A detailed survey of existing works on fractional-order nonlinear systems reveals the fact that practically no results exist on stability or any performance analysis of Markovian jumping fractional-order systems (FOSs) in general. The main reason is the theory of infinitesimal generator used to estimate the derivative of Lyapunov–Krasovskii Functional (LKF) is not well-developed in the fractional domain. This shortage, in theory, is focussed in this manuscript. In this work, we provide a lemma that aids in analyzing the stability of fractional-order delayed systems via integer-order derivative of LKF. Using this lemma, by constructing a new suitable LKF and employing known integral inequalities, linear matrix inequality (LMI)-based sufficient conditions that ensure stability along with H ∞/passive performance of the proposed fractional-order neural networks (FONNs) with Markovian jumping parameters are derived for the first time. Unlike the existing works, the results derived in the present study depend on the fractional order (FO) of the NNs. The importance of such order-dependent criteria is highlighted in numerical examples. Finally, the simulation results are given to show the reliability of the derived conditions.

中文翻译：

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分数阶中性延迟马尔可夫跳跃神经网络的混合 H∞/无源性能分析的稳定性

对分数阶非线性系统现有工作的详细调查表明，实际上几乎不存在关于马尔可夫跳跃分数阶系统 (FOS) 的稳定性或任何性能分析的结果。主要原因是用于估计Lyapunov-Krasovskii泛函（LKF）导数的无穷小生成器理论在分数域中没有得到很好的发展。理论上，这种短缺集中在这份手稿中。在这项工作中，我们提供了一个引理，有助于通过 LKF 的整数阶导数分析分数阶延迟系统的稳定性。使用这个引理，通过构造一个新的合适的 LKF 并采用已知的积分不等式，基于线性矩阵不等式 (LMI) 的充分条件，以确保稳定性以及H ∞首次推导了所提出的具有马尔可夫跳跃参数的分数阶神经网络（FONN）的被动性能。与现有工作不同，本研究得出的结果取决于 NN 的分数阶 (FO)。这种依赖于顺序的标准的重要性在数值示例中得到了强调。最后，给出了仿真结果，证明了推导条件的可靠性。