Stability of delayed systems with R-L fractional order and application to synchronization in networks with delayed impulses

This paper focuses on the $\theta$-exponential synchronization for fractional-order complex networks (FOCNs) with Riemann–Liouville (R-L) fractional order, hybrid couplings, and short memory under delayed impulse effects. Firstly, A new $\theta$-exponential stability criterion, which plays a crucial role for the network synchronization analysis later, is developed for delayed systems with R-L fractional order under delayed impulse effects, where the impulses can be stable or unstable, see Theorem 2.1. Secondly, the system model with respect to impulsive FOCNs with hybrid couplings and short memory under delayed impulse effects is established in the sense of R-L, which is more universal and practical. The novel feedback controller and adaptive feedback controller, are designed to realize the $\theta$-exponential synchronization objective, respectively. In addition, by utilizing Lyapunov functional approach, fractional calculus, matrix inequality analysis techniques and proposed stability criterion, the $\theta$-exponential synchronization conditions, which are associated with time delay and the order of systems, are derived in the form of linear matrix inequalities (LMIs). Finally, the numerical simulation result for Chua’s circuit networks, is provided to illustrate the validity of the obtained result and the effectiveness of the proposed control approach.