### Controllability of semilinear neutral fractional functional evolution equations with infinite delay

#### Abstract

This paper is concerned with the semilinear differential system of fractional order with infinite delay: $D^\alpha\,x(t)=Ax(t)+Bu(t)+f(t,x_t),~~t\in [0,T],$ $x(t)=\phi(t)$, $t\in ]-\infty,0]$, with $1\alpha2$. We prove that the system is controllable when $A$ generates an $\alpha$-resolvent family $(S_\alpha(t))_{t\geq 0}$ on a complex Banach space $\X$ and the control $u\in L^{2}([0,T];\X)$.