Exact and trajectory controllability of second‐order evolution systems with impulses and deviated arguments

Abstract

In this paper, we consider a control system represented by a second‐order evolution impulsive problems with delay and deviated arguments in a Banach space X. We used the strongly continuous cosine family of linear operators and fixed‐point method to study the exact controllability. Also, we study the trajectory controllability of the considered control problem. Finally, an example is provided to illustrate the application of the obtained abstract results.

Approximate Controllability of Neutral Measure Evolution Equations with Nonlocal Conditions

“…Recently, some authors had discussed existence, stability, and nonlocal controllability of the measure evolution equation [9,15,[19][20][21][22]. In the past few years, the existence and controllability of fractional abstract functional differential development systems with nonlocal conditions have been fully studied [2,[23][24][25][26][27][28][29][30][31]. However, the controllability problem of neutral measure evolution equations with nonlocal conditions is seldom studied.…”

Controllability of second‐order Sobolev‐type impulsive delay differential systems

“…The differential systems handling the instantaneous changes as impulsive conditions occur in many applications such as mechanical and biological models subject to shocks, biological systems, population dynamics, and radiation of electromagnetic waves. For more details on impulsive differential systems, we refer to several relevant books and papers . Moreover, the Sobolev‐type differential equations admit more adequate abstract representation to the partial differential equations arising in numerous applications such as in flow of fluid through fissured rocks, propagation of long waves of small amplitude, and thermodynamics.…”

“…For more details on impulsive differential systems, we refer to several relevant books [1][2][3] and papers. [4][5][6][7][8][9] Moreover, the Sobolev-type differential equations admit more adequate abstract representation to the partial differential equations arising in numerous applications such as in flow of fluid through fissured rocks, propagation of long waves of small amplitude, and thermodynamics.…”

“…In recent years, various kinds of nonlinear differential systems have been considered for the study of controllability using different kinds of approaches (eg, previous studies 4, [6][7][8][9]11,12 and the references therein). Particularly, Kang and Kwun 13 and Chang and Li 12 studied controllability to second-order differential inclusions in the space of continuous differential functions.…”

Complete controllability for a class of fractional evolution equations with uncertainty

“…It is well-known that many problems in control theory such as stabilization, optimal control or pole assignment can be solved under assumption that the considered system is controllable. After first introducing by Kalman in [20], the controllability of dynamic systems has attracted a lot of interest, see [5,11,16,18,31]. The concept of controllability can be separated into complete controllability and approximate controllability, where the concept of complete controllability is that the dynamical system can be steered exactly from one state to another state while the concept of approximate controllability means that the dynamical system can be steered to a small neighborhood of final state in a given time, that is to say that the complete controllability always implies the approximate controllability, but in general, the converse statement is not true excepting the case of finite dimensional system.…”