Topological properties of solution sets for Hilfer fractional nonlocal delay control systems and applications

The aim of this paper is to investigate the topological properties of mild solution sets for a
control system described by Hilfer fractional delay evolution equations with nonlocal initial conditions.
We firstly obtain the nonemptiness, the compactness and Rδ-property of the mild solution set by applying
Schauder’s fixed point theory, a fixed point theorem of nonconvex valued maps and the fractional calculus.
Then we apply this obtained result to show that the presented control problem has a reachable invariant
set under nonlinear perturbations. Furthermore, we also applly the obtained results to characterize the
approximate controllability of the presented control problem. Finally, we present an example to illustrate
the application of abstract results.