We consider the system
(
L)
y
′
(
t
)
=
A
y
(
t
)
+
∫
−
∞
t
B
(
t
−
s
)
y
(
s
)
d
s
,
y
(
t
)
=
f
(
t
)
,
t
⩽
0
({\text {L)}}y’(t) = Ay(t) + \int _{ - \infty }^t {B(t - s)y(s)ds,y(t) = f(t),t \leqslant 0}
where
y
(
t
)
y(t)
is an
n
n
-vector and
A
A
and
B
(
t
)
B(t)
are
n
×
n
n \times n
matrices. System
(
L)
({\text {L)}}
generates a semigroup given by
T
t
f
(
s
)
=
y
(
t
+
s
;
f
)
{T_t}f(s) = y(t + s;f)
for
f
f
bounded, continuous and having a finite limit at
−
∞
- \infty
. Under hypotheses concerning the roots of
det
(
λ
I
−
A
−
B
^
(
λ
)
)
\det (\lambda I - A - \hat B(\lambda ))
, where
B
^
(
λ
)
\hat B(\lambda )
is the Laplace transform, various results about the asymptotic behavior of
y
(
t
)
y(t)
are derived, generally after invoking the Hille-Yosida theorem. Two typical results are Theorem 1. If
B
(
t
)
∈
L
1
[
0
,
∞
)
B(t) \in {L^1}[0,\infty )
and
(
λ
I
−
A
−
B
^
(
λ
)
)
−
1
{(\lambda I - A - \hat B(\lambda ))^{ - 1}}
exists for
Re
λ
>
0
\operatorname {Re} \lambda > 0
, then for every
ϵ
>
0
\epsilon > 0
, there is an
M
ϵ
{M_{\epsilon }}
such that
|
|
T
t
f
|
|
⩽
M
ϵ
e
ϵ
t
|
|
f
|
|
||{T_t}f|| \leqslant {M_{\epsilon }}{e^{\epsilon t}}||f||
. Theorem 2. If
(
λ
I
−
A
−
B
^
(
λ
)
)
−
1
{(\lambda I - A - \hat B(\lambda ))^{ - 1}}
exists for
Re
λ
>
−
α
(
α
>
0
)
\operatorname {Re} \lambda > - \alpha (\alpha > 0)
and if
B
(
t
)
e
α
t
∈
L
1
[
0
,
∞
)
B(t){e^{\alpha t}} \in {L^1}[0,\infty )
, then the solution to
(
L)
({\text {L)}}
is exponentially asymptotically stable.