  In this research paper, we examine an optimal control problem involving a
dynamical system governed by a nonlinear Caputo fractional time-delay state
equation. The primary objective of this study is to obtain the necessary
conditions for optimality, both the first and second order, for the Caputo
fractional time-delay optimal control problem. We derive the first-order
necessary condition for optimality for the given fractional time-delay optimal
control problem. Moreover, we focus on a case where the Pontryagin maximum
principle degenerates, meaning that it is satisfied in a tivial manner.
Consequently, we proceed to derive the second order optimality conditions
specific to the problem under investigation. At the end illustrative examples
are provided.
