This paper concentrates on the stability and stabilization of short memory fractional differential equations with delayed impulses. The sufficient conditions for asymptotic stability of short memory fractional differential equations with two kinds of delayed impulses are derived, respectively. The results show that the delayed impulses in short memory fractional differential equations exhibit double effects on system performance. For an unstable system, one can stabilize the system by inputting delays in impulses; for a stable system, the stability would be destroyed if the delays were too long. Further, a class of fractional chaotic systems is presented to test the validity of the established theoretical results, some criteria for impulsive synchronization of fractional chaotic systems are derived, and the corresponding impulsive controllers are designed. Finally, a fractional Chua chaotic oscillator is presented to illustrate the practicability of the established impulsive controllers.

本文主要研究具有延迟脉冲的短记忆分数阶微分方程的稳定性和稳定性。分别推导了具有两种延迟脉冲的短记忆分数阶微分方程渐近稳定的充分条件。结果表明，短记忆分数阶微分方程中的延迟脉冲对系统性能表现出双重影响。对于不稳定的系统，可以通过输入脉冲延迟来稳定系统；对于一个稳定的系统，如果延迟时间过长，稳定性就会被破坏。进一步，提出了一类分数混沌系统来检验所建立理论结果的有效性，推导了分数混沌系统脉冲同步的一些判据，并设计了相应的脉冲控制器。最后，提出了分数Chua混沌振荡器来说明所建立的脉冲控制器的实用性。