Controllability of multi-term fractional-order impulsive dynamical systems with $$\varphi $$ -Caputo fractional derivative

Abstract

In this article, we consider a multi-term \(\varphi \)-Caputo fractional dynamical system with non-instantaneous impulses. Firstly, we derive the solution for the linear \(\varphi \)-Caputo fractional differential equation by using the generalized Laplace transform. Then, some necessary and sufficient conditions have been examined for the controllability of the linear multi-term \(\varphi \)-Caputo fractional dynamical system with non-instantaneous impulses. Further, we establish some sufficient conditions for the controllability of the nonlinear system by utilizing the Schauder’s fixed point theorem and Gramian matrix. Finally, a simulated example is used to validate the obtained results of this article.