Lyapunov function

Lyapunov functions arise in the study of equilibrium points of dynamical systems. In ${\displaystyle \mathbb {R} ^{n},}$ an arbitrary autonomous dynamical system can be written as

${\displaystyle {\dot {y}}=g(y)}$

for some smooth ${\displaystyle g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}.}$

An equilibrium point is a point ${\displaystyle y^{*}}$ such that ${\displaystyle g\left(y^{*}\right)=0.}$ Given an equilibrium point, ${\displaystyle y^{*},}$ there always exists a coordinate transformation ${\displaystyle x=y-y^{*},}$ such that:

${\displaystyle {\begin{cases}{\dot {x}}={\dot {y}}=g(y)=g\left(x+y^{*}\right)=f(x)\\f(0)=0\end{cases}}}$

Thus, in studying equilibrium points, it is sufficient to assume the equilibrium point occurs at ${\displaystyle 0}$.

By the chain rule, for any function, ${\displaystyle H:\mathbb {R} ^{n}\to \mathbb {R} ,}$ the time derivative of the function evaluated along a solution of the dynamical system is

${\displaystyle {\dot {H}}={\frac {d}{dt}}H(x(t))={\frac {\partial H}{\partial x}}\cdot {\frac {dx}{dt}}=\nabla H\cdot {\dot {x}}=\nabla H\cdot f(x).}$

A function ${\displaystyle H}$ is defined to be locally positive-definite function (in the sense of dynamical systems) if both ${\displaystyle H(0)=0}$ and there is a neighborhood of the origin, ${\displaystyle {\mathcal {B}}}$, such that:

${\displaystyle H(x)>0\quad \forall x\in {\mathcal {B}}\setminus \{0\}.}$

If the equilibrium is isolated, the Lyapunov-candidate-function ${\displaystyle V}$ is locally positive definite, and the time derivative of the Lyapunov-candidate-function is locally negative definite:

${\displaystyle {\dot {V}}(x)<0\quad \forall x\in {\mathcal {B}}\setminus \{0\}}$

for some neighborhood ${\displaystyle {\mathcal {B}}}$ of origin then the equilibrium is proven to be locally asymptotically stable.

If ${\displaystyle V}$ is a Lyapunov function, then the equilibrium is Lyapunov stable. The converse is also true, and was proved by José Luis Massera.

If the Lyapunov-candidate-function ${\displaystyle V}$ is globally positive definite, radially unbounded, the equilibrium isolated and the time derivative of the Lyapunov-candidate-function is globally negative definite:

${\displaystyle {\dot {V}}(x)<0\quad \forall x\in \mathbb {R} ^{n}\setminus \{0\},}$

then the equilibrium is proven to be globally asymptotically stable.

The Lyapunov-candidate function ${\displaystyle V(x)}$ is radially unbounded if

${\displaystyle \|x\|\to \infty \Rightarrow V(x)\to \infty .}$

(This is also referred to as norm-coercivity.)