All methods of stability studies developed by A. M. Lyapunov in [1] }} are divided by him into two methods (two categories). The first method includes all methods for studying stability, “which lead to the direct study of perturbed motion, and which are based on the search for general or particular solutions of differential equations. In general, these solutions will have to be sought under the guise of endless rows. . . This is the essence of the series, located in integer positive degrees of constant arbitrary. But further we will also meet with some rows of a different nature ” [1] . Sometimes the linearization method is also called the first Lyapunov method. However, this is not so: theorems on asymptotic stability and instability in a first approximation can be proved using methods of investigation of both the first and second Lyapunov methods. The second method by A. M. Lyapunov includes all methods for studying stability, which are based on the search for functions of the variables u, t “according to some given conditions that must be satisfied by their total derivatives with respect to t, made under the assumption that” u = u (t ) is a function satisfying the equation ẋ = F (x, t). (1) The second Lyapunov method is often called the direct method. It should be noted that methods for studying stability related to both the first and second methods before Lyapunov were used in particular cases of A. Poincare in [2] . As A. M. Lyapunov himself noted in his dissertation [1] : “Although Poincare is limited to very special cases, the methods that he uses allow much more general applications and can lead to many new results. "The ideas contained in the memoir [2] , I was guided by most of my research." The first method of A. M. Lyapunov allowed him to obtain a number of very deep and important results. As an example, we note the theory of conditional stability developed by him in his work on the basis of the first method [1] . One of the advantages of this method is that it works in the most delicate cases and allows not only to indicate a qualitative picture of the phenomenon being studied, but also to construct an explicit form of the solutions being studied. Lyapunov lays the foundation of his second method for several basic theorems established by him. These theorems turned out to be so effective that with their help it was possible to extremely simply solve the stability problem in a first approximation. At the same time, they allowed Lyapunov to consider some basic critical cases when the first approximation does not solve the stability problem. Currently, of the two methods, the direct Lyapunov method is most widely used due to its simplicity and effectiveness.
The direct (second) Lyapunov method of stability theorems
We give here theorems on the stability of the zero solution of a perturbed system in the space R n in the particular case when it is autonomous, that is, it has the form:
ư = f (u). (2) It is assumed that f (0) = 0, so that u 0 (t) = 0 is a solution to this equation. We arrive at this problem by studying the stability of the equilibrium of an autonomous system ẋ = F (x). (3)
For any continuously differentiable function V (u) defined in some neighborhood D of the point 0 ꞓ R n , we define V - the derivative of the function V (u) by virtue of differential equation (2), setting . (four)
If u (t) is any solution of equation (2), then the formula . (5) which confirms the appropriateness of definition (4).
- Definition 1.
A function V (u) is called sign-positive in a domain D if V (0) = 0, V (u) ≥ 0 for all and from the domain of its definition D (D is some neighborhood of zero in R n ).
- Definition 2.
A function V (u) is called definitely positive (or positive definite) if it is sign-positive u, moreover, V (u)> 0 for any u ꞓ D other than 0.
Similarly sign-negative and definitely negative functions are defined.
- Definition 3.
A function V is called sign-constant if it is sign-positive or sign-negative.
- Definition 4.
A function V is called sign-definite if it is positive definite or negative definite.
- Definition 5.
If the function V takes in the domain D values of both a positive sign and a negative one, then in this case V is called an alternating function.
In Theorems 1–4 below, it is assumed that V (u) is a continuously differentiable function defined in some neighborhood D of the point 0 ꞓ R n ; we use the notation V '(u), the derivative of the function V (u) by virtue of differential equation (2).
- Theorem 1 (Lyapunov stability theorem).
If there exists a definitely positive function V (u) whose derivative V '(u) is negative, then u 0 (t) is a stable solution to equation (2).
- Theorem 2 (Lyapunov's asymptotic stability theorem).
Let there exist a definitely positive function V (u) whose derivative V '(u) is definitely a negative function. Then u 0 (t) is an asymptotically stable solution to equation (2).
- Theorem 3 (Barbashin-Krasovsky theorem on asymptotic stability [3] [4] ).
If there exists a positive definite function V (u) such that V '(u) <0 outside M and V' (u) ≤ 0 on M, where M is a set that does not contain the entire trajectories of equation (2) except for the point zero, then the zero solution u 0 (t) of equation (2) is asymptotically stable.
- Definition 6.
A function V (u) is called infinitely large if, for any positive number K> 0, there exists R> 0 such that from | u | > R it follows that | V (u) | > K.
Theorem 4 (on asymptotic stability as a whole [3] ). If there exists a definitely positive infinitely large function V (u) whose derivative V '(u) is definitely a negative function in the whole space, then the zero solution u 0 (t) of equation (2) is asymptotically stable as a whole. Functions satisfying Theorems 1-2 of the direct Lyapunov method are called Lyapunov functions. The existence of an appropriate Lyapunov function is a sufficient condition for the stability or asymptotic stability of a solution.
Notes
- ↑ 1 2 3 4 Lyapunov A. M. The general problem of the stability of motion. - M .: Gostekhizdat, 1950.
- ↑ 1 2 Poincare A. On curves defined by differential equations. M .: Gostekhizdat, 1947
- ↑ 1 2 Barbashin E. A. Introduction to the theory of stability. - M., 1967.
- ↑ Krasovsky N. N. Some problems of the theory of stability of motion. - M .: Gostekhizdat, 1959.
Literature
- L.G. Kurakin, I.V. Ostrovskaya ELEMENTS OF THE THEORY OF SUSTAINABILITY - Rostov-on-Don: Southern Federal University, 2016. - 60 p.