Optimal control of a class of semilinear fractional elliptic equations

Abstract

In this paper, a class of semilinear fractional elliptic equations associated to the spectral fractional Dirichlet Laplace operator is considered. We establish the existence of optimal solutions as well as a minimum principle of Pontryagin type and the first order necessary optimality conditions of associated optimal control problems. Second order conditions for optimality are also obtained for $L^{\infty }$ and $L^2-$ local solutions under some structural assumptions.