Sufficient conditions for the asymptotic and uniform asymptotic stability of implicit differential systems in a neighborhood of the program manifold are established. Sufficient conditions of stability are also obtained for the known first integrals. A class of implicit systems for which it is possible to find the derivative of the Lyapunov function is selected.
References
V. N. Bajik, “Nonlinear function and stability of motions of implicit systems,” Int. J. Cont., 52, No. 5, 1167–1187 (1990).
A. O. Remizov, “On regular singular points of ordinary differential equations unsolved with respect to the derivatives,” Differents. Uravn., 38, No. 5, 622–630 (2002).
A. O. Remizov, “Implicit differential equations and vector fields with nonisolated singular points,” Mat. Sb., 193, No. 11, 105–124 (2002).
K. V. Kozerenko, “Stability of solutions of ordinary differential equations with finite states unsolved with respect to derivative,” Avtomat. Telemekh., No. 11, 85–93 (2000).
K. V. Kozerenko, “On the investigation of solutions of implicitly defined differential equations,” Zh. Vychisl. Mat. Mat. Fiz., 39, No. 2, 235–238 (1999).
A. M. Samoilenko and V. P. Yakovets’, “On the reducibility of a degenerate linear system to a central canonical form,” Dop. Nats. Akad. Nauk Ukr., No. 4, 10–15 (1993).
V. P. Yakovets’, “On some properties of degenerate linear systems,” Ukr. Mat. Zh., 49, No. 9, 1278–1296 (1997); English translation: Ukr. Math. J., 49, No. 9, 1442–1463 (1997).
V. P. Yakovets’, “On the structure of the general solution of a degenerate linear system of second-order differential equations,” Ukr. Mat. Zh., 50, No. 2, 292–298 (1998); English translation: Ukr. Math. J., 50, No. 2, 334–341 (1998)
K. Weierstrass, “Zur Theorie der bilinearen und quadratischen Formen,” Monatsh. Dtsch. Akad. Wiss. Berlin, 310–338 (1867).
L. Kronecker, “Algebraische Reduktion der Scharen bilinear er Formen,” Sitzungsber. Dtsch. Akad. Wiss. Berlin, 763–776 (1890).
S. S. Zhumatov, “Stability and attraction of the program manifold of implicit differential systems,” in: Proc. of the Third Internat. Conf. “Mathematical Simulation and Differential Equations” (Brest, September 17–22, 2012) [in Russian], Minsk (2012), pp. 143–151.
N. Rouche, P. Habets, and M. Laloy, Stability Theory by Liapunov’s Direct Method, Springer, New York (1977).
Author information
Authors and Affiliations
Additional information
Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 66, No. 4, pp. 558–565, April, 2014.
Rights and permissions
About this article
Cite this article
Zhumatov, S.S. Asymptotic Stability of Implicit Differential Systems in the Vicinity of Program Manifold. Ukr Math J 66, 625–632 (2014). https://doi.org/10.1007/s11253-014-0959-y
Received:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11253-014-0959-y