On the optimality conditions for a fractional diffusive equation with a nonlocal term

TL;DR

This paper investigates an optimal control problem for a fractional diffusive equation with a nonlocal time component, establishing well-posedness, existence, and optimality conditions, including local and global uniqueness of solutions.

Contribution

It introduces a comprehensive analysis of an optimal control problem involving a fractional Laplacian with nonlocal time effects, deriving new optimality conditions and uniqueness results.

Findings

Existence of at least one optimal control

Derivation of first-order optimality conditions

Conditions for local and global uniqueness

Abstract

We study a bilinear OCP for an evolution equation governed by the fractional Laplacian of order $0 < s < 1$, incorporating a nonlocal time component modeled by an integral kernel. After establishing well-posedness of the problem, we analyze the properties of the control-to-state operator. We prove the existence of at least one optimal control and derive both first-order and second-order optimality conditions, which ensure local uniqueness. Under further assumptions, we also demonstrate that global uniqueness of the optimal control can be achieved.