In the present paper, we give a necessary and sufficient condition for the zero solution of a linear retarded system d
x(
t)/d
t =
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 − βe
−z = 0 to have negative real parts. Here α is a complex number defined by α = τλ with an eigenvalue λ of
A, and β = τ
b. Our stability criterion is a natural extension of that for the widely-known case where α is a real number.
View full abstract