Lyapunov function

In the theory of ordinary differential equations (ODEs), Lyapunov functions are scalar functions that may be used to prove the stability of an equilibrium of an ODE. Named after the Russian mathematician Aleksandr Mikhailovich Lyapunov, Lyapunov functions (also called the Lyapunov’s second method for stability) are important to stability theory of dynamical systems and control theory. A similar concept appears in the theory of general state space Markov chains, usually under the name Foster–Lyapunov functions.

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 of a Lyapunov function

A Lyapunov function for an autonomous dynamical system

$g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$

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

with an equilibrium point at $y=0$ is a scalar function ${\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} }$ that is continuous, has continuous derivatives, is locally positive-definite, and for which ${\displaystyle -\nabla {V}\cdot g}$ is also locally positive definite. The condition that ${\displaystyle -\nabla {V}\cdot g}$ is locally positive definite is sometimes stated as ${\displaystyle \nabla {V}\cdot g}$ is locally negative definite.

A Lyapunov candidate-function for an autonomous dynamical system

$g:\mathbb {R} ^{n}\to \mathbb {R} ^{n}$

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

with an equilibrium point at $y=0$ is a scalar function ${\displaystyle V:\mathbb {R} ^{n}\to \mathbb {R} }$ that is continuous, has continuous derivatives, and is locally positive-definite. Thus, a Lyupanov function is a Lyupanov candidate-function for which ${\displaystyle -\nabla {V}\cdot g}$ is locally positive 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

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

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

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

${\dot {x}}={\dot {y}}=g(y)=g(x+y^{*})=f(x)\,$

${\displaystyle f(0)=0.}$

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

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

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

A function $H$ is defined to be locally positive-definite function if

${\displaystyle H(0)=0\,}$

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

Basic Lyapunov theorems for autonomous systems

Let

$x^{*}=0\,$

be an equilibrium of the autonomous system

${\dot {x}}=f(x).\,$

and use the notation ${\displaystyle {\dot {V}}(x)}$ to denote

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

which is the time derivative of the Lyapunov-candidate-function $V$.

Locally asymptotically stable equilibrium

If $V$ is a Lyapunov function, then the equilibrium is locally asymptotically stable.

The converse is also true, and was proved by J. L. Massera.

Stable equilibrium

If the Lyapunov-candidate-function $V$ is locally positive definite and the time derivative of the Lyapunov-candidate-function is locally negative semidefinite:

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

for some neighborhood ${\mathcal {B}}$ of ${\displaystyle 0}$, then the equilibrium is proven to be stable.

Globally asymptotically stable equilibrium

If the Lyapunov-candidate-function $V$ is globally positive definite, radially unbounded and the time derivative of the Lyapunov-candidate-function is globally negative definite:

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

then the equilibrium is proven to be globally asymptotically stable.

The Lyapunov-candidate function $V(x)$ is radially unbounded if

$\|x\|\to \infty \Rightarrow V(x)\to \infty .$

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

Example

Consider the following differential equation with solution $x$ on $\mathbb {R}$:

${\dot {x}}=-x.$

Considering that $x^{2}$ is always positive around the origin it is a natural candidate to be a Lyapunov function to help us study $x$. So let ${\displaystyle V(x)=x^{2}}$ on $\mathbb {R}$. Then,

${\displaystyle {\dot {V}}(x)=V'(x)f(x)=2x\cdot (-x)=-2x^{2}<0.}$

This correctly shows that the above differential equation, x, is asymptotically stable about the origin. Note that if using the same Lyapunov candidate one can show that the equilibrium is also globally asymptotically stable.

See also

• Lyapunov stability
• Ordinary differential equations
• Control-Lyapunov function
• Foster's theorem
• Lyapunov optimization

References

• Weisstein, Eric W. "Lyapunov Function". MathWorld.

• Khalil, H.K. (1996). Nonlinear systems. Prentice Hall Upper Saddle River, NJ.
• La Salle, Joseph; Lefschetz, Solomon (1961). Stability by Liapunov's Direct Method: With Applications. New York: Academic Press.
• This article incorporates material from Lyapunov function on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.

External links

• Example of determining the stability of the equilibrium solution of a system of ODEs with a Lyapunov function
• Some Lyapunov diagrams