Impulsive problems for fractional evolution equations and optimal controls in infinite dimensional spaces

Abstract

In this paper, a class of impulsive fractional evolution equations
and optimal controls in infinite dimensional spaces is considered.
A suitable concept of a $PC$-mild solution is introduced and
a suitable operator mapping is also constructed. By using a $PC$-type
Ascoli-Arzela theorem, the compactness of the operator mapping is
proven. Applying a generalized Gronwall inequality and
Leray-Schauder fixed point theorem, the existence and uniqueness of
the $PC$-mild solutions is obtained. Existence of optimal pairs for
system governed by impulsive fractional evolution equations is also
presented. Finally, an example illustrates the applicability of our
results.