Fractional order Lyapunov stability theorem and its applications in synchronization of complex dynamical networks

• The fractional order Lyapunov stability theorem is studied and proposed.
• The form of fractional order Lyapunov stability theorem is given.
• The association between fractional Lyapunov theorem and linear one is proved.
• The complex network model is described in both direct and undirected.
• The T–S fuzzy model pining control method with minimum controller is designed.

In this paper, we bring attention to the existence of fractional order Lyapunov stability theorem which is strictly descripted by mathematic formulas. Firstly, we introduce fractional-order Lyapunov function based on the definition of fractional calculation and integer order Lyapunov theory. By using the classical stability theorem of linear fractional order systems, we demonstrate convincingly the existence of fractional-order Lyapunov function and present a mathematical description of fractional-order Lyapunov stability theorem. Furthermore, as an example, we apply the presented fractional order Lyapunov stability theorem to synchronization of both direct and undirected complex networks with fractional order equation nodes. Based on the proposed theorem, a novel T–S fuzzy model pining controller with minimum control nodes is designed. Finally, numerical simulations are agreement with theoretical analysis, which both confirm that the correctness of the presented theory.