A Lyapunov stability criterion for a linear impulsive time-lagged differential system is formulated using the spectral theory of operators and assuming that the period between pulses is constant and equal to the time lag. Stability conditions for a mechanical system of two coupled pendulums subject to a periodic impulsive force and acceleration field that depends on the previous state of the system are established. The mechanical system is analyzed numerically, its stability domain is drawn and compared to that of the system without time lag
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Yu. M. Berezanskii, G. F. Us, and Z. G. Sheftel’, Functional Analysis [in Russian], Vyshcha Shkola; Kyiv (1990).
A. A. Boichuk, N. A. Perestyuk, and A. M. Samoilenko, “Periodic solutions of impulsive differential systems in critical cases,” Diff. Eq., 27, No. 9, 1516 – 1521 (1991).
V. S. Vladimirov, Distributions [in Russian], Nauka, Moscow (1979).
F. R. Gantmacher, Matrix Theory [in Russian], Nauka, Moscow (1966).
Yu. L. Daletskii and M. G. Krein, Stability of Solutions of Differential Equations in Banach Space, Amer. Math. Soc., Providence (1974).
A. A. Kirilov and A. D. Gvishiani, Theorems and Problems in Functional Analysis [in Russian], Nauka, Moscow (1988).
L. A. Lyusternik and L. A. Sobolev, Elements of Functional Analysis, Wiley–Hindustan, Delhi, India, (1974).
M. O. Perestyuk and V. Yu. Slyusarchuk, “Conditions for the existence of nonoscillating solutions of nonlinear differential equations with delay and pulse influence in a Banach space,” Ukr. Math. J., 55, No. 6, 956–965 (2003).
A. M. Samoilenko and N. A. Perestyuk, Impulsive Differential Equations, World Scientific, Singapore (1995).
A. M. Samoilenko and N. A. Perestyuk, “Stability of solutions of impulsive differential equations,” Diff. Eq., 13, 1981–1992 (1977).
V. I. Slyn’ko, “Sufficient practical-stability conditions for nonlinear impulsive systems,” Int. Appl. Mech., 40; No. 10, 1171–1174 (2004).
V. Yu. Slyusarchuk, Stability of Solutions of Differential Equations in Banach Space [in Ukrainian], Vyd. Ukr. Derzh. Univ. Vodn. Gospod. Pryrodokoryst., Rivne (2003).
J. K. Hale, Theory of Functional Differential Equations, Springer-Verlag, New York–Heidelberg–Berlin (1977).
N. G. Chebotarev and N. N. Meiman, Routh–Hurwitz Problem for Polynomials and Integer Functions, Nauka, Moscow–Leningrad (1949).
A. I. Dvirnyi and V. I. Slyn’ko, “Stability of impulsive nonholonomic mechanical systems,” Int. Appl. Mech., 44, No. 3, 353–360 (2008).
Z. E. Filer and A. I. Muzychenko, “Stability of linear mechanical systems with aftereffect,” Int. Appl. Mech., 46, No. 1, 103–112 (2010).
C. S. Hsu, “Nonlinear behavior of multibody systems under impulsive parametric exitation,” in: Proc. Dyn. Multibody Syst. Symp. (Munich, 1977), Berlin (1978), pp. 63–74.
I. L. Ivanov and V. I. Slyn’ko, “Conditions for the stability of an impulsive linear equation with pure delay,” Ukr. Math. J., 61, No. 9, 1417–1427 (2009).
A. A. Martynyuk and V. I. Slyn’ko, “On stability of motion with respect to two measures under uncertainty,” Int. Appl. Mech., 44, No. 1, 91–100 (2008).
Author information
Authors and Affiliations
Corresponding author
Additional information
Translated from Prikladnaya Mekhanika, Vol. 49, No. 6, pp. 120–131, November–December 2013.
Rights and permissions
About this article
Cite this article
Ivanov, I.L., Slyn’ko, V.I. A Stability Criterion for Autonomous Linear Time-Lagged Systems Subject to Periodic Impulsive Force. Int Appl Mech 49, 732–742 (2013). https://doi.org/10.1007/s10778-013-0607-5
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10778-013-0607-5