Fractional semilinear optimal control: optimality conditions, convergence, and error analysis

TL;DR

This paper investigates an optimal control problem involving a fractional semilinear elliptic PDE, establishing well-posedness, deriving optimality conditions, and analyzing convergence and error estimates for finite element discretizations.

Contribution

It introduces a comprehensive analysis of fractional semilinear PDE control problems, including existence, optimality conditions, and convergence of discretization schemes.

Findings

Existence of optimal solutions established

First and second order optimality conditions derived

Convergence and error estimates for finite element discretizations provided

Abstract

We adopt the integral definition of the fractional Laplace operator and analyze an optimal control problem for a fractional semilinear elliptic partial differential equation (PDE); control constraints are also considered. We establish the well-posedness of fractional semilinear elliptic PDEs and analyze regularity properties and suitable finite element discretizations. Within the setting of our optimal control problem, we derive the existence of optimal solutions as well as first and second order optimality conditions; regularity estimates for the optimal variables are also analyzed. We devise a fully discrete scheme that approximates the control variable with piecewise constant functions; the state and adjoint equations are discretized with continuous piecewise linear finite elements. We analyze convergence properties of discretizations and derive a priori error estimates.