Lyapunov stability

Stability

The equlibrium point is:

• Stable (in the sense of Lyapunov) if for each $\epsilon\geq0, \exists\delta(\epsilon)>0$ s.t: \begin{equation} \Vert x(0) \Vert < \delta \implies \Vert x(t) \Vert \leq \epsilon,\quad\forall t\geq 0 \end{equation}

• Unstable if it is not stable

• Asymptotically stable if it is stablem and $\exists \delta>0$ s.t. \begin{equation} \Vert x(0) \Vert < \delta \implies \lim_{t\to\infty} x(t) = 0 \end{equation}

• Exponentially stable if $\exists \delta, \alpha,\beta > 0$ s.t. \begin{equation} \Vert x(0) \Vert < \delta \implies \Vert x(t) \Vert < \beta e^{-\alpha t},\quad \forall t \geq 0 \end{equation}