Strong solution and optimal control problems for a class of fractional linear equations

In this paper, we examine the unique solvability (in the sense of strong solutions) of the Cauchy problem for a linear inhomogeneous equation in a Banach space solved with respect to the Caputo fractional derivative. We assume that the operator acting on the unknown function in the right-hand side of the equation generates an analytic resolving operator family for the corresponding homogeneous equation. We obtain a representation of a strong solution of the Cauchy problem and examine the solvability of optimal control problems with a convex, lower semicontinuous, lower bounded, coercive functional for the equation considered. The general results obtained are used to prove the existence of an optimal control in problems with specific functionals. Abstract results obtained for a control system described by an equation in a Banach space are illustrated by examples of optimal control problems for a fractional equation whose special cases are the subdiffusion equation and the diffusion wave equation.