Exploring a new discrete delayed Mittag–Leffler matrix function to investigate finite‐time stability of Riemann–Liouville fractional‐order delay difference systems

Abstract

In this paper, firstly, a new discrete delayed Mittag–Leffler matrix function is introduced, which generalizes the existing discrete delayed exponential matrix function. Secondly, based on it, the explicit formula of the solution of homogeneous Riemann–Liouville (RL) fractional‐order delay difference system is obtained. Thirdly, the explicit formulas of the solutions of nonhomogeneous RL fractional‐order delay difference systems are also derived in terms of the superposition principle and the new discrete delayed Mittag–Leffler matrix function. Furthermore, a finite‐time stability (FTS) criterion of nonhomogeneous RL fractional‐order delay difference system is given using the properties of the new discrete delayed Mittag–Leffler matrix function. Finally, three examples are presented to demonstrate the effectiveness of the proposed theorems.

Representation of solutions and finite‐time stability for fractional delay oscillation difference equations

“…However, it is verified that this method is not applicable to study the nonhomogeneous equation ( 2). In literature [33], the authors derived the solution of nonhomogeneous Riemann-Liouville-type FDDE via utilizing the superposition principle. Differently, we will present the exact solution of the nonhomogeneous equation ( 2) by the discrete Laplace transform method.…”

“…Du and Lu explored the solution of a homogeneous Caputo‐type FDDE with order $$$ 0<\mu <1 $$$ in [32]. The authors [33] also derived the solutions of homogeneous and nonhomogeneous Riemann–Liouville‐type fractional delay difference equations (FDDEs). In addition, Wu et al [18, 34] investigated the solutions of FDDEs of order $$$ 0<v<1 $$$.…”

Existence and Ulam‐type stability for impulsive oscillating systems with pure delay

“…In recent decades, there have been some important research results on seeking explicit formula of solution to delay differential/discrete systems by introducing continuous/discrete delayed exponential matrix functions [11][12][13][14][15][16][17]. In [18], the authors introduced the new concept of an impulsive delayed matrix function and obtained the representation of solutions to linear IDDSs.…”

Some remarks on the Gronwall integral inequality

“…Recently, fractional-order calculus has been a hot topic owing to its possible applications in science and engineering. 1,2 Stability analysis is one of most crucial issues in fractional-order systems. [3][4][5][6] The integer-order Gronwall integral inequality (IOGII) has been widely applied to study the stability of fractional-order systems.…”

“…Recently, fractional‐order calculus has been a hot topic owing to its possible applications in science and engineering 1,2 . Stability analysis is one of most crucial issues in fractional‐order systems 3–6 .…”