Hi,

Here is in a few words the spirit of the notion of Lyaponov's stability theory : If any movement of a system resulting from a sufficiently small neighborhood of a point of equilibrium Xe remains in the vicinity of this point, then Xe is said to be stable in the sense of Liapunov (strictly speaking, it is not a dynamic system which can be stable in the sense of Liapunov but a point of equilibrium of this system; some systems can have several points of equilibrium, some stable, others unstable). The central theorem of Alexander Liapunov says that an equilibrium point Xe is stable (in the sense of Liapunov) for a dynamic system (described by a differential equation of the type dot {x}=f(x,t) ) if and only s 'there is a function satisfying certain precise conditions and related to the function of the differential equation and to Xe. The problem of stability therefore boils down to looking for such a function (called Liapunov's function), often by trial and error.

For more details and information about this subject, i suggest you to see links and attached files on topic.

https://www.sciencedirect.com/topics/engineering/lyapunov-stability-theory

https://link.springer.com/content/pdf/10.1007%2F978-1-4757-3108-8_5.pdf

https://markcannon.github.io/assets/downloads/teaching/C21_Nonlinear_systems/nlc_notes.pdf

Article Control of nonlinear dynamic systems theory and applications

Best regards

It is usually bifold described as follows:

(1). The entropy of a nonlinear dynamical system related sensitively to initial conditions through the Lyapunov exponents;

(2). The instability dynamics of its spatial patterns considering the propagation of localized perturbations.

One of the explanations of the Lyapunov approach to prove the stability of any system is its relation with the energy of the system. This relates with positive definiteness of Lyapunov function. If any system loss or dissipates its energy it means the states of the system reach a final value or zero in asymtotic stability. Therefore the derivative of the Lyapunov equation must be smaller than zero for a stabiliy.

The general idea behind the Lyapunov stability theory Is quite simple. You take the system under consideration, take a small volume of the state space -- a tiny cube in 3D --, and observe its change in time. When the cube contact or stay the same, the system is stable. When it expands, it is unstable. When one dimension contract and the other expands, it is a saddle point.

From this idea, you can derive a lot of understanding of the system.

Let us take a similar example, a positive function. When its derivation decreases, the system goes to zero and the function decreases its value. When its derivation increases, the function increases its value.

For the system x'=f(x,t), the idea is to find an energy-like function, that we call the Lyapunov function V(x), which is zero at any equilibrium function and is positive when the system is not at an equilibrium point.

The important property is to compute the derivative of the Lyapunov function "along the trajectories of the system"

V'(x,t)=dV/dt=dV/dx dx/dt =dV/dx f(x,t).

If this derivative is negative, it implies that the Lyapunov function decreases "along the trajectories of the system."

Now, if the derivative would remain negative forever, the Lyapunov function would tend to reach minus infinite. Therefore, it must stop where the derivative is zero.

The equation V'(x,t)=0 thus tells us where the equilibrium points are.

The core idea, in my opinion, is the generalization of the powerful Hamiltonian and Lagrangian formulations of classical mechanics to systems where defining a function of physical energy is hard or even makes no sense (e.g. financial systems).

Given the ability to describe energy behavior with "non energy" functions the result is pretty simple to visualize: if an autonomous system loses energy over time then the system must eventually stop at some state, which is an attractor.

Don't forget Lypunov himself didn't actually apply his own result to engineering, it was Chetaev, about twenty years later to use Lyapunov's results to aviation, opening the doors of modern control theory.

To check the stability of a dynamic system, is only one role of Lyapunov stability theory.

One other meritorious role of Lyapunov function is to find a stabilizing control-law, particularly for nonlinear systems. This is known as control-Lyapunov function methodology (CLF), mainly due to Artstein and Sontag. For example, if you construct a CLF, V(x), with the proper requirements in the theory as V(x)=(x^2)/2, then

if the control system evolution ODE is:

xdot=f(x,u)=-x^3+u,

you can enforce the derivative of Lyapunov function to be negative in time:

Vdot=x*xdot=x*(-x^3+u)