An alternative method of demonstrating stability is given by Vasile Mihai POPOV, a great scientist of Romanian origin, who settled in the USA.

The theory of hyperstability (it has been renamed the theory of stability for positive systems) belongs exclusively to him ... (1965).

See Yakubovic-Kalman-Popov theorem, Popov-Belevitch-Hautus criterion, etc.

If the Liapunov (1892) method involves "guessing the optimal construction" of the Liapunov function to obtain a domain close to the maximum stability domain, Popov's stability criterion provides the maximum stability domain for nonlinearity parameters in the system (see Hurwitz , Aizerman hypothesis, etc.).

Hi Ali,

Depending on the system properties the are lots of theorem and lemma to prove the stability of a system such as LaSalle's theorem, Barbalat's Lemma, ...

I do suggest checking out the "Applied Nonlinear Control" by Slotine or/and " Nonlinear Systems" by Khalil.

Dear Armin Norouzi

All of these theories are based on the negative definiteness the derivative of a Lyapunov function so I believe they lie on the category of Lyapunov stability theory

Ali Namadchian This is somehow true, but some of the theorems have loosened definiteness of the derivative of the Lyapunov stability theorem. For example, what if the system is asymptotically stable but we only can prove that the derivative of a function in negative semi-definite? We have to use LaSalle's theorem. I think all of these theorems and lemmas come from the general definition of stability (bounded input-bounded output). But still, each of them has unique properties that make them useful to prove stability.

I have some publications about learning-based controller where I used a different method than usual to prove stability (in iteration domain), I suggest to check out that if you are interested.

Conference Paper Machine Learning-based Diesel Engine-Out NOx Reduction Using...

Article Integration of PD-type Iterative Learning Control with Adapt...

The properties of the spectrum can give the answer qualitative at least for example that some linear system is exponentially stabilizable.

In the case of finite dimensional linear continuous system, the system is stable if the spectrum of the matrix is strictly in the left half-plane. If we have discrete system then the spectrum of the matrix should be inside the unit open disc. In the case of the infinite dimensional systems the situation is a little different..Normally, it is assumed that the nonlinear system can be linearized. In infinite dimensional case, there is something that people call spectrum determined growth conditions ( see for example work of Greiner and Nagel, Tubingen, Germany, ¸1983 etc. ).

Lyapunov stability theory is not older than the classical stability theory. See, e.g., the basic stems from state of equilibrium, stationary state, saddle point, roots of the equation or equations, convergence, limit points which are very own concepts for analyzing behavioral approach to system dynamics.

Dear Ali Namadchian

We need to seek a Lyapunov function to test the stability of the equilibrium state, only when all real parts of the eigenvalues are negative, and some of them are zeros. On the other hand, you need to determine the solution of the system locally in the neighborhood of the equilibrium state using the Hartman- Grobman theorem, where you can linearize the system and find the solution explicitly. All that you have to do is to find the limit of the solution as t goes infinity. Best regards

Dear colleague Ali Namadchian ,

Depending on your system, you might find some methods available on the literature (the two brought by the colleague Armin Norouzi represents two classical ones to answer your question). For some specifics problems, you may find that "perturbation method" is a nice tool to evaluate linear stability. Although the method also represents an eigenvalue search method, it does in different way and bring specifics on important transition points of instabilities development. Try for example:

Feel free to contact me, in case you aid some further information.

Cheers,

Dear Ali Namadchian

Liapunov’s direct method has been widely used in establishing stability results for numerous varieties of differential equations. Yet, there is a large set of problems for which it has been ineffective. By means of various fixed point theorems (fixed point theory) a lot of this difficulties have been overcome.

For instance check :

Article Stability by fixed point theory or Liapunov's theory: A comparison

Best regards.

Dear Ali Namadchian:

Please consider the non-Lyapunov method " Necessary and Sufficient Condition for Asymptotic Stability of Nonlinear ODE’s ".

Thank you very much and best regards,

Andrés

In many cases, the stability of a dynamical system can be approached equivalently, as the stability of some operator between two normed function spaces. Miura, Miyajima and Takahasi introduced in

Hyers-Ulam stability of linear differential operator with constant coefficients, Math. Nachr. 258 (2003), 90-96

the notion of Hyers-Ulam stability of a mapping between normed spaces.

Myself, I was interested in the study of stability for some nonlinear equations or to obtain general stability results. Please find some papers below:

https://generalmathematics.ro/wp-content/uploads/2020/03/04_Teodorescu.pdf

or

1. Dinu Teodorescu, Hyers-Ulam Stability of a Semilinear Operator in a Hilbert Space, Journal of Nonlinear and Convex Analysis, 2016, Volume 17, Number 12, pp. 2529-2533, WOS:000393732300010

2. Mohammad Saeed Khan, Dinu Teodorescu, A note to establish the Hyers-Ulam stability for a nonlinear integral equation with Lipschitzian kernel, Tbilisi Mathematical Journal, Volume 11(3) 2018, pp. 41-45

I attach last two papers. These are not open acces.

Sorry! Now you can find the attached files:

LaSalle's invariance principle.

Matignon's stability criterion is useful for gauging the stability of integer or fractional order systems.

You can look at Optimal control mesh:

http://www.iaeng.org/publication/WCECS2012/WCECS2012_pp1110-1115.pdf

Hi Dear Ali,

Stabilty analysis of some dynamical systems may also be done by means of bifurcation diagrams. Neverthless it remains important to compute the largest Lyapunov exponent of the whole system to detect the onset of chaotic activities within the understudied system.

Dear Ali Namadchian,

You could try the frequency domain approach of stabilization theory.

Best regards,

Walid BOUGHAMDA

When you use an ODE , you fixed the behavior! But in general stability is not necessary to obey the ODE. For example Euler proposed an ODE fo study of linear elastic stability of structures. Since instability in most cases is not elastic and is not linear, the ODE is na.

Recently our research team detected and derived a new method called “change of state philosophy which is able to exactly determine the commence of instability state philosophy and Persian curves” without a need to use ODE or any other differential equation! Look for the title. It is interesting!

Dear all researchers,

There is an another linearization technique to test the stability of a system of ODE . I think it is a traditional approach for a nonlinear system. In my knowledge it would be better to use Lyapunov rather than linearization for global stability test.

You could use LaSalle's invariance principle.

But, I think, the Lyapunov method is the best, most powerful, and most flexible method.

Good Luck,

s, linearization and some invariance principle also good. but construction of lyapounv is most powerful, u may read some dynamical system books, that contains so many ideas

An alternative method of demonstrating stability is given by Vasile Mihai POPOV, a great scientist of Romanian origin, who settled in the USA.

The theory of hyperstability (it has been renamed the theory of stability for positive systems) belongs exclusively to him ... (1965).

See Yakubovic-Kalman-Popov theorem, Popov-Belevitch-Hautus criterion, etc.

If the Liapunov (1892) method involves "guessing the optimal construction" of the Liapunov function to obtain a domain close to the maximum stability domain, Popov's stability criterion provides the maximum stability domain for nonlinearity parameters in the system (see Hurwitz , Aizerman hypothesis, etc.).

There are Various approach of Stability analysis of a dynamical system which include Fixed point theorem which is pure functional approach. We can even use characteristic Equation by Linearization.

In general, there are several methods for analyzing the stability of dynamic system. Each of them is suitable for a specific application. for example:

-small gain theorem

-Input to state stability, and so on

You can use fixed point theorems for stability.

By Lyapunov stability, I'm assuming you mean Lyapunov's direct method where a Lyapunov function is used to identify stability.

Frankly, Lyapunov stability has stood the test of time by being arguably the most popular method of checking the stability of dynamical systems. However, Lyapunov stability can be insufficient for certain systems. If anything, addressing stability problems for dynamical systems where Lyapunov stability doesn't suffice rely on modifications or extensions of Lyapunov's direct method.

On example is the notion of invariant sets. Which can allow one to identify stability if a Lyapunov function cannot. Invariant set principles rely on the idea that Lyapunov functions are lower bounded meaning that a Lyapunov function, if decreasing, will eventually settle down to a minimum value. Correspondingly, trajectories will approach sets where the time derivative of a Lyapunov function no longer changes.

Another example is a technique to show asymptotic stability of nonautonomous (or time-varying) dynamical systems through Lyapunov-like functions. A Lyapunov-like function is one that has special restrictions on its properties. For instance, lower boundedness, uniformly continuous time derivative and a negative semi definite time derivative. This type of Lyapunov-like function can be paired with Barbalat's lemma which states that the time derivative of a bounded function with a uniformly continous time derivative will vanish over time. With a lower bounded Lyapunov-like function paired with Barbalat's lemma, one can still draw conclusions about asymptotic stability.

i totally agree with Dave Mandyam answer. thanks for you answer

Besides Liapunov method, global stsbility of a local stable equilibrium can be proved by excluding the existence of limit cycles by constructing proper Dulac function or by proving that if there exists a limit cycle surrounding the equilibrium thenit is stable.