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Proceedings of the American Mathematical Society

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

 

 

On the condition $ c^T A^{-1} b + r > 0$, in the Lurie problem


Author: Alfredo S. Somolinos
Journal: Proc. Amer. Math. Soc. 47 (1975), 432-441
MSC: Primary 34D25; Secondary 34H05
DOI: https://doi.org/10.1090/S0002-9939-1975-0357998-5
MathSciNet review: 0357998
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Abstract: The problem of Lurie consists in finding NASC's 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 < M$, for all $ \sigma $ and a certain $ M$, we obtain NASC's for the system to be absolutely stable. When $ f(\sigma )/\sigma < M$ as $ \vert\sigma \vert \to \infty $, we obtain conditions for ultimate uniform boundedness of the solutions of the system.


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DOI: https://doi.org/10.1090/S0002-9939-1975-0357998-5
Keywords: Differential equations, control theory, boundedness of solutions, absolute stability, problem of Lurie
Article copyright: © Copyright 1975 American Mathematical Society