eventually leads to sparsity of the control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing $L^2$-regularization parameter for the control. The associated convergence rate is estimated.

Mathematics Subject Classification: 49K20 35J61

Keywords: Sparse Controls, State Constraints, Stability Analysis, Second Order Conditions

Minisymposion: Stability, Sensitivity and Error Analysis for Optimal Control Problems