The LaSalle theorem for coupled systems of differential equations

The issue pertaining with the LaSalle theorem for coupled systems of differential equations (CSDEs) is considered in this paper. Through combining the method of Kirchhoff's Matrix Tree Theorem in graph theory and Lyapunov method, we proposed a systematic method to construct a global Lyapunov function for CSDEs. Furthermore, the LaSalle theorem for CSDEs is established. As main applications of this theorem, asymptotic stability as well as exponential stability of CSDEs is discussed with several sufficient criteria obtained according to the topology property of CSDEs. It shows the power of our new results when they are applied to the control problems for coupled systems. Finally, an example is presented to illustrate the feasibility and validity of our theoretical results.