In this paper, we examine the conditions of exact null controllability of fractional dynamical system with nonlocal initial condition in infinite dimensional setting. The fractional derivatives used in the system are in Caputo sense and order of the derivatives are taken as $r\in(0,1)$. Schauder's fixed point theorem is used to prove null controllability with the help of the null controllability of the associated linear.
Fractional dynamical systems (FDS's) gain more and more importance in the recent decades because of their ability to model real world problems in a more efficient way in comparison to integer order systems. The controllability problem of FDS's are studied by many authors in numerous articles. There are different types of controllability, namely, exact controllability, approximate controllability, null controllability, trajectory controllability etc. The null controllability of a dynamical system means that the system can be steered to zero state from an arbitrary initial state by means of some control inputs.