Sparse optimal control of a quasilinear elliptic PDE in measure spaces

Abstract

We considern a sparse optimal control problem governed by a quasilinear elliptic PDE in measure spaces. Under rather general assumptions we prove existence of optimal controls and derive first-order necessary optimality conditions. Under additional assumptions also second-order necessary and sufficient optimality conditions are obtained. The key technique of our analysis is the application of the so-called Kirchhoff transform, i.e. a nonlinear superposition operator that allows to transform the quasilinear state equation into a linear one.