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

Abstract

This manuscript mainly discusses the approximate controllability for certain fractional delay evolution equations in Banach spaces. We introduce a suitable complete space to deal with the disturbance due to the time delay. Compared with many related papers on this issue, the major tool we use is a set of differentiable properties based on resolvent operators, rather than the theory of C0-semigroup and the properties of some associated characteristic solution operators. By implementing an iterative method, some new controllability results of the considered system are derived. In addition, the system with non-local conditions and a parameter is also discussed as an extension of the original system. An instance is proposed to support the theoretical results.

Advances in Boundary Value Problems for Fractional Differential Equations

“…. , β m ), and the integrals from the boundary conditions (11) are Riemann-Stieltjes integrals with H i : [0, 1] → R, i = 0, . .…”

“…, m functions of bounded variation. By using the Guo-Krasnoselskii fixed point theorem of cone expansion and norm-type compression, they prove the existence and multiplicity of positive solutions for problem (10), (11).…”

“…Paper [11] is devoted to the fractional differential evolution equation in the Banach space X with a finite delay and a control…”

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

“…Controllability of control systems is an important component and research direction of control theory, as well as the foundation of optimal control and optimal estimation. In recent years, the controllability of various types of fractional dynamic systems, including fractional impulsive systems [9,10], delay syetems [11], stochastic systems [12,13], neutral systems [14], nonlocal systems [15], damped systems [16], integro-differential systems [17], measure evolution systems [18], etc., has been studied extensively and deeply. For example, in [10], the authors derived some new results of the total controllability (a type of exact controllability) for a fractional control system with non-instantaneous impulse by means of Krasnoselskii's fixed-point theorem.…”