The aim of this paper is to study the controllability of fractional systems involving the Atangana–Baleanu fractional derivative using the Caputo approach. In the first step, the solution of a linear fractional system is obtained. Then, based on the obtained solution, some necessary and sufficient conditions for the controllability of such a system will be presented. Afterwards, the controllability of a nonlinear fractional system will be analyzed, based on these results. Our tool for the presentation of the sufficient conditions of controllability in this part is Schauder fixed point theorem. In the last step, the analytical results are illustrated by numerical examples.

This paper discusses the controllability of continuous-time linear fractional systems with control delay. The Atangana-Baleanu fractional derivative with the Caputo approach is used. First, the solution expression for a linear fractional system is obtained. Then, the corresponding fractional delay controllability Gramian matrix is defined, and its non-singularity as necessary and sufficient conditions for the controllability is proved. Finally, another equivalent condition based on the matrix rank formed by the coefficients matrices of the original system is provided that is much easier to check.

We study a well-known problem concerning a random variable uniformly distributed between two independent random variables. Two different extensions, randomly weighted average on independent random variables and randomly weighted average on order statistics, have been introduced for this problem. For the second method, two-sided power random variables have been defined. By using classic method and power technical method, we study some properties for these random variables.