Global behavior in nonlinear systems with delayed feedback

Anatoli F. Ivanov; Manuel A. Pinto; Sergei I. Trofimchuk

doi:10.1109/CDC.2001.914602

Global behavior in nonlinear systems with delayed feedback

Anatoli F. Ivanov, Manuel A. Pinto, Sergei I. Trofimchuk

Penn State Wilkes-Barre

Research output: Contribution to journal › Article › peer-review

10 Scopus citations

Abstract

The problem of global stability in scalar delay differential equations of the form ẋ(t) = f(x(t - τ)) - g(x(t)) is studied. Functions f and g are continuous and such that the equation assumes a unique equilibrium. Two types of the sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size τ of the delay. Delay independent stability conditions make use of the global stability in the limiting (as τ → ∞) difference equation g(x_n+1) = f(x_n): the latter always implying the global stability in the differential equation for all values of the delay τ ≥ 0. The delay dependent conditions involve the global attractivity in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f and g, and the delay τ.

Original language	English (US)
Pages (from-to)	4420-4421
Number of pages	2
Journal	Proceedings of the IEEE Conference on Decision and Control
Volume	5
DOIs	https://doi.org/10.1109/CDC.2001.914602
State	Published - 2000

All Science Journal Classification (ASJC) codes

Control and Systems Engineering
Modeling and Simulation
Control and Optimization

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10.1109/CDC.2001.914602

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title = "Global behavior in nonlinear systems with delayed feedback",

abstract = "The problem of global stability in scalar delay differential equations of the form ẋ(t) = f(x(t - τ)) - g(x(t)) is studied. Functions f and g are continuous and such that the equation assumes a unique equilibrium. Two types of the sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size τ of the delay. Delay independent stability conditions make use of the global stability in the limiting (as τ → ∞) difference equation g(xn+1) = f(xn): the latter always implying the global stability in the differential equation for all values of the delay τ ≥ 0. The delay dependent conditions involve the global attractivity in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f and g, and the delay τ.",

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T1 - Global behavior in nonlinear systems with delayed feedback

AU - Ivanov, Anatoli F.

AU - Pinto, Manuel A.

AU - Trofimchuk, Sergei I.

PY - 2000

Y1 - 2000

N2 - The problem of global stability in scalar delay differential equations of the form ẋ(t) = f(x(t - τ)) - g(x(t)) is studied. Functions f and g are continuous and such that the equation assumes a unique equilibrium. Two types of the sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size τ of the delay. Delay independent stability conditions make use of the global stability in the limiting (as τ → ∞) difference equation g(xn+1) = f(xn): the latter always implying the global stability in the differential equation for all values of the delay τ ≥ 0. The delay dependent conditions involve the global attractivity in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f and g, and the delay τ.

AB - The problem of global stability in scalar delay differential equations of the form ẋ(t) = f(x(t - τ)) - g(x(t)) is studied. Functions f and g are continuous and such that the equation assumes a unique equilibrium. Two types of the sufficient conditions for the global asymptotic stability of the unique equilibrium are established: (i) delay independent, and (ii) conditions involving the size τ of the delay. Delay independent stability conditions make use of the global stability in the limiting (as τ → ∞) difference equation g(xn+1) = f(xn): the latter always implying the global stability in the differential equation for all values of the delay τ ≥ 0. The delay dependent conditions involve the global attractivity in specially constructed one-dimensional maps (difference equations) that include the nonlinearities f and g, and the delay τ.

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Global behavior in nonlinear systems with delayed feedback

Abstract

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