Lyapunov stability

Lyapunov stability is applicable to only unforced (no control input) dynamical systems. It is used to study the behaviour of dynamical systems under initial perturbations around equilibrium points.

Let us consider that the origin is the equilibrium point (EP) of the system and that two spheres of radius ε and δ surround the origin such that δ < ε. A system is said to be stable in the sense of Lyapunov if

$\|x(t_{o})\|\in \delta \quad =>\quad \|x(t)\|\in \epsilon \quad \forall t\in R_{+}$

The system is said to be asymptotically stable if as

$t\rightarrow \infty ,\quad \|x(t)\|\rightarrow 0(EP)$

Lyapunov stability theorems

Lyapunov stability theorems give only sufficient condition.

Lyapunov second theorem on stability

Consider a function V(x) : Rⁿ → R such that

• $V(x)>0:\forall {x}\neq 0$ (positive definite)
• ${\dot {V}}(x)<0$ (negative definite)

Then V(x) is called a Lyapunov function candidate and the system is asymptotically stable in the sense of Lyapunov.