In this paper, a finite-horizon optimal control problem involving a dynamical system described by a linear Caputo fractional differential equation and a quadratic cost functional is considered. An explicit formula for the value functional is given, which includes a solution of a certain Fredholm integral equation. A step-by-step feedback control procedure for constructing
-optimal controls with any accuracy
is proposed. The basis for obtaining these results is the study of a solution of the associated Hamilton–Jacobi–Bellman equation with so-called fractional coinvariant derivatives.