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On parameter dependence of exponential stability of equilibrium solutions in differential equations with a single constant delay. (English) Zbl 1354.34128

The paper investigates the transcendental equation \(\lambda-a-b e^{-\lambda\tau}=0\), which appears by the study of the linearized stability in scalar delay differential equations with one fixed delay \(\tau\). For the case of complex coefficients \(a\) and \(b\), necessary and sufficient conditions are given for the solutions \(\lambda\) to have all negative real parts. The analysis is performed using the Lambert W function and so-called “graph-like” expressions for this function. The obtained results are applied to the stabilization of an unstable equilibrium by the delayed feedback control and the stability condition of the synchronous state in oscillator networks.

MSC:

34K20 Stability theory of functional-differential equations
34K25 Asymptotic theory of functional-differential equations
93D15 Stabilization of systems by feedback
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References:

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