Controllability Results for a Class of Piecewise Nonlinear Impulsive Fractional Dynamic Systems

Abstract

We study the total controllability for a new class of piecewise nonlinear Langevin fractional dynamic equations with non-instantaneous impulses. New necessary and sufficient conditions are presented for the total controllability of the corresponding linear impulsive systems. Also, the nonlinear case is investigated and controllability criteria are established. We transform the controllability problem into the existence of a fixed point task by defining a nonlinear operator and a proper admissible piecewise control function on a Banach space. Fractional calculus, Mittag-Leffler functions, Gramian type matrices, and the Schauder’s fixed point theorem are employed to develop our main results. Finally, we illustrate the analytical outcome by providing a simulated example.