Sensitivity to infinitesimal delays in neutral equations

The stability of a steady state solution of a neutral functional differential equation can be sensitive to infinitesimal changes in the delays. This phenomenon is caused by the behavior of the essential spectrum and is determined by the roots of an exponential polynomial. Avellar and Hale [ J. Math. Anal. Appl., 73 ( 1980), pp. 434 452] have considered the case of multiple fixed and nonzero delays. In the first part of this paper their results are illustrated by means of spectral plots. In the second part we extend the theory of Avellar and Hale to the limit case whereby some of the delays are brought to zero, which may lead to characteristic roots with arbitrarily large real part. Necessary and sufficient conditions are provided. Using these results we show that the ratio of the delays plays a crucial role when several delays tend to zero simultaneously. As an illustration of the theory, we analyze the robustness of a boundary controlled PDE in the presence of a small feedback delay.