Fixed/predefined-time synchronization of complex-valued discontinuous delayed neural networks via non-chattering and saturation control

In this article, the fixed-time and predefined-time synchronization of fully complex-valued discontinuous delayed neural networks are studied with the non-separation approach. Above all, to accomplish specific control objectives, two novel types of complex-valued controllers are designed, one of which is non-chattering, and the other one is saturation. Subsequently, in light of the Filippov solution theory and the complex variable sign function theory, several novel concise and sufficient criteria are established to achieve fixed/predefined-time synchronization for the addressed systems based on the fixed/predefined-time stability theory respectively. Simultaneously, the settling time of fixed-time synchronization is explicitly reckoned. The predefined synchronization time is irrelevant to any parameter and any initial value, and the upper bound of the desired settling time can be set in advance for different applications. Especially, to simplify the theoretical analysis and derivation process, the direct analysis method is used instead of the ordinary separation method. Additionally, under the two different types of controllers, the chattering phenomenon is effectively eliminated or reduced. It is not only proved theoretically but also supported by numerical simulation results in this paper. However, in the previous works have only numerical simulations without theoretical proof. Eventually, two numerical examples with simulation results are provided to substantiate the effectiveness of the obtained theoretical results.