Approximate controllability of second-order differential equations with time delays in input and state

Abstract

This article examines the approximate controllability of a class of second-order ordinary differential equations, where the input and system state, both have non-zero delays. Here, the existence and uniqueness of the mild solution to the problem is derived using a generalized contraction principle and properties of the cosine family of bounded linear operators on a Banach space. Also, the approximate controllability of the system under consideration and the related linear system with delay are achieved under a set of suitable criteria. The methodology employed for establishing the outcomes guarantees that the nonlinear system's approximate controllability remains independent of the approximate controllability of the corresponding linear system with delay. Further, the results are valid whether or not the operators are noncompact.