Finite-Time Stability in Mean for Nabla Uncertain Fractional Order Linear Difference Systems

Abstract

In this paper, the finite-time stability in mean for the uncertain fractional order linear time-invariant discrete systems is investigated. First, the uncertain fractional order difference equations with the nabla operators are introduced. Then, some conditions of finite-time stability in mean for the systems driven by the nabla uncertain fractional order difference equations with the fractional order [Formula: see text] are obtained by the property of Riemann–Liouville-type nabla difference and the generalized Gronwall inequality. Furthermore, based on these conditions, the state feedback controllers are designed. Finally, some examples are presented to illustrate the effectiveness of the results.

Finite-Time Stability for Caputo Nabla Fractional-Order Switched Linear Systems

“…Remark 2. Since some of the conditions of the original version of the generalized Grönwall inequality do not need to be considered in this paper, they are discarded and modified in this paper, resulting in a slightly different citation from the original version, but this does not affect the validity of the inequality, and for a similar citation we can see ( [36], lemma 2.6).…”

“…As a result, the state trajectory of Caputo fractional-order discrete-time switched systems cannot follow the expression of integer-order discrete-time switched systems, which is also the most essential difference between fractional-order switched systems and integer-order switched systems. Because a fractional-order switched system has a different system matrix after each switching, its solution also cannot be obtained directly through the system equation and Mittag-Leffler function as in [35], and the study of its finite-time stability cannot be obtained simply through the Riemann-Liouville nabla properties and the generalized Grönwall inequality as in [36]. To our best knowledge, there are few systematic results on the stability of fractional discrete-time switched systems due to the above essential differences.…”

Novel Results on Finite‐Time Stability of Solutions for Stochastic Ψ‐Hilfer Fractional System

“…Since the groundbreaking work of Dorato [24], subsequently, the basic defnition of fnite-time stability in stochastic system was frst proposed by Kushner [25]. Te Grönwall approach was used to establish the fnite-time stability of stochastic fractional system in [26][27][28][29]. Based on the properties of nabla diference for Riemann-Liouvilletype and the generalized Grönwall inequality, Lu et al [26] researched fnite-time stability in the mean for the fractional diference equations with the nabla operator, in which contains uncertain term.…”

“…Te Grönwall approach was used to establish the fnite-time stability of stochastic fractional system in [26][27][28][29]. Based on the properties of nabla diference for Riemann-Liouvilletype and the generalized Grönwall inequality, Lu et al [26] researched fnite-time stability in the mean for the fractional diference equations with the nabla operator, in which contains uncertain term. With the aid of the Laplace transform and its inverse, Luo et al [27] researched two kinds of stochastic fractional delay systems, and the fnitetime stability results were established by applying the generalized Henry-Grönwall delay inequality.…”

Almost Surely Exponential Convergence Analysis of Time Delayed Uncertain Cellular Neural Networks Driven by Liu Process via Lyapunov–Krasovskii Functional Approach

“…Since its birth date, uncertainty theory has attracted extensive attentions in both the research and application communities. Up to now, uncertainty theory nearly has all theoretical results parallel to probability theory; see References [16][17][18][19][20][21][22][23][24][25][26][27][28][29]. In particular, the theory of uncertain processes and the uncertain calculus have been already developed well; see Reference [16] (see also [17,18,28], say).…”