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

TL;DR

This paper analyzes the semismooth Newton method for bilinear control problems in semilinear elliptic PDEs, proving convergence and optimality conditions, and demonstrating superlinear convergence under certain conditions.

Contribution

It provides a rigorous analysis of the semismooth Newton method, including convergence proofs and optimality conditions, for bilinear control problems in semilinear elliptic PDEs.

Findings

Proved existence, uniqueness, and regularity of solutions.

Derived first and second order optimality conditions.

Established superlinear convergence of the method.

Abstract

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