We discuss the controllability of nonlinear fractional control system with control delay. Firstly we obtain result about controllability of a linear fractional control system. After that, we give sufficient condition for the controllability of nonlinear fractional system with control delay. Our approach is based on Schauder fixed point theorem. At the end numerical example is constructed to support the result.

In this paper, by introducing the concepts of Ulam type stability for ODEs into the equations involving conformable fractional derivative, we utilize the technique of conformable fractional Laplace transform to investigate the Ulam–Hyers and Ulam–Hyers–Rassias stability for several kinds of linear differential equations in the frame of conformable fractional derivative.

This paper is concerned with controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel. First, the solution of fractional dynamical systems with a Mittag–Leffler kernel is given by Laplace transform. In addition, one necessary and sufficient condition for controllability of linear fractional dynamical systems with Mittag–Leffler kernel is established. On this basis, we obtain one sufficient condition to guarantee controllability of nonlinear fractional dynamical systems with a Mittag–Leffler kernel by fixed point theorem. Finally, an example is given to illustrate the applicability of our results.