Inter-Layer Noise-Based Topology of Complex-Valued Multi-Layer Networks: Almost Sure Stability via Time-Varying Hybrid Intermittent Pinning Control

This paper concentrates on stability of complex-valued multi-layer networks via time-varying hybrid intermittent pinning control (THIPC). A noise-based coupling that contains time-varying and nonlinear features is introduced first into complex-valued multi-layer networks. Different from existing literature on stochastic complex-valued systems that focus on moment stability, almost sure stability is explored in which noise can improve the stability of systems. Additionally, a novel THIPC strategy is proposed to achieve stability for a portion of nodes in the networks where the control gain is a sign-indefinite function and the control strategy designed integrates both stabilizing control and destabilizing input. For conquering the difficulties caused by time-varying gain, we develop the concept "average intermittent control gain" to quantify the control gain. Then the stability of remaining nodes is realized by virtue of the stability for pinning nodes and the conductive contribution by inter-layer noise-based topology. For the multi-layer networks we addressed, a stability analysis framework is constructed and we show the selection procedure of pinning nodes. Under this framework, employing Lyapunov method, stochastic analysis technique, almost sure stability criteria of complex-valued multi-layer networks are acquired. Finally, theoretical results are applied to inertial neural networks and cusp catastrophe models with numerical examples being provided.