  The paper deals with an optimal control problem in a dynamical system
described by a linear differential equation with the Caputo fractional
derivative. The goal of control is to minimize a Bolza-type cost functional,
which consists of two terms: the first one evaluates the state of the system at
a fixed terminal time, and the second one is an integral evaluation of the
control on the whole time interval. In order to solve this problem, we propose
to reduce it to some auxiliary optimal control problem in a dynamical system
described by a first-order ordinary differential equation. The reduction is
based on the representation formula for solutions to linear fractional
differential equations and is performed by some linear transformation, which is
called the informational image of a position of the original system and can be
treated as a special prediction of a motion of this system at the terminal
time. A connection between the original and auxiliary problems is established
for both open-loop and feedback (closed-loop) controls. The results obtained in
the paper are illustrated by examples.
