On the Existence of Sum‐Of‐Squares Lyapunov Functions for Polynomial Parameter‐Varying Systems With Locally Exponential Stability

ABSTRACT This article addresses a fundamental question regarding sum of squares approach, specifically, whether locally exponentially stable polynomial parameter‐varying systems admit a Lyapunov function with property of sum‐of‐squares. The Lyapunov function candidate is constructed through approximating the solution mapping by multiple Picard iterations. The obtained result is useful for searching a Lyapunov function in the analysis and synthesis of polynomial parameter‐varying systems by sum‐of‐squares theory. It can be concluded that the existence of sum‐of‐squares Lyapunov function is a necessary and sufficient condition for the locally exponential stability of polynomial parameter‐varying systems. The main result is also valid for linear parameter‐varying systems. Finally, the effectiveness of our result is demonstrated through three examples.