A system of linear functional differential equations with fractional derivative and aftereffect is considered. The question of the representation of solutions is investigated, the existence f the Cauchy matrix as the kernel of an integral representation is proved, and the main defining relationships for the Cauchy matrix are derived.The well-known definition of the fractional Caputo derivative of the order is used. The system under study includes, in addition to the Caputo derivative, a linear Volterra operator of a general form. Using the Riemann–Liouville fractional integration operator the initial system is reduced to a linear integral Volterra equation, for which the convergence of the Neumann series and the integral representation of the solution using a resolvent integral operator are established. It is shown that the Cauchy matrix is expressed explicitly through the resolvent kernel of this operator. In the case of a transition to the integer order of the derivative, the obtained defining relationship for the Cauchy matrix coincides with the known one. The use of the Cauchy matrix opens up wide possibilities for the study of fractional systems in terms of obtaining effective conditions of solvability of boundary value problems, control problems and the description of asymptotic behavior of solutions similar to how it is done for wide classes of systems with integer derivatives.All the constructions are based on the use of the basic statements of the theory of abstract functional differential equations developed by the heads of the Perm seminar, Professors N.V. Azbelev and L.F. Rakhmatullina.

functional differential equations, conditions of everywhere solvability, positive operators, periodic boundary value problem, modeling of dynamic systems.