Second-order and stability analysis for state-constrained elliptic optimal control problems with sparse controls

Abstract

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.