Analysis on Controllability Results for Wellposedness of Impulsive Functional Abstract Second-Order Differential Equation with State-Dependent Delay

Kulandhivel Karthikeyan,Dhatchinamoorthy Tamizharasan,Dimplekumar N Chalishajar

doi:10.3390/axioms10030188

Abstract

The functional abstract second order impulsive differential equation with state dependent delay is studied in this paper. First, we consider a second order system and use a control to determine the controllability result. Then, using Sadovskii’s fixed point theorem, we get sufficient conditions for the controllability of the proposed system in a Banach space. The major goal of this study is to demonstrate the controllability of an abstract second-order impulsive differential system with a state dependent delay mechanism. The wellposed condition is then defined. Next, we studied whether the defined problem is wellposed. Finally, we apply our results to examine the controllability of the second order state dependent delay impulsive equation.

Highlights

The investigation of this paper guarantees that the defined problem is controllable on any interval
We extend our research to investigate the controllability and wellposedness of an impulsive functional abstract second order differential equation with state dependent delay
The conditions for controllability and wellposedness of an abstract second-order differential system with state-dependent delay are investigated in this paper

Summary

Introduction

The investigation of this paper guarantees that the defined problem is controllable on any interval. Abstract differential equations and partial differential equations of second order were used in the research of state-dependent delay, as evidenced by a review of a few papers, see [5,30,31,32,33,34] Many of these articles use fixed point theorems to impart at least one mild solution using the condensing maps property, but the existence of exactly one solution to the given problem is not addressed, but except the paper cited [31]. We extend our research to investigate the controllability and wellposedness of an impulsive functional abstract second order differential equation with state dependent delay. The study of controllability and wellposedness of impulsive functional abstract second-order differential equations with state dependent delay is a nearly unexplored area in the literature, which is the motivating factor behind this paper. We illustrate few applications on impulsive differential equations of second order with delay condition

Basic Preliminaries

Main Results

Wellposedness

Examples

Conclusions