Optimal control of a class of semilinear fractional elliptic equations

In this paper, a class of semilinear fractional elliptic equations associated with the spectral or integral fractional Dirichlet Laplace operator was considered. Unlike most contributions to fractional optimal control, which assume that the control enters the state equation linearly, we addressed space-fractional equations in which the control appears in a nonlinear form. We proved existence and regularity results for the state equation and, using a measurable selection argument, established the existence of optimal controls. We then derived a pointwise Pontryagin-type minimum principle and first-order necessary optimality conditions. Second-order conditions for optimality are also obtained for $ L^{\infty} $, and $ L^2 $-local solutions under some structural assumptions.