# Robust Stability and Stability Radius for  Variational Control Systems

Authors: Bogdan Sasu
Journal: Abstract and Applied Analysis
 (2008)

---

## Abstract

We consider an integral variational control system on a Banach spaceX and we study the connections between its uniform exponential stability and the (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stability, where I and O are Banach function spaces. We identify the viable classes of input spaces and output spaces related to the exponential stability of systems and provide optimization techniques with respect to the input space. We analyze the robustness of exponential stability in the presence of structured perturbations. We deduce general estimations for the lower bound of the stability radius of a variational control system in terms of input-output operators acting on translation-invariant spaces. We apply the main results at the study of the exponential stability of nonautonomous systems and analyze in the nonautonomous case the robustness of this asymptotic property.


---

## Body

## 1. Introduction

In the past few years a significant progress was made in the asymptotic theory of dynamical systems and their   applications in control theory (see [1–41]). It is  well
known that if X is a Banach
space, Θ is a  compact
metric space, σ is a flow on Θ, and A
:
Θ
→
ℬ
(
X
) is a continuous
mapping, then Φ
(
θ
,
t
), the solution operator of the linear differential
equation(1.1)
x
˙
(
t
)
=
A
(
σ
(
θ
,
t
)
)
x
(
t
)
,
t
≥
0
,is a cocycle and the pair π
=
(
Φ
,
σ
) is a linear
skew-product flow. Often, (1.1) arises from the linearization of
nonlinear equations (see [9, 31] and the references therein). Equation (1.1) is the
starting point of our paper. We consider an integral model for the above type
of systems, defined in terms of linear skew-product flows and we study the exponential
stability of solutions.Input-output conditions in the study of the asymptotic
properties of evolution equations have a long and impressive history that goes
back to the work of Perron. In recent years, new ideas were developed in this
theory, providing input-output theorems for stability, expansiveness, and
dichotomy and also a number of applications in control theory (see [2, 7, 9, 11, 13, 15, 19–27, 29, 30, 32–41] and the references therein). The aim of our paper is
to establish input-output conditions for uniform exponential stability of
variational control systems, identifying the viable classes for input spaces
and output spaces, as well. We provide a new approach based on the fundamental
properties of Banach function spaces. Our attention is devoted to the
relationship between the uniform exponential stability of a variational system
and the stability of a pair of spaces which are translations invariant. We
establish the connections between (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stability and
uniform exponential stability of a variational control system on a Banach space X, when the input space I
(
ℝ
+
,
X
) and the output
space O
(
ℝ
+
,
X
) belong to
specific classes of Banach function spaces. By examples we motivate our
techniques and also discuss some optimization methods with respect to the input
space.In what follows, we consider a generalization of
systems described by differential equations of the form(1.2)
x
˙
(
t
)
=
A
(
σ
(
θ
,
t
)
)
x
(
t
)
+
B
(
σ
(
θ
,
t
)
)
u
(
t
)
,
y
(
t
)
=
C
(
σ
(
θ
,
t
)
)
x
(
t
)
,where σ is a flow on a
locally compact metric space Θ, A
(
θ
) are unbounded
operators on a Banach space X, and the operators B
(
θ
)
∈
ℬ
(
U
,
X
), C
(
θ
)
∈
ℬ
(
X
,
Y
), where U, Y are Banach
spaces. Roughly speaking, the family {
A
(
θ
)
}
θ
∈
Θ will be
subjected to additive structured perturbations, so that the perturbed system is x
˙
(
t
)
=
[
A
(
σ
(
θ
,
t
)
)
+
B
(
σ
(
θ
,
t
)
)
Δ
(
σ
(
θ
,
t
)
)
C
(
σ
(
θ
,
t
)
)
]
x
(
t
)
, which may be
interpreted as a system obtained by applying the feedback u
(
t
)
=
Δ
(
σ
(
θ
,
t
)
)
y
(
t
) to the
time-varying system (1.2). The main question is how general may be the
perturbation structure such that the property of exponential stability is preserved.
We will answer this question by determining a general lower bound for the
stability radius of such systems.In the last decades, stability radius became a subject
of large interest and various estimations for the lower bound of the stability
radius of systems were obtained (see [11, 17–20, 32, 33, 41]). This concept was
introduced by Hinrichsen, Ilchmann, and Pritchard (see [18]) and led to a systematic
study of the stability of linear infinite-dimensional systems under structured
time-varying perturbations. The main idea was to estimate the size of the
smallest disturbance operator under which the additively perturbed system loses
exponential stability. A significant theorem which gives a lower bound for the
stability radius of a system associated to a mild evolution operator in terms
of the norm of the input-output operator of the system has been obtained by
Hinrichsen and Pritchard (see [19, Theorem 3.2]). A distinct approach of this result was
given by Clark et al. in [11], employing an evolution semigroup technique. The
variational case was firstly treated in [32], where we obtained a lower bound for the stability
radius in terms of the Perron operators associated with a linear skew-product
semiflow. In [41],
Wirth and Hinrichsen introduced a concept of stability radius for the case of
discrete time-varying systems under structured perturbations of multi-output
feedback type. In the spirit of this theory, we have obtained in [33] an estimation for the lower
bound of the stability radius of a control system of difference equations on a
Banach space X, under structured infinite multiperturbations, in
terms of the norm of the input-output operator associated with a system on l
p
(
ℕ
,
X
) with p
∈
[
1
,
∞
]. An interesting approach for systems in
finite-dimensional spaces was presented by Jacob in [20]. There the author deduces a
formula for the stability radius of time-varying systems with coefficients in L
∞
loc
(
0
,
∞
;
K
n
×
n
), where K
∈
{
ℝ
,
C
}, in terms of the norms of a family of input-output
operators.In the fourth section of the present paper we propose
a new approach, deducing a lower bound for the stability radius of variational
systems in terms of input-output operators acting on rearrangement invariant
Banach function spaces. Until now, the most common function spaces used for
estimating the stability radius are theL
p-spaces
with p
∈
[
1
,
∞
), which satisfy in particular the requirements of the
classes introduced in this paper. Thus, our results provide a unified treatment
in a large class of function spaces, extending the above-mentioned
contributions. As particular cases, we obtain a lower bound for the stability
radius of a variational system in terms of L
p-spaces,
with p
∈
[
1
,
∞
], and generalize the main result in [32].In the last section, the central results of the paper
are applied at the study of the uniform exponential stability of nonautonomous
systems. We provide a complete analysis concerning the implications of the
variational case for nonautonomous systems. We point out several interesting
situations and deduce as particular cases the stability results due to Datko,
Clark, Latushkin, Montgomery-Smith, Randolph, Neerven, Megan, and many others.
Moreover, as an application we obtain a lower bound for the stability radius of
nonautonomous systems in terms of the norm of the input-output operators acting
on translations invariant spaces. Our results extend the existing contributions
in the literature on this topic.


## 2. Preliminary results on Banach function spaces

Letℳ
(
ℝ
+
,
ℝ
) be the linear
space of all Lebesgue measurable functions u
:
ℝ
+
→
ℝ, identifying the functions equal a.e.Definition 2.1.
A linear subspaceB of ℳ
(
ℝ
+
,
ℝ
) is called a normed function space if there is a
mapping |
·
|
B
:
B
→
ℝ
+ such that (i)
|
u
|
B
=
0 if and only if u
=
0 a.e.;
(ii)
|
α
u
|
B
=
|
α
|
|
u
|
B, for all (
α
,
u
)
∈
ℝ
×
B;
(iii)
|
u
+
v
|
B
≤
|
u
|
B
+
|
v
|
B, for all u
,
v
∈
B;
(iv)
if|
u
(
t
)
|
≤
|
v
(
t
)
| a.e. t
∈
ℝ
+ and v
∈
B, then u
∈
B and |
u
|
B
≤
|
v
|
B.
If(
B
,
|
·
|
B
) is complete,
then B is called a  Banach function space.Remark 2.2.
(i) If(
B
,
|
·
|
B
) is a Banach
function space and u
n
→
u in B, then there is a subsequence (
u
k
n
) such that u
k
n
→
u a.e. (see, e.g.,
[28]).
(ii) If(
B
,
|
·
|
B
) is a Banach
function space and u
∈
B, then |
u
(
·
)
|
∈
B.Definition 2.3.
A Banach function space(
B
,
|
·
|
B
) is said to be invariant to translations if for every u
:
ℝ
+
→
ℝ and every t
>
0, u
∈
B if and only if
the function(2.1)
u
t
:
ℝ
+
→
ℝ
,
u
t
(
s
)
=
{
u
(
s
−
t
)
,
s
≥
t
0
,
s
∈
[
0
,
t
)belongs to B and |
u
t
|
B
=
|
u
|
B.LetC
c
(
ℝ
+
,
ℝ
) be the linear
space of all continuous functions v
:
ℝ
+
→
ℝ with compact
support contained in ℝ
+ and let C
0
c
(
ℝ
+
,
ℝ
)
:
=
{
v
∈
C
c
(
ℝ
+
,
ℝ
)
:
v
(
0
)
=
0
}. Let L
loc
1
(
ℝ
+
,
ℝ
) be the linear
space of all locally integrable functions u
:
ℝ
+
→
ℝ.We denote by𝒯
(
ℝ
+
) the class of
all Banach function spaces B which are
invariant to translations and with the following properties:(i)
C
c
(
ℝ
+
,
ℝ
)
⊂
B
⊂
L
loc
1
(
ℝ
+
,
ℝ
);
(ii)
ifB
∖
L
1
(
ℝ
+
,
ℝ
)
≠
∅, then there is a continuous function δ
∈
B
∖
L
1
(
ℝ
+
,
ℝ
)
.For everyA
⊂
ℝ
+ we denote by χ
A the
characteristic function of the set A.Remark 2.4.
IfB
∈
𝒯
(
ℝ
+
), then χ
[
0
,
t
)
∈
B, for all t
>
0.Indeed, lett
>
0. If α
:
ℝ
+
→
[
0
,
1
] is a continuous
function with compact support such that α
(
s
)
=
1, for s
∈
[
0
,
t
] and α
(
s
)
=
0, for s
≥
t
+
1, then we have that χ
[
0
,
t
)
(
s
)
≤
α
(
s
), for all s
≥
0. Since α
∈
B, using (iv) from Definition 2.1, we deduce that χ
[
0
,
t
)
∈
B.Definition 2.5.
IfB
∈
𝒯
(
ℝ
+
), then the function F
B
:
(
0
,
∞
)
→
ℝ
+, F
B
(
t
)
=
|
χ
[
0
,
t
)
|
B is called the fundamental function of the space B.Remark 2.6.
The functionF
B is
nondecreasing.Example 2.7.
Letp
∈
[
1
,
∞
). Then the space L
p
(
ℝ
+
,
ℝ
)
=
{
u
∈
ℳ
(
ℝ
+
,
ℝ
)
:
∫
0
∞
|
u
(
τ
)
|
p
d
τ
<
∞
} with respect to
the norm ∥
u
∥
p
=
(
∫
0
∞
|
u
(
τ
)
|
p
d
τ
)
1
/
p is a Banach
function space which belongs to 𝒯
(
ℝ
+
).
Example 2.8.
LetL
∞
(
ℝ
+
,
ℝ
) be the linear
space of all essentially bounded functions u
∈
ℳ
(
ℝ
+
,
ℝ
). With respect to the norm ∥
u
∥
∞
:
=
ess
sup
t
≥
0
|
u
(
t
)
|, L
∞
(
ℝ
+
,
ℝ
) is a Banach
function space which belongs to 𝒯
(
ℝ
+
).
Example 2.9 (Orlicz spaces).
Letφ
:
ℝ
+
→
ℝ
+ be a
nondecreasing left-continuous function, which is not identically zero on (
0
,
∞
). The Young
function associated with φ is defined by Y
φ
(
t
)
=
∫
0
t
φ
(
s
)
d
s
. For every u
∈
ℳ
(
ℝ
+
,
ℝ
) we define M
φ
(
u
)
:
=
∫
0
∞
Y
φ
(
|
u
(
s
)
|
)
d
s
. The set O
φ of all u
∈
ℳ
(
ℝ
+
,
ℝ
) with the
property that there is k
>
0 such that M
φ
(
k
u
)
<
∞ is a linear
space. With respect to the norm |
u
|
φ
:
=
inf
{
k
>
0
:
M
φ
(
u
/
k
)
≤
1
}, O
φ is a Banach
space called the Orlicz space associated with φ.Remark 2.10.
It is easy to verify that the Orlicz spaces
introduced in Example2.9 belong to 𝒯
(
ℝ
+
).Remark 2.11.
The spacesL
p
(
ℝ
+
,
ℝ
), p
∈
[
1
,
∞
] are Orlicz
spaces, which may be obtained for φ
(
t
)
=
p
t
p
−
1, if p
∈
[
1
,
∞
) and for(2.2)
φ
(
t
)
=
{
0
,
t
∈
[
0
,
1
]
1
,
t
>
1
if
p
=
∞
.Lemma 2.12.
LetB
∈
𝒯
(
ℝ
+
) and ν
>
0. Then the function e
ν
:
ℝ
+
→
ℝ
+, e
ν
(
t
)
=
e
−
ν
t, belongs to B.Proof.
We have that(2.3)
e
ν
(
t
)
=
∑
n
=
0
∞
e
−
ν
t
χ
[
n
,
n
+
1
)
(
t
)
≤
∑
n
=
0
∞
e
−
ν
n
χ
[
n
,
n
+
1
)
(
t
)
,
∀
t
≥
0.This shows that e
ν
∈
B and |
e
ν
|
B
≤
|
χ
[
0
,
1
)
|
B
/
(
1
−
e
−
ν
)
.Definition 2.13.
Letu
,
v
∈
ℳ
(
ℝ
+
,
ℝ
). Say that u and v are equimeasurable if for every t
>
0 the sets {
s
∈
ℝ
+
:
|
u
(
s
)
|
>
t
} and {
s
∈
ℝ
+
:
|
v
(
s
)
|
>
t
} have the same
measure.
Definition 2.14.
A Banach function space(
B
,
|
·
|
B
) is rearrangement invariant if for every
equimeasurable functions u
,
v
:
ℝ
+
→
ℝ
+ with u
∈
B, one has v
∈
B and |
u
|
B
=
|
v
|
B.Remark 2.15.
The Orlicz spaces are rearrangement invariant (see
[4, Theorem 8.9]).We denote byℛ
(
ℝ
+
) the class of
all Banach function spaces B
∈
𝒯
(
ℝ
+
) which are
rearrangement invariant.Remark 2.16.
IfB
∈
ℛ
(
ℝ
+
), then B is an
interpolation space between L
1
(
ℝ
+
,
ℝ
) and L
∞
(
ℝ
+
,
ℝ
) (see [4, Theorem 2.2, page
106]).Lemma 2.17.
LetB
∈
𝒯
(
ℝ
+
) and ν
>
0. Then for every u
∈
B the function(2.4)
ψ
u
:
ℝ
+
→
ℝ
,
ψ
u
(
t
)
=
∫
0
t
e
−
ν
(
t
−
s
)
u
(
s
)
d
sbelongs to B. Moreover, there is λ
B
>
0 such that |
ψ
u
|
B
≤
λ
B
|
u
|
B, for all u
∈
B.Proof.
It is easy to see that(2.5)
D
ν
:
L
∞
(
ℝ
+
,
ℝ
)
→
L
∞
(
ℝ
+
,
ℝ
)
,
(
D
ν
(
u
)
)
(
t
)
=
∫
0
t
e
−
ν
(
t
−
s
)
u
(
s
)
d
sis a correctly defined bounded
linear operator. Moreover, the restriction D
ν
|
:
L
1
(
ℝ
+
,
ℝ
)
→
L
1
(
ℝ
+
,
ℝ
) is correctly
defined and is a bounded linear operator. Then from Remark 2.16 it follows that D
ν
|
B
:
B
→
B is a correctly
defined and bounded linear operator. Taking λ
B
=
∥
D
ν
|
B
∥, the proof is complete.We denote by𝒬
(
ℝ
+
) the class of
all Banach function spaces B
∈
𝒯
(
ℝ
+
) with the
property that sup
t
>
0
F
B
(
t
)
=
∞
.Remark 2.18.
Ifφ
(
t
)
∈
(
0
,
∞
), for all t
>
0, then the Orlicz space O
φ
∈
𝒬
(
ℝ
+
) (see, e.g.,
[24, Proposition
2.1]).LetC
0
(
ℝ
+
,
ℝ
) be the space of
all continuous functions u
:
ℝ
+
→
ℝ with lim
t
→
∞
u
(
t
)
=
0.
Lemma 2.19.
IfB
∈
𝒯
(
ℝ
+
)
∖
𝒬
(
ℝ
+
), then C
0
(
ℝ
+
,
ℝ
)
⊂
B.Proof.
Letu
∈
C
0
(
ℝ
+
,
ℝ
). Then there is an unbounded increasing sequence (
t
n
)
⊂
(
0
,
∞
) such that |
u
(
t
)
|
≤
1
/
(
n
+
1
), for all t
≥
t
n and n
∈
ℕ. We set u
n
=
u
χ
[
0
,
t
n
). According to our hypothesis, M
:
=
sup
t
>
0
F
B
(
t
)
<
∞. Then, we observe that(2.6)
|
u
n
+
p
−
u
n
|
B
≤
1
n
+
1
|
χ
[
t
n
,
t
n
+
p
)
|
B
≤
M
n
+
1
,
∀
p
∈
ℕ
*
,
∀
n
∈
ℕ
.Hence, the sequence (
u
n
) is fundamental
in B, so there is v
∈
B such that u
n
→
v in B. From Remark 2.2 there exists a subsequence (
u
k
n
) such that u
k
n
→
v a.e. This
implies that u
=
v a.e., so u
=
v in B. It follows that u
∈
B and the proof
is complete.We denote byℒ
(
ℝ
+
) the class of
all Banach function spaces B
∈
𝒯
(
ℝ
+
) with the
property that B
∖
L
1
(
ℝ
+
,
ℝ
)
≠
∅.Remark 2.20.
Using Remark2.2(ii) we have that if B
∈
ℒ
(
ℝ
+
), then there is a continuous function δ
:
ℝ
+
→
ℝ
+ with δ
∈
B
∖
L
1
(
ℝ
+
,
ℝ
).Lemma 2.21.
IfB
∈
𝒯
(
ℝ
+
), then B
∈
𝒬
(
ℝ
+
) or B
∈
ℒ
(
ℝ
+
).Proof.
Suppose by contrary that there existsB
∈
𝒯
(
ℝ
+
) such that B
∉
𝒬
(
ℝ
+
) and B
∉
ℒ
(
ℝ
+
). Then M
:
=
sup
t
>
0
F
B
(
t
)
<
∞ and B
⊂
L
1
(
ℝ
+
,
ℝ
). It follows that there is c
>
0 such that ∥
u
∥
1
≤
c
|
u
|
B, for all u
∈
B. In particular, for u
=
χ
[
0
,
t
) we deduce that t
≤
c
|
χ
[
0
,
t
)
|
B
=
c
F
B
(
t
)
≤
c
M, for all t
>
0, which is absurd.Let(
X
,
∥
·
∥
) be a real or
complex Banach space. We denote by C
c
(
ℝ
+
,
X
) the linear
space of all continuous functions v
:
ℝ
+
→
X with compact
support, let C
0
c
=
{
v
∈
C
c
(
ℝ
+
,
X
)
:
v
(
0
)
=
0
} and let L
loc
1
(
ℝ
+
,
X
) be the linear
space of all locally integrable functions u
:
ℝ
+
→
X.For everyB
∈
𝒯
(
ℝ
+
) we denote by B
(
ℝ
+
,
X
) the linear
space of all Bochner measurable functions u
:
ℝ
+
→
X with the
property that the mapping N
u
:
ℝ
+
→
ℝ
+, N
u
(
t
)
=
∥
u
(
t
)
∥ lies in B. Endowed with the norm ∥
u
∥
B
(
ℝ
+
,
X
)
:
=
|
N
u
|
B, B
(
ℝ
+
,
X
) is a Banach
space.


## 3. Stability of variational integral control systems

In the stability theory of evolution operators in
Banach spaces, an evolution family𝒰
=
{
U
(
t
,
s
)
}
t
≥
s
≥
0 is said to be (
E
(
ℝ
+
,
X
)
,
F
(
ℝ
+
,
X
)
) stable if for
every f
∈
E
(
ℝ
+
,
X
) the mapping P
f belongs to F
(
ℝ
+
,
X
) where(3.1)
P
f
(
t
)
=
∫
0
t
U
(
t
,
s
)
f
(
s
)
d
s
.The classical input-output
results concerning exponential stability of evolution families in Banach spaces
may be stated as follows.Let𝒰
=
{
U
(
t
,
s
)
}
t
≥
s
≥
0be an evolution
operator on a Banach space X andp
,
q
∈
[
1
,
∞
)withp
≤
q.  Then the following assertions are equivalent:(i)
𝒰
is uniformly exponentially stable;
(ii)
𝒰
is
(
C
0
(
ℝ
+
,
X
)
,
C
0
(
ℝ
+
,
X
)
)
stable;
(iii)
𝒰
is
(
C
0
(
ℝ
+
,
X
)
,
C
b
(
ℝ
+
,
X
)
)
stable;
(iv)
𝒰
is
(
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
)
stable.The implication (iv)⇒(i) was obtained by
Datko in 1973 (see [16]),
using one of his stability results contained in the same paper. In the last few
years, there were pointed out new and interesting methods of proving the above
theorem. The equivalences (i)⇔(ii)⇔(iv) were proved by
Neerven for the special case of C
0 semigroups of
linear operators (see [30]) and p
=
q. In [11], Clark et al. have provided an inedit technique of
proving the equivalences (i)⇔(ii)⇔(iv) for the case p
=
q. There the authors established important connections
between the asymptotic properties of evolution families and those of the
associated evolution semigroups with a large number of applications in control
theory. Roughly speaking the authors described the behavior of a nonautonomous
system in terms of the properties of an associated autonomous system.
Therefore, in their approach the input space must coincide with the output
space.The equivalence (i)⇔(ii)⇔(iii) has been also
proved by van Minh et al. (see [29]), employing an evolution semigroup technique. The
equivalence (i)⇔(ii) was also
treated by Buşe in
[7]. The case p
≠
q was completely
treated in [23], where
several new stability results of Perron type were established for exponential
stability of evolution families, generalizing the above equivalences and
obtaining that uniform exponential stability of an evolution family can be
expressed using boundedly locally dense subsets of C
b
(
ℝ
+
,
X
) and L
p
(
ℝ
+
,
X
), respectively. The main results in [23] were extended in [25] for the case of linear
skew-product flows, where the input-output operator was replaced with a family
of operators acting between the function spaces of the admissible pair. The
stability results obtained in [25] generalized some theorems from [7, 16, 23, 29, 30]. Moreover, the stability
theorems in [25] led
to several interesting consequences in control theory (see [26, 32]).In what follows, the input-output techniques in the
stability theory of variational equations will be treated from the perspective
of Banach function spaces arising from the interpolation theory. In this
section, our main purpose is to give a complete and unified study of the
exponential stability of variational systems via input-output methods,
providing the “structure” of the classes of input and output spaces,
respectively.LetX be a Banach
space, let (
Θ
,
d
) be a metric
space, and let ℰ
=
X
×
Θ. We denote by ℬ
(
X
) the Banach
algebra of all bounded linear operators on X. The norm on X and on ℬ
(
X
) will be denoted
by ∥
·
∥.
Definition 3.1.
LetJ
∈
{
ℝ
+
,
ℝ
}. A continuous mapping σ
:
Θ
×
J
→
Θ is called a flow on Θ if σ
(
θ
,0
)
=
θ and σ
(
θ
,
s
+
t
)
=
σ
(
σ
(
θ
,
s
)
,
t
), for all (
θ
,
s
,
t
)
∈
Θ
×
J
2.Definition 3.2.
A pairπ
=
(
Φ
,
σ
) is called a linear skew-product flow on ℰ
=
X
×
Θ if σ is a flow on Θ and Φ
:
Θ
×
ℝ
+
→
ℬ
(
X
) satisfies the
following conditions:(i)
Φ
(
θ
,0
)
=
I
d, the identity operator on X, for all θ
∈
Θ;
(ii)
Φ
(
θ
,
t
+
s
)
=
Φ
(
σ
(
θ
,
t
)
,
s
)
Φ
(
θ
,
t
), for all (
θ
,
t
,
s
)
∈
Θ
×
ℝ
+
2 (the
cocycle identity);
(iii)
(
θ
,
t
)
↦
Φ
(
θ
,
t
)
x is continuous,
for every x
∈
X
;
(iv)
there areM
≥
1 and ω
>
0 such that ∥
Φ
(
θ
,
t
)
∥
≤
M
e
ω
t, for all (
θ
,
t
)
∈
Θ
×
ℝ
+
.The mappingΦ given by
Definition 3.2 is called the cocycle associated to the linear skew-product flow π
=
(
Φ
,
σ
).Letπ
=
(
Φ
,
σ
) be a linear
skew-product flow on ℰ
=
X
×
Θ. In what follows, we consider the variational
integral control system(S)
x
θ
(
t
;
x
0
,
u
)
=
Φ
(
θ
,
t
)
x
0
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
(S)with u
∈
L
loc
1
(
ℝ
+
,
X
) and x
0
∈
X.
Definition 3.3.
The system (S) is said to be(i)
uniformly stable if there is L
>
0 such that(3.2)
∥
x
θ
(
t
;
x
0
,0
)
∥
≤
L
∥
x
0
∥
,
∀
(
θ
,
t
)
∈
Θ
×
ℝ
+
,
∀
x
0
∈
X
;
(ii)
uniformly exponentially stable if there
are K
,
ν
>
0 such that(3.3)
∥
x
θ
(
t
;
x
0
,0
)
∥
≤
K
e
−
ν
t
∥
x
0
∥
,
∀
(
θ
,
t
)
∈
Θ
×
ℝ
+
,
∀
x
0
∈
X
.Remark 3.4.
(i) The system (S) is uniformly
stable if and only if there is L
>
0 such that ∥
Φ
(
θ
,
t
)
∥
≤
L, for all (
θ
,
t
)
∈
Θ
×
ℝ
+.(ii) The system (S) is uniformly
exponentially stable if and only if there are K
,
ν
>
0 such that ∥
Φ
(
θ
,
t
)
∥
≤
K
e
−
ν
t, for all (
θ
,
t
)
∈
Θ
×
ℝ
+.Definition 3.5.
LetI, O be two Banach
function spaces with I
,
O
∈
𝒯
(
ℝ
+
). The system (S) is said to be (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
)stable  if the
following assertions hold:(i)
for everyu
∈
C
0
c
(
ℝ
+
,
X
) and every θ
∈
Θ the solution x
θ
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
)
;
(ii)
there isλ
>
0 such that ∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
λ
∥
u
∥
I
(
ℝ
+
,
X
), for all (
u
,
θ
)
∈
C
0
c
(
ℝ
+
,
X
)
×
Θ.I
(
ℝ
+
,
X
) is called the input space and O
(
ℝ
+
,
X
) is called the output space.We begin with a sufficient condition for uniform
stability.Theorem 3.6.
LetI
,
O
∈
𝒯
(
ℝ
+
). If the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then
the system (S) is uniformly
stable.Proof.
Letα
:
ℝ
+
→
[
0
,
2
] be a continuous
function with supp
α
⊂
(
0
,
1
) and ∫
0
1
α
(
τ
)
d
τ
=
1.
LetM
,
ω
∈
(
0
,
∞
) be given by
Definition 3.2 and let λ
>
0 be given by
Definition 3.5.
Let(
x
,
θ
)
∈
X
×
Θ. We consider the function u
:
ℝ
+
→
X, u
(
t
)
=
α
(
t
)
Φ
(
θ
,
t
)
x
. We have that u
∈
C
0
c
(
ℝ
+
,
X
) and ∥
u
(
t
)
∥
=
α
(
t
)
∥
Φ
(
θ
,
t
)
x
∥
≤
M
e
ω
∥
x
∥
α
(
t
), for all t
≥
0. This implies that(3.4)
∥
u
∥
I
(
ℝ
+
,
X
)
≤
M
e
ω
∥
x
∥
|
α
|
I
.From hypothesis we have that x
θ
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
) and(3.5)
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
λ
∥
u
∥
I
(
ℝ
+
,
X
)
.We observe that(3.6)
x
θ
(
t
;
0
,
u
)
=
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
=
(
∫
0
t
α
(
s
)
d
s
)
Φ
(
θ
,
t
)
x
=
Φ
(
θ
,
t
)
x
,
∀
t
≥
1. Then, for t
≥
2 we deduce that(3.7)
∥
Φ
(
θ
,
t
)
x
∥
χ
[
t
−
1,
t
)
(
s
)
≤
M
e
ω
∥
Φ
(
θ
,
s
)
x
∥
=
M
e
ω
∥
x
θ
(
s
;
0
,
u
)
∥
,
∀
s
≥
0which implies that(3.8)
∥
Φ
(
θ
,
t
)
x
∥
|
χ
[
t
−
1,
t
)
|
O
≤
M
e
ω
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
.Since O is invariant to
translations we have that |
χ
[
t
−
1,
t
)
|
O
=
|
χ
[
0
,
1
)
|
O
=
F
O
(
1
). Then, from relations (3.4)–(3.8) we obtain that(3.9)
∥
Φ
(
θ
,
t
)
x
∥
≤
λ
M
e
ω
F
O
(
1
)
∥
u
∥
I
(
ℝ
+
,
X
)
≤
λ
M
2
e
2
ω
|
α
|
I
F
O
(
1
)
∥
x
∥
,
∀
t
≥
2.For t
∈
[
0
,
2
] we have that ∥
Φ
(
θ
,
t
)
∥
≤
M
e
2
ω. Then, for   L
=
max
{
M
e
2
ω
,
(
λ
M
2
e
2
ω
|
α
|
I
)
/
F
O
(
1
)
} we deduce that ∥
Φ
(
θ
,
t
)
x
∥
≤
L
∥
x
∥, for all t
≥
0. Taking into account that L does not depend
on θ or x, it follows that ∥
Φ
(
θ
,
t
)
x
∥
≤
L
∥
x
∥, for all (
θ
,
t
)
∈
Θ
×
ℝ
+ and all x
∈
X and the proof
is complete.The first main result of this section is
the following.Theorem 3.7.
LetI
,
O
∈
𝒯
(
ℝ
+
). If I
∈
ℒ
(
ℝ
+
) and the system (S) is   (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then
the system (S) is uniformly
exponentially stable.Proof.
Letλ
>
0 be given by
Definition 3.5. From Theorem 3.6 we have that there is L
>
0 such that ∥
Φ
(
θ
,
t
)
∥
≤
L
, for all (
θ
,
t
)
∈
Θ
×
ℝ
+
.
SinceI
∈
ℒ
(
ℝ
+
) from Remark 2.20
it follows that there is a continuous function δ
:
ℝ
+
→
ℝ
+ such that δ
∈
I
∖
L
1
(
ℝ
+
,
ℝ
). Let h
>
0 be such that(3.10)
∫
0
h
δ
(
s
)
d
s
≥
2
e
λ
L
2
|
δ
|
I
F
O
(
1
)
.
For everyn
∈
ℕ
* let α
n
:
ℝ
+
→
[
0
,
1
] be a continuous
function with α
n
(
t
)
=
1, for t
∈
[
1
/
n
,
h
], α
n
(
0
)
=
0, and α
n
(
t
)
=
0, for t
≥
h
+
1. Since(3.11)
∫
0
h
α
n
(
s
)
δ
(
s
)
d
s
→
∫
0
h
δ
(
s
)
d
s
,
as
n
→
∞there is j
∈
ℕ
* such that(3.12)
a
:
=
∫
0
h
α
j
(
s
)
δ
(
s
)
d
s
≥
1
2
∫
0
h
δ
(
s
)
d
s
≥
e
λ
L
2
|
δ
|
I
F
O
(
1
)
.
Let(
x
,
θ
)
∈
ℰ
=
X
×
Θ. We consider the function(3.13)
u
:
ℝ
+
→
X
,
u
(
t
)
=
α
j
(
t
)
δ
(
t
)
Φ
(
θ
,
t
)
x
.We have that u
∈
C
0
c
(
ℝ
+
,
X
), so u
∈
I
(
ℝ
+
,
X
). In addition, ∥
u
(
t
)
∥
≤
L
∥
x
∥
δ
(
t
)
, for all t
≥
0. This implies that ∥
u
∥
I
(
ℝ
+
,
X
)
≤
L
∥
x
∥
|
δ
|
I
. We observe that(3.14)
∥
x
θ
(
t
;
0
,
u
)
∥
=
∥
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
∥
=
(
∫
0
t
α
j
(
s
)
δ
(
s
)
d
s
)
∥
Φ
(
θ
,
t
)
x
∥
≥
a
∥
Φ
(
θ
,
t
)
x
∥
,
∀
t
≥
h
. Then, we have that(3.15)
∥
Φ
(
θ
,
h
+
1
)
x
∥
≤
L
∥
Φ
(
θ
,
t
)
x
∥
≤
L
a
∥
x
θ
(
t
;
0
,
u
)
∥
,
∀
t
∈
[
h
,
h
+
1
)which implies that(3.16)
∥
Φ
(
θ
,
h
+
1
)
x
∥
χ
[
h
,
h
+
1
)
(
t
)
≤
L
a
∥
x
θ
(
t
;
0
,
u
)
∥
,
∀
t
≥
0.Since O is invariant to
translations we have that |
χ
[
h
,
h
+
1
)
|
O
=
|
χ
[
0
,
1
)
|
O
=
F
O
(
1
). Then from (3.16) we deduce that(3.17)
∥
Φ
(
θ
,
h
+
1
)
x
∥
F
O
(
1
)
≤
L
a
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
.According to our hypothesis(3.18)
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
λ
∥
u
∥
I
(
ℝ
+
,
X
)
.Using relations (3.17) and (3.18)
it follows that(3.19)
a
∥
Φ
(
θ
,
h
+
1
)
x
∥
≤
λ
L
F
O
(
1
)
∥
u
∥
I
(
ℝ
+
,
X
)
≤
λ
L
2
|
δ
|
I
F
O
(
1
)
∥
x
∥
.We set r
=
h
+
1. From (3.19) and (3.12) we obtain that ∥
Φ
(
θ
,
r
)
x
∥
≤
(
1
/
e
)
∥
x
∥. Since r does not depend
on θ or x we deduce that ∥
Φ
(
θ
,
r
)
x
∥
≤
(
1
/
e
)
∥
x
∥, for all (
x
,
θ
)
∈
ℰ.
Letν
=
1
/
r and K
=
L
e. Let t
≥
0. Then there is n
∈
ℕ and s
∈
[
0
,
r
) such that t
=
n
r
+
s. It follows that ∥
Φ
(
θ
,
t
)
∥
≤
L
∥
Φ
(
θ
,
n
r
)
∥
≤
L
e
−
n
≤
K
e
−
ν
t. This implies that the system (S) is uniformly
exponentially stable.The second main result of this section is
the following.Theorem 3.8.
LetI
,
O
∈
𝒯
(
ℝ
+
). If O
∈
𝒬
(
ℝ
+
) and the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then (S) is uniformly
exponentially stable.Proof.
Letλ
>
0 be given by
Definition 3.5 and let L
>
0 be given by
Theorem 3.6. Let h
>
0 be such that(3.20)
F
O
(
h
)
≥
2
e
λ
L
2
F
I
(
1
)
.Let α
:
ℝ
+
→
[
0
,
2
] be a continuous
function with supp
α
⊂
(
0
,
1
) and ∫
0
1
α
(
τ
)
d
τ
=
1.
Let(
x
,
θ
)
∈
ℰ
=
X
×
Θ. We consider the function(3.21)
u
:
ℝ
+
→
X
,
u
(
t
)
=
α
(
t
)
Φ
(
θ
,
t
)
x
.Since u
∈
C
0
c
(
ℝ
+
,
X
) we have that u
∈
I
(
ℝ
+
,
X
). Moreover, ∥
u
(
t
)
∥
≤
L
∥
x
∥
α
(
t
)
≤
2
L
∥
x
∥
χ
[
0
,
1
)
(
t
), for all t
≥
0, which implies that(3.22)
∥
u
∥
I
(
ℝ
+
,
X
)
≤
2
L
∥
x
∥
F
I
(
1
)
.We observe that x
θ
(
t
,0,
u
)
=
Φ
(
θ
,
t
)
x, for all t
≥
1. Then, we have that(3.23)
∥
Φ
(
θ
,
h
+
1
)
x
∥
χ
[
1
,
h
+
1
)
(
t
)
≤
L
∥
Φ
(
θ
,
t
)
x
∥
χ
[
1
,
h
+
1
)
(
t
)
≤
L
∥
x
θ
(
t
;
0
,
u
)
∥
,
∀
t
≥
0 Since O
∈
𝒯
(
ℝ
+
) we have that |
χ
[
1
,
h
+
1
)
|
O
=
|
χ
[
0
,
h
)
|
O
=
F
O
(
h
). Then from (3.23) and (3.22) we deduce that(3.24)
∥
Φ
(
θ
,
h
+
1
)
x
∥
F
O
(
h
)
≤
L
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
λ
L
∥
u
∥
I
(
ℝ
+
,
X
)
≤
2
λ
L
2
F
I
(
1
)
∥
x
∥
. From relations (3.20) and (3.24)
it follows that ∥
Φ
(
θ
,
h
+
1
)
x
∥
≤
(
1
/
e
)
∥
x
∥. Taking into account that h does not depend
on θ or x we obtain that ∥
Φ
(
θ
,
h
+
1
)
x
∥
≤
(
1
/
e
)
∥
x
∥, for all (
x
,
θ
)
∈
ℰ
=
X
×
Θ. Using similar arguments as in Theorem 3.7 we obtain
that the system (S) is uniformly
exponentially stable.The central result of this section is
the following.Theorem 3.9.
LetI
,
O
∈
𝒯
(
ℝ
+
) be such that I
∈
ℒ
(
ℝ
+
) or O
∈
𝒬
(
ℝ
+
). Then, the following assertions hold:(i)
if the
system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then (S) is uniformly
exponentially stable;
(ii)
ifI
⊂
O and one of the
spaces I, O belongs to the
class ℛ
(
ℝ
+
), then the system (S) is uniformly
exponentially stable if and only if the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable.Proof.
(i) This follows from Theorems3.7 and 3.8.
(ii)Necessity. Let K
,
ν
>
0 be such that ∥
Φ
(
θ
,
t
)
x
∥
≤
K
e
−
ν
t
∥
x
∥, for all (
x
,
θ
)
∈
ℰ and all t
≥
0. Since I
⊂
O, there is γ
>
0 such that(3.25)
|
w
|
O
≤
γ
|
w
|
I
,
∀
w
∈
I
.
Letu
∈
C
0
c
(
ℝ
+
,
X
) and let(3.26)
ψ
u
:
ℝ
+
→
ℝ
,
ψ
u
(
t
)
=
∫
0
t
e
−
ν
(
t
−
s
)
∥
u
(
s
)
∥
d
s
.Then for every θ
∈
Θ we have that(3.27)
∥
x
θ
(
t
;
0
,
u
)
∥
≤
K
ψ
u
(
t
)
,
t
≥
0.Case 1.
I
∈
ℛ
(
ℝ
+
) Let λ
I
>
0 be given by
Lemma 2.17. Since u
∈
I
(
ℝ
+
,
X
), from Lemma 2.17 we have that ψ
u
∈
I and |
ψ
u
|
I
≤
λ
I
∥
u
∥
I
(
ℝ
+
,
X
). Then from (3.27) it follows that x
θ
(
·
;
0
,
u
)
∈
I
(
ℝ
+
,
X
), so x
θ
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
). Moreover, from (3.25) and (3.27) we have that(3.28)
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
γ
∥
x
θ
(
·
;
0
,
u
)
∥
I
(
ℝ
+
,
X
)
≤
γ
K
λ
I
∥
u
∥
I
(
ℝ
+
,
X
)
.Taking λ
=
γ
K
λ
I and observing
that λ does not depend
on u or θ we deduce that
the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
)stable.Case 2.
O
∈
ℛ
(
ℝ
+
) Let λ
O
>
0 be given by
Lemma 2.17. From u
∈
O
(
ℝ
+
,
X
), using Lemma 2.17 we have that ψ
u
∈
O and |
ψ
u
|
O
≤
λ
O
∥
u
∥
O
(
ℝ
+
,
X
). Then, from (3.25) and (3.27) we deduce that x
θ
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
) and(3.29)
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
K
λ
O
∥
u
∥
O
(
ℝ
+
,
X
)
≤
γ
K
λ
O
∥
u
∥
I
(
ℝ
+
,
X
)
.Taking λ
=
γ
K
λ
O we conclude
that the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable.Sufficiency. It follows from (i).
In what follows, we prove that the main result given
by Theorem3.9 is the most general in this topic. Specifically, we will show
that if I
∉
ℒ
(
ℝ
+
) and O
∉
𝒬
(
ℝ
+
), then the (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stability of
the system (S) does not imply
the uniform exponential stability of (S).Example 3.10.
LetΘ
=
ℝ
+ and let σ
:
Θ
×
ℝ
+
→
Θ, σ
(
θ
,
t
)
=
θ
+
t. Let X
=
C
0
(
ℝ
+
,
ℝ
) and for every (
θ
,
t
)
∈
Θ
×
ℝ
+, let(3.30)
Φ
(
θ
,
t
)
:
X
→
X
,
(
Φ
(
θ
,
t
)
x
)
(
s
)
=
x
(
s
+
t
)
.Then π
=
(
Φ
,
σ
) is a linear
skew-product flow on ℰ
=
X
×
Θ. We consider  the system(S)
x
θ
(
t
;
x
0
,
u
)
=
Φ
(
θ
,
t
)
x
0
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
(S)with u
∈
L
loc
1
(
ℝ
+
,
X
) and x
0
∈
X. Observing that ∥
Φ
(
θ
,
t
)
∥
=
1, for all (
θ
,
t
)
∈
Θ
×
ℝ
+, we deduce that the system (S) is not
uniformly exponentially stable.
LetI
,
O
∈
𝒯
(
ℝ
+
) with I
∉
ℒ
(
ℝ
+
) and O
∉
𝒬
(
ℝ
+
). Then I
⊂
L
1
(
ℝ
+
,
ℝ
) and from Lemma
2.19 we have that C
0
(
ℝ
+
,
ℝ
)
⊂
O. In what follows, we prove that the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable.
Letu
∈
L
1
(
ℝ
+
,
X
) and θ
∈
Θ. Let ε
>
0. Since u
∈
L
1
(
ℝ
+
,
X
), there is h
>
0 such that ∫
h
∞
∥
u
(
s
)
∥
d
s
<
ε
/
2. Let v
:
=
∫
0
h
Φ
(
σ
(
θ
,
τ
)
,
h
−
τ
)
u
(
τ
)
d
τ
. From v
∈
X we have that
there is δ
>
0 such that |
v
(
τ
)
|
<
ε
/
2, for all t
≥
δ. Then, for t
≥
δ
+
h, we deduce that(3.31)
∥
∫
0
h
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
∥
=
∥
Φ
(
σ
(
θ
,
h
)
,
t
−
h
)
v
∥
=
sup
r
≥
0
|
v
(
t
−
h
+
r
)
|
<
ε
2
.This implies that(3.32)
∥
x
θ
(
t
;
0
,
u
)
∥
≤
∥
∫
0
h
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
∥
+
∫
h
t
∥
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
∥
∥
u
(
s
)
∥
d
s
<
ε
2
+
∫
h
t
∥
u
(
s
)
∥
d
s
<
ε
,
∀
t
≥
δ
+
h so x
θ
(
·
;
0
,
u
)
∈
C
0
(
ℝ
+
,
X
). It follows that for every (
u
,
θ
)
∈
L
1
(
ℝ
+
,
X
)
×
Θ we have that x
θ
(
·
;
0
,
u
)
∈
C
0
(
ℝ
+
,
X
).
It makes sense to define the linear operator(3.33)
P
:
L
1
(
ℝ
+
,
X
)
→
C
0
(
ℝ
+
,
X
)
,
P
(
u
)
=
x
0
(
·
;
0
,
u
)
.It is easy to verify that P is closed, so
from the closed graph theorem we obtain that P is bounded.
Setting γ
=
∥
P
∥ we have that(3.34)
∥
x
0
(
·
;
0
,
u
)
∥
C
0
(
ℝ
+
,
X
)
≤
γ
∥
u
∥
L
1
(
ℝ
+
,
X
)
,
∀
u
∈
L
1
(
ℝ
+
,
X
)
.Since I
⊂
L
1
(
ℝ
+
,
ℝ
) there is α
>
0 such that(3.35)
∥
u
∥
1
≤
α
|
u
|
I
,
∀
u
∈
I
.Since C
0
(
ℝ
+
,
ℝ
)
⊂
O there is β
>
0 such that(3.36)
|
w
|
O
≤
β
∥
w
∥
C
0
(
ℝ
+
,
ℝ
)
,
∀
w
∈
C
0
(
ℝ
+
,
ℝ
)
.
Let(
u
,
θ
)
∈
C
0
c
(
ℝ
+
,
X
)
×
Θ. Since x
θ
(
·
;
0
,
u
)
∈
C
0
(
ℝ
+
,
X
) and   C
0
(
ℝ
+
,
X
)
⊂
O
(
ℝ
+
,
X
) it follows that x
θ
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
). It is easy to observe that x
θ
(
·
;
0
,
u
)
=
x
0
(
·
;
0
,
u
). Setting λ
=
α
β
γ, from relations (3.34)–(3.36) we deduce that(3.37)
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
=
∥
x
0
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
β
∥
x
0
(
·
;
0
,
u
)
∥
C
0
(
ℝ
+
,
X
)
≤
β
γ
∥
u
∥
L
1
(
ℝ
+
,
X
)
≤
λ
∥
u
∥
I
(
ℝ
+
,
X
) Taking into account that λ does not depend
on u or θ we conclude
that the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable. But,
for all that (S) is not
uniformly exponentially stable.Remark 3.11.
The input-output characterizations for asymptotic
properties of systems have a wide applicability area if the input space is as
small as possible and the output space is very general. We note that in the
main result given by Theorem3.9 the input functions belong to C
0
c
(
ℝ
+
,
X
), while the output space is a general function space.
Moreover, the class𝒯
(
ℝ
+
) is closed to
finite intersections. If   I
1
,
I
2
,
…
,
I
n
∈
𝒯
(
ℝ
+
), then we may consider the space I
:
=
I
1
∩
I
2
∩
⋯
∩
I
n with respect to
the norm(3.38)
|
u
|
I
:
=
max
{
|
u
|
I
1
,
|
u
|
I
2
,
…
,
|
u
|
I
n
}which is a Banach function space
in 𝒯
(
ℝ
+
). Now, if we analyze condition (ii) in Definition 3.5,
it is obvious that if the input space is I
(
ℝ
+
,
X
), then in the right member of the inequality we have a
“larger” norm and thus the estimation is more permissive.Letq
∈
[
1
,
∞
], n
∈
ℕ
*, and p
1
,
…
,
p
n
∈
(
1
,
∞
). We consider the space(3.39)
I
p
1
,
…
,
p
n
,
q
(
ℝ
+
,
ℝ
)
=
L
p
1
(
ℝ
+
,
ℝ
)
∩
⋯
∩
L
p
n
(
ℝ
+
,
ℝ
)
∩
L
q
(
ℝ
+
,
ℝ
)which is a Banach space with
respect the norm(3.40)
∥
u
∥
p
1
,
…
,
p
n
,
q
=
max
{
∥
u
∥
p
1
,
…
,
∥
u
∥
p
n
,
∥
u
∥
q
}
.Corollary 3.12.
The
system (S) is uniformly
exponentially stable if and only if the system (S) is (
I
p
1
,
…
,
p
n
,
q
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable.Proof.
Ifq
∈
[
1
,
∞
), then L
q
(
ℝ
+
,
ℝ
)
∈
𝒬
(
ℝ
+
). If q
=
∞, then I
∈
ℒ
(
ℝ
+
). By applying Theorem 3.9 we obtain the
conclusion.Corollary 3.13.
LetW
∈
ℛ
(
ℝ
+
). The system (S) is uniformly
exponentially stable if and only if (S) is (
W
(
ℝ
+
,
X
)
,
W
(
ℝ
+
,
X
)
) stable.Proof.
This follows from Theorem3.9 and Lemma 2.21.Lemma 3.14.
Letν
>
0 and p
,
q
∈
[
1
,
∞
] with p
≤
q. For every u
∈
L
p
(
ℝ
+
,
ℝ
), the function(3.41)
f
u
:
ℝ
+
→
ℝ
,
f
u
(
t
)
=
∫
0
t
e
−
ν
(
t
−
s
)
u
(
s
)
d
sbelongs to L
q
(
ℝ
+
,
ℝ
). In addition, there is γ
>
0 such that ∥
f
u
∥
q
≤
γ
∥
u
∥
p, for all u
∈
L
p
(
ℝ
+
,
X
).Proof.
Using Hölder's inequality it follows that for everyu
∈
L
p
(
ℝ
+
,
ℝ
+
) the function f
u
∈
L
q
(
ℝ
+
,
ℝ
+
). Then, from(3.42)
|
f
u
(
t
)
|
≤
∫
0
t
e
−
ν
(
t
−
s
)
|
u
(
s
)
|
d
s
,
∀
t
≥
0,
∀
u
∈
L
p
(
ℝ
+
,
ℝ
)we deduce that for every u
∈
L
p
(
ℝ
+
,
ℝ
) the function f
u belongs to L
q
(
ℝ
+
,
ℝ
). Hence, it makes sense to define the linear operator Γ
:
L
p
(
ℝ
+
,
ℝ
)
→
L
q
(
ℝ
+
,
ℝ
), Γ
(
u
)
=
f
u. It is easy to see that Γ is closed, so
it is bounded. Setting γ
=
∥
Γ
∥ we obtain the
conclusion.Corollary 3.15.
Letp
,
q
∈
[
1
,
∞
] with (
p
,
q
)
≠
(
1
,
∞
). The following assertions hold:(i)
if the system (S) is (
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable, then (S) is uniformly
exponentially stable;
(ii)
ifp
≤
q, then the system (S) is uniformly
exponentially stable if and only if it is (
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable.Proof.
(i) It follows from Theorem3.9(i).
(ii)Necessity. Let γ
>
0 be given by
Lemma 3.14. Let K
,
ν
>
0 be such that ∥
Φ
(
θ
,
t
)
x
∥
≤
K
e
−
ν
t
∥
x
∥, for all (
x
,
θ
)
∈
ℰ and all t
≥
0. Let (
u
,
θ
)
∈
C
0
c
(
ℝ
+
,
X
)
×
Θ. Then(3.43)
∥
x
θ
(
t
;
0
,
u
)
∥
≤
K
∫
0
t
e
−
ν
(
t
−
s
)
∥
u
(
s
)
∥
d
s
,
∀
t
≥
0.Using Lemma 3.14 we obtain that x
θ
(
·
;
0
,
u
)
∈
L
q
(
ℝ
+
,
X
) and(3.44)
∥
x
θ
(
·
;
0
,
u
)
∥
L
q
(
ℝ
+
,
X
)
≤
K
γ
∥
u
∥
L
p
(
ℝ
+
,
X
)
.Taking into account that γ and K do not depend
on u or θ, it follows that the system (S) is (
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable.
Sufficiency. This follows from (i).Definition 3.16.
LetI, O be two Banach
function spaces with I
,
O
∈
𝒯
(
ℝ
+
). The system (S) is said to be completely(
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
)stable if the following assertions hold:(i)
for everyu
∈
I
(
ℝ
+
,
X
) and every θ
∈
Θ the solution x
θ
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
)
;
(ii)
there isλ
>
0 such that ∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
λ
∥
u
∥
I
(
ℝ
+
,
X
), for all (
u
,
θ
)
∈
I
(
ℝ
+
,
X
)
×
Θ.
Theorem 3.17.
LetI
,
O
∈
𝒯
(
ℝ
+
) be such that I
∈
ℒ
(
ℝ
+
) or O
∈
𝒬
(
ℝ
+
). Then, the following assertions hold:(i)
if the system (S) is completely (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then (S) is uniformly
exponentially stable;
(ii)
ifI
⊂
O and one of the
spaces I, O  belongs to the
class ℛ
(
ℝ
+
), then the system (S) is uniformly
exponentially stable if and only if the system (S) is completely (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable.Proof.
(i) This follows from Theorem3.9(i).
(ii)Necessity follows using similar arguments as in the necessity part of Theorem 3.9(ii).
Sufficiency is given by (i).Corollary 3.18.
Letq
∈
[
1
,
∞
], n
∈
ℕ
* and p
1
,
…
,
p
n
∈
(
1
,
∞
). The system (S) is uniformly
exponentially stable if and only if the system (S) is completely (
I
p
1
,
…
,
p
n
,
q
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable.Proof.
This follows from Corollary3.12.Corollary 3.19.
LetW
∈
ℛ
(
ℝ
+
). The system (S) is uniformly
exponentially stable if and only if (S) is completely (
W
(
ℝ
+
,
X
)
,
W
(
ℝ
+
,
X
)
) stable.Remark 3.20.
LetW
∈
ℛ
(
ℝ
+
). If the system (S) is uniformly
exponentially stable, then for every θ
∈
Θ the linear
operator(3.45)
P
W
θ
:
W
(
ℝ
+
,
X
)
→
W
(
ℝ
+
,
X
)
,
(
P
W
θ
u
)
(
t
)
=
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
sis correctly defined and
bounded. Moreover, if λ
>
0 is given by
Definition 3.16, then we have that sup
θ
∈
Θ
∥
P
W
θ
∥
≤
λ.Corollary 3.21.
Letp
,
q
∈
[
1
,
∞
] with (
p
,
q
)
≠
(
1
,
∞
). The following assertions hold:(i)
if the system (S) is completely (
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable, then (S) is uniformly
exponentially stable;
(ii)
ifp
≤
q, then the system (S) is uniformly
exponentially stable if and only if it is completely (
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
)stable.Proof.
(i) This follows from Corollary3.15(i).
(ii)Necessity follows using similar arguments
as in the necessity of Corollary 3.15(ii). Sufficiency follows from (i).Remark 3.22.
A distinct proof for Corollary3.21(i) was given in
[27] (see Theorem
5.2).LetC
b
(
ℝ
+
,
ℝ
) be the space of
all bounded continuous functions u
:
ℝ
+
→
ℝ and C
00
(
ℝ
+
,
ℝ
)
=
{
u
∈
C
0
(
ℝ
+
,
ℝ
)
:
u
(
0
)
=
0
}.
Definition 3.23.
LetU
,
Y
∈
{
C
b
(
ℝ
+
,
ℝ
)
,
C
0
(
ℝ
+
,
ℝ
)
,
C
00
(
ℝ
+
,
ℝ
)
}. The system (S) is said to be completely(
U
(
ℝ
+
,
X
)
,
Y
(
ℝ
+
,
X
)
)stable if the following assertions hold:(i)
for everyu
∈
U
(
ℝ
+
,
X
) and every θ
∈
Θ the solution x
θ
(
·
;
0
,
u
)
∈
Y
(
ℝ
+
,
X
)
;
(i)
there isλ
>
0 such that ∥
x
θ
(
·
;
0
,
u
)
∥
Y
(
ℝ
+
,
X
)
≤
λ
∥
u
∥
U
(
ℝ
+
,
X
), for all (
u
,
θ
)
∈
U
(
ℝ
+
,
X
)
×
Θ.Corollary 3.24.
LetU
,
Y
∈
{
C
b
(
ℝ
+
,
ℝ
)
,
C
0
(
ℝ
+
,
ℝ
)
,
C
00
(
ℝ
+
,
ℝ
)
} with U
⊂
Y. The system (S) is uniformly
exponentially stable if and only if (S) is completely (
U
(
ℝ
+
,
X
)
,
Y
(
ℝ
+
,
X
)
) stable.Proof.
Necessity is a simple exercise.
Sufficiency. If the system (S) is completely (
U
(
ℝ
+
,
X
)
,
Y
(
ℝ
+
,
X
)
) stable, then it
is (
L
∞
(
ℝ
+
,
X
)
,
L
∞
(
ℝ
+
,
X
)
) stable. By
applying Corollary 3.15 we deduce that (S) is uniformly
exponentially stable.Remark 3.25.
A different proof for Corollary3.24 for the cases   (
U
,
Y
)
=
(
C
00
(
ℝ
+
,
ℝ
)
,
C
00
(
ℝ
+
,
ℝ
)
) and (
U
,
Y
)
=
(
C
00
(
ℝ
+
,
ℝ
)
,
C
b
(
ℝ
+
,
ℝ
)
) was given in
[27] (see Theorem 5.1).


## 4. Stability radius

In this section, we will obtain a lower bound for the
stability radius of linear skew-product flows in terms of input-output
operators acting on Banach function spaces which belong to a class of
rearrangement invariant spaces.LetU, Y be Banach
spaces and let Θ be a locally
compact metric space. We denote by ℬ
(
U
,
Y
) the space of
all bounded linear operators from U into Y and by 𝒞
s
(
Θ
,
ℬ
(
U
,
Y
)
) the space of
all continuous bounded mappings H
:
Θ
→
ℬ
(
U
,
Y
). With respect to the norm ∥
|
H
∥
|
:
=
sup
θ
∈
Θ
∥
H
(
θ
)
∥, 𝒞
s
(
Θ
,
ℬ
(
U
,
Y
)
) is a Banach
space.Remark 4.1.
Ifπ
=
(
Φ
,
σ
) is a linear
skew-product flow on ℰ
=
X
×
Θ and P
∈
𝒞
s
(
Θ
,
ℬ
(
X
)
), then there exists a unique linear skew-product flow
denoted π
P
=
(
Φ
P
,
σ
) on X
×
Θ such that(4.1)
Φ
P
(
θ
,
t
)
x
=
Φ
(
θ
,
t
)
x
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
P
(
σ
(
θ
,
s
)
)
Φ
P
(
θ
,
s
)
x
d
sfor all (
x
,
θ
,
t
)
∈
X
×
Θ
×
ℝ
+ (see [26, Theorem 2.1]).LetX be a Banach
space, let Θ be a locally
compact metric space, and let π
=
(
Φ
,
σ
) be a linear
skew-product flow on ℰ
=
X
×
Θ. We consider the variational integral control system(S)
x
θ
(
t
;
x
0
,
u
)
=
Φ
(
θ
,
t
)
x
0
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
(S)with u
∈
L
loc
1
(
ℝ
+
,
X
) and x
0
∈
X.For everyP
∈
𝒞
s
(
Θ
,
ℬ
(
X
)
) we consider the
perturbed system(S,P)
x
θ
(
t
;
x
0
,
u
)
=
Φ
P
(
θ
,
t
)
x
0
+
∫
0
t
Φ
P
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
(S,P)with u
∈
L
loc
1
(
ℝ
+
,
X
) and x
0
∈
X.LetU
,
Y be Banach
spaces, let B
∈
𝒞
s
(
Θ
,
ℬ
(
U
,
X
)
) and let C
∈
𝒞
s
(
Θ
,
ℬ
(
X
,
Y
)
). We consider the system (
S
,
B
,
C
) described by
the following integral model:(4.2)
x
θ
(
t
,
x
0
,
u
)
=
Φ
(
θ
,
t
)
x
0
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
B
(
σ
(
θ
,
s
)
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
,
y
θ
(
t
,
x
0
,
u
)
=
C
(
σ
(
θ
,
t
)
)
x
θ
(
t
,
x
0
,
u
)
,
t
≥
0,
θ
∈
Θ
,where x
0
∈
X and u
∈
L
loc
1
(
ℝ
+
,
U
).Throughout this section, we suppose that (S) is uniformly
exponentially stable. The stability
radius of (S) with respect to
the perturbation structure (
B
,
C
) is defined by(4.3)
r
stab
(
S
,
B
,
C
)
=
sup
{
r
≥
0
:
∀
Δ
∈
𝒞
s
(
Θ
,
ℬ
(
Y
,
U
)
)
with
∥
|
Δ
∥
|
≤
r
⇒
the system
(
S
,
B
Δ
C
)
is uniformly exponentially stable
}
.Let𝒱
(
ℝ
+
) be the class of
all Banach function spaces B
∈
ℛ
(
ℝ
+
) with the
property that for every u
∈
B, |
u
χ
[
0
,
n
)
|
B
→
|
u
|
B as n
→
∞.Remark 4.2.
It is easy to
verify that the Orlicz spaces belong to the class𝒱
(
ℝ
+
).LetV
∈
𝒱
(
ℝ
+
). Since the system (S) is uniformly
exponentially stable from Remark 3.20 we have that for every θ
∈
Θ, the linear operators(4.4)
P
V
θ
:
V
(
ℝ
+
,
X
)
→
V
(
ℝ
+
,
X
)
,
(
P
V
θ
u
)
(
t
)
=
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
sare bounded and sup
θ
∈
Θ
∥
P
V
θ
∥
<
∞.
Lemma 4.3.
For everyθ
∈
Θ, the linear operators,(4.5)
B
V
θ
:
V
(
ℝ
+
,
U
)
→
V
(
ℝ
+
,
X
)
,
(
B
V
θ
u
)
(
t
)
=
B
(
σ
(
θ
,
t
)
)
u
(
t
)
,
C
V
θ
:
V
(
ℝ
+
,
X
)
→
V
(
ℝ
+
,
Y
)
,
(
C
V
θ
u
)
(
t
)
=
C
(
σ
(
θ
,
t
)
)
u
(
t
)
, are bounded. Moreover, sup
θ
∈
Θ
∥
B
V
θ
∥
≤
∥
|
B
∥
| and sup
θ
∈
Θ
∥
C
V
θ
∥
≤
∥
|
C
∥
|.Proof.
Letθ
∈
Θ. For every u
∈
V
(
ℝ
+
,
U
) we have that(4.6)
∥
(
B
V
θ
u
)
(
t
)
∥
≤
∥
B
(
σ
(
θ
,
t
)
)
∥
∥
u
(
t
)
∥
≤
∥
|
B
∥
|
∥
u
(
t
)
∥
,
∀
t
≥
0which implies that ∥
B
V
θ
u
∥
V
(
ℝ
+
,
X
)
≤
∥
|
B
∥
|
∥
u
∥
V
(
ℝ
+
,
U
)
. This shows that ∥
B
V
θ
∥
≤
∥
|
B
∥
|, for all θ
∈
Θ. Similarly, we obtain that ∥
C
V
θ
∥
≤
∥
|
C
∥
|, for all θ
∈
Θ.Definition 4.4.
For everyθ
∈
Θ, we consider the bounded linear operators(4.7)
L
V
θ
:
V
(
ℝ
+
,
U
)
→
V
(
ℝ
+
,
Y
)
,
L
V
θ
=
C
V
θ
P
V
θ
B
V
θ
.The family {
L
V
θ
}
θ
∈
Θ is called the family of input-output operators associated to the system (
S
,
B
,
C
). In all what follows we denote(4.8)
α
V
(
S
,
B
,
C
)
:
=
sup
θ
∈
Θ
∥
L
V
θ
∥
.Remark 4.5.
Using Remark3.20 and Lemma 4.3 we have that α
V
(
S
,
B
,
C
)
<
∞.For everyΔ
∈
𝒞
s
(
Θ
,
ℬ
(
Y
,
U
)
) and every θ
∈
Θ, we consider the bounded linear operator(4.9)
Δ
V
θ
:
V
(
ℝ
+
,
Y
)
→
V
(
ℝ
+
,
U
)
,
(
Δ
V
θ
u
)
(
t
)
=
Δ
(
σ
(
θ
,
t
)
)
u
(
t
)
.Then we have that ∥
Δ
V
θ
∥
≤
∥
|
Δ
∥
|, for all θ
∈
Θ.In what follows we suppose that there ism
>
0 such that ∥
C
(
θ
)
x
∥
Y
≥
m
∥
x
∥
X, for all (
x
,
θ
)
∈
X
×
Θ.
Theorem 4.6.
LetΔ
∈
𝒞
s
(
Θ
,
ℬ
(
Y
,
U
)
) with ∥
|
Δ
∥
|
<
1
/
α
V
(
S
,
B
,
C
). Then, for every (
x
,
θ
)
∈
X
×
Θ, the function(4.10)
f
x
,
θ
:
ℝ
+
→
X
,
f
x
,
θ
(
t
)
=
Φ
B
Δ
C
(
θ
,
t
)
xbelongs to V
(
ℝ
+
,
X
).Proof.
Letγ
V
=
α
V
(
S
,
B
,
C
)
∥
|
Δ
∥
|. Let K
,
ν
>
0 be such that ∥
Φ
(
θ
,
t
)
∥
≤
K
e
−
ν
t, for all (
θ
,
t
)
∈
Θ
×
ℝ
+.
Let(
x
,
θ
)
∈
ℰ. For every n
∈
ℕ
* let(4.11)
f
n
:
ℝ
+
→
X
,
f
n
(
t
)
=
χ
[
0
,
n
)
(
t
)
f
x
,
θ
(
t
)
,
g
n
:
ℝ
+
→
X
,
g
n
(
t
)
=
C
(
σ
(
θ
,
t
)
)
f
n
(
t
)
. Using Remark 4.1, we have that f
n
(
t
)
=
χ
[
0
,
n
)
(
t
)
Φ
(
θ
,
t
)
x
+
(
P
V
θ
B
V
θ
Δ
V
θ
g
n
)
(
t
), for all t
∈
[
0
,
n
), which implies that(4.12)
g
n
(
t
)
=
χ
[
0
,
n
)
(
t
)
C
(
σ
(
θ
,
t
)
)
Φ
(
θ
,
t
)
x
+
(
L
V
θ
Δ
V
θ
g
n
)
(
t
)
,
∀
t
∈
[
0
,
n
)
.Then, for t
∈
[
0
,
n
) we have that(4.13)
∥
g
n
(
t
)
∥
Y
≤
∥
|
C
∥
|
K
∥
x
∥
e
−
ν
t
+
∥
L
V
θ
∥
∥
Δ
V
θ
∥
∥
g
n
(
t
)
∥
Y
≤
∥
|
C
∥
|
K
∥
x
∥
e
−
ν
t
+
γ
V
∥
g
n
(
t
)
∥
Y Since g
n
(
t
)
=
0 for t
≥
n, we deduce that(4.14)
∥
g
n
(
t
)
∥
Y
≤
∥
|
C
∥
|
K
∥
x
∥
e
−
ν
t
+
γ
V
∥
g
n
(
t
)
∥
Y
,
∀
t
≥
0.
Lete
ν
:
ℝ
+
→
ℝ
+, e
ν
(
t
)
=
e
−
ν
t. From V
∈
𝒱
(
ℝ
+
) and using Lemma
2.12 we have that e
ν
∈
V. Then, from (4.14) it follows that(4.15)
∥
g
n
∥
V
(
ℝ
+
,
Y
)
≤
δ
x
|
e
ν
|
V
+
γ
V
∥
g
n
∥
V
(
ℝ
+
,
Y
)
,where δ
x
=
∥
|
C
∥
|
K
∥
x
∥
. The above inequality shows that(4.16)
∥
g
n
∥
V
(
ℝ
+
,
Y
)
≤
δ
x
|
e
ν
|
V
1
−
γ
V
,
∀
n
∈
ℕ
*
.Since V
∈
𝒱
(
ℝ
+
) from (4.16) we
deduce that the function g
x
,
θ
:
ℝ
+
→
Y, g
x
,
θ
(
t
)
=
C
(
σ
(
θ
,
t
)
)
f
x
,
θ
(
t
) belongs to V
(
ℝ
+
,
Y
). Then, from m
∥
f
x
,
θ
(
t
)
∥
X
≤
∥
g
x
,
θ
(
t
)
∥
Y, for all t
≥
0, we obtain that f
x
,
θ
∈
V
(
ℝ
+
,
X
).Theorem 4.7.
The
following estimation holds:(4.17)
r
stab
(
S
,
B
,
C
)
≥
1
α
V
(
S
,
B
,
C
)
.Proof.
LetΔ
∈
𝒞
s
(
Θ
,
ℬ
(
Y
,
U
)
) be such that ∥
|
Δ
∥
|
<
1
/
α
V
(
S
,
B
,
C
). Let γ
V
=
α
V
(
S
,
B
,
C
)
∥
|
Δ
∥
|.
We prove that the perturbed system(
S
,
B
Δ
C
) is completely   (
V
(
ℝ
+
,
X
)
,
V
(
ℝ
+
,
X
)
) stable. Let v
∈
V
(
ℝ
+
,
X
) and θ
∈
Θ. The corresponding solution of the system (
S
,
B
Δ
C
) is(4.18)
x
θ
(
·
;
0
,
v
)
:
ℝ
+
→
X
,
x
θ
(
t
;
0
,
v
)
=
∫
0
t
Φ
B
Δ
C
(
σ
(
θ
,
s
)
,
t
−
s
)
v
(
s
)
d
s
.Let n
∈
ℕ
*. We set v
n
=
v
χ
[
0
,
n
) and x
n
=
x
θ
(
n
;
0
,
v
). Then, we have that(4.19)
x
θ
(
t
;
0
,
v
n
)
=
Φ
B
Δ
C
(
σ
(
θ
,
n
)
,
t
−
n
)
x
n
,
∀
t
≥
n
.From Theorem 4.6 we have that
the function(4.20)
φ
:
ℝ
+
→
X
,
φ
(
t
)
=
Φ
(
σ
(
θ
,
n
)
,
t
)
x
nbelongs to V
(
ℝ
+
,
X
). Since V is invariant to
translations, it follows that(4.21)
ψ
:
ℝ
+
→
X
,
ψ
(
t
)
=
{
Φ
(
σ
(
θ
,
n
)
,
t
−
n
)
x
n
,
t
≥
n
,
0
,
t
∈
[
0
,
n
)
,belongs to V
(
ℝ
+
,
X
).
Sincex
θ
(
·
;
0
,
v
n
) is continuous,
setting M
n
=
sup
t
∈
[
0
,
n
]
∥
x
θ
(
t
;
0
,
v
n
)
∥ and using (4.19)
we deduce that(4.22)
∥
x
θ
(
t
;
0
,
v
n
)
∥
≤
M
n
χ
[
0
,
n
)
(
t
)
+
∥
ψ
(
t
)
∥
,
∀
t
≥
0which implies that x
θ
(
·
;
0
,
v
n
)
∈
V
(
ℝ
+
,
X
).
Using Remark4.1 and Fubini's theorem we deduce that(4.23)
(
C
V
θ
x
θ
(
·
;
0
,
v
n
)
)
(
t
)
=
(
C
V
θ
P
V
θ
v
n
)
(
t
)
+
(
L
V
θ
Δ
V
θ
C
V
θ
x
θ
(
·
;
0
,
v
n
)
)
(
t
)
,
∀
t
≥
0.This implies that(4.24)
∥
C
V
θ
x
θ
(
·
;
0
,
v
n
)
∥
V
(
ℝ
+
,
Y
)
≤
∥
|
C
∥
|
sup
θ
∈
Θ
∥
P
V
θ
∥
∥
v
∥
V
(
ℝ
+
,
X
)
+
γ
V
∥
C
V
θ
x
θ
(
·
;
0
,
v
n
)
∥
V
(
ℝ
+
,
Y
)
. Setting δ
=
∥
|
C
∥
|
sup
θ
∈
Θ
∥
P
V
θ
∥ it follows that(4.25)
∥
C
V
θ
x
θ
(
·
;
0
,
v
n
)
∥
V
(
ℝ
+
,
Y
)
≤
δ
1
−
γ
V
∥
v
∥
V
(
ℝ
+
,
X
)
.From ∥
C
V
θ
x
θ
(
·
;
0
,
v
n
)
∥
V
(
ℝ
+
,
Y
)
≥
m
∥
x
θ
(
·
;
0
,
v
n
)
∥
V
(
ℝ
+
,
X
), using (4.25) we deduce that(4.26)
∥
x
θ
(
·
;
0
,
v
n
)
∥
V
(
ℝ
+
,
X
)
≤
δ
m
(
1
−
γ
V
)
∥
v
∥
V
(
ℝ
+
,
X
)
.Taking into account that x
θ
(
t
;
0
,
v
)
=
x
θ
(
t
;
0
,
v
n
), for all t
∈
[
0
,
n
), from (4.26) it follows that(4.27)
∥
x
θ
(
·
;
0
,
v
)
χ
[
0
,
n
)
∥
V
(
ℝ
+
,
X
)
≤
δ
m
(
1
−
γ
V
)
∥
v
∥
V
(
ℝ
+
,
X
)
,
∀
n
∈
ℕ
*
.Since V
∈
𝒱
(
ℝ
+
) as n
→
∞ in (4.27), we
obtain that(4.28)
∥
x
θ
(
·
;
0
,
v
)
∥
V
(
ℝ
+
,
X
)
≤
δ
m
(
1
−
γ
V
)
∥
v
∥
V
(
ℝ
+
,
X
)
.Since δ
,
m
,
γ
V do not depend
on θ or v, we deduce that the system (
S
,
B
Δ
C
) is completely (
V
(
ℝ
+
,
X
)
,
V
(
ℝ
+
,
X
)
) stable. By
applying Corollary 3.19, we have that the system (
S
,
B
Δ
C
) is uniformly
exponentially stable and the proof is complete.The main result of this section is  the
following.Theorem 4.8.
The following estimation holds:(4.29)
r
stab
(
S
,
B
,
C
)
≥
sup
V
∈
𝒱
(
ℝ
+
)
1
α
V
(
S
,
B
,
C
)
.Proof.
This follows from Theorem4.7.A lower bound for the stability radius of (S) with respect to
the perturbation structure (
B
,
C
) in terms of L
p-spaces
is given by the following.Corollary 4.9.
The following estimation holds:(4.30)
r
stab
(
S
,
B
,
C
)
≥
sup
p
∈
[
1
,
∞
]
1
α
L
p
(
S
,
B
,
C
)
.


## 5. Applications for nonautonomous systems

In this section, we apply the main results of this
paper in order to obtain characterizations for uniform exponential stability of
nonautonomous systems and deduce lower bound for the stability radius of such
systems.LetX be a Banach
space and let I
d be the identity
operator on X.
Definition 5.1.
A family𝒰
=
{
U
(
t
,
s
)
}
t
≥
s
≥
0
⊂
ℬ
(
X
) is called an evolution family if the following
properties hold:(i)
U
(
t
,
t
)
=
I
d and U
(
t
,
s
)
U
(
s
,
t
0
)
=
U
(
t
,
t
0
), for all t
≥
s
≥
t
0
≥
0;
(ii)
there areM
≥
1 and ω
>
0 such that ∥
U
(
t
,
s
)
∥
≤
M
e
ω
(
t
−
s
), for all t
≥
s
≥
0;
(iii)
for everyx
∈
X the mapping (
t
,
s
)
↦
U
(
t
,
s
)
x is
continuous.Let𝒰
=
{
U
(
t
,
s
)
}
t
≥
s
≥
0 be an evolution
family on X. We consider  the nonautonomous integral control
system(S
U)
x
s
(
t
;
x
0
,
u
)
=
U
(
t
,
s
)
x
0
+
∫
s
t
U
(
t
,
τ
)
u
(
τ
)
d
τ
,
t
≥
s
,
s
≥
0with u
∈
L
loc
1
(
ℝ
+
,
X
) and x
0
∈
X.
Definition 5.2.
The system (S
U) is said to be uniformly exponentially stable if there
are K
,
ν
>
0 such that ∥
x
s
(
t
;
x
0
,0
)
∥
≤
K
e
−
ν
(
t
−
s
)
∥
x
0
∥, for all t
≥
s
≥
0 and all x
0
∈
X.Definition 5.3.
LetI, O be two Banach
function spaces with I
,
O
∈
𝒯
(
ℝ
+
). The system (S
U) is said to be (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
)stable if the following properties hold:(i)
for everyu
∈
C
0
c
(
ℝ
+
,
X
) the solution x
0
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
);
(ii)
there isλ
>
0 such that ∥
x
0
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
λ
∥
u
∥
I
(
ℝ
+
,
X
), for all u
∈
C
0
c
(
ℝ
+
,
X
).Lemma 5.4.
LetB
∈
𝒯
(
ℝ
+
), u
∈
B, and t
>
0. Then, the function u
t
:
ℝ
+
→
ℝ, u
t
(
s
)
=
u
(
s
+
t
) belongs to B and |
u
t
|
B
≤
|
u
|
B.Proof.
Let(5.1)
v
:
ℝ
+
→
ℝ
,
v
(
s
)
=
{
u
(
s
)
,
s
≥
t
,
0
,
s
∈
[
0
,
t
)
.Then v
∈
B and |
v
|
B
≤
|
u
|
B. Using the invariance to translations of B, we obtain that u
t
∈
B and |
u
t
|
B
=
|
v
|
B, which concludes the proof.The first main result of this section is as
follows.Theorem 5.5.
LetI
,
O
∈
𝒯
(
ℝ
+
) be such that I
∈
ℒ
(
ℝ
+
) and O
∈
𝒬
(
ℝ
+
). Then, the following assertions hold:(i)
if the system (S
U) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then (S
U) is uniformly
exponentially stable;
(ii)
ifI
⊂
O and one of the
spaces I or O belongs to ℛ
(
ℝ
+
), then the system (S
U) is uniformly
exponentially stable if and only if the system (S
U) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable.Proof.
(i) Letλ
>
0 be given by
Definition 5.3.
LetΘ
=
ℝ
+ and let σ
:
Θ
×
ℝ
+
→
Θ, σ
(
θ
,
t
)
=
θ
+
t. For every (
θ
,
t
)
∈
Θ
×
ℝ
+, let Φ
(
θ
,
t
)
=
U
(
t
+
θ
,
θ
). Then π
=
(
Φ
,
σ
) is a linear
skew-product flow on X
×
Θ. We consider the variational integral control system(S)
x
θ
(
t
;
x
0
,
u
)
=
Φ
(
θ
,
t
)
x
0
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
.
(S)We prove that (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable.
Indeed, letu
∈
C
0
c
(
ℝ
+
,
X
) and let θ
∈
Θ. We consider the function(5.2)
u
θ
:
ℝ
+
→
X
,
u
θ
(
s
)
=
{
u
(
s
−
θ
)
,
s
≥
θ
,
0
,
s
∈
[
0
,
θ
)
.Then u
θ
∈
C
0
c
(
ℝ
+
,
X
) and from I
∈
𝒯
(
ℝ
+
) we have that ∥
u
θ
∥
I
(
ℝ
+
,
X
)
=
∥
u
∥
I
(
ℝ
+
,
X
). From hypothesis we have that the function(5.3)
x
0
(
t
;
0
,
u
θ
)
=
∫
0
t
U
(
t
,
τ
)
u
θ
(
τ
)
d
τ
,
t
≥
0belongs to O
(
ℝ
+
,
X
). Because O
(
ℝ
+
,
X
)
∈
𝒯
(
ℝ
+
), using Lemma 5.4 we have that the function(5.4)
φ
:
ℝ
+
→
X
,
φ
(
t
)
=
x
0
(
t
+
θ
;
0
,
u
θ
)belongs to O
(
ℝ
+
,
X
) and ∥
φ
∥
O
(
ℝ
+
,
X
)
≤
∥
x
0
(
·
;
0
,
u
θ
)
∥
O
(
ℝ
+
,
X
). Observing that(5.5)
φ
(
t
)
=
∫
0
t
+
θ
U
(
t
+
θ
,
τ
)
u
θ
(
τ
)
d
τ
=
∫
θ
t
+
θ
U
(
t
+
θ
,
τ
)
u
(
τ
−
θ
)
d
τ
=
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
=
x
θ
(
t
;
0
,
u
)
,
∀
t
≥
0, it follows that(5.6)
∥
x
θ
(
·
;
0
,
u
)
∥
O
(
ℝ
+
,
X
)
≤
∥
x
0
(
·
;
0
,
u
θ
)
∥
O
(
ℝ
+
,
X
)
≤
λ
∥
u
θ
∥
I
(
ℝ
+
,
X
)
=
λ
∥
u
∥
I
(
ℝ
+
,
X
)
.Since u
∈
C
0
c
(
ℝ
+
,
X
) and θ
∈
Θ were arbitrary,
we deduce that the system (S) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable. By
applying Theorem 3.9(i) we deduce that the system (S) is uniformly
exponentially stable, so there are K
,
ν
>
0 such that(5.7)
∥
Φ
(
θ
,
t
)
x
0
∥
≤
K
e
−
ν
t
∥
x
0
∥
,
∀
t
≥
0,
∀
x
0
∈
X
.This implies that(5.8)
∥
x
s
(
t
;
x
0
,0
)
∥
=
∥
U
(
t
,
s
)
x
0
∥
=
∥
Φ
(
s
,
t
−
s
)
x
0
∥
≤
K
e
−
ν
(
t
−
s
)
∥
x
0
∥
,
∀
t
≥
s
≥
0,
∀
x
0
∈
X
. In conclusion, the system (S
U) is uniformly
exponentially stable.
(ii)Necessity follows using similar arguments
as those used in the proof of Theorem 3.9. Sufficiencyfollows from (i).Remark 5.6.
From Example3.10 we deduce
that the result given by Theorem 5.5 is the most general in this topic.
Precisely, if I
∉
ℒ
(
ℝ
+
) or O
∉
𝒬
(
ℝ
+
), then the (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stability of
the system (S
U) does not assure
the uniform exponential stability of (S
U).
Definition 5.7.
LetI
,
O
∈
𝒯
(
ℝ
+
). The system (S
U) is said to be completely(
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
)stable if for every u
∈
I
(
ℝ
+
,
X
), the solution x
0
(
·
;
0
,
u
)
∈
O
(
ℝ
+
,
X
).Remark 5.8.
If the system (S
U) is completely (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then it
makes sense to consider the linear operator(5.9)
𝒫
:
I
(
ℝ
+
,
X
)
→
O
(
ℝ
+
,
X
)
,
𝒫
(
u
)
=
x
0
(
·
;
0
,
u
)
.It is easy to see that 𝒫 is closed, so it is bounded.Remark 5.9.
If the system (S
U) is completely (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then (S
U) is (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable (with λ
=
∥
𝒫
∥).As an application of Theorem5.5 and using Remark 5.9
we obtain the following. Theorem 5.10.
LetI
,
O
∈
𝒯
(
ℝ
+
) be such that I
∈
ℒ
(
ℝ
+
) and O
∈
𝒬
(
ℝ
+
). Then, the following assertions hold:(i)
if the system (S
U) is completely (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable, then (S
U) is uniformly
exponentially stable;
(ii)
ifI
⊂
O and one of the
spaces I or O belongs to ℛ
(
ℝ
+
), then the system (S
U) is uniformly
exponentially stable if and only if the system (S
U) is completely (
I
(
ℝ
+
,
X
)
,
O
(
ℝ
+
,
X
)
) stable.Corollary 5.11.
IfW
∈
ℛ
(
ℝ
+
), then the system (S
U) is uniformly
exponentially stable if and only if the system (S
U) is completely   (
W
(
ℝ
+
,
X
)
,
W
(
ℝ
+
,
X
)
) stable.Corollary 5.12.
Letp
,
q
∈
[
1
,
∞
] with (
p
,
q
)
≠
(
1
,
∞
). The following assertions hold:(i)
if the system (S
U) is completely (
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable, then (S
U) is uniformly
exponentially stable;
(ii)
ifp
≤
q, then the system (S
U) is uniformly
exponentially stable if and only if it is completely (
L
p
(
ℝ
+
,
X
)
,
L
q
(
ℝ
+
,
X
)
) stable.Remark 5.13.
Forp
,
q
∈
[
1
,
∞
), Corollary 5.12(i) was firstly proved  by Datko in
[16]. Recently, for p
=
q
∈
[
1
,
∞
), Corollary 5.11 was proved by Clark et al. in
[11], using evolution
semigroups techniques. A different proof for this corollary was given by Megan
et al. in [23]. The
autonomous case was treated by van Neerven in [30], for p
=
q
∈
[
1
,
∞
).Corollary 5.14.
LetU
,
Y
∈
{
C
b
(
ℝ
+
,
ℝ
)
,
C
0
(
ℝ
+
,
ℝ
)
,
C
00
(
ℝ
+
,
ℝ
)
} with U
⊂
Y. The system (S
U) is uniformly
exponentially stable if and only if the system (S
U) is completely (
U
(
ℝ
+
,
X
)
,
Y
(
ℝ
+
,
X
)
) stable.Proof.
Necessity is immediate.
Sufficiency follows from Theorem 5.10 observing that the complete (
U
(
ℝ
+
,
X
)
,
Y
(
ℝ
+
,
X
)
) stability
implies the (
L
∞
(
ℝ
+
,
X
)
,
L
∞
(
ℝ
+
,
X
)
)stability of (S
U).Remark 5.15.
A distinct proof of Corollary5.14 was given in [23]. The case U
=
Y
=
C
0
(
ℝ
+
,
ℝ
) was obtained in
[11], reducing the
nonautonomous case to the autonomous case and using evolution semigroups.An interesting application of the results obtained in
the fourth section is the study of the persistence of stability properties for
the case of nonautonomous systems. We mention that the nonautonomous case was
treated by Hinrichsen et al. in valuable papers (see [17–19, 41]), which represent the
starting point for many research studies in this area.In what follows, we will deduce a lower bound for the
stability radius of nonautonomous systems, as a consequence of the behavior in
the variational case, providing the connections between these cases.LetX be a Banach
space and let 𝒰
=
{
U
(
t
,
s
)
}
t
≥
s
≥
0 be an evolution
family on X.Remark 5.16.
For everyP
∈
𝒞
s
(
ℝ
+
,
ℬ
(
X
)
) (see, e.g.,
[14]) there is a
unique evolution family 𝒰
P
=
{
U
P
(
t
,
s
)
}
t
≥
s
≥
0 such that(5.10)
U
P
(
t
,
s
)
x
=
U
(
t
,
s
)
x
+
∫
s
t
U
(
t
,
τ
)
P
(
τ
)
U
P
(
τ
,
s
)
x
d
τ
,
∀
t
≥
s
≥
0,
∀
x
∈
X
.We consider the nonautonomous integral control system(S
U)
x
s
(
t
;
x
0
,
u
)
=
U
(
t
,
s
)
x
0
+
∫
s
t
U
(
t
,
τ
)
u
(
τ
)
d
τ
,
t
≥
s
,
s
≥
0with u
∈
L
loc
1
(
ℝ
+
,
X
), x
0
∈
X, and respectively, for every P
∈
𝒞
s
(
ℝ
+
,
B
(
X
)
) we consider the
perturbed system(S
U
,
P)
x
s
(
t
;
x
0
,
u
)
=
U
P
(
t
,
s
)
x
0
+
∫
s
t
U
P
(
t
,
τ
)
u
(
τ
)
d
τ
,
t
≥
s
,
s
≥
0with u
∈
L
loc
1
(
ℝ
+
,
X
), x
0
∈
X.LetU, Y be Banach
spaces, let B
∈
𝒞
s
(
ℝ
+
,
ℬ
(
U
,
X
)
) and let C
∈
𝒞
s
(
ℝ
+
,
ℬ
(
X
,
Y
)
). We consider the system (
S
𝒰
,
B
,
C
) described by
the following integral model(5.11)
x
s
(
t
;
x
0
,
u
)
=
U
(
t
,
s
)
x
0
+
∫
s
t
U
(
t
,
τ
)
B
(
τ
)
u
(
τ
)
d
τ
,
t
≥
s
,
s
≥
0,
y
s
(
t
;
x
0
,
u
)
=
C
(
t
)
x
s
(
t
;
x
0
,
u
)
,
t
≥
s
,
s
≥
0,with u
∈
L
loc
1
(
ℝ
+
,
U
), x
0
∈
X.In what follows we suppose that (S
U) is uniformly
exponentially stable. The stability
radius of (S
U) with respect to
the perturbation structure (
B
,
C
) is defined by(5.12)
r
stab
(
S
𝒰
,
B
,
C
)
=
sup
{
r
≥
0
:
∀
Δ
∈
𝒞
s
(
ℝ
+
,
ℬ
(
Y
,
U
)
)
with
∥
|
Δ
∥
|
≤
r
⇒
(
S
𝒰
,
B
Δ
C
)
is uniformly exponentially stable
}
.LetV
∈
𝒱
(
ℝ
+
). Since the system (S
U) is uniformly
exponentially stable, we have that (S
U) is completely (
V
(
ℝ
+
,
X
)
,
V
(
ℝ
+
,
X
)
) stable, so it
makes sense to consider the linear operator(5.13)
𝒫
V
:
V
(
ℝ
+
,
X
)
→
V
(
ℝ
+
,
X
)
,
(
𝒫
V
u
)
(
t
)
=
∫
0
t
U
(
t
,
τ
)
u
(
τ
)
d
τ
.It is easy to verify that 𝒫
V is closed, so it is bounded.We consider the operators(5.14)
ℬ
V
:
V
(
ℝ
+
,
U
)
→
V
(
ℝ
+
,
X
)
,
(
ℬ
V
u
)
(
t
)
=
B
(
t
)
u
(
t
)
,
𝒞
V
:
V
(
ℝ
+
,
U
)
→
V
(
ℝ
+
,
X
)
,
(
𝒞
V
u
)
(
t
)
=
C
(
t
)
u
(
t
)
, and we note that ℬ
V, 𝒞
V are bounded
linear operators with ∥
ℬ
V
∥
≤
∥
|
B
∥
| and ∥
𝒞
V
∥
≤
∥
|
C
∥
|.
Definition 5.17.
The operator(5.15)
ℒ
V
:
V
(
ℝ
+
,
U
)
→
V
(
ℝ
+
,
Y
)
,
ℒ
V
(
u
)
=
(
𝒞
V
𝒫
V
ℬ
V
)
(
u
)
,is called the input-output operator associated with
the system (
S
𝒰
,
B
,
C
).In what follows, we suppose that there ism
>
0 such that ∥
C
(
t
)
x
∥
Y
≥
m
∥
x
∥, for all x
∈
X and all t
≥
0.The second main result of this section is
the following.Theorem 5.18.
The
following estimation holds:(5.16)
r
stab
(
S
𝒰
,
B
,
C
)
≥
1
∥
ℒ
V
∥
.Proof.
LetΔ
∈
𝒞
s
(
ℝ
+
,
ℬ
(
Y
,
U
)
) be such that ∥
|
Δ
∥
|
∥
ℒ
V
∥
<
1.
LetΘ
=
ℝ
+
,
σ
:
Θ
×
ℝ
+
→
Θ
,
σ
(
θ
,
t
)
=
θ
+
t and let Φ
(
θ
,
t
)
=
U
(
t
+
θ
,
θ
), for all θ
,
t
≥
0. Then π
=
(
Φ
,
σ
) is a linear
skew-product flow. We consider the variational integral control system(S)
x
θ
(
t
;
x
0
,
u
)
=
Φ
(
θ
,
t
)
x
0
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
(S)with u
∈
L
loc
1
(
ℝ
+
,
X
) and x
0
∈
X and,
respectively, the system (
S
,
B
,
C
) given
by(5.17)
x
θ
(
t
,
x
0
,
u
)
=
Φ
(
θ
,
t
)
x
0
+
∫
0
t
Φ
(
σ
(
θ
,
s
)
,
t
−
s
)
B
(
σ
(
θ
,
s
)
)
u
(
s
)
d
s
,
t
≥
0,
θ
∈
Θ
,
y
θ
(
t
,
x
0
,
u
)
=
C
(
σ
(
θ
,
t
)
)
x
θ
(
t
,
x
0
,
u
)
,
t
≥
0,
θ
∈
Θ
,where x
0
∈
X and u
∈
L
loc
1
(
ℝ
+
,
U
).
For everyθ
∈
Θ we associate
with the system (
S
,
B
,
C
) the operators P
V
θ, B
V
θ, C
V
θ, and L
V
θ according to
the definitions in Section 4. We denote by α
V
(
S
,
B
,
C
)
:
=
sup
θ
∈
Θ
∥
L
V
θ
∥.
We prove thatα
V
(
S
,
B
,
C
)
≤
∥
ℒ
V
∥. Indeed, let θ
∈
Θ and let u
∈
V
(
ℝ
+
,
U
). Since V is invariant to
translations, the function(5.18)
u
θ
:
ℝ
+
→
U
,
u
θ
(
t
)
=
{
u
(
t
−
θ
)
,
t
≥
θ
,
0
,
t
∈
[
0
,
θ
)
,belongs to V
(
ℝ
+
,
U
) and ∥
u
θ
∥
V
(
ℝ
+
,
U
)
=
∥
u
∥
V
(
ℝ
+
,
U
). In addition, using Lemma 5.4 we have that the
function(5.19)
φ
:
ℝ
+
→
Y
,
φ
(
t
)
=
(
ℒ
V
u
θ
)
(
t
+
θ
)
,belongs to V
(
ℝ
+
,
Y
) and ∥
φ
∥
V
(
ℝ
+
,
Y
)
≤
∥
ℒ
V
(
u
θ
)
∥
V
(
ℝ
+
,
Y
). We observe that(5.20)
L
V
θ
(
u
)
(
t
)
=
C
(
θ
+
t
)
∫
0
t
U
(
θ
+
t
,
θ
+
s
)
B
(
θ
+
s
)
u
(
s
)
d
s
=
C
(
θ
+
t
)
∫
θ
θ
+
t
U
(
θ
+
t
,
τ
)
B
(
τ
)
u
(
τ
−
θ
)
d
τ
=
C
(
θ
+
t
)
∫
0
θ
+
t
U
(
θ
+
t
,
τ
)
B
(
τ
)
u
θ
(
τ
)
d
τ
=
[
ℒ
V
(
u
θ
)
]
(
t
+
θ
)
=
φ
(
t
)
,
∀
t
≥
0.  This implies that(5.21)
∥
L
V
θ
(
u
)
∥
V
(
ℝ
+
,
Y
)
=
∥
φ
∥
V
(
ℝ
+
,
Y
)
≤
∥
ℒ
V
(
u
θ
)
∥
V
(
ℝ
+
,
Y
)
≤
∥
ℒ
V
∥
∥
u
θ
∥
V
(
ℝ
+
,
U
)
=
∥
ℒ
V
∥
∥
u
∥
V
(
ℝ
+
,
U
) Since u
∈
V
(
ℝ
+
,
U
) and θ
∈
Θ were arbitrary,
from the above inequality we obtain that ∥
L
V
θ
∥
≤
∥
ℒ
V
∥, for all θ
∈
Θ, so α
V
(
S
,
B
,
C
)
≤
∥
ℒ
V
∥.
Since∥
|
Δ
∥
|
∥
ℒ
V
∥
<
1, we have that ∥
|
Δ
∥
|
α
V
(
S
,
B
,
C
)
<
1. Then, from Theorem 4.7 it follows that the perturbed
system (
S
,
B
Δ
C
) is uniformly
exponentially stable, so there are K
,
ν
>
0 such that(5.22)
∥
Φ
B
Δ
C
(
θ
,
t
)
∥
≤
K
e
−
ν
t
,
∀
t
≥
0,
∀
θ
∈
Θ
.But(5.23)
Φ
B
Δ
C
(
θ
,
t
)
x
=
U
(
t
+
θ
,
θ
)
x
+
∫
0
t
U
(
θ
+
t
,
θ
+
s
)
(
B
Δ
C
)
(
θ
+
s
)
Φ
B
Δ
C
(
θ
,
s
)
x
d
s
,
∀
θ
,
t
≥
0,
∀
x
∈
X
. which is equivalent to(5.24)
Φ
B
Δ
C
(
θ
,
t
−
θ
)
x
=
U
(
t
,
θ
)
x
+
∫
θ
t
U
(
t
,
τ
)
(
B
Δ
C
)
(
τ
)
Φ
B
Δ
C
(
θ
,
τ
−
θ
)
x
d
τ
,
∀
t
≥
θ
≥
0,
∀
x
∈
X
. On the other hand, we have that(5.25)
U
B
Δ
C
(
t
,
θ
)
x
=
U
(
t
,
θ
)
x
+
∫
θ
t
U
(
t
,
τ
)
(
B
Δ
C
)
(
τ
)
U
B
Δ
C
(
τ
,
θ
)
x
d
τ
,
∀
t
≥
θ
≥
0,
∀
x
∈
X
.
Letθ
≥
0. For every T
>
0 and x
∈
X, let(5.26)
φ
x
,
T
:
[
θ
,
θ
+
T
]
→
X
,
φ
x
,
T
(
τ
)
=
Φ
B
Δ
C
(
θ
,
τ
−
θ
)
x
−
U
B
Δ
C
(
τ
,
θ
)
x
.If M
,
ω
>
0 are such that ∥
U
(
t
,
s
)
∥
≤
M
e
ω
(
t
−
s
), for all t
≥
s
≥
0, then from (5.24) and (5.25) it follows that(5.27)
∥
φ
x
,
T
(
t
)
∥
≤
K
∫
θ
t
∥
φ
x
,
T
(
τ
)
∥
d
τ
,
∀
t
∈
[
θ
,
θ
+
T
]
,where K
=
M
e
ω
T
∥
|
B
Δ
C
∥
|. From (5.27), using Gronwall's lemma, we deduce that φ
x
,
T
(
τ
)
=
0, for all τ
∈
[
θ
,
θ
+
T
]. This implies that(5.28)
Φ
B
Δ
C
(
θ
,
τ
−
θ
)
x
=
U
B
Δ
C
(
τ
,
θ
)
x
,
∀
τ
∈
[
θ
,
θ
+
T
]
,
∀
x
∈
X
.Since T
>
0, x
∈
X, and θ
∈
Θ were arbitrary,
we deduce that(5.29)
Φ
B
Δ
C
(
θ
,
τ
−
θ
)
=
U
B
Δ
C
(
τ
,
θ
)
,
∀
τ
≥
θ
≥
0.Then, from (5.22) we obtain that(5.30)
∥
U
B
Δ
C
(
τ
,
θ
)
∥
≤
K
e
−
ν
(
τ
−
θ
)
,
∀
τ
≥
θ
≥
0.In conclusion, the perturbed
system (
S
𝒰
,
B
Δ
C
) is uniformly
exponentially stable and the proof is complete.Concluding the above estimations, we deduce
the following.Theorem 5.19.
The
following estimation holds:(5.31)
r
stab
(
S
𝒰
,
B
,
C
)
≥
sup
V
∈
𝒱
(
ℝ
+
)
1
∥
ℒ
V
∥
.A lower bound for the stability radius of
nonautonomous systems in terms ofL
p-spaces
is given by the following.
Corollary 5.20.
The
following estimation holds:(5.32)
r
stab
(
S
𝒰
,
B
,
C
)
≥
sup
p
∈
[
1
,
∞
]
1
∥
ℒ
L
p
∥
.Remark 5.21.
For the casep
∈
[
1
,
∞
), the result given by Corollary 5.20 was firstly
proved in [19] (see
[19, Theorem 3.2]) for
a very general class of nonautonomous systems described by mild evolution
families. A distinct proof for Corollary 5.20 was presented in [11], for p
∈
[
1
,
∞
), using evolution semigroups techniques (see [11, Theorem 4.2]).Remark 5.22.
A distinct approach for the
estimation of the stability radius was given by Jacob in [20] for the case of
time-varying systems in finite dimensional spaces. There, the author considered
a time-varying system (
A
,
B
,
C
) with A
∈
L
loc
∞
(
[
0
,
∞
)
,
K
n
×
n
), B
∈
L
∞
(
[
0
,
∞
)
,
K
n
×
m
), C
∈
L
∞
(
[
0
,
∞
)
,
K
q
×
n
), where K
∈
{
ℝ
,
C
}. The family of input-output operators {
L
t
0
}
t
0
≥
0 is defined by(5.33)
(
L
t
0
)
u
(
t
)
:
=
C
(
t
)
∫
t
0
t
Φ
A
(
t
,
τ
)
B
(
τ
)
u
(
τ
)
d
τ
,
t
≥
t
0
≥
0,
u
∈
L
p
(
[
t
0
,
∞
)
,
K
m
)
,where {
Φ
A
(
t
,
s
)
}
t
≥
s
≥
0 denotes the
evolution family generated by A. In this case, the author proved that(5.34)
r
stab
(
A
,
B
,
C
)
=
sup
t
0
≥
0
1
∥
L
t
0
∥
.



---

*Source: 381791-2008-04-24.xml*