Total Controllability of the Second Order Semi-Linear Differential Equation with Infinite Delay and Non-Instantaneous Impulses

Abstract

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.

Exact and Nonstandard Finite Difference Schemes for Coupled Linear Delay Differential Systems

“…corresponding to the three terms in expression (5). Then, expanding the binomial terms and rearranging and renaming indices, one has…”

“…Due to the presence of time lags in the dynamics of most real systems, delay differential equations (DDE) have become basic instruments in the mathematical modelling of a wide range of problems in science and engineering, such as in population biology, physiology, epidemiology, economics, and control problems (see, e.g., [1][2][3][4][5], and references therein), and special methods have been developed to compute numerical solutions for DDE [6]. In the case of differential problems without delay, exact schemes have been defined for different particular problems, and the use of nonstandard finite difference (NSFD) numerical schemes has gained increasing interest in the last years [7][8][9].…”

Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces

“…Since B is a bounded linear operator from H to X, the above convergence also implies 13) for all t ∈ J. Here, we used the weak convergence given in (3.12) and the compactness of the operator (Qf )(…”

“…Controllability plays a vital role in designing and analyzing control systems. The theory of controllability of the first and second order deterministic or stochastic systems in infinite dimensional spaces is well developed, and has been studied extensively, see for instance [6,7,8,10,13,16,28,31,42], etc. Controllability means that a dynamical system can be steered to the desired final state constrained to an admissible class of controls.…”

Numerical approach to the controllability of fractional order impulsive differential equations

“…It deals with whether or not the state of a state-space dynamic system can be controlled from the input. Many authors dealt with controllability problems that can be found in many recently published papers [19][20][21][22].…”