Existence and Controllability of Conformable Impulsive Equations

In this study, the focus is directed towards the explanation of existence and uniqueness phenomena in impulsive dynamical systems governed by conformable fractional nonlinear differential equations. Concurrently, the analysis encompasses the controllability of systems delineated by linear and semilinear conformable fractional impulsive control mechanisms. By utilizing the mathematical apparatus of conformable fractional derivatives, salient constructs such as the conformable controllability operator and the conformable controllability Gramian matrix are introduced. These instrumentalities facilitate the derivation of both necessary and sufficient conditions that are requisite for achieving comprehensive controllability in linear impulsive systems within the framework of conformable fractional calculus. Moreover, an assemblage of rigorously formulated sufficient criteria is prefered to ascertain the controllability of semilinear impulsive systems in the domain of conformable fractional calculus.