To see the invitation, please click the link https://www.researchgate.net/post/This_is_the_invitation_to_the_discussion_of_a_general_method_for_constructing_Lyapunov_functions_presented_in_http_arxivorg_abs_14035761
The questions can be divided into three groups. The first group consists of the questions, to which the author has the answers directly resulting from the paper. The second group is composed of the questions, for which he has only conjectures or guesstimates. The third group represents the questions, the answers to which the author has no ideas about. The questions that interest me particularly as the author are as follows:
1. What is the mathematical nature (algebraic, geometrical, topological, etc.) of Lyapunov functions? What physical interpretations can be given to them?
2. Are there any direct or indirect relations between Lyapunov functions, first integrals and the right-hand sides of systems of differential equations? If yes, then what kinds they are?
3. How to approach a nonlinear non-autonomous system of the most general form by means of the second Lyapunov method? What and why do we need in the very beginning to know and how to get on with the system from this initial point further using the general procedure of utilization of Lyapunov functions? Is the procedure workable enough to crack the concrete practical problems of stability despite the presence of general nonlinearity, non-autonomousness, structural and coefficient uncertainties?
4. What are advantages and disadvantages of the utilization of Lyapunov functions at the investigation of the stability of nonlinear non-autonomous systems in the light of the results of the paper?
5. What role if any does the Lyapunov concept of stability play for quantum-mechanical and biochemical physical processes? Can it be considered one of the fundamental principles of the creation, formation and existence of living and nonliving matter?
Lyapunov function is positive definite therefore if its derivative is negative definite then the problem is stable
Dear Adil AL-Rammahi, Sorry for the belated response. That's right. But the key problem of the Lyapunov second method is in the lack of a general method of construction of Lyapunov functions. In my paper I've proposed certain solution to this very old problem. I've found the direct relation between the first integrals governing a given dynamical systems of general form and Lyapunov functions. It has deep geometric-topological nature. This is definitely a big breakthrough in Stability Theory. But this is not the end because usually the given dynamical system is described by ODEs and we are unable to find the first integrals using them. We can always solve the inverse problem, namely to find the ODEs using the first integrals. The second step in this research should be building the bridge between my result and solutions to the linear PDEs associated with systems of ODEs. Most of my previous questions imply it.
Of course there is a way of constructing Lyapunov functions! Here is one way to do it:
Take x(t;x0) as the solution of dx/dt = f(x), x(0) = x0. Pick a small neighbourhood U of x* (the globally stable equilibrium) and set V(x0) = inf { t>0 | x(t;x0) in U }.
Did you want to construct a Lyapunov function in order to prove x* is globally stable, or find an analytical formula for V? Well then you're out of luck.
David, I am awfully sorry for the very belated reply. Only today I've found your comment. First of all, the main objects of my research are NOT equilibrium or singular points. They have been studied quite well. But what badly lacks the attention of researchers are non-singular orbits describing various transient and unsteady processes, for example, hypersonic airplane maneuvering in dense atmospheric layers. In geometric terms its flight trajectory is an arbitrary curve. Second, the goal of the research is NOT to invent a general method of constructing analytical formulae for V. The prime objective is to show what is the intrinsic mathematical nature of Lyapunov functions in the structure of a given system of ODE and how they are related to its first integrals, what are their weak points and what advantages give us Poincare's approach over Lyapunov functions in stability problems. As to "global", the concept of foliations is used in the paper. This means that by the definition of an one-codimensional foliation, which leaves are the elementary “bricks” of phase spaces, that is by default we tackle the problems of stability globally. If you have any further questions, please don't hesitate to ask. Once again, please accept my apologies for the very belated reply.
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