Stabilization Analysis via Discrete-Time Feedback for Hybrid Stochastic Neural Networks with Non-Differentiable Delays and Lévy Noise under Mixed Growth Conditions

This paper endeavors to systematically develop a comprehensive class of hybrid stochastic neural networks (SNNs) that incorporate both time-delay dynamics and Lévy noise. In terms of system settings, Markov switching capturing the complex transitions between different operational modes is used to effectively characterize the inherent hybrid nature, and the activation functions stored in each mode exhibit mixed growth rather than a single linear or polynomial growth. Meanwhile, the time-delay is non-differentiable including piece-wise constant or distributed, which is more general, and Lévy noise as a jump disturbance is processed by Khasminskii-type conditions. Considering the explosion difficulties caused by switching and Lévy noise, proofs of global existence and uniqueness of solutions and moment boundedness of the original system are detailed completed. Furthermore, discrete-time feedback control based on switching-mode observers in Markov chains is introduced to stabilize the original system. By using the Lyapunov functional, the generalized Itô’s formula, the M-matrix theory, the sufficient conditions for three stabilization modes are obtained. Moreover, we refined the unclear exponential decay rates near zero in previous studies and expanded the determination methods. Finally, a numerical example illustrating the feasibility of theoretical results is provided. Contrary to previous studies, which employed symmetric Lévy noise, our analysis of existing literature confirms and substantiates the benefits of asymmetric noise.