Optimal control of a class of semilinear fractional elliptic equations

TL;DR

This paper investigates optimal control problems for a class of semilinear fractional elliptic equations, establishing existence, optimality conditions, and second order optimality criteria under certain assumptions.

Contribution

It introduces new optimality conditions and second order criteria for controlling semilinear fractional elliptic equations involving spectral fractional Laplace operators.

Findings

Existence of optimal solutions proven.

First order necessary optimality conditions derived.

Second order conditions established for local solutions.

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.