  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.
