Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

A differential equation of Lurie type


Author: T. A. Burton
Journal: Proc. Amer. Math. Soc. 36 (1972), 491-496
MSC: Primary 34C10
DOI: https://doi.org/10.1090/S0002-9939-1972-0310334-X
MathSciNet review: 0310334
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The problem of Lurie consists of finding necessary and sufficient conditions for all solutions of the system $ \{ x' = Ax + bf(\sigma ),\sigma ' = {c^T}x - rf(\sigma )\} $ to tend to zero as $ t \to \infty $ under appropriate conditions on the functions involved. When $ f(\sigma )/\sigma \to 0$ as $ \vert\sigma \vert \to \infty $, we obtain NASC's for all solutions to be uniformly ultimately bounded.


References [Enhancements On Off] (What's this?)

  • [1] M. A. Aĭzerman and F. R. Gantmacher, Absolute stability of regulator systems, Izdat. Akad. Nauk SSSR, Moscow, 1963; English transl., Holden-Day, San Francisco, Calif., 1964; German transl., Oldenbourg Verlag, München-Wien, 1965. MR 32 #1036. MR 0183556 (32:1036)
  • [2] J. K. Hale, Ordinary differential equations, Wiley, New York, 1969. MR 0419901 (54:7918)
  • [3] J. P. La Salle, Complete stability of a nonlinear control system, Proc. Nat. Acad. Sci. U.S.A. 48 (1962), 600-603. MR 25 #304. MR 0136843 (25:304)
  • [4] -, Stability theory for ordinary differential equations, J. Differential Equations 4 (1968), 57-65. MR 36 #5454. MR 0222402 (36:5454)
  • [5] S. Lefschetz, Stability of nonlinear control systems, Math. in Sci. and Engineering, vol. 13, Academic Press, New York, 1965. MR 30 #5860. MR 0175676 (30:5860)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 34C10

Retrieve articles in all journals with MSC: 34C10


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0310334-X
Keywords: Problem of Lurie, differential equation, boundedness of solutions, control theory
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society