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" (i.s.L.) if

<math>

\|x(t_o)\| \in \delta \quad => \quad \|x(t)\| \in \epsilon \quad \forall t \in R_+<math>

The system is said to be asymptotically stable if as

<math> t \rightarrow \infty, \quad \|x(t)\| \rightarrow 0 (EP) <math>

Contents

1 Lyapunov stability theorems 1.1 Lyapunov second theorem on stability 2 Stability for state space models 3 See also

Lyapunov stability theorems

Lyapunov stability theorems give only sufficient condition.

Lyapunov second theorem on stability

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

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

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

Stability for state space models

A state space model <math>\dot{\textbf{x}} = A\textbf{x}<math> is asymptotically stable iff

<math>A^{T}M + MA + N = 0<math>

has a solution where <math>N = N^{T} > 0<math> and <math>M = M^{T} > 0<math> (positive definite matrices).

See also

• Table of mathematical symbols