  Distributed-order fractional non-local operators have been introduced and
studied by Caputo at the end of the 20th century. They generalize fractional
order derivatives/integrals in the sense that such operators are defined by a
weighted integral of different orders of differentiation over a certain range.
The subject of distributed-order non-local derivatives is currently under
strong development due to its applications in modeling some complex real world
phenomena. Fractional optimal control theory deals with the optimization of a
performance index functional subject to a fractional control system. One of the
most important results in classical and fractional optimal control is the
Pontryagin Maximum Principle, which gives a necessary optimality condition that
every solution to the optimization problem must verify. In our work, we extend
the fractional optimal control theory by considering dynamical systems
constraints depending on distributed-order fractional derivatives. Precisely,
we prove a weak version of Pontryagin's maximum principle and a sufficient
optimality condition under appropriate convexity assumptions.
