Lyapunov stability via differential moments

Author:
R. J. Charron

Journal:
Quart. Appl. Math. **49** (1991), 447-452

MSC:
Primary 34D20

DOI:
https://doi.org/10.1090/qam/1121677

MathSciNet review:
MR1121677

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Abstract | References | Similar Articles | Additional Information

Abstract: We show how differential moments can be used to construct Lyapunov functions for general, autonomous and nonautonomous, second- and third-degree ordinary differential equations. In certain instances, one can extend classical results to sequences of Lyapunov functions.

**[1]**E. A. Barbashin,*On the stability of solution of a third order nonlinear differential equation*, Prik. Math. Mekh.**16**, 629-632 (1952)**[2]**J. O. C. Ezeilo,*On the stability of solutions of certain differential equations of the third order*, Quart. J. Math.**11**, 64-69 (1960) MR**0117394****[3]**G. K. Kulev and D. D. Bainov,*On the asymptotic stability of systems with impulses by the direct method of Lyapunov*, J. Math. Anal. Appl.**140**, 324-340 (1989) MR**1001859****[4]**H. Leipholz,*Stability Theory*, Academic Press, New York, 1970 MR**0359445****[5]**A. M. Lyapunov,*Problème générale de la stabilité du mouvement*, Annals of Math. Studies, Vol. 17, Princeton University Press, Princeton, New Jersey, 1949**[6]**G. J. Michael,*Explicit stability criteria for the damped Mathieu equation*, IEEE Trans. Automat. Control**AC-12**, 337-338 (1967)**[7]**K. S. Narendra and J. H. Taylor,*Stability of the damped Mathieu equation*, IEEE Trans. Automat. Control**AC-13**, 726 (1968)**[8]**K. S. Narendra and R. M. Goldwyn,*Stability of certain nonlinear differential equations*, IEEE Trans. Automat. Control**AC-8**, 381-382 (1963)**[9]**P. C. Parks,*Comments on 'Explicit stability criteria for the damped Mathieu equation*', IEEE Trans. Automat. Control**AC-13**, 129 (1968)**[10]**K. P. Persidski,*On the stability of motion in first approximation*, Mat. Sb.**40**, 284-293 (1933)**[11]**P. J. Ponzo,*On the Stability of certain nonlinear differential equations*, IEEE Trans. Automat. Control**AC-10**, 470-472 (1965) MR**0188554****[12]**P. J. Ponzo,*Some stability conditions for linear differential equations*, IEEE Trans. Automat. Control**AC-13**, 721-722 (1968) MR**0276568****[13]**S. Ramarajan and S. N. Rao,*An improved stability criteria for the damped Mathieu equation*, IEEE Trans. Automat. Control**AC-16**, 363-364 (1971)**[14]**N. Rouche, P. Habets, and M. Laloy,*Stability Theory by Lyapunov's Direct Method*, Applied Mathematical Sciences, Vol. 22, Springer-Verlag, New York, 1977 MR**0450715****[15]**S. N. Simanov,*On the stability of solution of a nonlinear equation of the third order*, Akad. Nauk. SSSR, Prikl. Mat. Mekh.**17**, 369-372 (1953) (in Russian) MR**0055523**

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Additional Information

DOI:
https://doi.org/10.1090/qam/1121677

Article copyright:
© Copyright 1991
American Mathematical Society