Geometric translation of a theorem about stability of equilibrium point

Question

In the book Nonlinear Systems by Hassan Khalil, there is a theorem about the stability of equilibrium point ‎ which asserts that :

Theorem :‎ Let‎ ‎$X=0$ ‎be an equilibrium point for‎ ‎$‎\dot x = f(x)‎$‎ where‎ ‎$f : D ‎\rightarrow ‎R^n‎$‎ is a locally Lipschitz map from a domain‎ ‎$D \subset R^n$‎ into ‎ ‎$R^n$‎‎ and‎ ‎$D $ containing‎ ‎$x = 0 $‎ . Let‎ ‎$V : D ‎\rightarrow R ‎‎‎$‎ be a continuously differentiable function, such that ‎$‎V(0) = 0‎$‎ and ‎$‎V(x) \gt 0‎$‎ in ‎$‎D - ‎\lbrace0\rbrace‎‎‎‎$‎ ‎$‎‎\dot V(x) ‎\leq ‎0‎‎$‎ in ‎$‎D‎$‎ . Then ‎$‎x = 0‎$‎ is stable. Moreover, if ‎$‎‎\dot V(x) \lt 0‎$‎ in ‎$‎D - ‎\lbrace0‎\rbrace‎‎$ then $‎x = 0‎$‎ is asymptotically stable.

‎ My question is that what is the geometric representation (i.e. geometric translation or visualization) of the function ‎$V$ ‎in ‎this ‎theorem.‎ Thanks in advance for any forthcoming idea/comment.

Answer 1

V shall be the energy of your system.

Answer 2

The $V$ function, also known as a Lyapunov function, can be considered as a function approximation to the true potential function of the system, which usually means energy. Or, to put it another way, $V$ doesn't represent the energy of a system, but a function that approximates it.

To determine a $V$ function, we consider Lyapunov function candidates which need to satisfy at least $V\ge0$ and $\dot V\le0$, and usually we hope to bound the actual potential function, $P_V$ say from above, i.e. $|P_V(x)|\le|V(x)|,\;\forall x\in D$. This allows for a reasonable determination of the behaviour of a system.

A Lyapunov function has Lyapunov surfaces given by $V_c=\{x|V(x)=c\}$ which show in a geometric sense the 'potential' of $V$ (chapter $3$, page $102$ of 'Nonlinear Systems - Hassan Khalil') as contours on a relief map.