Different from the existing multiple Mittag-Leffler stability or multiple asymptotic stability, the multiple exponential stability, which has explicit and faster convergence rate, is investigated in this paper for fractional-order impulsive control Cohen–Grossberg neural networks. First, by using the definition of Dirac delta function, the fractional-order control Cohen–Grossberg neural networks are translated into fractional-order impulsive neural networks, in which pulse effect relies on the fractional order of the addressed system. Then, based on maximum norm, norm and general norm , a series of novel criteria are obtained respectively to ensure that such neuron neural networks can have total equilibrium points and locally exponentially stable equilibrium points, by utilizing the known fixed point theorem, the method of average impulsive interval, the theory of fractional-order differential equations, and the method of Lyapunov function. This paper’s investigation reveals the effects of impulsive function, impulsive interval, and fractional order on the dynamic behaviors. Finally, theoretical results are shown to be effective by four illustrative examples.

中文翻译：

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分数阶脉冲控制 Cohen-Grossberg 神经网络的多个平衡点的共存和局部指数稳定性

与现有的多重Mittag-Leffler稳定性或多重渐近稳定性不同，本文研究了分数阶脉冲控制Cohen-Grossberg神经网络的多重指数稳定性，它具有显式且更快的收敛速度。首先，通过使用狄拉克δ函数的定义，将分数阶控制Cohen-Grossberg神经网络转化为分数阶脉冲神经网络，其中脉冲效应依赖于所处理系统的分数阶。然后，基于最大范数、范数和一般范数，利用已知的不动点定理，分别得到一系列新颖的准则，以保证此类神经元神经网络能够具有总平衡点和局部指数稳定平衡点，方法为平均脉冲区间、分数阶微分方程理论、Lyapunov函数方法。本文的研究揭示了脉冲函数、脉冲区间和分数阶对动态行为的影响。最后，通过四个说明性例子证明理论结果是有效的。