Lyapunov Function
Lyapunov's Direct Method (Second Method)¶
In the direct method, we define a continuous scalar function \(V(x,t): \mathbb{R}^n \times \mathbb{R}_+ \to \mathbb{R}\) which acts as a generalized measure of "energy" in the system. We let \(B_\epsilon\) be a ball of radius \(\epsilon\) around the origin, i.e., \(B_\epsilon = \{x \in \mathbb{R}^n : \|x\| < \epsilon\}\).
We classify \(V(x,t)\) using strictly increasing functions \(\alpha\) and \(\beta\) (where \(\alpha(0)=\beta(0)=0\)): * Locally Positive Definite (lpdf): \(V(0,t)=0\) and \(V(x,t) \ge \alpha(\|x\|)\) for all \(x \in B_\epsilon, t \ge 0\). * Globally Positive Definite (pdf): \(V\) is lpdf, and additionally \(\alpha(p) \to \infty\) as \(p \to \infty\) (it is radially unbounded). * Decrescent: \(V(x,t) \le \beta(\|x\|)\) for all \(x \in B_\epsilon, t \ge 0\).
We then take the time derivative of \(V\) along the system trajectory \(f(x,t)\):
\(-\dot{V}(x,t)\) represents the energy dissipation along the trajectory. By evaluating \(V(x,t)\) and its negative derivative \(-\dot{V}(x,t)\), we can determine the stability of the system based on the following criteria:
| \(V(x,t)\) | \(-\dot{V}(x,t)\) | Conclusion |
|---|---|---|
| lpdf | \(\ge 0\) locally | Stable |
| lpdf, decrescent | \(\ge 0\) locally | Uniformly stable |
| lpdf, decrescent | lpdf | Uniformly asymptotically stable |
| pdf, decrescent | Globally uniformly asymptotically stable |