  We consider a Bolza-type optimal control problem for a dynamical system
described by a fractional differential equation with the Caputo derivative of
an order $\alpha \in (0, 1)$. The value of this problem is introduced as a
functional in a suitable space of histories of motions. We prove that this
functional satisfies the dynamic programming principle. Based on a new notion
of coinvariant derivatives of the order $\alpha$, we associate the considered
optimal control problem with a Hamilton-Jacobi-Bellman equation. Under certain
smoothness assumptions, we establish a connection between the value functional
and a solution to this equation. Moreover, we propose a way of constructing
optimal feedback controls. The paper concludes with an example.
