A class of fractional evolution equations and optimal controls

This paper concerns the existence of mild solutions for semilinear fractional evolution equations and optimal controls in the αα-norm. A suitable αα-mild solution of the semilinear fractional evolution equations is introduced. The existence and uniqueness of αα-mild solutions are proved by means of fractional calculus, singular version Gronwall inequality and Leray–Schauder fixed point theorem. The existence of optimal pairs of system governed by fractional evolution equations is also presented. Finally, an example is given for demonstration.