In order to state the main theorem, we need to introduce the notion of positive definiteness for a function.
Definition 1.1 Let
A scalar function
It is positive definite on
Similarly, it is negative semidefinite (negative definite) on
Summary: This module describes the direct method of Lyapunov for determining the stability of the equilibrium to an Ordinary Differetial Equation.
Read more about ordinary differential equations, initial value problem and stability of equilibrium.
Determining if an equilibrium of a differential equation is stable or unstable is of great interest in many practical applications. In his memoir [1], Lyapunov developed a method to determine the stability of an equilibrium without having to solve the differential equation nor having to directly apply the definitions of stability. This method, known as Lyapunov's direct method, is related to the notion of energy of the dynamical system.
Let
The function
In order to state the main theorem, we need to introduce the notion of positive definiteness for a function.
Definition 1.1 Let
A scalar function
It is positive definite on
Similarly, it is negative semidefinite (negative definite) on
We can now introduce the theorem of Lyapunov which gives sufficient conditions for an equilibrium to be stable and asymptotically stable.
Theorem 2.1
If there exists an open set
If, in addition,
Intuitively, the Lyapunov function
The chain rule is an important tool for the study of differential equations. As an example, notice that
As a consequence, the direct method does not require to solve for a solution to the differential equation Equation 1 to determine the stability of the equilibrium.
Notice that the theorem is stated in the case where
where we set
Consider the following differential equation, for some
Consider the following candidate Lyapunov function:
It is clear that
As a consequence, if