Abstract
In this manuscript, a stronger concept of exact controllability called Total Controllability has been introduced. Sufficient conditions have been established for the total controllability of the proposed problem. The proposed control problem is a second-order semi-linear differential equation with infinite delay and non-instantaneous impulses. The tools for study include the strongly continuous cosine family and Sadovskii’s fixed point theorem. The cosine family and the nonlinear function associated with the system are assumed to be non-compact. In addition, the total controllability of an integrodifferential problem has been investigated. Finally, an example is provided to illustrate the analytical findings.
Highlights
A number of evolving processes such as shocks, harvesting, and natural disasters are generally subjected to abrupt changes
Abrupt changes may stay for finite time intervals
Such impulses are known as non-instantaneous impulses
Summary
A number of evolving processes such as shocks, harvesting, and natural disasters are generally subjected to abrupt changes. None of the research papers have so far discussed the controllability of the non-instantaneous impulsive differential equation with infinite delay. This manuscript is devoted to the study of total controllability for the following second-order semi-linear differential equation with infinite delay and non-instantaneous impulses in a Banach space X:. The theory of strongly continuous cosine family is used to study the second-order abstract differential equations Many authors utilize this strongly continuous cosine family tool to investigate the existence and uniqueness of the solutions and various types of controllability for second-order nonlinear in abstract spaces [19,20,21].
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