A SEMISMOOTH NEWTON METHOD FOR A CLASS OF SEMILINEAR OPTIMAL CONTROL PROBLEMS WITH BOX AND VOLUME CONSTRAINTS

Abstract. In this paper we consider optimal control problems subject to a semilinear elliptic state equation together with the control constraints 0 ≤ u ≤ 1 and u = m. Optimality conditions for this problem are derived and reformulated as a nonlinear, nonsmooth equation which is solved using a semismooth Newton method. A regularization of the nonsmooth equation is necessary to obtain the superlinear convergence of the semismooth Newton method. We prove that the solutions of the regularized problems converge to a solution of the original problem and a path-following technique is used to ensure a constant decrease rate of the residual. We show that, in certain situations, the optimal controls take 0 − 1 values, which amounts to solving a topology optimization problem with volume constraint. 1