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

J. Sprekels, E. Valdinoci

Research output: Contribution to journalArticle

11 Citations (Scopus)

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 sth 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 suffcient 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 identiffcation problems: while usually in identification problems the type of the differential operator is prescribed and one or several of its coeffcient 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 diffculty that with changing fractional parameter s also the domain of deffnition, and thus the underlying function space, of the fractional operator changes.
Original languageEnglish
Pages (from-to)70-93
Number of pages24
JournalSIAM Journal on Control and Optimization
Volume55
Issue number1
DOIs
Publication statusPublished - 2017
Externally publishedYes

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Nonlocal Equations
Identification Problem
Fractional Order
Evolution Equation
Fractional
Mathematical operators
Differential operator
Operator
Mathematical Biology
Second-order Optimality Conditions
Discrete Spectrum
Differentiability
Positive definite
Function Space
Control Parameter
Minimization Problem
Linear equation
Existence and Uniqueness
First-order
Minimise

Cite this

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