Lyapunov stability via differential moments

Charron, R. J.

doi:10.1090/qam/1121677

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: 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.

E. A. Barbashin, On the stability of solution of a third order nonlinear differential equation, Prik. Math. Mekh. 16, 629–632 (1952)

J. O. C. Ezeilo, On the stability of solutions of certain differential equations of the third order, Quart. J. Math. Oxford Ser. (2) 11 (1960), 64–69. MR 117394, DOI https://doi.org/10.1093/qmath/11.1.64
G. K. Kulev and D. D. Baĭnov, On the asymptotic stability of systems with impulses by the direct method of Lyapunov, J. Math. Anal. Appl. 140 (1989), no. 2, 324–340. MR 1001859, DOI https://doi.org/10.1016/0022-247X%2889%2990067-X
Horst Leipholz, Stability theory. An introduction to the stability of dynamic systems and rigid bodies, Academic Press, New York-London, 1970. Translated from the German by Scientific Translation Service. MR 0359445

A. M. Lyapunov, Problème générale de la stabilité du mouvement, Annals of Math. Studies, Vol. 17, Princeton University Press, Princeton, New Jersey, 1949

G. J. Michael, Explicit stability criteria for the damped Mathieu equation, IEEE Trans. Automat. Control AC-12, 337–338 (1967)

K. S. Narendra and J. H. Taylor, Stability of the damped Mathieu equation, IEEE Trans. Automat. Control AC-13, 726 (1968)

K. S. Narendra and R. M. Goldwyn, Stability of certain nonlinear differential equations, IEEE Trans. Automat. Control AC-8, 381–382 (1963)

P. C. Parks, Comments on ’Explicit stability criteria for the damped Mathieu equation’, IEEE Trans. Automat. Control AC-13, 129 (1968)

K. P. Persidski, On the stability of motion in first approximation, Mat. Sb. 40, 284–293 (1933)

Peter J. Ponzo, On the stability of certain nonlinear differential equations, IEEE Trans. Automatic Control AC-10 (1965), 470–472. MR 0188554, DOI https://doi.org/10.1109/tac.1965.1098187
Peter J. Ponzo, Some stability conditions for linear differential equations, IEEE Trans. Automatic Control AC-13 (1968), 721–722. MR 0276568, DOI https://doi.org/10.1109/tac.1968.1099036

S. Ramarajan and S. N. Rao, An improved stability criteria for the damped Mathieu equation, IEEE Trans. Automat. Control AC-16, 363–364 (1971)

Nicolas Rouche, P. Habets, and M. Laloy, Stability theory by Liapunov’s direct method, Springer-Verlag, New York-Heidelberg, 1977. Applied Mathematical Sciences, Vol. 22. MR 0450715
S. N. Šimanov, On stability of solution of a nonlinear equation of the third order, Akad. Nauk SSSR. Prikl. Mat. Meh. 17 (1953), 369–372 (Russian). MR 0055523