Lyapunov function

For certain classes of ODEs, the existence of Lyapunov functions is a necessary and sufficient condition for stability. Whereas there is no general technique for constructing Lyapunov functions for ODEs, in many specific cases the construction of Lyapunov functions is known. For instance, quadratic functions suffice for systems with one state, the solution of a particular linear matrix inequality provides Lyapunov functions for linear systems, and conservation laws can often be used to construct Lyapunov functions for physical systems.

Definition

A Lyapunov function for an autonomous dynamical system

${\displaystyle \{\begin{array}{ll}g:{\mathbb{R}}^n\to {\mathbb{R}}^n& \\ \dot{y}=g(y)\end{array}}$

with an equilibrium point at ${\displaystyle y=0}$ is a scalar function ${\displaystyle V:{\mathbb{R}}^n\to \mathbb{R}}$ that is continuous, has continuous first derivatives, is strictly positive for ${\displaystyle y\ne 0}$, and for which the time derivative ${\displaystyle \dot{V}=\mathrm{\nabla }V\cdot g}$ is non positive (these conditions are required on some region containing the origin). The (stronger) condition that ${\displaystyle -\mathrm{\nabla }V\cdot g}$ is strictly positive for ${\displaystyle y\ne 0}$ is sometimes stated as ${\displaystyle -\mathrm{\nabla }V\cdot g}$ is locally positive definite, or ${\displaystyle \mathrm{\nabla }V\cdot g}$ is locally negative definite.

Further discussion of the terms arising in the definition

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

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

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

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

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:

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