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The author presents a method for determining the stability exponent and eigenvalue abscissas of a linear delay system. The method is based on examining the endpoint values of the solution to a functional equation occurring in the Lyapunov theory of delay equations. The question of existence of the solution to this functional equation is examined in more detail than in the previous delay systems literature. Numerical examples are given, including one in which we show that the decay rate of a feedback system can be improved by delay feedback.

of f~~ ( s ) in the closed right half plane {Re ( 8 ) 2 0). The system ($1 is said to be stable independent of delay (i.0.d.) if it is asymptotically stable for every /I E H . In this case we say that the system is Example 2.1: Consider the delay-differential system (*) s'(t) = James Louisell absolutely stable. -x ( t ) -~( t h ) -0.5:r(t -2h). the system used by Datko [41Abs~cl-Tbeaulhorconsidersmatrlxdelay-di~eranUaIsyste~which as an example,the while is simply the delay operator acting on scalar functions. The characteristic function is f h ( 8 ) = 8 + 1 + e-"' + 0.5~-'*'', and one can easily show that for each h E H, the function flE ( 8 ) has no zeros in (Re(8) 2 0). Thus the system (*) is stable i.0.d.Dalko,s example one considers the systems ( * a ) ,rr(t) = -r ( t ) -z ( t -/I.) -0 . 5 z ( t -(~+ E ) / I ) ,where E > 0 is a perturbation ape polynomial In several delay operators. Using a neceasary and sulRclent cdterion for stability independent or delay, or absolute anlhor shows thatsystemstabilltyforaUv~lupaofthedelay vector lyingin a sector wlll imply absolute stability. We then show that absolute stability is Ill-ped with respect to arbitrarily small perturbations Of the delay ratios if a cerlaln extended delay.diffen?ntlal system which Is formed fmm the original Is not also absolutely stable.

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We present a new approach to determining the imaginary axis eigenvalues of a matrix delay equation. With a full rank delay coefficient matrix, the approach requires computation of the generalized eigenvalues of a pair of matrices which are a quarter of the size used in currently known matrix-based or operator approaches. These matrices have square of n for both row and column length, improved from twice those lengths, for an equation having matrix n × n coefficients with rank n delay matrix. Considering problem dimension and computational demands, this brings matrix-based approaches more in synchrony with the well-known scalar substitution approach. Given the symmetries evident in our approach, problems previously done with computation alone can now be considered with insight, and sometimes completed with simple calculations. Some of the computational and mathematical features of the approach are displayed in a section of examples. Introduction.In this paper we consider the question of determining the imaginary axis eigenvalues of a neutral or retarded autonomous matrix delay system with a single delay. We will present a new matrix-and operator-based approach which has some noteworthy advantages in terms of dimensionality and simplicity.The topic of oscillation frequencies and stability for autonomous time-delay equations is fundamental and has a long history. It is interesting, then, that this topic continues to draw attention in engineering and the mathematical sciences. For a small sample, we mention Kokame et al. [KHKM] for delay equations occurring from difference feedback; Engelborghs and Roose [ER], motivated by bifurcation numerics; Zhao [Z] on aircraft flutter; Chen and Aihara [CA] for genetic regulatory networks; and Yuksel [Y] on economic cycles. Also noteworthy is Michiels, Verheyden, and Niculescu [MVN] for theoretical work on systems with rapid periodic time dependence, where autonomous operators enter from an averaging process. Interest is also clearly evident in recent books on time-delay control systems by Gu, Kharitonov, and Chen [GKC] and Chiasson and Loiseau [CL], and in the area of delay differential equations by Balachandran, Kalmar-Nagy, and Gilsinn [BKG]. Considering this, it is clear that the stability of autonomous matrix delay equations still constitutes a large share of core knowledge in the general area of time-delay phenomena.The oscillation eigenvalues of a delay system occur in stability analysis especially in determining stability changing lag parameters, since the oscillation frequencies give us direct, simple trigonometric formulas for these parameters. The approach to stability analysis germinating in this observation is well established and will be the approach on display when we give examples.