The controllability of nonlinear implicit fractional delay dynamical systems

Abstract

This paper is concerned with the controllability of nonlinear fractional delay dynamical systems with implicit fractional derivatives for multiple delays and distributed delays in control variables. Sufficient conditions are obtained by using the Darbo fixed point theorem. Further, examples are given to illustrate the theory.

Existence and uniqueness results for the coupled systems of implicit fractional differential equations with periodic boundary conditions

“…Recently, some scholars have considered very interesting aspects of IVPs and BVPs for the implicit FDEs (see [29][30][31][32][33][34][35][36][37][38][39]). For example, Nieto, Ouahab, and Venktesh [32] investigated a class of implicit FIVP:…”

Frequency Response Based Curve Fitting Approximation of Fractional–Order PID Controllers

“…In fractional calculus, the fractional-order differintegral operator, a combined fractional-order differentiator and integrator, which generalizes the notation for the differentiator (Re(γ) > 0) and the integrator (Re(γ) < 0) for the function x(t) (Kaczorek, 2018;Joice Nirmala and Balachandran, 2017), is defined as…”

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

“…For the sake of using fixed-point theorem, the controllability problem of nonlinear systems is transformed to a fixed-point problem of corresponding nonlinear operator in a appropriate function space. Frequently used fixed-point theorems include Banach's fixed-point theorem [12], Schauder's fixed-point theorem [21,29,129], Darbo's fixed-point theorem [16,26], Schaefer's fixed-point theorem [16], Krasnoselskii's fixed-point theorem [59,100,116], Sadovskii's fixed-point theorem [40,68,122], Mönch's fixed-point theorem [14,32,54,123,125], etc. It should be particularly noted that the controllability of fractional evolution systems (FESs) is an important issue for lots of practical problems since the fractional calculus can derive better results than integeral order one.…”

“…Controllability of some basic systems. During the past four decades, controllability problems of various dynamical systems, including integeral order and fractional order derivatives, have been widely investigated in finite-dimensional and infinite-dimensional spaces [3,25,26,54,60,69,122]. Among such controllability problems, the two most fundamental types are exact controllability and approximate controllability.…”

“…For instance, authors in [25] established some necessary and sufficient conditions for the exact controllability of certain linear fractional finite delay systems by using the Laplace transformation techniques and Mittag-Leffler function, and also presented a sufficient condition for the exact controllability of nonlinear fractional finite delay system via Schauder's fixed-point theorem. [26] investigated a implicit fractional finite delay system with multiple delays in control. Under the hypothesis that control Gramian matrix W is positive definite and nonlinear term is Lipschitz continuous and bounded, some sufficient conditions ensuring the exact controllability are obtained by Darbo's fixed-point theorem [26].…”