Abstract
This paper concerns the fractional finite time delay evolution systems and optimal controls in infinite-dimensional spaces. A suitable mild solution of the fractional finite time delay evolution systems is introduced. Using the singular version of the Gronwall inequality with finite time delay, we obtain some sufficient conditions for the existence, uniqueness and continuous dependence of mild solutions of these control systems. A formulation for the fractional Lagrange problem is introduced. The existence of optimal pairs of fractional-time-delay evolution systems is also presented. Finally, an example is given for demonstration.