Existence of Optimal Controls for a Semilinear Composite Fractional Relaxation Equation

We consider a control system governed by a semilinear composite fractional relaxation equation in Banach space X . Under suitable conditions on the data, we prove that the system has a mild solution. Then, we investigate the existence of an optimal state-control pair, solution to the associated Lagrange optimal control problem. An example is also given to illustrate our results.