Uniformly stable equilibrium
Graduate (time-varying systems)
Consider a time-varying system $\dot{x} = f(t, x)$ with an equilibrium at $0$, and its solutions $x(t)$ starting at $x(t_0)$ at the initial time $t_0$. The equilibrium is uniformly stable if, for every $\varepsilon > 0$, there is a $\delta > 0$, independent of $t_0$, such that
\[\norm{x(t_0)} < \delta \implies \norm{x(t)} < \varepsilon \quad \text{for all } t \geq t_0 \text{ and every } t_0 \geq 0.\]Caveat. For a Lyapunov stable equilibrium of a time-varying system, $\delta$ may depend on $t_0$.
Properties
- For a time-invariant system, every stable equilibrium is uniformly stable, because shifting the initial time does not change the solutions.