On the impulsive stabilization of some differential and integro-differential equations of second order

The aim of this master thesis is to study the impulsive exponential stabilization of some problems of specific nonlinear impulsive delay differential and impulsive delay integro-differential equations. In the present thesis, two mathematical models such as nonlinear impulsive delay differential and impulsive delay integro-differential equations are considered. Here, we consider new impulsive mathematical models, which have more general forms than that can be found in a paper of the present literature, and generalize the results of that paper. In this thesis project, to achieve the aim of the present thesis, two new Lyapunov-Krasovskiĭ functionals are defined as the main tools. Hence, Lyapunov – Krasovskiĭ functional method is used to prove the new results of the thesis. This thesis study involves six chapters. The first chapter of the present thesis is given under the title "Introduction". In this chapter, some brief information is given with regard to the significance of impulsive differential equations, their historical development, the importance of impulsive differential equations in applications, impulsive problems to be discussed in the thesis and the method to be used throughout this thesis. The 2. Chapter of this thesis is arranged under the title as "Literature Review". In this chapter, some papers and basic books, which are related to the content of the thesis and can be reached in the pertaining literature are given. In 3. Chapter of the present thesis, the materials and methods to be utilized in this thesis are given. In Chapter 4 of the present thesis, some basic definitions in relation to impulsive differential, some theorems pertaining to the existence and uniqueness of solutions and fundamental information are presented. In Chapter 5 of this thesis, a nonlinear impulsive differential equation including multiple constant delays is considered. Here, a new theorem which includes sufficient conditions for the exponentially stabilization of solutions of the considered equation is proved by using the Lyapunov–Krasovskiĭ functional method. Lastly, in Chapter 6 of this thesis, a nonlinear impulsive integro-differential equation including multiple constant delays is taken under consideration. In this chapter, a new theorem which includes sufficient conditions for the exponentially stabilization of solutions of the considered integro-differential equation is verified by help the Lyapunov–Krasovskiĭ functional method. In order to achieve the proofs of the results in Chapter 5 and Chapter 6 of this thesis, two new Lyapunov – Krasovskiĭ functionals are defined and thereby these Lyapunov – Krasovskiĭ functionals are utilized as the main tools in the proofs of the new results of this thesis. Keywords: Exponential stabilization, Impulsive differential equation, Impulsive integro-differential equation, Lyapunov – Krasovskiĭ functional, Second order