Finite-time synchronization control for a class of delayed neural networks: an improved two-step control method

In this paper, a two-step control method is proposed, leveraging the generalized Halanay inequality and existing finite-time stability theorems, to achieve finite-time synchronization for a class of neural networks with bounded time-varying delay. In the first step, the system state is attenuated from V(t₀) to γV(t₀) using the generalized Halanay inequality, where 0 < γ ⩽ 1 is a free parameter. In the second step, by applying existing finite-time stability theorems, the system state further decays from γV(t₀) to 0. Building on the above ideas, two novel finite-time stability lemmas for the error system are presented, and the convergence rate as well as the settling time is estimated. Furthermore, the value of γ that results in the shortest settling time for the error system is also provided. With the help of the derived lemmas, several sufficient algebraic criteria are established to achieve finite-time synchronization between the considered delayed neural networks. The results of this paper not only improve the existing two-step control method but also overcome the limitations of certain one-step finite-time control approaches. Finally, the validity and practical applicability of the obtained theoretical results are demonstrated through two numerical examples and an image protection experiment.