Controllability of nonlinear fractional evolution systems in Banach spaces: A survey

Abstract

<p style='text-indent:20px;'>This paper presents a survey for some recent research on the controllability of nonlinear fractional evolution systems (FESs) in Banach spaces. The prime focus is exact controllability and approximate controllability of several types of FESs, which include the basic systems with classical initial and nonlocal conditions, FESs with time delay or impulsive effect. In addition, controllability results via resolvent operator are reviewed in detail. At last, the conclusions of this work and the research prospect are presented, which provides a reference for further study.</p>

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

“…Further investigation about the nonlocal controllability for a class of fractional impulsive integrodifferential evolution inclusions with time delay and nonlocal conditions will be carried on: (18) where g t : C([−b, a], E) → E is given function. Compared with the classical initial condition x(0) = x 0 and other nonlocal items [25,34,35], this nonlocal condition has better application effect in physics.…”

“…The results are obtained by utilizing Banach contraction mapping theorem due to the Lipschitz conditions of the systems. In addition, some excellent results of exact controllability for various fractional differential equations have also been established recently [7,[10][11][12][13][14][15][16][17][18][19][20][21][22][23], but the limitation is also that the functions in the systems are either Lipschitz continuous, compact or satisfy some special growth suppositions. Although the exact controllability studied in [13] does not require the nonlinear term to satify Lipschitz condition, the considered evolution system in [13] have no effects of time delay and impulse.…”

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].…”

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

“…Fractional differential systems and evolution systems have been studied extensively owing to its widespread backgrounds of some scientific and engineering realms, such as signal processing, finance, anomalous diffusion phenomena, heat conduction, etc. We refer readers to [1][2][3][4] for further detailed information. On the other side, controllability has gained a lot of importance and interest, and it plays a significant role in the description of various dynamical problems [5][6][7][8].…”