Abstract
It is proposed the very simple and quick method for estimation of the asymptotic stability of any nonlinear dynamic systems, in particular, of the high-dimensional systems for which Tailor series of the right-hand sides of the differential equations converge very slowly. In such problems, the sum of terms of the order of smallness higher than two can substantially exceed the value of any term of second order. In this case, Lyapunov’s methods cannot guarantee correct stability estimate at all. The new method does not use the notion of Liapunov function and, therefore, one has no numerous shortcomings of all Liapunov methods. In this paper, it is proposed to replace the very complex problem of the searching for Liapunov function with a very simple problem of the searching maximum of the function of n coordinates (that is of the velocity of variation in metrics of the perturbed state space). However, one is not intended for the linear systems.
Highlights
IntroductionIt is proposed such broadening of the Smol’yakov method [1, 2] which permits quickly to solve the question about existence or absence of the asymptotical stability even in very complex cases when the solution could not be found
It is proposed such broadening of the Smol’yakov method [1, 2] which permits quickly to solve the question about existence or absence of the asymptotical stability even in very complex cases when the solution could not be found.The working out of the problem of the movement steadiness began in the end of 19 century in several works of E
Among shortcomings of the classic theory [3,4,5,6,7,8,9,10,11,12,13,14], it is necessary to notice, that, firstly, the searching for the desirable function V (x) is very complex, secondly, it is required to expand the perturbed equations in rows, thirdly, method of Liapunov functions is not correct because the different Liapunov functions define absolutely different asymptotical stability sets in any concrete problem
Summary
It is proposed such broadening of the Smol’yakov method [1, 2] which permits quickly to solve the question about existence or absence of the asymptotical stability even in very complex cases when the solution could not be found. Among shortcomings of the classic theory [3,4,5,6,7,8,9,10,11,12,13,14], it is necessary to notice, that, firstly, the searching for the desirable function V (x) is very complex, secondly, it is required to expand the perturbed equations in rows, thirdly, method of Liapunov functions is not correct because the different Liapunov functions define absolutely different asymptotical stability sets in any concrete problem (see example 1 in [1]). The proposed new method is based on the necessary conditions and one allows always to find the stable motion when it exists
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