Necessary Optimality Conditions for Singular Controls of Caputo Fractional Systems with Delay in Control

In this paper, we consider the optimal control of nonlinear systems of fractional-order with control delay, where the fractional derivatives are expressed in the Caputo sense. For this problem, we first obtained the necessary first-order optimality condition in the form of the Pontryagin maximum principle. For a control that is singular in the sense of the Pontryagin maximum principle, a higher-order necessary optimality condition is obtained. Further, in one particular case, a multipoint necessary second-order optimality condition is obtained. The effectiveness of the results obtained is illustrated by specific examples.