Approximate controllability of nonlocal impulsive neutral integro‐differential equations with finite delay

Abstract

In this paper, we apply the resolvent operator theory and an approximating technique to derive the existence and controllability results for nonlocal impulsive neutral integro-differential equations with finite delay in a Hilbert space. To establish the results, we take the impulsive functions as a continuous function only, and we assume that the nonlocal initial condition is Lipschitz continuous function in the first case and continuous functions only in the second case. The main tools applied in our analysis are semigroup theory, the resolvent operator theory, an approximating technique, and fixed-point theorems. Finally, we illustrate the main results with the help of two examples.

New exploration on approximate controllability of fractional neutral‐type delay stochastic differential inclusions with non‐instantaneous impulse

“…Dynamical systems having such sudden changes in their state can be modeled by impulsive differential equations. When the duration of impulsive action is very small in comparison with the total duration of the whole process, then it is named as instantaneous impulse, whereas when the impulsive action starts abruptly at a certain moment of time and remains active on a finite time interval, then it is known as non-instantaneous impulse [1][2][3][4][5][6][7]. On the other hand, in real life, there are many problems in which the current state at any time depends upon the prior time.…”

“…Controllability is the qualitative property of the dynamical control system which shows the existence of a control function which can steer the dynamical system from an initial state to any desired final state. In recent years, many researchers have extensively investigated controllability results for various dynamical systems [4,8,[12][13][14][15][16]. Furthermore, the deterministic models often experience variations that are random, or that at least look to be random due to environmental noise.…”