Convergence Analysis of the Semismooth Newton Method for Sparse Control Problems Governed by Semilinear Elliptic Equations

We show that a second order sufficient condition for local optimality, along with a strict complementarity condition, is enough to get the superlinear convergence of the semismooth Newton method for an optimal control problem governed by a semilinear elliptic equation. The objective functional may include a sparsity promoting term and we allow for box control constraints.