Stability

Definition

The term stability as used in this entry is a property of an equilibrium of a dynamic system. There are various definitions of stability which describe how a dynamic system being at an equilibrium reacts on a small disturbance (Plaschko and Brod 1995). In machining science, the Lyapunov stability and the asymptotic stability are of high interest. Consider the continuous dynamic system:

$$ \dot{x}=f\left(x(t)\right),\kern0.5em x(0)={x}_0. $$

(1)

We suppose that the system has an equilibrium at

$$ x(t)={x}_e. $$

(2)

Lyapunov Stability

The above equilibrium is called Lyapunov stable, if

$$ \forall \epsilon >0\kern0.5em \exists \delta =\delta \left(\epsilon \right)>0 $$

(3)

such that

$$ \left|{x}_0-{x}_e\right|<\delta \Rightarrow \forall t\ge 0\;\left|x(t)-{x}_e\right|<\epsilon . $$

(4)

This means that solutions starting within a range of δ from the equilibrium will forever stay within a range of ϵ to the equilibrium.

Asymptotic Stability

An equilibrium is called asymptotically...