Recently, several research articles have investigated the existence of
solution of dynamical systems with fractional-order and as well as
expounded controllability. Nevertheless, very little attention has been
given to observability of such dynamical systems. In the present work,
we explore the outcomes of controllability and observability regarding a
differential system of fractional order with input delay. Laplace and
inverse Laplace transforms along with the Mittage-Leffler matrix
function are applied to the proposed dynamical system in Caputo's sense
and obtain a general solution in the form of an integral equation. Then
we set out conditions for the controllability of the underlying model,
regarding the linear case. We then expound controllability conditions
for the nonlinear case with the aid of fixed point theorem of Schaefer
and the Arzola-Ascoli theorem. After converting the problem considered
to a fixed point problem, we prove the observability of the linear case
using the observability Grammian matrix. The necessary and sufficient
conditions, for the nonlinear case, are investigated with the help of
the Banach contraction mapping theorem. Finally, we add some examples to
elaborate our work.