This paper mainly deals with the problem of exponential and adaptive synchronization for a type of inertial complex-valued neural networks via directly constructing Lyapunov functionals without utilizing standard reduced-order transformation for inertial neural systems and common separation approach for complex-valued systems. At first, a complex-valued feedback control scheme is designed and a nontrivial Lyapunov functional, composed of the complex-valued state variables and their derivatives, is proposed to analyze exponential synchronization. Some criteria involving multi-parameters are derived and a feasible method is provided to determine these parameters so as to clearly show how to choose control gains in practice. In addition, an adaptive control strategy in complex domain is developed to adjust control gains and asymptotic synchronization is ensured by applying the method of undeterminated coefficients in the construction of Lyapunov functional and utilizing Barbalat Lemma. Lastly, a numerical example along with simulation results is provided to support the theoretical work.

中文翻译：

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惯性复值神经网络的指数和自适应同步：一种非约简和非分离方法。

本文主要通过直接构造Lyapunov泛函而无需利用惯性神经系统的标准降阶变换和复杂值系统的通用分离方法，来直接构造一类惯性复值神经网络的指数和自适应同步问题。首先，设计了一种复数值反馈控制方案，并提出了一个由复数值状态变量及其导数组成的非平凡的Lyapunov函数来分析指数同步。推导了涉及多参数的一些准则，并提供了一种确定这些参数的可行方法，以清楚地表明实际中如何选择控制增益。此外，在Lyapunov泛函的构造中，利用Barbalat引理，采用不确定系数的方法，开发了一种复杂域的自适应控制策略，以调节控制增益，并确保渐近同步。最后，提供了一个数值例子和仿真结果来支持理论工作。