Construction of Lyapunov functions using Helmholtz-Hodge decomposition

Abstract

The Helmholtz-Hodge decomposition (HHD) is applied to the construction of Lyapunov functions. It is shown that if a stability condition is satisfied, such a decomposition can be chosen so that its potential function is a Lyapunov function. In connection with the Lyapunov function, vector fields with strictly orthogonal HHD are analyzed. It is shown that they are a generalization of gradient vector fields and have similar properties. Finally, to examine the limitations of the proposed method, planar vector fields are analyzed.