Bilinear control of semilinear elliptic PDEs: convergence of a semismooth Newton method

Abstract

In this paper, we carry out the analysis of the semismooth Newton method for control constrained bilinear control problems of semilinear elliptic PDEs. We prove existence, uniqueness and regularity for the solution of the state equation, as well as differentia bility properties of the control to state mapping. Then, first and second order optimality conditions are obtained. Finally, we prove the superlinear convergence of the semis mooth Newton method to local solutions satisfying no-gap second order sufficient optimality conditions as well as a strict complementarity condition.