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

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.