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May 11, 2026Open Access

Relative Controllability of Caputo Fractional‐Order Stochastic Delay System Driven by Lévy Noise

YZYun ZhongGuizhou University MLMengmeng LiGuizhou University

Key Points

The aim is to explore the relative controllability of fractional stochastic delay systems influenced by Lévy noise.
Derived the solution of linear fractional stochastic delay systems using the delayed Mittag-Leffler matrix function.
Discussed relative controllability using the Grammian matrix for linear systems.
Utilized Cauchy-Schwarz and Jensen's inequalities along with Krasnoselskii’s theorem for nonlinear systems.
The theoretical results are validated via a practical example.
Relative controllability conditions for both linear and nonlinear systems are established.
The innovative approach highlights the role of Lévy noise in controlling fractional-order systems.

Abstract

In this paper, we study the relative controllability of fractional stochastic delay system (FSDS) driven by Lévy noise. First, we derive the solution of linear FSDS by using the delayed Mittag–Leffler matrix function. Then, using Grammian matrix, we discuss the relative controllability of linear FSDS. In addition, the relative controllability of nonlinear FSDS is discussed using the Cauchy–Schwarz inequality, Jensen’s inequality, It’s formula, and Krasnoselskii fixed‐point theorem. Finally, an example is given to verify the theoretical results.

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