  The current article examines the approximate controllability problem for
non-instantaneous impulsive fractional evolution equations of order
$1<\alpha<2$ with state-dependent delay in separable reflexive Banach spaces.
In order to establish sufficient conditions for the approximate controllability
of our problem, we first formulate the linear-regulator problem and obtain the
optimal control in feedback form. By using this optimal control, we deduce the
approximate controllability of the linear fractional control system of order
$1<\alpha<2$. Further, we derive sufficient conditions for the approximate
controllability of the nonlinear problem. Finally, we provide a concrete
example to validate the efficiency of the derived results.
