A study on controllability of impulsive fractional evolution equations via resolvent operators

Abstract

In this article, we study the controllability for impulsive fractional integro-differential evolution equation in a Banach space. The discussions are based on the Mönch fixed point theorem as well as the theory of fractional calculus and the $(\alpha ,\beta )$ ( α , β ) -resolvent operator, we concern with the term $u'(\cdot )$ u ′ ( ⋅ ) and finding a control v such that the mild solution satisfies $u(b)=u_{b}$ u ( b ) = u b and $u'(b)=u'_{b}$ u ′ ( b ) = u b ′ . Finally, we present an application to support the validity study.

Iterative Positive Solutions to a Coupled Riemann-Liouville Fractional q -Difference System with the Caputo Fractional q -Derivative Boundary Conditions

“…Consequently, there exist ðu * , v * Þ and ðw * , z * Þ in P l which are nonnegative extremal solutions of BVP ( 5) and (6). Since f k ðt, 0, 0Þ≡0, ∀t ∈ ½0, 1, for k = 1, 2; then, zero is not a solution of problem (5). It is obvious that ðu * , v * ÞðtÞ > 0, ðw * , z * ÞðtÞ > 0.…”

“…Over the next several hundred years, many mathematicians have studied the properties of fractional calculus. From these studies, it is not difficult to conclude that fractional calculus has some very nice properties and can be applied to various fields, such as random processes [3], non-Newtonian fluid mechanics [4], control theory [5], economics [6], and medical [7]. Especially in the modeling of dynamic problems, fractional differential equations are quite common.…”

The Existence Results of Solutions to the Nonlinear Coupled System of Hilfer Fractional Differential Equations and Inclusions

“…However, fractional calculus has only been revitalized in the last few decades, thanks to the development of various fields that promote research in fractional calculus. The new study of fractional calculus can also offer novel ideas for solving challenging problems in various fields, see [2][3][4][5][6][7].…”

Controllability of stochastic fractional systems involving state-dependent delay and impulsive effects

“…Moreover, controllability analysis of second-order systems with SDD and impulses was proved in [3]. Numerous authors have emphasized how controllability systems are increasingly common and adequate in applications in [12,[38][39][40].…”