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This article mainly dedicates on the issue of finite-time stabilization of complex-valued neural networks with proportional delays and inertial terms via directly constructing Lyapunov functions without separating the original complex-valued neural networks into two real-valued subsystems equivalently. First of all, in order to facilitate the analysis of the second-order derivative caused by the inertial term, two intermediate variables are introduced to transfer complex-valued inertial neural networks (CVINNs) into the first-order differential equation form. Then, under the finite-time stability theory, some new criteria with less conservativeness are established to ensure the finite-time stabilizability of CVINNs by a newly designed complex-valued feedback controller. In addition, for reducing expenses of the control, an adaptive control strategy is also proposed to achieve the finite-time stabilization of CVINNs. At last, numerical examples are given to demonstrate the validity of the derived results. [ABSTRACT FROM AUTHOR]

• In this paper, the p-norm fixed-time synchronization of delayed ICVNNs is investigated for the first time, in which the settling time estimation in terms of p-norm is derived. Our results are more flexible and adjustable than these results [36–38]. • Compared with the works [3,13,36-38,42], what we get herein is fixed-time synchronization, and these results are also applied to image encryption. • Different from the real-valued INNs [5–7,19,21] and first-order CVNNs [13,39-42]. In this paper, the neural model is defined in the complex field and has an inertial term, which broadens the application range of INNs and makes up for some previous literature results. This paper researches the problem of p-norm fixed-time synchronization for a class of delayed inertial complex-valued neural networks (ICVNNs). By using reduced-order transformation and separating real and imaginary parts of complex-valued parameters , the second-order ICVNNs can be converted into the form of first-order real-valued differential equations. Then some new flexible and adjustable algebraic criteria to ensure the fixed-time synchronization of ICVNNs are established by means of the non-smooth Lyapunov function and inequality analytical techniques. Moreover, the settling time of fixed-time synchronization is theoretically estimated, which does not depend on the initial value of systems. Finally, simulation examples and applications are presented to illustrate the validity and availability of the obtained results. [ABSTRACT FROM AUTHOR]