MR3221838 Hiroyuki NAKAJIMA Hiroyuki NAKAJIMA Funkcialaj Ekvacioj. Serio Internacia 57 2014 43--56 http://fe.math.kobe-u.ac.jp/FE/FullPapers/57-1/57_43.pdf http://www.ams.org/mathscinet-getitem?mr=MR3221838 In the present paper, we give a necessary and sufficient condition for the zero solution of a linear retarded system, $dx(t)/dt=Ax(t)+Bx(t-\tau)$, to be asymptotically stable. Here $A$ is a real-valued $n\times n$ matrix and $B=bI$, where $b$ is a scalar parameter and $I$ is the $n\times n$ unit matrix.
The stability analysis is reduced to deriving a necessary and sufficient condition for all the roots of a characteristic equation, $z-\alpha-\beta e^{-z}=0$, to have negative real parts. Here $\alpha$ is a complex number defined by $\alpha =\tau\lambda$ with an eigenvalue $\lambda$ of $A$, and $\beta=\tau b$. Our stability criterion is a natural extension of that for the widely-known case where $\alpha$ is a real number. Linear retarded differential-difference equation, Stability, Zero solution, Characteristic equation. 34K06, 34K20. 57-43 2014 Hiroyuki NAKAJIMA Hiroyuki NAKAJIMA 1 Sakata, S. Funkcial. Ekvac. 41 1998 435-449 MR1676882 1676882 2 Diekmann, O.; van Gils, S. A.; Verduyn Lunel, S. M.; Walther, H.-O. Delay Equations Applied Mathematical Sciences, 110, Springer-Verlag, New York 1995 MR1345150

3 Braddock, R. D.; van den Driessche, P. J. Austral. Math. Soc. Ser. B 19 1976 358-370 MR0447748