Simple and Effective Theory of Movement Steadiness

Smol’yakov Eduard Rimovich

doi:doi:10.11648/j.ijtam.20190506.15

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Simple and Effective Theory of Movement Steadiness

Smol’yakov Eduard Rimovich

Published in International Journal of Theoretical and Applied Mathematics (Volume 5, Issue 6)

Received: 10 November 2019 Accepted: 2 December 2019 Published: 11 December 2019

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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.

Published in	International Journal of Theoretical and Applied Mathematics (Volume 5, Issue 6)
DOI	10.11648/j.ijtam.20190506.15
Page(s)	113-117
Creative Commons	This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited.
Copyright	Copyright © The Author(s), 2024. Published by Science Publishing Group

Keywords

Nonlinear Dynamical Systems, Movement Steadiness, New Theory

References

[1]	Smol’ yakov E. R. Absolutely New, Simple and Effective Theory of Movement Steadiness//International Journal of Theoretical and Applied Mathematics, 2018, Vo. 4, no. 4, pp. 35-39.
[2]	Smol’yakov E. R. An Effective Method of Stability Analysis for Highly Nonlinear Dynamic Systems// Cybernetics and Systems Analysis, 2019, Vol. 55, no. 4, pp. 15-23.
[3]	Merkin D. A. Introduction into theory of movement stability. Moscow: Nauka, 1987.
[4]	Liapunov A. M. “Collected works”, Vol. 1-5, Moscow-Leningrad (1954–1965).
[5]	Routh E. J. The Advanced Part of a treatise on the Dynamics of a System of rigid bodies. London, 1884.
[6]	Ziegler H. Linear Elastic Stability. Critical Analysis of Methods// ZAMP. Basel – Zurich, IV, F–2. 1953.
[7]	Hagedorn P. Uber die instabilitat konservativer systeme mit girosropischen Kraften//RationalMech and Anal. 1975, Bd. 58, № 1.
[8]	Herrman G. Stability of Equilibrium of Elastic Systems Subjested to Nonconservative Fordes//Applied Mechanics Reviews. 1967. V.20. № 2.
[9]	Karapetyan A. V. Stability of Stationary movements. Moscow: URSS, 1998.
[10]	Chetaev N. G. Stability of movement. The works on the analytical mechanics. Moscow: RAN SSSR, 1962.
[11]	Krasovskii N. N. Some problems of the movement stability theory. Moscow: Fizmatgis, 1959.
[12]	Demidovich B. P. Lecture on mathematical theory of stability. Moscow: MGU, 1998.
[13]	La Salle J., Lefschetz S., Stability by Lyapunov's direct method with applications, Acad. Press (1961).
[14]	Lin C. C., The theory of hydrodynamic stability, Cambridge Univ. Press (1955).
[15]	Bliss G. A. Lecture on the variational calculation.. Moscow.: IL, 1950.

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Smol’yakov Eduard Rimovich. (2019). Simple and Effective Theory of Movement Steadiness. International Journal of Theoretical and Applied Mathematics, 5(6), 113-117. https://doi.org/10.11648/j.ijtam.20190506.15

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ACS Style

Smol’yakov Eduard Rimovich. Simple and Effective Theory of Movement Steadiness. Int. J. Theor. Appl. Math. 2019, 5(6), 113-117. doi: 10.11648/j.ijtam.20190506.15

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Smol’yakov Eduard Rimovich. Simple and Effective Theory of Movement Steadiness. Int J Theor Appl Math. 2019;5(6):113-117. doi: 10.11648/j.ijtam.20190506.15

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@article{10.11648/j.ijtam.20190506.15,
  author = {Smol’yakov Eduard Rimovich},
  title = {Simple and Effective Theory of Movement Steadiness},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {5},
  number = {6},
  pages = {113-117},
  doi = {10.11648/j.ijtam.20190506.15},
  url = {https://doi.org/10.11648/j.ijtam.20190506.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20190506.15},
  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.},
 year = {2019}
}

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TY  - JOUR
T1  - Simple and Effective Theory of Movement Steadiness
AU  - Smol’yakov Eduard Rimovich
Y1  - 2019/12/11
PY  - 2019
N1  - https://doi.org/10.11648/j.ijtam.20190506.15
DO  - 10.11648/j.ijtam.20190506.15
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 113
EP  - 117
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20190506.15
AB  - 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.
VL  - 5
IS  - 6
ER  -

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Author Information

Smol’yakov Eduard Rimovich

Department of Mathematics, Lomonosov Moscow State University, Moscow, Russia

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APA Style

Smol’yakov Eduard Rimovich. (2019). Simple and Effective Theory of Movement Steadiness. International Journal of Theoretical and Applied Mathematics, 5(6), 113-117. https://doi.org/10.11648/j.ijtam.20190506.15

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ACS Style

Smol’yakov Eduard Rimovich. Simple and Effective Theory of Movement Steadiness. Int. J. Theor. Appl. Math. 2019, 5(6), 113-117. doi: 10.11648/j.ijtam.20190506.15

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AMA Style

Smol’yakov Eduard Rimovich. Simple and Effective Theory of Movement Steadiness. Int J Theor Appl Math. 2019;5(6):113-117. doi: 10.11648/j.ijtam.20190506.15

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@article{10.11648/j.ijtam.20190506.15,
  author = {Smol’yakov Eduard Rimovich},
  title = {Simple and Effective Theory of Movement Steadiness},
  journal = {International Journal of Theoretical and Applied Mathematics},
  volume = {5},
  number = {6},
  pages = {113-117},
  doi = {10.11648/j.ijtam.20190506.15},
  url = {https://doi.org/10.11648/j.ijtam.20190506.15},
  eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.ijtam.20190506.15},
  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.},
 year = {2019}
}

Copy | Download

TY  - JOUR
T1  - Simple and Effective Theory of Movement Steadiness
AU  - Smol’yakov Eduard Rimovich
Y1  - 2019/12/11
PY  - 2019
N1  - https://doi.org/10.11648/j.ijtam.20190506.15
DO  - 10.11648/j.ijtam.20190506.15
T2  - International Journal of Theoretical and Applied Mathematics
JF  - International Journal of Theoretical and Applied Mathematics
JO  - International Journal of Theoretical and Applied Mathematics
SP  - 113
EP  - 117
PB  - Science Publishing Group
SN  - 2575-5080
UR  - https://doi.org/10.11648/j.ijtam.20190506.15
AB  - 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.
VL  - 5
IS  - 6
ER  -

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