Lyapunov Exponents Are Continuous in a Smooth Family
Lyapunov functions, titled after Aleksandr Lyapunov, are scalar functions that can be used to verify the stability of equilibrium of an ordinary differential equation in the concept of ordinary differential equations (ODEs). Lyapunov functions (also known as Lyapunov's second method for stability) are crucial in dynamical system stability and control theory. A concept comparable to Foster–Lyapunov functions is found in the theory of general state-space Markov chains. Let's take a look at the definition, applications, and examples of Lyapunov functions in this article.
Table of Contents:
- Definition
- Lyapunov Stability Theorems
- Lyapunov Instability Theorem
- Advantages and Disadvantages
- Example
- FAQs
What are Lyapunov Functions?
A Lyapunov function is a scalar function established on phase space that can be used to show an equilibrium point's stability.
Suppose V(X) be a continuously differentiable function in the origin's neighbourhood U. If the following requirements are satisfied, the function V(X) is known as the Lyapunov function for an autonomous system X' = f(x).
- V(X) >0 for all X ∈ U\ {0}
- (dV/dt) ≤ 0 for all X ∈ U
- V(0) = 0
Let's look at an example to better grasp the Lyapunov function.
To examine the stability of various differential equations and systems, the Lyapunov function method is used. We will limit ourselves to autonomous systems in the next sections.
X' = f(x) or
(dxi)/dt = fi (x1, x2, …xn)
Here, i = 1, 2, …n
With x≡0, which is the zero equilibrium
Assume we're given a continuously differentiable function V(X) = V(x1, x2, …, xn) in the origin's neighbourhood U. Letting V(X) be the value of all X ∈ U\ {0} and V(0) in the origin. These are functions of the kind, for example,
V(x1, x2) = ax1 2+bx2 2, V(x1, x2) = ax1 2+bx2 4, a, b>0
The total derivative of the function V(X) with respect to time t is found as follows:
\(\begin{array}{l}\frac{dV}{dt}= \frac{\partial V}{\partial x_{1}}\frac{dx_{1}}{dt}+ \frac{\partial V}{\partial x_{2}}\frac{dx_{2}}{dt}+…+\frac{\partial V}{\partial x_{n}}\frac{dx_{n}}{dt}\end{array} \)
Like a scalar (dot) product of two vectors, this expression can be represented as:
\(\begin{array}{l}\frac{dV}{dt}=\left ( grad\ V, \frac{dX}{dt} \right )\end{array} \)
Where,
\(\begin{array}{l}grad\ V = \left ( \frac{\partial V}{\partial x_{1}}, \frac{\partial V}{\partial x_{2}}, …, \frac{\partial V}{\partial x_{n}} \right )\end{array} \)
\(\begin{array}{l}\frac{dX}{dt}=\left ( \frac{dx_{1}}{dt}, \frac{dx_{2}}{dt}, …\frac{dx_{n}}{dt} \right )\end{array} \)
The first vector is the gradient of V(X), which means it's always pointing in the direction of the largest rise in V(X). V(x) is a function that typically increases with distance from the origin, i.e. given |X|→∞. The velocity vector is the second vector in the scalar product. It is tangent to the phase trajectory at all times.
Consider the scenario where the derivative of V(X) in the origin's neighbourhood U is negative:
\(\begin{array}{l}\frac{dV}{dt}= \left ( grad\ V, \frac{dX}{dt} \right ) <0\end{array} \)
This indicates that the angle φ between the gradient and velocity vectors is larger than 90 degrees.
In the following diagrams, a function with two variables is depicted schematically:
If a derivative dV/dt along a phase trajectory is always negative, the trajectory will tend to the origin, indicating that the system is stable. Whenever the derivative dV/dt is positive, the system is unstable because the trajectory goes away from the origin.
Lyapunov Stability Theorems
The Lyapunov Stability Theorems are as follows:
Stability Theorem in the Lyapunov Sense
If a Lyapunov function V(X) exists in the neighbourhood U of an autonomous system's zero solution X = 0, the system's equilibrium point X = 0 is Lyapunov stable.
Asymptotic Stability Theorem
If a Lyapunov function V(X) with a negative definite derivative (dV/dt) < 0 for all X ∈ U\ {0} exists in the neighbourhood U of an autonomous system's zero solution X = 0, then the system's equilibrium point X = 0 is asymptotically stable.
As can be observed, for the asymptotic stability of the zero solution, the total derivative dV/dt should be strictly negative in the neighbourhood of the origin.
Lyapunov Instability Theorem
Assume that a continuously differentiable function V(x) exists in the neighbourhood U of the zero solution X =0, with
- V(0) = 0
- dV/dt > 0
If there are points in the neighbourhood U where V(X) > 0, the zero solution X =0 will be unstable.
Advantages and Disadvantages of Lyapunov Functions
Advantage:
Lyapunov functions can be used to identify whether a system is stable or unstable. This method has the advantage of not requiring us to know the actual solution x(t). Furthermore, this method can be used to investigate the stability of equilibrium points in non-rough systems. For instance, suppose the equilibrium point is a centre.
Disadvantage:
There is no generic method for building Lyapunov functions, which is a drawback. The Lyapunov function can be calculated as a quadratic form in the situation of homogeneous autonomous systems with constant coefficients.
Solved Example on Lyapunov Functions
Example:
Examine the stability of the system's zero solution: dx/dt = -2x, dy/dt = x-y
Solution:
This system has constant coefficients and is a linear homogeneous system. The quadratic form is used as a Lyapunov function.
V(x) = V(x, y) = ax2 + by2
where a and b are the coefficients, which have to be determined
Except at the origin, where it is zero, the function V(x, y) is obviously positive everywhere. Here, the total derivative of the function V(x, y) is calculated.
Hence,
dV/dt = (∂V/∂x)(dx/dt) + (∂V/∂y)(dy/dt)
dV/dt = 2ax (-2x) + 2by(x-y)
dV/dt = -4ax2 +2bxy – 2by2
dV/dt = -2b [(4a/2b)x2 – xy + y2]
dV/dt = -2b [(2a/b)x2 – xy + y2]
If the preceding condition is met, the expression in brackets can be transformed to a square of the difference:
2a/b = ¼
It can also be written as;
8a = b
We can use any acceptable combination, such as a = 1 and b = 8. The derivative then becomes
dV/dt = -16 [(x2/4) – xy + y2]
dV/dt = -16[(x/2) – y]2 < 0.
Thus, a Lyapunov function exists for the given system, with a negative derivative everywhere except at the origin. As a result, the system's zero solution is asymptotically stable.
Frequently Asked Questions on Lyapunov Functions
What are Lyapunov functions?
Lyapunov functions are scalar functions that can be used to demonstrate the stability of an Ordinary Differential Equation's equilibrium.
Give the application of the Lyapunov function.
The stability features of equilibrium points of linear and nonlinear systems can be examined using Lyapunov functions.
What are the advantages of Lyapunov functions?
A system's stability or instability can be determined using Lyapunov functions. This method has the advantage of not requiring us to know the exact solution. This method can also be used to investigate the stability of non-rough equilibrium points.
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