短记忆分数阶微分方程的稳定性分析
Stability Analysis of Short Memory Fractional Differential Equations
摘要：
本文定义了一种具有短时记忆性质的分数阶导数，它可以看作是Caputo分数阶导数的推广。然后讨论了短记忆分数阶导数的一些性质。给出了一类短记忆分数阶系统的比较定理，并通过该定理建立了短记忆分数阶系统与Caputo分数阶系统之间的关系.利用比较定理和李雅普诺夫直接法，得到了保证一类短记忆分数阶方程渐近稳定性的充分判据。此外，还得到了一个特殊的结果，利用该结果可以直接判断某些特殊系统的稳定性。最后，通过三个算例验证了主要结果的有效性.
英文摘要：
In this paper, a fractional derivative with short-term memory properties is defined, which can be viewed as an extension of Caputo fractional derivative. Then, some properties of the short memory fractional derivative are discussed. Also, a comparison theorem for a class of short memory fractional systems is shown, via which some relationship between short memory fractional systems and Caputo fractional systems can be established. By applying the comparison theorem and Lyapunov direct method, some sufficient criteria are obtained, which can ensure the asymptotic stability of some short memory fractional equations. Moreover, a special result is presented, by which the stability of some special systems can be judged directly. Finally, three examples are provided to demonstrate the effectiveness of the main results.
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