Lyapunov’s theorem

Suppose that we can find a continuously differentiable, real-valued function $L(\boldsymbol{x})$ such that $L(\boldsymbol{x})>L(\bar{\boldsymbol{x}})$ for all $\boldsymbol{x}$ in some neighborhood $U$ of $\bar{\boldsymbol{x}}$.
If $\dot{L}(\boldsymbol{x})\leq 0$ for all $\boldsymbol{x}\in U$, then $\bar{\boldsymbol{x}}$ is stable.
If $\dot{L}(\boldsymbol{x})< 0$ for all $\boldsymbol{x}\in U\backslash\{\bar{\boldsymbol{x}}\}$, then $\bar{\boldsymbol{x}}$ is asymptotically stable.
If $\dot{L}(\boldsymbol{x})> 0\thinspace \text{for all}\thinspace \boldsymbol{x}\in U\backslash\{\bar{\boldsymbol{x}}\}$, then $\bar{\boldsymbol{x}}$ is unstable.