Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
NONUNIFORM
POLYNOMIAL
BEHAVIOR
FOR
LINEAR
RANDOM
ONE-SIDED
DISCRETE-TIME
SYSTEMS
∗
Oana
Albescu
†
Larisa
Elena
Biri¸
s
‡
Traian
Ceau¸
su
§
Nicolae
Marian
Seimeanu
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.105
Abstract
This
paper
investigates
the
concepts
of
nonuniform
polynomial
sta-
bility
and
instability
for
linear
random
one-sided
discrete-time
systems.
We
establish
necessary
and
sufficient
conditions
for
these
properties
and
provide
Lyapunov-type
characterizations
in
terms
of
Lyapunov
functions
and
Lyapunov
norms.
Keywords:
polynomial
stability,
polynomial
instability,
Lyapunov
func-
tions,
Lyapunov
norms.
MSC:
34D05,
93C55,
93D05.
∗
Accepted
for
publication
on
March
17,
2026
†
oana.albescu@unitbv.ro
,
Faculty
of
Mathematics
and
Computer
Science,
Transilva-
nia
University
of
Brasov,
Romania
‡
larisa.biris@e-uvt.ro
,
West
University
of
Timisoara,
Department
of
Mathematics,
Timisoara,
Romania
§
traian.ceausu@e-uvt.ro
,
West
University
of
Timisoara,
Department
of
Mathemat-
ics,
Timisoara,
Romania
nicusei@yahoo.com
,
West
University
of
Timisoara,
Department
of
Mathematics,
Timisoara,
Romania
105
Nonuniform
polynomial
behavior
106
1
Introduction
Time-varying
linear
systems
and
non-autonomous
difference
equations
play
a
fundamental
role
in
the
theory
of
discrete
dynamical
systems;
see,
for
in-
stance,
[
1
],
[
2
],
[
15
],
[
19
],
[
21
].
In
the
deterministic
framework,
qualitative
properties
such
as
exponential
stability,
splitting,
and
dichotomy
have
been
extensively
investigated.
The
skew-product
approach
has
proved
particu-
larly
effective
in
this
context,
providing
a
natural
setting
for
the
analysis
of
non-invertible
systems
and
non-autonomous
dynamics
(see
[
4
],
[
5
],
[
17
],
[
22
]).
Polynomial
dichotomies
and
related
asymptotic
behaviors
for
discrete
systems
on
the
half-line
were
further
studied
in
[
6
],
[
7
],
[
12
]
extending
the
classical
exponential
theory.
The
stochastic
extension
of
hyperbolic
theory
is
grounded
in
the
multiplicative
ergodic
theorem
of
Oseledets,
which
char-
acterizes
asymptotic
behavior
in
terms
of
Lyapunov
exponents
and
lays
the
foundation
for
nonuniform
exponential
phenomena.
Building
on
this
frame-
work,
Arnold
[
3
]
developed
the
theory
of
random
dynamical
systems
using
the
skew-product
formulation,
which
has
become
central
in
the
study
of
lin-
ear
random
cocycles.
Subsequent
contributions
have
addressed
Lyapunov
exponents,
topological
dynamics,
and
qualitative
properties
of
random
sys-
tems(see
[
8
],
[
10
]).
Exponential
stability
and
tempered
exponential
dichotomies
for
random
difference
equations
and
cocycles
were
investigated
in
[
11
],
[
20
],
[
25
].
While
the
nonuniform
exponential
theory
is
by
now
well
developed,
con-
siderably
less
attention
has
been
devoted
to
polynomial
rates
of
growth
and
decay
in
the
random
discrete
setting.
In
many
cases,
exponential
di-
chotomies
fail
to
occur,
whereas
polynomial
behavior
naturally
describes
the
asymptotic
dynamics.
In
the
deterministic
case,
nonuniform
polynomial
instability
and
Lyapu-
nov-type
techniques
have
been
analyzed
in
[
7
],
[
18
].
However,
a
system-
atic
treatment
of
nonuniform
polynomial
stability
and
instability
for
dis-
crete
random
semi-dynamical
systems,
formulated
within
the
skew-product
framework,
is
still
lacking.
The
present
paper
is
devoted
to
the
study
of
the
aforementioned
problem
within
this
framework.
We
investigate
nonuniform
polynomial
stability
and
nonuniform
polynomial
instability
for
linear
random
one-sided
discrete-time
systems.
We
establish
necessary
and
sufficient
conditions
for
these
proper-
ties
and
provide
Zabczyk-type
as
well
as
Lyapunov-type
characterizations
in
terms
of
Lyapunov
functions
and
Lyapunov
norms.
We
note
that,
for
polynomial
instability,
only
necessary
conditions
are
obtained.
Moreover,
we
analyze
the
corresponding
properties
for
the
adjoint
and
inverse
systems
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
107
associated
with
the
original
one.
The
results
obtained
here
broaden
the
scope
of
the
exponential
theory
and
contribute
to
a
deeper
understanding
of
nonuniform
behavior
in
random
dynamical
systems.
2
Preliminaries
Let
Z
+
be
the
set
of
all
positive
integers
and
(
X,
||
·
||
)
a
Banach
space.
By
B
(
X
)
we
denote
the
Banach
algebra
of
all
bounded
linear
operators
acting
from
X
into
X
.
Also,
by
(Ω
,
F
,
P
)
we
denote
a
probability
space.
We
consider
a
measurable
application
θ
n
:
Ω
→
Ω
,
for
all
n
∈
Z
+
.
This
application
preserves
the
probability
measure,
i.e.
P
◦
θ
n
=
P
,
thus
P
(
θ
n
B
)
=
P
(
B
)
,
for
all
B
∈
F
.
Moreover,
we
have
that
θ
0
=
I
Ω
and
θ
n
◦
θ
m
=
θ
n
+
m
=
θ
n
+
m
=
θ
m
◦
θ
n
,
for
all
n,
m
∈
Z
+
.
Finally,
we
point
out
that
the
application
Z
+
×
Ω
(
n,
ω
)
→
θ
n
ω
∈
Ω
is
measurable
for
all
n
∈
Z
+
.
For
simplicity,
the
family
of
applications
(
θ
n
)
n
∈
Z
+
will
be
denoted
by
θ
.
In
this
case,
for
the
measurable
family
θ
:
Ω
→
Ω
we
consider
the
metric
semidynamical
system
which
is
denoted
by
(Ω
,
F
,
Z
+
,
θ
).
The
application
ϕ
:
Ω
→
(0
,
+
∞
)
is
called
θ
−
invariant
if
verifies
the
property
ϕ
(
θ
t
ω
)
=
ϕ
(
ω
),
for
all
n
∈
Z
+
and
ω
∈
Ω.
We
consider
the
measurable
application
φ
:
Z
+
×
Ω
→
B
(
X
)
which
satisfies
the
properties:
(a)
φ
(0
,
ω
)
=
I
X
,
for
all
ω
∈
Ω;
(b)
φ
(
n
+
m,
ω
)
=
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
),
for
all
n,
m
∈
Z
+
and
ω
∈
Ω.
Obviously,
the
relation
(
b
)
implies
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
)
=
φ
(
n
+
m,
ω
)
=
φ
(
m
+
n,
ω
)
=
φ
(
m,
θ
n
ω
)
φ
(
n,
ω
)
,
for
all
n,
m
∈
Z
+
and
all
ω
∈
Ω.
Throughout
this
work,
the
pair
(
θ,
φ
)
is
used
to
denote
a
linear
random
one-sided
discrete-time
system.
3
Nonuniform
polynomial
stability
Definition
1.
We
say
that
a
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuniformly
polynomially
stable
if
there
exist
a
θ
-invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
and
a
function
N
:
Ω
→
[1
,
+
∞
)
such
that
||
φ
(
n,
ω
)
x
||
≤
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
,
(1)
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Nonuniform
polynomial
behavior
108
Proposition
1.
The
following
statements
are
equivalent:
(a)
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuni-
formly
polynomially
stable;
(b)
there
exist
a
function
N
:
Ω
→
[1
,
+
∞
)
and
a
θ
-
invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
such
that
||
φ
(
n,
ω
)
||
≤
N
(
ω
)(
n
+
1)
−
α
(
ω
)
,
(2)
for
all
(
n,
ω
)
∈
Z
+
×
Ω
;
(c)
there
exist
a
function
N
:
Ω
→
[1
,
+
∞
)
and
a
θ
-
invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
such
that
||
φ
(
n
+
m,
ω
)
x
||
≤
N
(
θ
m
ω
)(
n
+
1)
−
α
(
ω
)
||
φ
(
m,
ω
)
x
||
,
(3)
for
all
(
m,
n,
ω,
x
)
∈
Z
+
×
Z
+
×
Ω
×
X
;
Proof.
Let
(
n,
ω
)
∈
Z
+
×
Ω.
First,
one
can
easily
see
that
the
implication
(
a
)
⇒
(
b
)
can
be
verified
for
||
x
||
=
1.
We
have
||
φ
(
n,
ω
)
||
=
sup
||
x
||
=1
||
φ
(
n,
ω
)
x
||
≤
sup
||
x
||
=1
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
=
N
(
ω
)(
n
+
1)
−
α
(
ω
)
.
For
the
second
implication
(
b
)
⇒
(
a
),
from
(
2
),
we
obtain
||
φ
(
n,
ω
)
x
||
≤
||
φ
(
n,
ω
)
||
·
||
x
||
≤
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
,
for
all
x
∈
X.
To
prove
(
c
)
⇒
(
a
),
let
m
=
0
in
(
3
).
It
follows
that
||
φ
(
n,
ω
)
x
||
=
||
φ
(
n
+
0
,
ω
)
x
||
≤
N
(
θ
0
ω
)(
n
+
1)
−
α
(
ω
)
||
φ
(0
,
ω
)
x
||
=
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
.
For
the
implication
(
a
)
⇒
(
c
).
From
(
a
),
we
obtain
||
φ
(
n,
θ
m
ω
)
x
||
≤
N
(
θ
m
ω
)(
n
+
1)
−
α
(
θ
m
ω
)
||
x
||
.
Since
the
variable
α
is
θ
-invariant,
i.e.
α
(
θ
m
ω
)
=
α
(
ω
),
it
follows
that:
||
φ
(
n,
θ
m
ω
)
x
||
≤
N
(
θ
m
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
.
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
109
From
the
previous
inequality,
we
have
||
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
)
x
||
≤
N
(
θ
m
ω
)(
n
+
1)
−
α
(
ω
)
||
φ
(
m,
ω
)
x
||
.
Using
the
hypothesis,
it
results
||
φ
(
n
+
m,
ω
)
x
||
=
||
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
)
x
||
≤
N
(
θ
m
ω
)(
n
+
1)
−
α
(
ω
)
||
φ
(
m,
ω
)
x
||
,
for
all
(
m,
n,
ω,
x
)
∈
Z
+
×
Z
+
×
Ω
×
X
.
3.1
Zabczyk
type
theorems
Theorem
1.
Let
(
θ,
φ
)
be
a
linear
random
one-sided
discrete-time
system.
Then
(
θ,
φ
)
is
nonuniformly
polynomially
stable
if
and
only
if
there
exist
a
θ
-invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
and
two
functions
M,
N
:
Ω
→
[1
,
+
∞
)
such
that
+
∞
n
=
m
(
n
−
m
+
1)
β
(
ω
)
−
1
N
(
θ
m
ω
)
−
1
||
φ
(
n,
ω
)
x
||
dt
≤
M
(
ω
)
||
φ
(
m,
ω
)
x
||
,
(4)
for
all
(
m,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Proof.
Necessity:
Let
N
and
α
as
in
Definition
1
.
Let
β
:
Ω
→
(0
,
+
∞
)
a
θ
-invariant
random
variable
such
that
0
<
β
<
α
.
For
every
n
≥
m
we
have
||
φ
(
n,
ω
)
x
||
=
||
φ
(
n
−
m
+
m,
ω
)
x
||
=
||
φ
(
n
−
m,
θ
m
ω
)
φ
(
m,
ω
)
x
||
≤
N
(
θ
m
ω
)(
n
−
m
+
1)
−
α
(
ω
)
||
φ
(
m,
ω
)
x
||
.
Therefore
+
∞
n
=
m
(
n
−
m
+
1)
β
(
ω
)
−
1
N
(
θ
m
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
+
∞
n
=
m
(
n
−
m
+
1)
β
(
ω
)
−
1
−
α
(
ω
)
||
φ
(
m,
ω
)
x
||
=
||
φ
(
m,
ω
)
x
||
+
∞
n
=
m
1
(
n
−
m
+
1)
1+
α
(
ω
)
−
β
(
ω
)
=
M
(
ω
)
||
φ
(
m,
ω
)
x
||
.
Nonuniform
polynomial
behavior
110
Sufficiency:
We
assume
that
there
exist
the
functions
M,
N
:
Ω
→
[1
,
+
∞
)
and
a
θ
-invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
such
that
the
inequality
given
in
the
statement
holds.
For
m
=
0
we
have
+
∞
n
=0
(
n
+
1)
β
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
M
(
ω
)
||
x
||
.
We
aim
to
establish
the
existence
of
a
function
D
:
Ω
→
[1
,
+
∞
)
and
a
θ
-invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
such
that
||
φ
(
n,
ω
)
x
||
≤
D
(
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
holds,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
The
argument
is
proved
by
contradiction.
For
any
function
D
:
Ω
→
[1
,
+
∞
)
and
any
θ
-invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
we
have
||
φ
(
n,
ω
)
x
||
>
D
(
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
In
particular,
taking
D
(
ω
)
=
1
and
α
(
ω
)
=
β
(
ω
)
we
obtain
M
(
ω
)
||
x
||
≥
+
∞
n
=0
(
n
+
1)
β
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
>
>
D
(
ω
)
||
x
||
+
∞
n
=0
1
n
+
1
=
+
∞
.
This
leads
to
a
contradiction,
and
therefore
we
conclude
that
there
exist
a
function
D
:
Ω
→
[1
,
+
∞
)
and
a
θ
-invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
such
that
the
inequality
(
1
)
holds.
Theorem
2.
Let
(
θ,
φ
)
be
a
linear
random
one-sided
discrete-time
system.
Then
(
θ,
φ
)
is
nonuniformly
polynomially
stable
if
and
only
if
there
exist
a
θ
-invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
and
a
function
M
:
Ω
→
[1
,
+
∞
)
such
that
+
∞
n
=0
(
n
+
1)
β
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
M
(
ω
)
||
x
||
,
(5)
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
111
Proof.
Necessity:
We
consider
β
:
Ω
→
(0
,
+
∞
)
such
that
0
<
β
<
α
.
Then
+
∞
n
=0
(
n
+
1)
β
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
+
∞
n
=0
(
n
+
1)
β
(
ω
)
−
1
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
x
||
=
N
(
ω
)
||
x
||
+
∞
n
=0
1
(
n
+
1)
1+
α
(
ω
)
−
β
(
ω
)
=
M
(
ω
)
||
x
||
.
Sufficiency:
The
proof
is
analogous
to
that
of
the
sufficiency
part
of
the
previous
theorem
and
is
therefore
omitted.
3.2
Lyapunov
functions
Definition
2.
We
say
that
L
:
Z
+
×
Ω
×
X
→
R
+
is
a
Lyapunov
function
for
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
if
there
exist
a
function
D
:
Ω
→
[1
,
+
∞
)
and
a
θ
-invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
such
that
L
(
n,
ω,
x
)
≤
D
(
ω
)(1
+
n
)
−
α
(
ω
)
||
x
||
,
(6)
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Theorem
3.
The
following
statements
are
equivalent:
(a)
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuni-
formly
polynomially
stable;
(b)
there
exist
a
Lyapunov
function
L
:
Z
+
×
Ω
×
X
→
R
+
and
a
θ
-invariant
random
variable
η
:
Ω
→
(0
,
+
∞
)
such
that
L
(0
,
ω,
x
)
−
L
(
n,
ω,
x
)(1
+
n
)
η
(
ω
)
≥
n
−
1
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
,
(7)
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X
;
(c)
there
exist
a
Lyapunov
function
L
1
:
Z
+
×
Ω
×
X
→
R
+
and
a
θ
-
invariant
random
variable,
η
:
Ω
→
(0
,
+
∞
)
such
that
L
1
(
n,
ω,
x
)
+
n
−
1
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
L
1
(0
,
ω,
x
)
,
(8)
Nonuniform
polynomial
behavior
112
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X
.
Proof.
To
prove
the
implication
(
a
)
⇒
(
b
),
we
consider
the
θ
-invariant
ran-
dom
variable
β
:
Ω
→
(0
,
+
∞
)
with
0
<
β
<
α
.
We
define
L
:
Z
+
×
Ω
×
X
→
R
+
by
L
(
m,
ω,
x
)
=
(1
+
m
)
−
β
(
ω
)
+
∞
k
=
m
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
,
for
all
(
m,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Then
L
(
n,
ω,
x
)
=
(1
+
n
)
−
β
(
ω
)
+
∞
k
=
n
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
(1
+
n
)
−
β
(
ω
)
+
∞
k
=
n
(1
+
k
)
β
(
ω
)
−
1
N
(
ω
)(1
+
k
)
−
α
(
ω
)
||
x
||
=
(1
+
n
)
−
β
(
ω
)
N
(
ω
)
||
x
||
+
∞
k
=
n
1
(1
+
k
)
1+
α
(
ω
)
−
β
(
ω
)
≤
(1
+
n
)
−
β
(
ω
)
N
(
ω
)
||
x
||
+
∞
k
=0
1
(1
+
k
)
1+
α
(
ω
)
−
β
(
ω
)
=
D
(
ω
)(1
+
n
)
−
β
(
ω
)
||
x
||
.
We
obtain
that
L
(
n,
ω,
x
)(1
+
n
)
β
(
ω
)
=
+
∞
k
=
n
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X
.
Hence
L
(
n,
ω,
x
)(1
+
n
)
β
(
ω
)
+
n
−
1
k
=0
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
=
=
+
∞
k
=0
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
=
L
(0
,
ω,
x
)
.
To
prove
the
converse
implication
(
b
)
⇒
(
a
),
we
deduce
from
(
7
)
that
n
−
1
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
L
(0
,
ω,
x
)
.
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
113
Taking
the
limit
as
n
→
+
∞
we
conclude
that
+
∞
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
L
(0
,
ω,
x
)
.
Using
(
6
),
this
yields
to
+
∞
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
L
(0
,
ω,
x
)
≤
D
(
ω
)
||
x
||
.
According
to
Theorem
2
it
follows
that
linear
random
one-sided
discrete-
time
system
(
θ,
φ
)
is
nonuniformly
polynomially
stable.
To
prove
the
implication
(
a
)
⇒
(
c
),
we
consider
β
:
Ω
→
(0
,
+
∞
)
such
that
0
<
2
β
<
α
.
We
define
L
1
:
Z
+
×
Ω
×
X
→
R
+
by
L
1
(
m,
ω,
x
)
=
+
∞
k
=
m
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
,
for
all
(
m,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Then
L
1
:
Z
+
×
Ω
×
X
→
R
+
is
a
Lyapunov
function
since
L
1
(
n,
ω,
x
)
=
(1
+
n
)
−
β
(
ω
)
+
∞
k
=
n
(1
+
n
)
β
(
ω
)
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
(1
+
n
)
−
β
(
ω
)
+
∞
k
=
n
(1
+
n
)
β
(
ω
)
(1
+
k
)
β
(
ω
)
−
1
N
(
ω
)(1
+
k
)
α
(
ω
)
||
x
||
=
N
(
ω
)
||
x
||
(1
+
n
)
−
β
(
ω
)
+
∞
k
=
n
(1
+
k
)
2
β
(
ω
)
1
(1
+
k
)
1+
α
(
ω
)
≤
N
(
ω
)
||
x
||
(1
+
n
)
−
β
(
ω
)
+
∞
k
=0
1
(1
+
k
)
1+
α
(
ω
)
−
2
β
(
ω
)
≤
D
(
ω
)
||
x
||
(1
+
n
)
−
β
(
ω
)
.
The
function
L
1
:
Z
+
×
Ω
×
X
→
R
satisfies
(
8
)
since
L
1
(
n,
ω,
x
)
+
+
∞
k
=0
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
=
=
+
∞
k
=
n
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
+
n
−
1
k
=0
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
Nonuniform
polynomial
behavior
114
=
+
∞
k
=0
(1
+
k
)
β
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
=
L
1
(0
,
ω,
x
)
.
Finally,
we
prove
(
c
)
⇒
(
a
).
From
(
8
)
we
have
that
n
−
1
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
L
1
(0
,
ω,
x
)
.
Taking
the
limit
as
n
→
+
∞
we
conclude
that
+
∞
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
L
1
(0
,
ω,
x
)
.
Using
(
6
),
this
yields
to
+
∞
k
=0
(1
+
k
)
η
(
ω
)
−
1
||
φ
(
k,
ω
)
x
||
≤
L
1
(0
,
ω,
x
)
≤
D
(
ω
)
||
x
||
.
According
to
Theorem
2
it
follows
that
the
linear
random
one-sided
discrete-
time
system
(
θ,
φ
)
is
nonuniformly
polynomially
stable.
3.3
Lyapunov
norms
Definition
3.
We
say
that
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
has
polynomial
growth
if
there
exist
a
θ
-invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
and
a
function
M
:
Ω
→
[1
,
+
∞
)
such
that
||
φ
(
n,
ω
)
x
||
≤
M
(
ω
)(
n
+
1)
β
(
ω
)
||
x
||
,
(9)
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Proposition
2.
The
following
statements
are
equivalent:
(a)
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
has
polynomial
growth;
(b)
there
exist
a
function
M
:
Ω
→
[1
,
+
∞
)
and
a
θ
-
invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
such
that
||
φ
(
n,
ω
)
||
≤
M
(
ω
)(
n
+
1)
β
(
ω
)
,
(10)
for
all
(
n,
ω
)
∈
Z
+
×
Ω
;
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
115
(c)
there
exist
a
function
M
:
Ω
→
[1
,
+
∞
)
and
a
θ
-
invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
such
that
||
φ
(
n
+
m,
ω
)
x
||
≤
M
(
θ
m
ω
)(
n
+
1)
β
(
ω
)
||
φ
(
m,
ω
)
x
||
,
(11)
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Proof.
Let
(
n,
ω
)
∈
Z
+
×
Ω.
Now
we
prove
the
implication
(
a
)
⇒
(
b
).
If
||
φ
(
n,
ω
)
x
||
≤
M
(
ω
)(
n
+
1)
β
(
ω
)
||
x
||
,
then
by
taking
the
supremum
over
||
x
||
=
1
we
obtain
||
φ
(
n,
ω
)
||
=
sup
||
x
||
=1
||
φ
(
n,
ω
)
x
||
≤
sup
||
x
||
=1
M
(
ω
)(
n
+
1)
β
(
ω
)
||
x
||
=
M
(
ω
)(
n
+
1)
β
(
ω
)
.
For
(
b
)
⇒
(
a
),
if
(
10
)
holds,
then
||
φ
(
n,
ω
)
x
||
≤
||
φ
(
n,
ω
)
||
·
||
x
||
≤
M
(
ω
)(
n
+
1)
β
(
ω
)
||
x
||
,
for
all
x
∈
X
.
We
establish
now
(
c
)
⇒
(
a
).
Let
m
=
0
in
(
11
).
It
follows
that
||
φ
(
n
+
0
,
ω
)
x
||
=
||
φ
(
n,
ω
)
x
||
≤
M
(
θ
0
ω
)(
n
+
1)
β
(
ω
)
||
φ
(0
,
ω
)
x
||
=
M
(
ω
)(
n
+
1)
β
(
ω
)
||
x
||
.
For
the
implication
(
a
)
⇒
(
c
),
we
note
that
||
φ
(
n,
θ
m
ω
)
x
||
≤
M
(
θ
m
ω
)(
n
+
1)
β
(
θ
m
ω
)
||
x
||
.
Since
the
random
variable
β
is
θ
-invariant,
we
obtain
that
||
φ
(
n,
θ
m
ω
)
x
||
≤
M
(
θ
m
ω
)(
n
+
1)
β
(
ω
)
||
x
||
.
From
the
previous
inequality,
we
have
||
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
)
x
||
≤
M
(
θ
m
ω
)(
n
+
1)
β
(
ω
)
||
φ
(
m,
ω
)
x
||
.
Finally,
using
φ
(
n
+
m,
ω
)
x
=
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
)
,
Nonuniform
polynomial
behavior
116
it
follows
that
||
φ
(
n
+
m,
ω
)
x
||
=
||
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
)
x
||
≤
M
(
θ
m
ω
)(
n
+
1)
β
(
ω
)
||
φ
(
m,
ω
)
x
||
,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Definition
4.
Let
(
θ,
φ
)
be
a
linear
random
one-sided
discrete-time
system
with
polynomial
growth.
For
every
ω
∈
Ω
,
the
mapping
||
·
||
:
X
→
R
+
defined
for
all
x
∈
X
by
||
x
||
ω
=
sup
n
∈
N
(1
+
n
)
−
β
(
ω
)
||
φ
(
n,
ω
)
x
||
is
called
the
Lyapunov
norm
generated
by
the
linear
random
one-sided
discrete-
time
system
(
θ,
φ
)
.
Remark
1.
The
following
statements
are
true:
(a)
if
||
φ
(0
,
ω
)
x
||
≥
||
x
||
then
||
x
||
≤
||
x
||
ω
≤
M
(
ω
)
||
x
||
.
(b)
if
the
linear
random
one-sided
discrete-time
system
is
nonuniformly
polynomially
stable,
with
M
=
N
and
β
=
α
,
then
||
x
||
≤
||
x
||
ω
≤
N
(
ω
)
||
x
||
.
(c)
the
topology
generated
by
the
family
of
norms
{||·||
ω
:
ω
∈
Ω
}
does
not
depend
on
the
θ
-invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
.
Indeed,
let
x
∈
X
and
||
x
||
ω,β
=
sup
n
∈
N
(1
+
n
)
−
β
(
ω
)
||
φ
(
n,
ω
)
x
||
.
Now,
let
β
1
,
β
2
:
Ω
→
(0
,
+
∞
)
be
two
θ
-invariant
random
variables
such
that
β
1
≤
β
2
.
From
(1
+
n
)
−
β
2
(
ω
)
≤
(1
+
n
)
−
β
1
(
ω
)
it
follows
that
||
x
||
ω,β
2
≤
||
x
||
ω,β
1
.
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
117
Banach’s
theorem
for
equivalent
norms
implies
that
||·||
ω,β
1
and
||·||
ω,β
2
are
indeed
equivalent.
We
now
assume
that
the
θ
-invariant
random
variables
β
1
:
Ω
→
(0
,
+
∞
)
and
β
2
:
Ω
→
(0
,
+
∞
)
are
arbitrary.
We
consider
θ
-invariant
random
variable
β
3
=
max
{
β
1
,
β
2
}
:
Ω
→
(0
,
+
∞
)
.
Then,
from
β
1
≤
β
3
we
obtain
the
equivalence
of
the
norms
||·||
ω,β
1
and
||·||
ω,β
3
.
Morever,
from
β
2
≤
β
3
we
have
the
equivalence
of
the
norms
||
·
||
ω,β
2
and
||
·
||
ω,β
3
.
Thus,
the
norms
||
·
||
ω,β
1
and
||
·
||
ω,β
2
are
equivalent.
Consequently,
the
norms
||
·
||
ω,β
are
equivalent
and
this
equivalence
is
independent
of
the
θ
-invariant
random
variable
β
:
Ω
→
(0
,
+
∞
)
.
Theorem
4.
If
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
has
polynomial
growth,
then
||
φ
(
m
0
,
ω
)
x
||
θ
m
0
ω
≤
(1
+
m
0
)
β
(
ω
)
||
x
||
ω
,
for
all
(
m
0
,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Proof.
A
direct
computation
yields
||
φ
(
m
0
,
ω
)
x
||
θ
m
0
ω
=
sup
k
∈
N
(1
+
k
)
−
β
(
θ
m
0
ω
)
||
φ
(
k,
θ
m
0
ω
)
φ
(
m
0
,
ω
)
x
||
=
sup
k
∈
N
(1
+
k
)
−
β
(
ω
)
||
φ
(
k
+
m
0
,
ω
)
x
||
=
sup
n
≥
m
0
(1
+
n
−
m
0
)
−
β
(
ω
)
||
φ
(
n,
ω
)
x
||
=
sup
n
≥
m
0
1
+
n
1
+
n
−
m
0
β
(
ω
)
(1
+
n
)
−
β
(
ω
)
||
φ
(
n,
ω
)
x
||
≤
sup
n
≥
m
0
(1
+
m
0
)
β
(
ω
)
(1
+
n
)
−
β
(
ω
)
||
φ
(
n,
ω
)
x
||
≤
(1
+
m
0
)
β
(
ω
)
sup
n
∈
N
(1
+
n
)
−
β
(
ω
)
||
φ
(
n,
ω
)
x
||
=
(1
+
m
0
)
β
(
ω
)
||
x
||
ω
,
since
1
+
n
1
+
n
−
m
0
≤
1
+
m
0
,
for
all
n
≥
m
0
.
Remark
2.
The
next
result
is
a
theorem
of
Zabczyk-type
for
Lyapunov
norms.
Nonuniform
polynomial
behavior
118
Theorem
5.
Let
(
θ,
φ
)
be
a
linear
random
one-sided
discrete-time
sys-
tem
with
polynomial
growth.
Then
the
system
(
θ,
φ
)
is
nonuniformly
poly-
nomially
stable
if
and
only
if
there
exist
a
θ
-invariant
random
variable
η
:
Ω
→
(0
,
+
∞
)
and
a
function
D
:
Ω
→
[1
,
+
∞
)
such
that
+
∞
n
=0
(1
+
n
)
η
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
θ
n
ω
≤
D
(
ω
)
||
x
||
ω
,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Proof.
Necessity.
There
exist
a
θ
-invariant
random
variable
α
:
Ω
→
(0
,
+
∞
)
and
a
function
N
:
Ω
→
[1
,
+
∞
)
satisfying
||
φ
(
n,
ω
)
x
||
≤
N
(
ω
)(1
+
n
)
−
α
(
ω
)
||
x
||
,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
We
obtain
||
φ
(
n,
ω
)
x
||
θ
n
ω
=
sup
k
∈
N
(1
+
k
)
−
β
(
ω
)
||
φ
(
k,
θ
n
ω
)
φ
(
n,
ω
)
x
||
=
sup
k
∈
N
(1
+
k
)
−
β
(
ω
)
||
φ
(
k
+
n,
ω
)
x
||
≤
sup
k
∈
N
(1
+
k
)
−
β
(
ω
)
N
(
ω
)
||
x
||
(
k
+
n
+
1)
−
α
(
ω
)
=
N
(
ω
)
||
x
||
sup
m
≥
n
(1
+
m
−
n
)
−
β
(
ω
)
(
m
+
1)
−
α
(
ω
)
≤
N
(
ω
)
||
x
||
(
n
+
1)
−
α
(
ω
)
.
Therefore,
for
any
θ
-invariant
random
variable
η
:
Ω
→
(0
,
+
∞
)
with
0
<
η
<
α
we
have
that
+
∞
n
=0
(1
+
n
)
η
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
θ
n
ω
≤
≤
N
(
ω
)
||
x
||
·
+
∞
n
=0
1
(1
+
n
)
1+
α
(
ω
)
−
η
(
ω
)
=
=
D
(
ω
)
||
x
||
≤
D
(
ω
)
||
x
||
ω
,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Sufficiency.
By
direct
computation
we
have
that
+
∞
n
=0
(1
+
n
)
η
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
+
∞
n
=0
(1
+
n
)
η
(
ω
)
−
1
||
φ
(
n,
ω
)
x
||
θ
n
ω
≤
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
119
≤
D
(
ω
)
||
x
||
ω
≤
D
(
ω
)
M
(
ω
)
||
x
||
=
K
(
ω
)
||
x
||
.
Finally,
using
Theorem
2
,
it
follows
that
the
linear
random
one-sided
discrete-
time
system
(
θ,
φ
)
is
nonuniformly
polynomially
stable.
4
Nonuniform
polynomial
instability
Definition
5.
We
say
that
the
linear
random
one-sided
discrete-time
sys-
tem
(
θ,
φ
)
is
nonuniformly
polynomially
unstable
if
there
exist
a
θ
-invariant
random
variable
σ
:
Ω
→
(0
,
+
∞
)
and
a
function
T
:
Ω
→
[1
,
+
∞
)
such
that
T
(
ω
)
||
φ
(
n,
ω
)
x
||
≥
(1
+
n
)
σ
(
ω
)
||
x
||
(12)
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X.
Remark
3.
From
the
previous
inequality,
by
taking
the
supremum
over
||
x
||
=
1
we
obtain
the
inequality
T
(
ω
)
||
φ
(
n,
ω
)
||
≥
(1
+
n
)
σ
(
ω
)
.
Note
that
this
inequality
does
not
imply
the
inequality
given
in
the
previous
definition.
Proposition
3.
The
following
statements
are
equivalent:
(a)
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuni-
formly
polynomially
unstable;
(b)
there
exist
a
function
T
:
Ω
→
[1
,
+
∞
)
and
a
θ
-
invariant
random
variable
σ
:
Ω
→
(0
,
+
∞
)
such
that
T
(
θ
m
ω
)
||
φ
(
m
+
n,
ω
)
x
||
≥
(1
+
n
)
σ
(
ω
)
||
φ
(
m,
ω
)
x
||
,
for
all
m,
n
∈
Z
+
,
ω
∈
Ω
and
x
∈
X
;
(c)
there
exist
a
function
T
:
Ω
→
[1
,
+
∞
)
and
a
θ
-
invariant
random
variable
σ
:
Ω
→
(0
,
+
∞
)
such
that
T
(
θ
n
ω
)
||
φ
(
n
+
m,
ω
)
x
||
≥
(1
+
m
)
σ
(
ω
)
||
φ
(
n,
ω
)
x
||
,
for
all
n,
m
∈
Z
+
,
ω
∈
Ω
and
x
∈
X
;
Nonuniform
polynomial
behavior
120
(d)
there
exist
a
function
T
:
Ω
→
[1
,
+
∞
)
and
a
θ
-
invariant
random
variable
σ
:
Ω
→
(0
,
+
∞
)
such
that
T
(
ω
)
||
φ
(
n
+
m,
ω
)
x
||
≥
(1
+
n
+
m
)
σ
(
ω
)
||
x
||
,
for
all
n,
m
∈
Z
+
,
ω
∈
Ω
and
x
∈
X
.
Proof.
Let
T
and
σ
be
given
as
in
Definition
5
.
Then
T
(
θ
m
ω
)
||
φ
(
m
+
n,
ω
)
x
||
=
T
(
θ
m
ω
)
||
φ
(
n,
θ
m
ω
)
φ
(
m,
ω
)
x
||
≥
≥
(1
+
n
)
σ
(
θ
m
ω
)
||
φ
(
m,
ω
)
x
||
=
(1
+
n
)
σ
(
ω
)
||
φ
(
m,
ω
)
x
||
.
The
implication
(
a
)
⇒
(
c
)
follows
by
an
analogous
argument.
The
im-
plication
(
b
)
⇒
(
a
)
holds
for
m
=
0.
For
n
=
0
we
obtain
the
implica-
tion
(
c
)
⇒
(
a
).
The
implication
(
a
)
⇒
(
d
)
is
immediate.
The
implication
(
d
)
⇒
(
a
)
follows
for
m
=
0.
Theorem
6.
If
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuniformly
polynomially
unstable
then
exist
a
θ
-invariant
random
variable
δ
:
Ω
→
(0
,
+
∞
)
and
two
functions
P,
Q
:
Ω
→
[1
,
+
∞
)
such
that
m
n
=0
(
m
−
n
+
1)
δ
(
ω
)
−
1
P
(
θ
n
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
Q
(
ω
)
||
φ
(
m,
ω
)
x
||
,
(13)
for
all
(
m,
ω,
x
)
∈
Z
+
×
Ω
×
X
.
Proof.
Let
T
and
σ
be
given
as
in
Definition
5
.
Then
||
φ
(
m,
ω
)
x
||
=
||
φ
(
m
−
n
+
n,
ω
)
x
||
=
=
||
φ
(
m
−
n,
θ
n
ω
)
φ
(
n,
ω
)
x
||
≥
T
(
θ
n
ω
)
−
1
(
m
−
n
+
1)
σ
(
θ
n
ω
)
||
φ
(
n,
ω
)
x
||
=
T
(
θ
n
ω
)
−
1
(
m
−
n
+
1)
σ
(
ω
)
||
φ
(
n,
ω
)
x
||
,
for
all
(
m,
ω,
x
)
∈
Z
+
×
Ω
×
X
.
We
consider
P
=
T
:
Ω
→
[1
,
+
∞
)
a
θ
-invariant
random
variable
δ
:
Ω
→
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
121
(0
,
+
∞
)
such
that
0
<
δ
<
σ
.
Then
m
n
=0
(
m
−
n
+
1)
δ
(
ω
)
−
1
P
(
θ
n
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
≤
m
n
=0
(
m
−
n
+
1)
δ
(
ω
)
−
1
−
σ
(
ω
)
||
φ
(
m,
ω
)
x
||
=
||
φ
(
m,
ω
)
x
||
·
m
+1
k
=1
1
k
1+
σ
(
ω
)
−
δ
(
ω
)
≤
||
φ
(
m,
ω
)
x
||
·
+
∞
k
=1
1
k
1+
σ
(
ω
)
−
δ
(
ω
)
≤
Q
(
ω
)
||
φ
(
m,
ω
)
x
||
,
were
Q
(
ω
)
=
1
+
+
∞
k
=1
1
k
1+
σ
(
ω
)
−
δ
(
ω
)
.
Definition
6.
We
say
that
L
2
:
Z
+
×
Ω
×
X
→
R
+
is
a
Lyapunov
function
for
the
linear
random
one-sided
discrete-time
system
if
there
exist
a
θ
-invariant
random
variable
ξ
:
Ω
→
(0
,
+
∞
)
and
a
function
N
:
Ω
→
[1
,
+
∞
)
such
that
L
2
(0
,
ω,
x
)+
m
n
=0
(
m
−
n
+1)
ξ
(
ω
)
−
1
N
(
θ
n
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
L
2
(
m,
ω,
x
)
,
(14)
for
all
(
m,
ω,
x
)
∈
Z
+
×
Ω
×
X
.
Theorem
7.
If
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuniformly
polynomially
unstabile,
then
there
exist
a
Lyapunov
function
L
2
:
Z
+
×
Ω
×
X
→
R
+
and
a
function
K
:
Ω
→
[1
,
+
∞
)
such
that
L
2
(
n,
ω,
x
)
≤
K
(
ω
)
||
φ
(
n,
ω
)
x
||
,
for
all
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X
.
Proof.
If
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuni-
formly
polynomially
unstable
then
from
Theorem
6
there
exist
a
θ
-invariant
Nonuniform
polynomial
behavior
122
random
variable
δ
:
Ω
→
(0
,
+
∞
)
and
two
functions
P,
Q
:
Ω
→
[1
,
+
∞
)
such
that
m
n
=0
(
m
−
n
+
1)
δ
(
ω
)
−
1
P
(
θ
n
ω
)
−
1
||
φ
(
n,
ω
)
x
||
≤
Q
(
ω
)
||
φ
(
m,
ω
)
x
||
.
Let
N
=
P
:
Ω
→
[1
,
+
∞
)
and
ξ
=
δ
:
Ω
→
(0
,
+
∞
).
We
define
L
2
(
m,
ω,
x
)
=
0
,
m
=
0
m
n
=0
(
m
−
n
+
1)
ξ
(
ω
)
−
1
N
(
θ
n
ω
)
−
1
||
φ
(
n,
ω
)
x
||
,
m
>
0
Then
L
2
:
Z
+
×
Ω
×
X
→
R
+
is
a
Lyapunov
function
that
satisfies
the
conditions
of
the
statement
with
K
=
Q
:
Ω
→
[1
,
+
∞
)
.
Remark
4.
For
nonuniform
polynomial
instability,
both
Zabczyk-type
and
Lyapunov-type
criteria
provide
only
necessary
conditions.
Sufficiency
is
still
an
open
problem.
5
Applications
to
the
adjoint
and
inverse
system
Given
a
linear
random
one-sided
discrete-time
system
(
θ,
φ
),
two
associated
linear
random
one-sided
discrete-time
systems
can
be
defined:
the
inverse
system
(
θ,
φ
−
1
)
and
the
adjoint
system
(
θ,
φ
∗
).
Let
(
n,
ω
)
∈
Z
+
×
Ω.
Then
φ
(
n,
ω
)
−
1
:
X
→
X
satisfies
P
(
n,
ω
)
φ
(
n,
ω
)
−
1
=
φ
(
n,
ω
)
−
1
φ
(
n,
ω
)
x
=
x,
for
all
x
∈
X
.
If
X
∗
is
the
topological
dual
of
the
space
X
,
then
φ
(
n,
ω
)
∗
:
X
∗
→
X
∗
satisfies
φ
(
n,
ω
)
∗
y
∗
(
x
)
=
y
∗
φ
(
n,
ω
)
x
,
for
all
y
∗
∈
X
∗
.
Proposition
4.
The
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuniformly
polinomially
stable
if
and
only
if
the
adjoint
system
(
θ,
φ
∗
)
is
nonuniformly
polinomially
stable.
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
123
Proof.
Necessity.
Let
(
n,
ω,
x
)
∈
Z
+
×
Ω
×
X
and
y
∗
∈
X
∗
.
Let
N
and
α
as
in
Definition
1
.
We
deduce
that
||
φ
(
n,
ω
)
∗
y
∗
(
x
)
||
=
||
y
∗
φ
(
n,
ω
)
x
||
≤
≤
||
y
∗
||
·
||
φ
(
n,
ω
)
x
||
≤
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
y
∗
||
·
||
x
||
.
Taking
the
supremum
over
||
x
||
=
1
we
obtain
that
||
φ
(
n,
ω
)
∗
y
∗
||
≤
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
y
∗
||
.
This
proves
that
(
θ,
φ
∗
)
is
nonuniformly
polynomially
stable.
Sufficiency.
We
assume
that
(
θ,
φ
∗
)
is
nonuniformly
polynomially
stable.
Let
N
:
Ω
→
[1
,
+
∞
)
and
α
:
Ω
→
(0
,
+
∞
)
a
θ
-invariant
random
variable
such
that
||
φ
(
n,
ω
)
∗
y
∗
||
≤
N
(
ω
)(
n
+
1)
−
α
(
ω
)
||
y
∗
||
.
Taking
the
supremum
over
||
y
∗
||
=
1
we
obtain
that
||
φ
(
n,
ω
)
||
=
||
φ
(
n,
ω
)
∗
||
≤
N
(
ω
)(
n
+
1)
−
α
(
ω
)
.
Now
from
Proposition
1
we
deduce
that
the
system
(
θ,
φ
)
is
nonuniformly
polynomially
stable.
Theorem
8.
The
following
statements
are
equivalent:
(a)
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuni-
formly
polynomially
stable
if
and
only
if
the
inverse
system
(
θ,
φ
−
1
)
is
nonuniformly
polynomially
unstable;
(b)
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuni-
formly
polynomially
stable
if
and
only
if
the
inverse
system
(
θ,
φ
−
1
)
is
nonuniformly
polynomially
stable.
Proof.
(
a
)
Let
(
n,
ω
)
∈
Z
+
×
Ω
and
y
∈
X
.
There
exists
a
unique
x
∈
X
such
that
φ
(
n,
ω
)
x
=
y
⇔
x
=
φ
(
n,
ω
)
−
1
y.
If
N
and
α
are
as
in
Definition
1
,
then
||
y
||
=
||
φ
(
n,
ω
)
x
||
≤
N
(
ω
)(1
+
n
)
−
α
(
ω
)
||
x
||
=
N
(
ω
)(1
+
n
)
−
α
(
ω
)
||
φ
(
n,
ω
)
−
1
y
||
.
Therefore,
N
(
ω
)
||
φ
(
n,
ω
)
−
1
y
||
≥
(1
+
n
)
α
(
ω
)
||
y
||
.
Nonuniform
polynomial
behavior
124
Conversely,
let
(
n,
ω
)
∈
Z
+
×
Ω
and
x
∈
X
.
There
exists
a
unique
y
∈
X
such
that
φ
(
n,
ω
)
−
1
y
=
x
⇔
φ
(
n,
ω
)
x
=
y.
We
assume
that
the
function
S
:
Ω
→
[1
,
+
∞
)
and
the
θ
-invariant
random
variable
ξ
:
Ω
→
(0
,
+
∞
)
satisfy
S
(
ω
)
||
φ
(
n,
ω
)
−
1
y
||
≥
(1
+
n
)
ξ
(
ω
)
||
y
||
.
Hence
S
(
ω
)
||
x
||
≥
(1
+
n
)
ξ
(
ω
)
||
φ
(
n,
ω
)
x
||
,
which
is
equivalent
to
||
φ
(
n,
ω
)
x
||
≤
S
(
ω
)(1
+
n
)
−
ξ
(
ω
)
||
x
||
.
(
b
)
Let
(
n,
ω
)
∈
Z
+
×
Ω
and
let
y
∈
X
.
There
exists
a
unique
x
∈
X
such
that
φ
(
n,
ω
)
x
=
y
⇔
x
=
φ
(
n,
ω
)
−
1
y.
Let
T
:
Ω
→
[1
,
+
∞
)
and
σ
:
Ω
→
(0
,
+
∞
)
be
as
in
Definition
5
.
It
follows
that
T
(
ω
)
||
y
||
=
T
(
ω
)
||
φ
(
n,
ω
)
x
||
≥
(1
+
n
)
σ
(
ω
)
||
x
||
=
=
(1
+
n
)
σ
(
ω
)
||
φ
(
n,
ω
)
−
1
y
||
.
Hence
||
φ
(
n,
ω
)
−
1
y
||
≤
T
(
ω
)(1
+
n
)
−
σ
(
ω
)
||
y
||
.
Conversely,
let
(
n,
ω
)
∈
Z
+
×
Ω
and
x
∈
X
.
There
exists
a
unique
y
∈
X
such
that
φ
(
n,
ω
)
−
1
y
=
x
⇔
φ
(
n,
ω
)
x
=
y.
Consider
the
function
S
:
Ω
→
[1
,
+
∞
)
and
a
θ
-invariant
random
variable
ξ
:
Ω
→
(0
,
+
∞
)
such
that
||
φ
(
n,
ω
)
−
1
y
||
≤
S
(
ω
)(1
+
n
)
−
ξ
(
ω
)
||
y
||
.
Hence
||
x
||
≤
S
(
ω
)(1
+
n
)
−
ξ
(
ω
)
||
φ
(
n,
ω
)
x
||
,
which
is
equivalent
to
S
(
ω
)
||
φ
(
n,
ω
)
x
||
≥
(1
+
n
)
ξ
(
ω
)
||
x
||
.
O.
Albescu,
L.E.
Biri¸
s,
T.
Ceau¸
su,
N.M.
Seimeanu
125
Theorem
9.
The
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuniformly
polynomially
unstable
if
and
only
if
the
adjoint
system
(
θ,
φ
∗
)
is
nonuniformly
polynomially
unstable.
Proof.
We
have
that
the
linear
random
one-sided
discrete-time
system
(
θ,
φ
)
is
nonuniformly
polynomially
unstable
if
and
only
if
the
system
(
θ,
φ
−
1
)
is
nonuniformly
polynomially
stable.
Next,
the
system
(
θ,
φ
−
1
)
is
nonuniformly
polynomially
stable
if
and
only
if
the
system
θ,
(
φ
−
1
)
∗
=
θ,
(
φ
∗
)
−
1
is
nonuniformly
polynomially
stable.
Finally,
the
system
θ,
(
φ
∗
)
−
1
is
nonuni-
formly
polynomially
stable
if
and
only
if
the
system
θ,
(
φ
∗
)
−
1
−
1
=
(
θ,
φ
∗
)
is
nonuniformly
polynomially
unstable,
which
concludes
the
proof.
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