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In the theory ofordinary differential equations (ODEs),Lyapunov functions, named afterAleksandr Lyapunov, are scalar functions that may be used to prove the stability of anequilibrium of an ODE. Lyapunov functions (also called Lyapunov’s second method for stability) are important tostability theory ofdynamical systems andcontrol theory. A similar concept appears in the theory of general state-spaceMarkov 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 particularlinear matrix inequality provides Lyapunov functions for linear systems, andconservation laws can often be used to construct Lyapunov functions forphysical systems.

A Lyapunov function for an autonomousdynamical system

with an equilibrium point at${\displaystyle y=0}$ is ascalar 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${\displaystyle -\mathrm{\nabla }V\cdot g}$ islocally positive definite, or${\displaystyle \mathrm{\nabla }V\cdot g}$ islocally negative definite.

Lyapunov functions arise in the study of equilibrium points of dynamical systems. In${\displaystyle {\mathbb{R}}^n,}$ an arbitrary autonomousdynamical 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 locallypositive-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:

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

Let${\displaystyle x^{\ast }=0}$ be an equilibrium point of the autonomous system

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

and use the notation${\displaystyle \dot{V}(x)}$ to denote the time derivative of the Lyapunov-candidate-function${\displaystyle V}$:

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

If the equilibrium point 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:

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

If${\displaystyle V}$ is a Lyapunov function, then the equilibrium isLyapunov stable.

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:

then the equilibrium is proven to beglobally asymptotically stable.

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

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

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

The converse is also true,^[1] and was proved byJosé Luis Massera (see alsoMassera's lemma).

Consider the following differential equation on${\displaystyle \mathbb{R}}$:

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

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

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

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

• 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 onPlanetMath, which is licensed under theCreative Commons Attribution/Share-Alike License.

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

• Stability theory

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