Abstract
This paper is concerned with the stability of fractional-order systems with randomly timevarying parameters. Two approaches are provided to check the stability of such systems in mean sense. The first approach is based on suitable Lyapunov functionals to assess the stability, which is of vital importance in the theory of stability. By an example one finds that the stability conditions obtained by the first approach can be tabulated for some special cases. For some complicated linear and nonlinear systems, the stability conditions present computational difficulties. The second alternative approach is based on integral inequalities and ingenious mathematical method. Finally, we also give two examples to demonstrate the feasibility and advantage of the second approach. Compared with the stability conditions obtained by the first approach, the stability conditions obtained by the second one are easily verified by simple computation rather than complicated functional construction. The derived criteria improve the existing related results.
Highlights
In recent years, the increasing interest of the scientific community towards fractional calculus experienced an exceptional boost and its applications can be found in a variety of real world problems, for example, viscous material [6], random and disordered media [21, 23], finance [16, 26], electrical circuits [14], automatic control system [18, 30] and so on
We provided two approaches to assess the stability of fractional-order systems with randomly time-varying parameters
We can construct suitable Lyapunov functionals satisfying some conditions to discuss the stability of such systems
Summary
The increasing interest of the scientific community towards fractional calculus experienced an exceptional boost and its applications can be found in a variety of real world problems, for example, viscous material [6], random and disordered media [21, 23], finance [16, 26], electrical circuits [14], automatic control system [18, 30] and so on. Fractional differential equations are considered as useful tools as they can model many physical systems. Other mathematical strategies are mainly based on the expressions of the solutions to the systems and integral inequalities. Fractional stochastic differential systems often arise in applications [1, 3, 7, 13, 22, 26, 30]. A lot of important results based on the existence and uniqueness and stability of the solutions to the systems were obtained. To the authors’ knowledge, the stability of solution of fractional-order differential system with randomly time-varying coefficients have yet to be reported. We plan to investigate the stability of the following fractionalorder differential system with randomly time-varying parameters:.
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