Stability analysis of nonlinear systems using higher order derivatives of Lyapunov function candidates

The Lyapunov method for stability analysis of an equilibrium state of a nonlinear dynamic system requires a Lyapunov function v(t,x)v(t,x) having the following properties: (1) vv is a positive definite function, and (2) v̇ is at least a negative semi-definite function. Finding such a function is a challenging task. The first theorem presented in this paper simplifies the second property for a Lyapunov function candidate, i.e. this property is replaced by negative definiteness of some weighted average of the higher order time derivatives of vv. This generalizes the well-known Lyapunov theorem. The second theorem uses such weighted average of the higher order time derivatives of a Lyapunov function candidate to obtain a suitable Lyapunov function for nonlinear systems’ stability analysis. Even if we have a suitable Lyapunov function then this theorem can be used to prove a bigger region of attraction. The approach is illustrated by some examples.