Funkcialaj Ekvacioj
Print ISSN : 0532-8721
On the Stability of a Linear Retarded Differential-Difference Equation
Hiroyuki Nakajima
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2014 Volume 57 Issue 1 Pages 43-56

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Abstract

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 − τ) to be asymptotically stable. Here A is a real-valued n × n matrix and B = bI, where b is a scalar parameter and I is the n × n unit matrix.
The stability analysis is reduced to deriving a necessary and sufficient condition for all the roots of a characteristic equation z − α − βez = 0 to have negative real parts. Here α is a complex number defined by α = τλ with an eigenvalue λ of A, and β = τb. Our stability criterion is a natural extension of that for the widely-known case where α is a real number.

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© 2014 by the Division of Functional Equations, The Mathematical Society of Japan
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