A new type of identification problems: optimizing the fractional order in a nonlocal evolution equation

TL;DR

This paper investigates a novel class of identification problems involving the optimization of the fractional order in a nonlocal evolution equation, with applications in mathematical biology, establishing existence, uniqueness, and optimality conditions.

Contribution

It introduces a new identification problem where the fractional order of a diffusion operator is optimized, including analysis of existence, differentiability, and optimality conditions.

Findings

Proved existence and uniqueness of solutions for the fractional evolution equation.

Derived first-order necessary and second-order sufficient optimality conditions.

Established the differentiability of solutions with respect to the fractional parameter.

Abstract

In this paper, we consider a rather general linear evolution equation of fractional type, namely a diffusion type problem in which the diffusion operator is the $s$th power of a positive definite operator having a discrete spectrum in $\R^+$. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter $s$. These results are then employed to derive existence as well as first-order necessary and second-order sufficient optimality conditions for a minimization problem, which is inspired by considerations in mathematical biology. In this problem, the fractional parameter $s$ serves as the "control parameter" that needs to be chosen in such a way as to minimize a given cost functional. This problem constitutes a new class of identification problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coefficient functions need to be identified, in the present case one has to determine the type of the differential operator itself. This problem exhibits the inherent analytical difficulty that with changing fractional parameter $s$ also the domain of definition, and thus the underlying function space, of the fractional operator changes.