Exponential stability

Remarks

An exponentially stable equilibrium is asymptotically stable. If $f$ is of class $C^2$, then $x^*$ is exponentially stable if and only if every eigenvalue of $Df(x^*)$ has negative real part.

Example: $x' = -x$ on $\R$ with $x^* = 0$. The solution is $x_0 e^{-t}$, so clause 2 holds with $C = 1$ and $\alpha = 1$, for any $\delta$.

Non-example: $x' = -x^3$ on $\R$ with $x^* = 0$. The solution is $x(t) = x_0 / \sqrt{1 + 2 x_0^2 t}$, since $\frac{d}{dt} x^{-2} = 2$. It tends to $0$, but $x(t) e^{\alpha t} \to \infty$ for each $\alpha > 0$ and $x_0 \ne 0$, so clause 2 fails.

For a time-dependent system $x' = f(t, x)$ with $f(t, 0) = 0$, the origin is uniformly exponentially stable if $\norm{x(t)} \le C e^{-\alpha (t - t_0)} \norm{x(t_0)}$ for all $t \ge t_0$, all initial times $t_0$ and all $\norm{x(t_0)} < \delta$.