A new approach to optimal control of ill-posed semilinear elliptic equations in the absence of convexity

An optimal control problem governed by a class of ill-posed semilinear elliptic equations in the absence of convexity is studied in this paper. Firstly, by using a variant of the penalization method, the Pontryagin’s maximum principle for the initial problem is obtained. Secondly, an existence result is also proved.