A Study on Controllability of a Class of Impulsive Fractional Nonlinear Evolution Equations with Delay in Banach Spaces

Abstract

Under a new generalized definition of exact controllability we introduced and with a appropriately constructed time delay term in a special complete space to overcome the delay-induced-difficulty, we establish the sufficient conditions of the exact controllability for a class of impulsive fractional nonlinear evolution equations with delay by using the resolvent operator theory and the theory of nonlinear functional analysis. Nonlinearity in the system is only supposed to be continuous rather than Lipschitz continuous by contrast. The results obtained in the present work are generalizations and continuations of the recent results on this issue. Further, an example is presented to show the effectiveness of the new results.

New Discussion on Approximate Controllability for Semilinear Fractional Evolution Systems with Finite Delay Effects in Banach Spaces via Differentiable Resolvent Operators

“…φ ∈ L 1 ([−b, 0]; X). The main tools we are about to use here can be the theory of differentiable resolvent operators or analytic resolvent operators [25,30,31]. Furthermore, evolutionary fractional behavior is more accurately captured by variable-order fractional calculus.…”

Approximate Controllability for a Class of Semi-Linear Fractional Integro-Differential Impulsive Evolution Equations of Order 1 < α < 2 with Delay

“…Currently, the theory of fractional calculus has been extensively employed in disciplines such as viscoelasticity and rheology, physics, signal processing, control engineering, etc. For further information regarding these studies, please go through [1][2][3][4][5][6][7][8].…”

Controllability of a Class of Impulsive ψ-Caputo Fractional Evolution Equations of Sobolev Type

“…In 2021, Zhao [29] studied the exact controllability of a class of impulsive fractional nonlinear evolution equations with delay in Banach spaces:…”

“…where γ ∈ (0, 1), A : D ⊂ X → X is a closed the linear unbounded operator on X with dense domain D. In Ref. [29], Zhao defined the mild solution of system (2) as follows:…”