A new type of identification problems: Optimizing the fractional order in a nonlocal evolution equation
- Sprekels, Jürgen
- Valdinoci, Enrico
2010 Mathematics Subject Classification
- 49K21 35S11 49R05 47A60
- Fractional operators, identification problems, first-order necessary and second-order sufficient optimality conditions, existence, uniqueness, regularity
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 power of a positive definite operator having a positive and discrete spectrum. We prove existence, uniqueness and differentiability properties with respect to the fractional parameter. 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 classof 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 also the domain of definition, and thus the underlying function space, of the fractional operator changes.
- SIAM J. Control Optim., 55 (2017), pp. 70--93.