Towards more general stability analysis of systems with delay-dependent coefficients

This paper proposes a systematic method to analyse the stability of systems with single delay with coefficient polynomial depending on the delay. Such models often arise the describing dynamics in Life Science and Engineering systems. A method was presented by Beretta and Kuang in 2002. Their work extends their results to the general case with the exception of some degenerate cases. The interval of interest for the delay is partitioned to smaller intervals so that the magnitude condition generate a fixed number of frequencies as functions of the delay within each interval. The crossing conditions are expressed in a general form, and a simplified derivation for the first order derivative criterion is obtained.

[1]  A. Wilmot-Smith,et al.  # 2006. The American Astronomical Society. All rights reserved. Printed in U.S.A. A TIME DELAY MODEL FOR SOLAR AND STELLAR DYNAMOS , 2006 .

[2]  Boris Polyak,et al.  D-decomposition technique state-of-the-art , 2008 .

[3]  F. G. Boese.  Stability with Respect to the Delay: On a Paper of K. L. Cooke and P. van den Driessche , 1998 .

[4]  Wim Michiels,et al.  Stability, Control, and Computation for Time-Delay Systems - An Eigenvalue-Based Approach (2. ed.) , 2014, Advances in design and control.

[5]  J. Bence,et al.  Space-limited recruitment in open systems: the importance of time delays , 1989 .

[6]  D. D. Perlmutter,et al.  Stability of time‐delay systems , 1972 .

[7]  Yang Kuang,et al.  Geometric Stability Switch Criteria in Delay Differential Systems with Delay Dependent Parameters , 2002, SIAM J. Math. Anal..

[8]  F. Crauste.  Global asymptotic stability and hopf bifurcation for a blood cell production model. , 2006, Mathematical biosciences and engineering : MBE.

[9]  Peilin Fu,et al.  High-Order Analysis Of Critical Stability Properties of Linear Time-Delay Systems , 2007, 2007 American Control Conference.

[10]  C. Nett,et al.  A new method for computing delay margins for stability of linear delay systems , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[11]  Kirk S. Walton,et al.  Direct method for TDS stability analysis , 1987 .

[12]  N. Young.  An identity which implies Cohn's theorem on the zeros of a polynomial , 1979 .

[13]  S. Niculescu.  Delay Effects on Stability: A Robust Control Approach , 2001 .

[14]  Keqin Gu,et al.  A Review of Some Subtleties of Practical Relevance , 2012 .

[15]  Konrad Knopp,et al.  Theory of functions : parts 1 and 2 , 1996 .

[16]  C. Hsu,et al.  On the $\tau $-Decomposition Method of Stability Analysis for Retarded Dynamical Systems , 1969 .

[17]  K. Cooke,et al.  On zeroes of some transcendental equations , 1986 .

[18]  John L. Casti,et al.  Introduction to the theory and application of differential equations with deviating arguments , 1973 .