A study of instantaneous and non-instantaneous impulsive differential equations with applications in control problems (PhD)

Abstract

The present research work deals with the investigation of various kinds of impulsive differential models with instantaneous as well as non-instantaneous impulses in abstract spaces. This work provides insight into the different types of controllability, existence and stability of solutions of various second order impulsive models with and without deviated argument. We establish necessary and sufficient conditions for existence, uniqueness and stability of solutions to second order nonlinear differential equations with non-instantaneous impulses. Periodicity is also investigated of an impulsive model described by second order nonlinear differential equations with non-instantaneous impulses. Furthermore, we obtain some significant results on various kinds of controllability for the second order evolution systems with instantaneous impulses and deviated argument. Also, the existence of solution and controllability to second order neutral differential equation with non-instantaneous impulses have been investigated. Finally, the controllability of fractional differential equation of order α ∈ (1,2] with non-instantaneous impulses has been established, which has not been proposed so far to the best of our knowledge. This work leads to the motivation for the study of total controllability of the given problem which is a stronger notion of exact controllability. Several examples have been provided in order to make our theoretical analysis more concrete. Also, some numerical results have been carried out to support the analytical findings. In this work, the main used techniques are Banach fixed point theorem, strongly continuous cosine family of bounded linear operators, fractional cosine family, Poincare´ operator and evolution operator.