Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
4/2026
THE
MEAN-SQUARE
STABILITY
AND
MEAN-SQUARE
STABILIZABILITY
OF
THE
DISCRETE-TIME
MEAN-FIELD
STOCHASTIC
LINEAR
SYSTEMS
WITH
PERIODIC
COEFFICIENTS
∗
Vasile
Dr˘
agan
†
DOI 10.56082/annalsarscimath.2026.4.339
Abstract
In
this
work,
several
aspects
of
the
mean-square
stability
and
of
the
mean-square
stabilizability
of
a
class
of
discrete-time
mean-field
stochastic
linear
systems
(MFSLS)
with
periodic
coefficients
is
ad-
dressed.
To
a
discrete-time
MFSLS
one
associates
in
a
natural
way
a
companion
standard
discrete-time
stochastic
linear
system
(DTSLS).
Our
aim
is
to
compare
the
mean-field
exponential
stability
in
mean-
square
(MFESMS)
property
of
the
MFSLS
under
consideration
with
the
exponential
stability
in
mean-square
(ESMS)
property
of
its
com-
panion
DTSLS.
Even
if
in
some
already
published
works
is
asserted
that
the
two
concepts
of
exponential
stability
are
equivalent,
we
show
that
this
equivalence
is
not
true.
We
also
show
that
the
MF
mean-
square
stabilizability
of
a
discrete-time
controlled
MFSLS
is
not
equiv-
alent
to
the
mean-square
stabilizability
of
its
controlled
companion
DTSLS.
∗
Accepted
for
publication
on
October
19,
2026
†
Vasile.Dragan@imar.ro
,
Simion
Stoilow
Institute
of
Mathematics
of
the
Romanian
Academy,
P.O.Box
1-764,
RO-014700,
Bucharest,
Romania;
Academy
of
Romanian
Sci-
entists,
Str.
Ilfov,
3,
Bucharest,
Romania
339
On
mean-square
stability
and
stabilizability
340
Keywords:
mean-field
stochastic
linear
systems,
mean-square
stability,
mean-square
stabilizability,
periodic
coefficients.
MSC:
93E15,
93B05,
93B07,
93D15,
47B65.
1
Introduction
In
recent
years,
mean-field
control
theory
has
received
increasing
interest,
both
from
a
theoretical
perspective
as
well
as
from
engineering
and
finance
applications.
The
main
distinctive
feature
of
the
mean-field
setting
when
compared
with
classical
stochastic
optimal
control
is
that
the
system
dy-
namics
involves
state
and
control,
as
well
as
their
expected
values.
Since
the
pioneering
works
[
10
,
12
]
there
have
been
several
important
contri-
butions
related
to
this
type
of
stochastic
systems
from
the
control
perspec-
tive
[
1
–
5
,
8
,
11
,
14
,
16
].
In
the
present
work,
we
consider
a
class
of
discrete-time
mean-field
con-
trolled
stochastic
linear
systems
with
periodic
coefficients.
For
this
kind
of
stochastic
systems
we
introduce
the
concept
of
mean-field
exponential
stabil-
ity
in
mean-square
(MF-ESMS)
(Definition
1
)
and
the
concept
of
mean-field
stabilizability
in
mean-square
by
linear
state
feedback
(Definition
2
).
The
MF-ESMS
property
is
characterized
by
exponential
decay
of
the
trajec-
tories
with
random
initial
states
of
the
mean-field
stochastic
system
under
consideration.
We
show
that
without
loss
of
generality,
we
can
character-
ize
the
MF-ESMS
property
only
by
exponential
decay
of
the
trajectories
starting
from
deterministic
initial
states
(Proposition
1
).
To
a
mean-field
stochastic
linear
system
one
may
associate,
in
a
natural
way,
a
compan-
ion
discrete-time
stochastic
linear
system
with
state
multiplicative
and
/or
control
multiplicative
independent
random
disturbances.
In
some
already
published
papers
(
[
13
],
[
15
]),
is
asserted
that
the
MF-ESMS
property
of
a
mean-field
stochastic
linear
system
is
equivalent
to
the
standard
exponential
stability
in
mean-square
(ESMS)
of
its
discrete-time
companion
stochastic
linear
system.
Through
Example
1
,
we
show
that
this
equivalence
is
not
always
true.
However,
it
remains
true
that
the
ESMS
property
of
the
com-
panion
discrete-time
linear
stochastic
system
implies
the
MF-ESMS
prop-
erty
of
the
mean-field
stochastic
linear
system
under
consideration.
The
ex-
ponential
stability
in
mean-square
of
the
mean-field
stochastic
linear
system
induced
by
the
ESMS
property
of
its
companion
discrete-time
linear
stochas-
tic
system
is
named,
in
this
work,
”mean-field
strong
exponential
stability
in
mean-square
(MF-SESMS)”.
The
Lyapunov
type
criteria
for
exponential
stability
in
mean-square
of
the
companion
discrete-time
linear
stochastic
V.
Dr˘
agan
341
system
provide
necessary
and
sufficient
conditions
for
MF-SESMS
property
but
only
sufficient
conditions
for
the
MF-ESMS
property
of
the
mean-field
stochastic
linear
system
under
consideration.
By
the
Example
2
,
we
show
that
the
mean-field
stabilizability
in
mean-square
of
a
mean-field
controlled
stochastic
linear
system
is
not
equivalent
to
the
standard
stochastic
stabi-
lizability
in
mean-square
by
a
decentralized
linear
state
feedback.
The
mean-field
stabilizability
of
the
mean-field
controlled
stochastic
linear
system
induced
by
the
stochastic
stabilizability
by
a
decentralized
linear
state
feedback
of
its
companion
controlled
discrete-time
stochastic
linear
system
is
named
MF
strong
stabilizability
in
mean-square
of
the
mean-field
controlled
stochastic
linear
system.
This
property
may
be
used
in
future
research,
in
order
to
establish
the
existence
of
the
maximal
solution
and
of
the
stabilizing
solution
of
a
Riccati
type
equation
involved
in
the
solution
of
some
optimal
control
problems
for
mean-field
controlled
stochastic
linear
systems.
The
developments
from
sections
3
and
4,
of
the
present
work,
are
the
discrete-time
counterpart
of
the
results
from
the
recent
paper
[
7
].
2
The
problem
setting
2.1
Notation
In
this
work
(Ω
,
F
,
P
)
is
a
probability
space
where
a
sequence
of
r
-dimensional
independent
random
vectors
{
w
(
t
)
}
t
∈
Z
+
,
w
(
t
)
=
(
w
1
(
t
)
w
2
(
t
)
.
.
.
w
r
(
t
))
is
defined.
Here
and
in
the
sequel
Z
+
=
{
0
,
1
,
2
....
}
.
We
assume
that
the
following
assumption
holds:
H1
).
E
[
w
(
t
)]
=
0
and
E
[
w
(
t
)
w
(
t
)]
=
I
r
for
all
t
∈
Z
+
,
where
I
r
is
the
identity
matrix
of
size
r
×
r
.
Throughout
this
work
E
[
·
]
denotes
the
mathematical
expectation
of
a
random
variable,
while
the
superscript
stands
for
the
transpose
of
a
matrix
or
a
vector.
For
each
t
∈
Z
+
,
F
t
:=
{
σ
(
w
(
s
))
,
0
≤
s
≤
t
}
⊂
F
denotes
the
smallest
σ
-algebra
with
respect
to
which
the
random
vectors
(
w
(
s
),
0
≤
s
≤
t
,
are
measurable.
If
t
0
∈
Z
+
,
2
F
{
t
0
,
∞
;
R
d
}
denotes
the
linear
space
of
all
sequences
{
v
(
t
)
}
t
≥
t
0
of
R
d
valued
random
vectors
with
the
properties:
a)
for
each
t
≥
t
0
,
v
(
t
)
are
F
t
-measurable;
b)
∞
t
=
t
0
E
[
|
v
(
t
)
|
2
]
<
∞
.
Here
and
in
the
sequel
|
·
|
denotes
the
Euclidean
norm
of
a
vector
i.e.
|
v
|
=
(
v
v
)
1
/
2
for
all
v
∈
R
d
.
As
usual
R
m
×
n
denotes
the
linear
space
of
m
×
n
real
matrices.
For
p
≥
2,
On
mean-square
stability
and
stabilizability
342
S
p
⊂
R
p
×
p
stands
for
the
space
of
symmetric
matrices
of
size
p
×
p
.
S
+
p
⊂
S
p
denotes
the
closed,
solid,
normal
convex
cone
of
the
positive
semidefinite
matrices
of
size
p
×
p
.
If
t
0
∈
Z
+
,
we
denote
X
t
0
the
set
of
all
n
-dimensional
random
vectors
x
0
which
are
independent
of
all
random
vectors
w
(
t
),
t
>
t
0
and
are
satisfying
E
[
|
x
0
|
2
]
<
∞
.
Remark
1.
The
set
X
t
0
contains
the
n
-dimensional
random
vectors
x
0
with
finite
second
order
moments
which
are
either
F
t
0
-measurable,
or,
are
inde-
pendent
of
all
random
vectors
w
(
t
)
,
t
∈
Z
+
.
2.2
Model
description
We
consider
the
discrete-time
controlled
stochastic
linear
system
having
the
state
space
representation
of
the
form:
x
(
t
+
1)
=
A
0
(
t
)
x
(
t
)
+
A
0
(
t
)
E
[
x
(
t
)]
+
B
0
(
t
)
u
(
t
)
+
B
0
(
t
)
E
[
u
(
t
)]
+
(1)
r
k
=1
(
A
k
(
t
)
x
(
t
)
+
A
k
(
t
)
E
[
x
(
t
)]
+
B
k
(
t
)
u
(
t
)
+
B
k
(
t
)
E
[
u
(
t
)])
w
k
(
t
)
,
t
≥
t
0
,
x
(
t
0
)
=
x
0
∈
X
t
0
,
where
x
(
t
)
∈
R
n
are
the
vectors
of
the
state
parameters
and
u
(
t
)
∈
R
m
are
the
vectors
of
the
control
parameters.
In
(
1
),
{
A
k
(
t
)
}
t
∈
Z
,
{
A
k
(
t
)
}
)
t
∈
Z
are
matrix
valued
sequences
from
R
n
×
n
and
{
B
k
(
t
)
}
t
∈
Z
,
{
B
k
(
t
)
}
t
∈
Z
,
0
≤
k
≤
r
are
matrix
valued
sequences
in
R
n
×
m
which
are
periodic
of
period
θ
≥
1.
The
set
U
ad
(
t
0
)
of
the
admissible
controls
consists
of
all
sequences
{
u
(
t
)
}
t
≥
t
0
of
R
m
valued
random
vectors
with
the
properties:
a)
for
each
t
≥
t
0
,
u
(
t
)
is
F
t
-measurable;
b)
E
[
|
u
(
t
)
|
2
]
<
∞
.
The
controlled
stochastic
linear
system
(
1
)
is
the
discrete-time
controlled
linear
system
of
McKean-Vlasov
type
which
is
also
known
as
a
discrete-time
mean-field
controlled
stochastic
linear
system
(MFCSLS).
The
study
of
the
stochastic
differential
equations
of
McKean-Vlasov
type
was
initiated
in
[
10
,
12
].
Among
the
most
recent
papers
which
studied
the
control
problems
involving
discrete-time
stochastic
linear
equations
of
McKean-Vlasov
type,
are
[
13
,
15
].
For
each
0
≤
t
0
<
t
∈
Z
+
,
G
t
0
,t
⊂
F
stands
for
the
smallest
σ
-algebra
with
respect
to
which
the
random
vectors
x
0
∈
X
t
0
and
w
(
s
),
t
0
≤
s
≤
t
−
1
are
measurable.
The
next
lemma
collects
several
properties
of
the
solutions
of
(
1
).
Their
proofs
may
be
done
recursively
with
respect
to
t
≥
t
0
.
V.
Dr˘
agan
343
Lemma
1.
For
each
t
0
∈
Z
+
,
x
0
∈
X
t
0
and
u
=
{
u
(
t
)
}
t
≥
t
0
∈
U
ad
(
t
0
)
,
the
problem
with
given
initial
values
(IVP)
(
1
)
has
a
unique
solution
x
(
t
;
t
0
,
x
0
,
u
)
,
with
the
properties:
a)
x
(
t
;
t
0
,
x
0
,
u
)
is
G
t
0
,t
-measurable;
b)
E
[
|
x
(
t
;
t
0
,
x
0
,
u
)
|
2
]
<
∞
;
c)
x
(
t
0
;
t
0
,
x
0
,
u
)
=
x
0
.
Bearing
in
mind
the
assumption
H1
)
and
Lemma
1
a),
we
obtain
that
the
random
vectors
x
(
t
;
t
0
,
x
0
,
u
)
and
w
(
t
)
are
mutually
independent
for
any
t
≥
t
0
.
This
allows
us
to
obtain
from
(
1
)
that
if
x
(
t
)
x
(
t
;
t
0
,
x
0
,
u
),
t
≥
t
0
,
{
E
[
x
(
t
)]
}
t
≥
t
0
is
satisfying
the
following
IVP:
E
[
x
(
t
+
1)]
=
(
A
0
(
t
)
+
A
0
(
t
))
E
[
x
(
t
)]
+
(
B
0
(
t
)
+
B
0
(
t
))
E
[
u
(
t
)]
(2a)
E
[
x
(
t
0
)]
=
E
[
x
0
]
.
(2b)
Subtracting
(
2
)
from
(
1
),
we
obtain:
x
(
t
+
1)
−
E
[
x
(
t
+
1)]
=
A
0
(
t
)(
x
(
t
)
−
E
[
x
(
t
)])
+
B
0
(
t
)(
u
(
t
)
−
E
[
u
(
t
)])
+
r
k
=1
[
A
k
(
t
)(
x
(
t
)
−
E
[
x
(
t
)])
+
(
A
k
(
t
)
+
A
k
(
t
))
E
[
x
(
t
)]
(3a)
+
B
k
(
t
)(
u
(
t
)
−
E
[
u
(
t
)])
+
(
B
k
(
t
)
+
B
k
(
t
))
E
[
u
(
t
)]]
w
k
(
t
)
,
t
≥
t
0
,
x
(
t
0
)
−
E
[
x
(
t
0
)]
=
x
0
−
E
[
x
0
]
.
(3b)
We
set
x
1
(
t
)
x
(
t
)
−
E
[
x
(
t
)]
(4a)
x
2
(
t
)
E
[
x
(
t
)]
(4b)
u
1
(
t
)
u
(
t
)
−
E
[
u
(
t
)]
(5a)
u
2
(
t
)
E
[
u
(
t
)]
(5b)
t
≥
t
0
.
With
these
notations,
we
obtain
from
(
2
)
and
(
3
)
the
following
IVP:
x
1
(
t
+
1)
x
2
(
t
+
1)
=
A
0
(
t
)
x
1
(
t
)
x
2
(
t
)
+
B
0
(
t
)
u
1
(
t
)
u
2
(
t
)
+
r
k
=1
(
A
k
(
t
)
x
1
(
t
)
x
2
(
t
)
+
B
k
(
t
)
u
1
(
t
)
u
2
(
t
)
)
w
k
(
t
)
,
t
≥
t
0
(6a)
x
1
(
t
0
)
x
2
(
t
0
)
=
x
0
−
E
[
x
0
]
E
[
x
0
]
(6b)
On
mean-square
stability
and
stabilizability
344
with:
A
0
(
t
)
=
A
0
(
t
)
0
0
A
0
(
t
)
+
A
0
(
t
)
(7a)
A
k
(
t
)
=
A
k
(
t
)
A
k
(
t
)
+
A
k
(
t
)
0
0
,
1
≤
k
≤
r,
(7b)
B
0
(
t
)
=
B
0
(
t
)
0
0
B
0
(
t
)
+
B
0
(
t
)
(8a)
B
k
(
t
)
=
B
k
(
t
)
B
k
(
t
)
+
B
k
(
t
)
0
0
,
1
≤
k
≤
r.
(8b)
Along
with
the
IVP
(
6
),
we
introduce
the
following
IVP
on
the
space
R
n
×
R
n
:
x
(
t
+
1)
=
A
0
(
t
)
x
(
t
)
+
B
0
(
t
)
u
(
t
)
+
r
k
=1
(
A
k
(
t
)
x
(
t
)
+
B
k
(
t
)
u
(
t
))
w
k
(
t
)
,
(9a)
x
(
t
0
)
=
x
0
=
x
1
0
x
2
0
(9b)
where
x
(
t
)
=
x
1
(
t
)
x
2
(
t
)
∈
R
n
×
R
n
and
u
(
t
)
=
u
1
(
t
)
u
2
(
t
)
∈
R
m
×
R
m
,
x
k
0
∈
X
t
0
,
k
=
1
,
2.
The
IVP
(
6
)
is
the
particular
case
of
the
IVP
(
9
),
when
x
1
0
=
x
0
−
E
[
x
0
]
and
x
2
0
=
E
[
x
0
].
In
the
sequel,
the
discrete-time
controlled
linear
stochastic
system,
having
the
state
space
representation
described
by
(
6a
)
(or
(
9a
),
respectively)
will
be
named
the
companion
controlled
differ-
ence
stochastic
linear
system
(CDSLS),
associated
to
the
MFCSLS
(
1
).
When
B
k
(
t
)
=
B
k
(
t
)
=
0,
t
∈
Z
,
0
≤
k
≤
r
,
the
MFCSLS
(
1
)
reduces
to
the
uncontrolled
mean-field
stochastic
linear
system
(MFSLS):
x
(
t
+
1)
=
A
0
(
t
)
x
(
t
)
+
A
0
(
t
)
E
[
x
(
t
)]
+
r
k
=1
(
A
k
(
t
)
x
(
t
)
+
A
k
(
t
)
E
[
x
(
t
)])
w
k
(
t
)
,
(10a)
x
(
t
0
)
=
x
0
∈
X
t
0
.
(10b)
The
IVPs
(
6
)
and
(
9
)
take
the
form:
x
1
(
t
+
1)
x
2
(
t
+
1)
=
A
0
(
t
)
x
1
(
t
)
x
2
(
t
)
+
r
k
=1
A
k
(
t
)
x
1
(
t
)
x
2
(
t
)
w
k
(
t
)
,
(11a)
x
1
(
t
0
)
x
2
(
t
0
)
=
x
0
−
E
[
x
0
]
E
[
x
0
]
(11b)
V.
Dr˘
agan
345
x
(
t
+
1)
=
A
0
(
t
)
x
(
t
)
+
r
k
=1
A
k
(
t
)
x
(
t
)
w
k
(
t
)
,
t
≥
t
0
(12a)
x
0
(
t
0
)
=
x
0
=
x
1
0
x
2
0
(12b)
where
x
k
0
∈
X
t
0
,
k
=
1
,
2.
The
discrete-time
stochastic
linear
system
(
11a
)
((
12a
),
respectively),
will
be
named
the
companion
discrete-time
stochastic
linear
system
(DTSLS)
associated
to
the
MFSLS
(
10a
).
2.3
Basic
definitions
and
the
problem
setting
Definition
1.
We
say
that
the
zero
solution
of
the
MFSLS
(
10a
)
or
equiv-
alently
the
MFSLS
(
10
)
is
mean-field
exponentially
stable
in
mean
square
(MF-ESMS)
if
for
all
initial
time
t
0
∈
Z
+
and
any
initial
state
x
0
∈
X
t
0
,
the
corresponding
solution
x
(
·
;
t
0
,
x
0
)
of
(
10a
)
has
a
decay
of
the
form
E
[
|
x
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
E
[
|
x
0
|
2
]
(13)
for
all
t
≥
t
0
,
where
β
≥
1
,
δ
∈
(0
,
1)
are
constants
not
depending
upon
t,
t
0
and
x
0
.
Remark
2.
Unlike
the
nonlinear
case,
where
the
stability
property
refers
to
a
given
solution,
in
the
case
of
a
linear
system,
the
stability
property
of
a
solution
is
equivalent
to
the
stability
of
all
solutions
of
that
system.
That
is
why,
we
may
say
that
the
MFSLS
(
10a
)
is
MF-ESMS
if
its
solutions
are
satisfying
(
13
).
Definition
2.
We
say
that
the
MFCSLS
(
1
)
is
mean-field
stabilizable
in
mean-square,
or
equivalently
mean-field
stabilizable
by
linear
state
feedback
if
there
exist
θ
-periodic
R
m
×
n
valued
matrix
sequences
{
F
(
t
)
}
t
∈
Z
,
{
F
(
t
)
}
t
∈
Z
with
the
property
that
the
corresponding
closed-loop
MFSLS
x
(
t
+
1)
=
(
A
0
(
t
)
+
B
0
(
t
)
F
(
t
))
x
(
t
)
+
[
A
0
(
t
)
+
(
B
0
(
t
)
+
B
0
(
t
))
F
(
t
)
−
B
0
(
t
)
F
(
t
)]
E
[
x
(
t
)]
+
r
k
=1
{
(
A
k
(
t
)
+
B
k
(
t
)
F
(
t
))
x
(
t
)
+
[
A
k
(
t
)
+
(
B
k
(
t
)
+
B
k
(
t
))
F
(
t
)
−
B
k
(
t
)
F
(
t
)]
E
[
x
(
t
)]
}
w
k
(
t
)
,
t
≥
t
0
(14a)
x
(
t
0
)
=
x
0
∈
X
t
0
(14b)
is
MF-ESMS
in
the
sense
of
Definition
1
.
On
mean-square
stability
and
stabilizability
346
The
previous
definition
is
a
time-varying
version
of
the
Definition
3.2
from
[
15
].
Our
aim
is
to
highlight
several
useful
properties
of
the
MF-ESMS
of
a
MFSLS
(
10
)
and
of
the
mean-field
stabilizability
in
mean-square
of
a
MFC-
SLS
of
the
form
1
).
We
shall
emphasize
the
relationship
between
the
MF-
ESMS
property
of
the
MFSLS
(
10
)
and
ESMS
property
of
the
corresponding
companion
DTSLS
(
12a
).
We
shall
see
that
the
ESMS
property
of
the
companion
DTSLS
(
12
)
implies
the
MF-ESMS
of
the
MFSLS
(
10
)
but
the
converse
implication
is
not
always
true.
Also,
we
shall
show
that
the
stochastic
stabilizability
in
mean-
square
of
the
corresponding
companion
discrete-time
controlled
stochastic
linear
system
implies
the
mean-field
stabilizability
in
mean-square
of
the
MFCSLS
(
1
),
but
the
converse
implication
is
not
always
true.
Based
on
the
particular
form
of
the
matrix
coefficients
of
the
discrete-
time
stochastic
linear
system
(
12a
)
given
in
(
7
)-(
8
),
we
obtain
Lyapunov
type
criteria
for
the
ESMS
property
of
the
companion
DTSLS
(
12
)
of
lower
dimension
than
those
obtained
applying
directly
the
already
existing
results
from
the
literature.
3
The
mean-field
exponential
stability
in
mean-square
3.1
An
equivalent
characterization
of
the
MF-ESMS
property
First
we
show
that
without
loss
of
generality,
we
can
characterize
the
MF-
ESMS
property
of
the
MFSLS
(
10a
)
based
only
on
exponential
decay
of
its
solutions
starting
from
deterministic
initial
states.
Proposition
1.
Under
the
assumption
H1
)
the
following
are
equivalent:
(i)
the
MFSLS
(
10a
)
is
MF-ESMS
in
the
sense
of
Definition
1
;
(ii)
E
[
|
x
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
|
x
0
|
2
(15)
for
all
t
≥
t
0
≥
0
,
x
0
∈
R
n
,
where
β
≥
1
,
δ
∈
(0
,
1)
are
constants
not
depending
upon
t,
t
0
,
x
0
.
Proof.
The
implication
(
i
)
⇒
(
ii
)
is
obvious
because
R
n
⊂
X
t
0
for
any
t
0
∈
Z
+
.
Further,
we
proof
(
ii
)
⇒
(
i
).
Let
t
0
∈
Z
+
and
˜
x
0
∈
X
t
0
be
arbitrary
but
fixed.
We
have
to
show
that
the
solution
x
(
·
;
t
0
,
˜
x
0
)
of
the
MFSLS
(
10a
)
V.
Dr˘
agan
347
corresponding
to
this
initial
pair
has
an
exponential
decay
as
that
from
(
13
).
For
each
1
≤
j
≤
n
,
we
consider
ξ
j
(
·
,
t
0
)
the
solution
of
the
following
IVP:
ξ
j
(
t
+
1)
=
A
0
(
t
)
ξ
j
(
t
)
+
A
0
(
t
)
E
[
ξ
j
(
t
)]
+
r
k
=1
(
A
k
(
t
)
ξ
j
(
t
)
+
A
k
(
t
)
E
[
ξ
j
(
t
)])
w
k
(
t
)
,
t
≥
t
0
(16a)
ξ
j
(
t
0
)
=
e
j
(16b)
where
e
j
∈
R
n
is
the
j
-th
column
of
the
identity
matrix
I
n
.
Comparing
the
IVP
(
16
)
to
the
IVP
(
10
),
we
obtain
based
on
the
unicity
of
the
solution
of
an
IVP,
that
ξ
j
(
t,
t
0
)
=
x
(
t
;
t
0
,
e
j
)
for
all
t
≥
t
0
.
If
(ii)
holds,
we
obtain
from
(
15
)
applied
in
the
case
of
the
previous
equality
that
E
[
|
ξ
j
(
t
;
t
0
)
|
2
]
≤
βδ
t
−
t
0
(17)
for
all
t
≥
t
0
.
We
define
the
matrix
Ξ(
t
;
t
0
)
=
ξ
1
(
t,
t
0
)
ξ
2
(
t,
t
0
)
...
ξ
n
(
t,
t
0
)
.
Bear-
ing
in
mind
(
16
)
we
may
infer
that
{
Ξ(
t
;
t
0
)
}
t
≥
t
0
is
the
matrix
valued
solution
of
the
following
IVP:
Ξ(
t
+
1)
=
A
0
(
t
)Ξ(
t
)
+
A
0
E
[Ξ(
t
)]
+
r
k
=1
(
A
k
(
t
)Ξ(
t
)
+
A
k
(
t
)
E
[Ξ(
t
)])
w
k
(
t
)
,
(18a)
Ξ(
t
0
;
t
0
)
=
I
n
.
(18b)
Starting
from
the
definition
of
the
euclidian
norm
of
a
symmetric
positive
semidefinite
matrix,
we
may
write
successively:
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)]
2
=
λ
max
(
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)])
≤
Tr
[
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)]]
=
n
j
=1
E
[
|
ξ
j
(
t,
t
0
)
|
2
]
(19)
where
λ
max
(
·
)
denotes
the
maximal
eigenvalue
of
a
positive
semidefinite
symmetric
matrix
and
Tr
[
·
]
stands
for
the
trace
of
a
matrix.
Further,
(
17
)
and
(
19
)
yield
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)]
2
≤
nβδ
t
−
t
0
(20)
On
mean-square
stability
and
stabilizability
348
for
all
t
≥
t
0
≥
0.
Let
˜
x
(
t
)
:=
Ξ(
t
;
t
0
)˜
x
0
.
Employing
(
18
),
we
deduce
that
˜
x
(
t
)
is
the
solu-
tion
of
the
IVP
(
10
)
corresponding
to
the
initial
pair
(
t
0
,
˜
x
0
).
Hence,
x
(
t
;
t
0
,
˜
x
0
)
=
Ξ(
t,
t
0
)˜
x
0
(21)
for
all
t
≥
t
0
≥
0,
˜
x
0
∈
X
t
0
.
Let
σ
[˜
x
0
]
⊂
F
be
the
smallest
σ
-algebra
with
respect
to
which
the
random
vector
˜
x
0
is
measurable.
Based
on
the
properties
of
the
conditional
expectation,
we
obtain
from
(
21
)
that
E
[
|
x
(
t
;
t
0
,
˜
x
0
)
|
2
|
σ
[˜
x
0
]]
=
E
[˜
x
T
0
Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)˜
x
0
|
σ
[˜
x
0
]]
=
˜
x
T
0
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)
|
σ
[˜
x
0
]]˜
x
0
.
(22)
Bearing
in
mind
the
properties
of
the
random
vectors
from
X
t
0
,
we
may
infer
via
(
18
)
that
the
components
of
the
matrix
Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)
are
random
vectors
independent
with
respect
to
the
σ
-algebra
σ
[˜
x
0
].
Hence
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)
|
σ
[˜
x
0
]]
=
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)]
a.s.
(23)
Since
0
≤
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)]
≤
E
[Ξ
T
(
t,
t
0
)Ξ(
t,
t
0
)]
2
I
n
we
obtain
via
(
20
),
(
22
),
(
23
)
that
E
[
|
x
(
t
;
t
0
,
˜
x
0
)
|
2
|
σ
[˜
x
0
]]
≤
nβδ
t
−
t
0
|
˜
x
0
|
2
a.s.
for
all
t
≥
t
0
≥
0,
˜
x
0
∈
X
t
.
Taking
the
expectation
in
the
previous
inequality,
we
may
conclude
that
E
[
|
x
(
t
;
t
0
,
˜
x
0
)
|
2
]
≤
nβδ
t
−
t
0
E
[
|
˜
x
0
|
2
]
for
all
t
≥
t
0
≥
0,
˜
x
0
∈
X
t
0
.
Thus
the
proof
ends.
3.2
The
connection
between
the
MF-ESMS
property
of
the
MFSLS
and
the
ESMS
property
of
its
companion
DTSLS
Let
x
(
·
;
t
0
,
x
0
)
be
the
solution
of
the
MFSLS
(
10a
)
corresponding
to
an
arbitrary
initial
pair
(
t
0
,
x
0
),
where
t
0
∈
Z
+
and
x
0
∈
X
t
0
.
We
set
x
1
(
t
;
t
0
,
x
0
)
x
(
t
;
t
0
,
x
0
)
−
E
[
x
(
t
;
t
0
,
x
0
)]
(24a)
x
2
(
t
;
t
0
,
x
0
)
E
[
x
(
t
;
t
0
,
x
0
)]
(24b)
V.
Dr˘
agan
349
t
≥
t
0
and
obtain
the
solution
of
the
IVP
(
11
)
generated
by
the
solution
x
(
·
;
t
0
,
x
0
)
of
the
IVP
(
10
).
Conversely,
if
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
0
,
x
0
)
is
a
solution
of
the
IVP
(
11
),
then
x
(
·
,
t
0
,
x
0
)
defined
by
x
(
t
;
t
0
,
x
0
)
x
1
(
t
;
t
0
,
x
0
)
+
x
2
(
t
;
t
0
,
x
0
)
,
t
≥
t
0
(25)
is
the
solution
of
the
IVP
(
10
).
Employing
the
equalities
(
24
)
and
(
25
)
we
stabilize
an
equivalence
be-
tween
the
MF-ESMS
property
of
the
MFSLS
(
10a
)
and
the
exponential
decay
of
the
solutions
of
the
IVP
(
11
)
for
an
arbitrary
x
0
∈
X
t
0
.
Proposition
2.
Under
the
assumption
H1
)
,
the
following
are
equivalent:
(i)
the
MFSLS
(
10a
)
is
MF-ESMS
in
the
sense
of
the
Definition
1
;
(ii)
for
each
initial
time
t
0
∈
Z
+
and
any
initial
state
x
0
∈
X
t
0
,
the
solution
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
0
,
x
0
)
of
the
corresponding
IVP
(
11
)
has
a
decay
of
the
form:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
˜
β
˜
δ
t
−
t
0
(
E
[
|
x
0
−
E
[
x
0
]
|
2
]
+
|
E
[
x
0
]
2
)
,
(26)
where
˜
β
≥
1
,
˜
δ
∈
(0
,
1)
are
constants
not
depending
upon
t,
t
0
,
x
0
.
Proof:
Assuming
that
(
i
)
holds,
we
obtain
from
(
13
),
(
24b
),
that:
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
E
[
|
x
0
|
2
]
(27)
for
all
t
≥
t
0
∈
Z
+
and
x
0
∈
X
t
0
,
with
β
≥
1,
δ
∈
(0
,
1)
not
depending
upon
t,
t
0
,
x
0
.
Further,
(
25
)
gives
E
[
|
x
(
t
;
t
0
,
x
0
)
|
2
]
=
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
(28)
for
all
t
≥
t
0
∈
Z
+
because
E
[
x
1
(
t
;
t
0
,
x
0
)(
x
2
(
t
;
t
0
,
x
0
))
T
]
=
0
for
all
t
≥
t
0
.
Employing
(
13
),
(
24a
),
(
27
),
(
28
)
we
deduce
that:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
≤
2
βδ
t
−
t
0
E
[
x
0
|
2
]
.
Combining
this
inequality
to
(
27
),
we
obtain:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
3
βδ
t
−
t
0
E
[
|
x
0
|
2
]
(29)
for
all
t
≥
t
0
∈
Z
+
,
x
0
∈
X
t
0
.
Bearing
in
mind
that
E
[
|
x
0
|
2
]
=
E
[
|
x
0
−
E
[
x
0
]
|
2
]
+
|
E
[
x
0
]
|
2
we
may
conclude
that
(
29
)
coincides
to
(
26
),
when
˜
β
=
3
β
,
˜
δ
=
δ
.
Thus
the
validity
of
the
implication
(
i
)
⇒
(
ii
)
is
confirmed.
In
order
to
check
the
validity
of
the
implication
(
ii
)
⇒
(
i
)
we
remark
that
(
28
)
allows
us
to
deduce
that
(
13
)
holds
if
(
26
)
is
true.
Thus
the
proof
ends.
On
mean-square
stability
and
stabilizability
350
Remark
3.
a)
Even
if
x
2
(
t
;
t
0
,
x
0
)
are
deterministic
vectors,
we
preferred
to
use
the
notation
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
in
(
26
)
in
order
to
preserve
the
unit
of
the
presentation.
b)
From
(
24b
)
we
obtain,
via
(
2
)
written
for
B
0
(
t
)
=
B
0
(
t
)
=
0
for
all
t
∈
Z
,
that
{
x
2
(
t
;
t
0
,
x
0
)
}
t
≥
t
0
solves
the
deterministic
discrete-time
linear
equation
(DTLE)
on
the
euclidian
space
R
n
:
x
2
(
t
+
1)
=
(
A
0
(
t
)
+
A
0
(
t
))
x
2
(
t
)
.
(30)
Invoking
the
implication
(
i
)
⇒
(
ii
)
from
Proposition
2
we
may
infer
that
the
exponential
stability
of
the
zero
solution
of
the
deterministic
DTLE
(
30
)
is
a
necessary
condition
for
the
MF-ESMS
property
of
the
MFSLS
(
10a
).
For
the
reader
convenience,
we
recall
here
the
definition
of
the
ESMS
property
of
the
DTSLS
(
12a
).
Definition
3.
We
say
that
the
zero
solution
of
the
DTSLS
(
12a
)
or
equiv-
alently,
the
system
(
12a
)
is
exponentially
stable
in
mean-square
(ESMS)
if
its
solutions
have
a
decay
of
the
form
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
β
1
δ
t
−
t
0
1
(
E
[
|
x
1
0
|
2
]
+
E
[
|
x
2
0
|
2
])
(31)
for
all
t
≥
t
0
∈
Z
+
and
any
initial
states
x
0
=
x
1
0
x
2
0
∈
X
t
0
×
X
t
0
.
In
(
31
),
β
1
≥
1
,
δ
1
∈
(0
,
1)
are
constants
not
depending
upon
t,
t
0
,
x
k
0
,
k
=
1
,
2
.
Remark
4.
Bearing
in
mind
that
the
initial
states
(
11
)
are
particular
case
of
(
12b
)
we
may
conclude
via
the
implication
(
ii
)
⇒
(
i
)
from
Proposition
2
that
the
MFSLS
(
10a
)
is
MF-ESMS
if
its
companion
DTSLS
(
12a
)
is
ESMS.
The
next
example
shows
that
the
converse
implication
is
not
always
true.
Example
1.
We
consider
the
1-dimensional
discrete-time,
time-invariant
MFSLS:
x
(
t
+
1)
=
a
0
x
(
t
)
+
a
0
E
[
x
(
t
)]
+
r
k
=1
(
a
k
x
(
t
)
+
a
k
E
[
x
(
t
)])
w
k
(
t
)
,
t
≥
t
0
(32a)
x
(
t
0
)
=
x
0
∈
R
(32b)
where
a
k
,
a
k
,
0
≤
k
≤
r
are
real
numbers
subject
to
the
following
constraints:
c
1
)
a
k
+
a
k
=
0
,
1
≤
k
≤
r
;
c
2
)
|
a
0
+
a
0
|
<
1
;
V.
Dr˘
agan
351
c
3
)
r
k
=0
a
2
k
≥
1
,
{
w
k
(
t
)
}
t
≥
0
,
1
≤
k
≤
r
,
are
sequences
of
random
variables
which
are
satis-
fying
the
conditions
from
the
assumption
H
1)
.
According
to
Proposition
1
,
for
the
study
of
the
MF-ESMS
property
of
the
MFSLS
(
32
)
we
may
assume
without
loss
of
generality
that
the
initial
states
x
0
are
deterministic.
Under
the
constraint
c
1
)
,
the
corresponding
IVP
(
11
)
becomes:
x
1
(
t
+
1)
=
(
a
0
+
r
k
=1
a
k
w
k
)
x
1
(
t
)
,
t
≥
t
0
(33)
x
1
(
t
0
)
=
0
x
2
(
t
+
1)
=
(
a
0
+
a
0
)
x
2
(
t
)
,
t
≥
t
0
(34)
x
2
(
t
0
)
=
x
0
.
Based
on
the
unicity
of
the
solution
of
an
IVP
we
obtain
that
the
unique
solution
of
the
IVP
(
33
)
is
x
1
(
t
)
=
0
a.s.
t
≥
t
0
and
the
unique
solution
of
(
34
)
is
x
2
(
t
)
=
(
a
0
+
a
0
)
t
−
t
0
x
0
,
t
≥
t
0
.
Invoking
c
2
)
we
may
infer
that
|
x
2
(
t
)
|
2
≤
δ
t
−
t
0
|
x
0
|
2
for
all
t
≥
t
0
∈
Z
+
,
where
δ
∈
(0
,
1)
.
In
this
case,
the
inequality
(
26
)
becomes:
E
[
|
x
1
(
t
)
|
2
]
+
E
[
|
x
2
(
t
)
|
2
]
≤
δ
t
−
t
0
|
x
0
|
2
,
∀
t
≥
t
0
.
The
implication
(
ii
)
⇒
(
i
)
from
Proposition
2
applied
in
this
particular
case,
allows
us
to
conclude
that
the
MFSLS
(
32
)
is
MF-ESMS.
The
com-
panion
discrete-time
stochastic
linear
system
of
the
form
(
12
),
associated
to
(
32
)
is:
x
1
(
t
+
1)
=
a
0
x
1
(
t
)
+
r
k
=1
a
k
x
1
(
t
)
w
k
(
t
)
,
t
≥
t
0
(35)
x
1
(
t
0
)
=
x
1
0
x
2
(
t
+
1)
=
(
a
0
+
a
0
)
x
2
(
t
)
,
t
≥
t
0
(36)
x
2
(
t
0
)
=
x
2
0
where
x
k
0
∈
X
t
0
,
k
=
1
,
2
.
Hence,
the
companion
DTSLS
associated
to
the
MFSLS
(
32
)
is
ESMS
if
and
only
if
the
discrete-time
stochastic
linear
equations
(
35
)
and
(
36
)
are
simultaneously
ESMS.
Employing
a
Lyapunov-type
criterion
for
the
characterization
of
the
ESMS
property
of
a
discrete-time
stochastic
linear
equation
(see
section
3.5.2
from
On
mean-square
stability
and
stabilizability
352
[
6
])
we
obtain
that
the
DTSLS
(
35
)
is
ESMS
if
and
only
if
its
coefficients
are
satisfying
the
condition
r
k
=0
a
2
k
<
1
.
This
is
in
contradiction
to
c
3
)
.
Thus,
we
may
conclude
that
the
MFSLS
(
32
)
is
MF-ESMS
but
its
com-
panion
discrete-time
stochastic
linear
system
is
not
ESMS.
This
contradicts
Theorem
2.1
from
[
13
]
where
is
asserted
that
the
MF-ESMS
property
of
a
MFSLS
is
equivalent
to
the
ESMS
property
of
its
companion
discrete-time
standard
stochastic
linear
system.
Let
us
consider
the
analogous
for
the
MFSLS
(
32
)
of
the
operator
L
defined
in
(
24
)
from
[
15
].
This
is
L
X
=
(
a
0
+
a
0
)
2
x
0
0
r
k
=0
a
2
k
x
for
all
X
=
x
0
0
x
.
One
sees
that
λ
=
r
k
=0
a
2
k
is
the
eigenvalue
of
the
operator
L
corresponding
to
the
eigenvector
X
=
0
0
0
1
.
Bearing
in
mind
the
constraint
c
3
)
we
deduce
that
λ
≥
1
.
Hence,
the
spectral
radius
of
the
operator
L
satisfies
ρ
(
L
)
≥
1
.
So,
the
MFSLS
(
32
)
is
MF-ESMS,
but
the
spectral
radius
of
the
Lyapunov-type
operator
defined
by
its
companion
standard
discrete-time
stochastic
linear
system
is
not
strictly
less
than
1.
This
shows
that
the
equivalence
stated
in
Theorem
3.1
from
[
15
]
is
not
always
true.
In
the
next
definition,
we
introduce
a
new
kind
of
mean-field
exponential
stability
in
mean-square.
Definition
4.
We
say
that
the
MFSLS
(
10
)
is
mean-field
strong
exponen-
tially
stable
in
mean-square
(MF-SESMS)
if
its
companion
discrete-time
SLS
(
12
)
is
ESMS
in
the
sense
of
Definition
3
.
Remark
5.
a)
From
Remark
4
and
Definition
4
,
we
may
infer
that
the
MF-SESMS
property
of
the
MFSLS
(
10
)
implies
its
MF-ESMS
property.
The
Example
1
and
Definition
4
shows
that
the
converse
implication
is
not
always
true.
b)
Bearing
in
mind
that
for
the
characterization
of
the
ESMS
property
of
a
discrete-time
stochastic
linear
system
there
exist
a
lot
of
spectral
criteria
and
Lyapunov-type
criteria,
we
may
infer
via
a)
that
these
criteria
provide
a
set
of
useful
sufficient
conditions
for
characterization
of
the
MF-ESMS
property
of
a
MFSLS.
Employing
the
special
structure
of
the
matrix
coefficients
of
the
DTSLS
described
in
(
7
),
we
shall
derive
in
the
next
section,
a
set
of
necessary
and
V.
Dr˘
agan
353
sufficient
conditions
for
the
MF-SESMS
property
of
the
MFSLS
(
10
)
of
lower
dimension
than
those
obtained
applying
directly
the
already
existing
results
from
the
literature
(see
[
6
],
[
17
]).
4
Criteria
for
MF-SESMS
The
main
result
of
this
section
is
in
the
next
theorem.
Theorem
1.
Under
the
assumption
H1
)
the
following
are
equivalent:
(i)
the
MFSLS
(
10
)
is
MF-SESMS;
(ii)
the
discrete-time
stochastic
linear
equation
(DTSLE):
x
(
t
+
1)
=
(
A
0
(
t
)
+
r
k
=1
A
k
(
t
)
w
k
(
t
))
x
(
t
)
(37)
(obtained
from
(
10a
)
taking
A
k
(
t
)
=
0
,
t
∈
Z
,
0
≤
k
≤
r
)
is
ESMS
and
the
discrete-time
deterministic
linear
equation
(DTDLE):
x
(
t
+
1)
=
(
A
0
(
t
)
+
A
0
(
t
))
x
(
t
)
,
t
∈
Z
(38)
(obtained
from
(
10
)
taking
A
k
(
t
)
=
A
k
(
t
)
=
0
,
t
∈
Z
,
1
≤
k
≤
r
)
is
exponentially
stable;
(iii)
there
exist
θ
-periodic
matrix
valued
sequences
{
x
(
t
)
}
t
∈
Z
⊂
S
+
n
,
{
X
(
t
)
}
t
∈
Z
⊂
S
+
n
which
are
satisfying
the
backward
time-varying
linear
ma-
trix
inequalities
(LMIs):
X
(
t
)
−
r
k
=0
A
k
(
t
)
X
(
t
+
1)
A
k
(
t
)
>
0
(39)
X
(
t
)
>
0
,
t
∈
Z
,
X
(
t
)
−
(
A
0
(
t
)
+
A
0
(
t
))
X
(
t
+
1)(
A
0
(
t
)
+
A
0
(
t
))
>
0
(40)
X
(
t
)
>
0
,
t
∈
Z
;
(iv)
there
exist
θ
-periodic
matrix
valued
sequences
{
Y
(
t
)
}
t
∈
Z
⊂
S
+
n
,
{
Y
(
t
)
}
t
∈
Z
⊂
S
+
n
solving
the
following
forward
time-varying
LMIs:
Y
(
t
+
1)
−
r
k
=0
A
k
(
t
)
Y
(
t
)
A
k
(
t
)
>
0
(41)
Y
(
t
)
>
0
,
On
mean-square
stability
and
stabilizability
354
t
∈
Z
Y
(
t
+
1)
−
(
A
0
(
t
)
+
A
(
0
t
))
Y
(
t
)(
A
0
(
t
)
+
A
0
(
t
))
>
0
(42)
Y
(
t
)
>
0
,
t
∈
Z
.
Proof.
Let
us
assume
that
the
MFSLS
(
10
)
is
MF-SESMS.
This
means
that
the
solutions
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
o
,
x
0
)
of
the
IVP
(
12
)
have
a
decay
rate
of
the
form
(
31
)
for
all
initial
time
t
0
∈
Z
+
and
initial
state
x
0
=
x
1
0
x
2
0
where
x
k
0
∈
X
t
0
,
k
=
1
,
2.
In
the
particular
case
x
0
=
x
0
0
with
arbitrary
x
0
∈
R
n
we
have
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
0
,
x
0
)
=
x
1
(
·
;
t
0
,
x
0
)
0
where
x
1
(
·
;
t
0
,
x
0
)
is
the
solution
of
the
DTSLE
(
37
)
corresponding
to
the
initial
pair
(
t
0
,
x
0
)
∈
Z
+
×
R
n
.
In
this
case,
(
31
)
gives:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
≤
β
1
δ
t
−
t
0
1
|
x
0
|
2
(43)
∀
t
≥
t
0
∈
Z
+
,
x
0
∈
R
n
,
β
1
≥
1,
δ
1
∈
(0
,
1)
are
not
depending
upon
t,
t
0
,
x
0
.
On
the
other
hand,
in
the
particular
case
when
x
0
=
0
x
0
with
arbitrary
x
0
∈
R
n
,
we
have:
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
0
,
x
0
)
=
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
0
,
x
0
)
where
x
2
(
·
;
t
0
,
x
0
)
is
the
solution
of
the
DTDLE
(
38
)
corresponding
to
the
initial
pair
(
t
0
,
x
0
)
∈
Z
+
×
R
n
.
In
this
case,
(
31
)
becomes:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
β
1
δ
t
−
t
0
1
|
x
0
|
2
for
all
t
≥
t
0
∈
Z
+
,
x
0
∈
R
n
.
Hence,
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
β
1
δ
t
−
t
0
1
|
x
0
|
2
(44)
for
all
t
≥
t
0
∈
Z
+
,
x
0
∈
R
n
.
Bearing
in
mind
(
43
)
and
(
44
),
we
may
infer
that
(
ii
)
holds
if
(
i
)
is
true.
Further,
we
show
that
the
implication
(
ii
)
⇒
(
i
)
V.
Dr˘
agan
355
holds.
Based
on
(
7
)
we
rewrite
(
12a
)
as:
x
1
(
t
+
1)
=
A
0
(
t
)
x
1
(
t
)
+
r
k
=1
(
A
k
(
t
)
x
1
(
t
)
+
f
k
(
t
))
w
k
(
t
)
(45a)
x
2
(
t
+
1)
=
(
A
0
(
t
)
+
A
0
(
t
))
x
2
(
t
)
(45b)
x
1
(
t
0
)
=
x
1
0
,
x
2
(
t
0
)
=
x
2
0
,
(45c)
where
x
k
0
∈
X
t
0
,
k
=
1
,
2,
and
f
k
(
t
)
(
A
k
(
t
)
+
A
k
(
t
))
x
2
(
t
)
(46)
t
∈
Z
+
,
1
≤
k
≤
r
.
If
(
ii
)
holds,
then,
(
43
)
and
(
44
)
hold
too.
In
this
case,
the
solutions
of
(
45b
)
satisfy
E
[
|
x
2
(
t
;
t
0
,
x
2
0
)
|
2
]
≤
˜
β
˜
δ
t
−
t
0
E
[
|
x
2
0
|
2
]
(47)
for
all
t
≥
t
0
∈
Z
+
,
x
2
0
∈
X
t
0
.
Combining
(
46
)
and
(
47
)
we
deduce
that
the
forcing
terms
from
(
45a
)
are
satisfying
E
[
|
f
k
(
t
)
|
2
]
≤
γ
k
˜
δ
t
−
t
0
E
[
|
x
2
0
|
2
]
(48)
for
all
t
≥
t
0
∈
Z
+
,
where
γ
k
=
˜
β
sup
t
∈
Z
A
k
(
t
)
+
A
k
(
t
)
and
˜
δ
∈
(0
,
1).
Bearing
in
mind
(
43
),
we
may
infer
via
Corollary
3.9
(iii)
from
[
6
]
applied
in
the
case
of
non-homogeneous
DTSLE
(
45a
)
that
its
solutions
are
satisfying:
lim
t
→∞
E
[
|
x
1
(
t
;
t
0
,
x
1
0
)
|
2
]
=
0
(49)
for
any
initial
time
t
0
∈
Z
+
and
the
initial
state
x
1
0
∈
X
t
0
.
Employing
(
48
)
and
(
49
)
we
may
conclude
that
the
solutions
of
the
IVP
(
12
)
are
satisfying
lim
t
→∞
(
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
=
0
for
all
t
0
∈
Z
+
and
any
x
0
=
x
1
0
x
2
0
with
x
k
0
∈
X
t
0
,
k
=
1
,
2.
Since
(
12a
)
is
a
discrete-time
linear
equation
with
periodic
coefficients,
we
may
deduce
via
the
equivalence
(
i
)
⇔
(
v
)
from
Theorem
3.10
in
[
6
],
applied
in
the
case
of
the
equation
(
12a
),
that
its
solutions
are
satisfying
(
31
).
According
to
Definition
4
we
may
conclude
that
the
MFSLS
(
10
)
is
MF-SESMS.
Thus
the
validity
of
the
implication
(
ii
)
⇔
(
i
)
is
confirmed.
The
equivalences
(
ii
)
⇔
(
iii
)
and
(
ii
)
⇔
(
iv
)
follow
applying
Theorem
3.11
from
[
6
]
in
the
case
of
the
DTSLE
(
37
)
and
DTDLE
(
38
),
respectively.
Thus
the
proof
is
complete.
On
mean-square
stability
and
stabilizability
356
Remark
6.
The
Lyapunov-type
criteria
(
39
)-(
40
)
and
(
41
)-(
42
),
respec-
tively,
are
based
on
the
solutions
of
some
time-varying
LMIs
with
periodic
coefficients
with
lower
dimension
than
those
obtained
applying
directly,
for
example,
Theorem
3.11
from
[
6
]
in
the
case
of
the
DTSLE
(
12a
).
Based
on
Remark
5
we
may
infer
that
(
39
)-(
40
)
and
(
41
)-(
42
)
provide
sets
of
sufficient
conditions
to
test
if
a
MFSLS
is
MF-ESMS.
The
next
proposition
shows
that
the
conclusion
of
the
Example
1
is
still
valid
also
in
the
time-varying
context.
Proposition
3.
We
assume:
a)
the
assumption
H1
)
holds;
b)
A
k
(
t
)
+
A
k
(
t
)
=
0
,
t
∈
Z
,
1
≤
k
≤
r
;
c)
the
DTDLS
(
38
)
is
exponentially
stable,
but
the
DTSLDE
(
37
)
is
not
ESMS.
Under
these
conditions,
the
MFSLS
(
10
)
is
MF-ESMS
but
it
is
not
MF-
SESMS.
Proof.
According
to
Proposition
1
,
without
loss
of
generality,
we
may
assume
that
x
0
∈
R
n
because
R
n
⊂
X
t
0
for
any
t
0
∈
Z
+
.
Under
the
assump-
tion
b
),
the
IVP
(
12
)
becomes:
x
1
(
t
+
1)
=
A
0
(
t
)
x
1
(
t
)
+
r
k
=1
A
k
(
t
)
x
1
(
t
)
w
k
(
t
)
,
t
≥
t
0
,
(50)
x
1
(
t
0
)
=
0
,
x
2
(
t
+
1)
=
(
A
0
(
t
)
+
A
0
(
t
))
x
2
(
t
)
,
t
≥
t
0
(51)
x
2
(
t
0
)
=
x
0
.
Based
on
the
unicity
of
the
solution
of
an
IVP,
we
may
conclude
that
the
unique
solution
of
the
IVP
(
50
)
is
x
1
(
t
;
t
0
,
x
0
)
=
0
a.s.
t
≥
t
0
and
the
unique
solution
of
the
IVP
(
51
)
satisfies
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
|
x
0
|
2
.
(52)
In
this
case,
(
52
)
gives
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
|
x
0
|
2
(53)
for
all
t
≥
t
0
≥
0,
x
0
∈
R
n
.
Since,
E
[
|
x
(
t
;
t
0
,
x
0
)
|
2
]
=
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
we
obtain
from
(
53
)
that
E
[
|
x
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
|
x
0
|
2
V.
Dr˘
agan
357
for
all
t
≥
t
0
∈
Z
+
,
x
0
∈
R
n
,
where
β
≥
1,
δ
∈
(0
,
1)
are
not
depending
upon
t,
t
0
,
x
0
.
Hence,
the
implication
(
ii
)
⇒
(
i
)
from
Proposition
2
allows
us
to
conclude
that
under
the
considered
assumptions
the
MFSLS
(
10
)
is
MF-
ESMS.
It
can
not
be
MF-SESMS
because
the
implication
(
i
)
⇒
(
ii
)
from
Theorem
1
would
imply
that
the
DTSLDE
(
37
)
would
be
ESMS,
which
contradicts
the
assumption
c
).
Thus
the
proof
ends.
5
The
mean-fields
stabilizability
in
mean-square
In
this
section
we
shall
use
the
results
derived
in
the
previous
sections
to
characterize
the
MF
stabilizability
in
mean-square
of
the
MFCSLS
of
the
form
(
1
).
Remark
7.
Based
on
Proposition
1
applied
in
the
case
of
the
closed-loop
MFSLS
(
14
),
we
may
infer
that,
without
loss
of
generality,
in
order
to
define
the
MF
stabilizability
in
mean-square
property
of
the
MFCSLS
(
1
)
we
may
assume
that
its
initial
states
x
0
are
deterministic
vectors.
Taking
the
expectation
in
(
14
)
we
obtain:
E
[
x
(
t
+
1)]
=
[
A
0
(
t
)
+
A
0
(
t
)
+
(
B
0
(
t
)
+
B
0
(
t
))
F
(
t
)]
E
[
x
(
t
)]
(54a)
E
[
x
(
t
0
)]
=
E
[
x
0
]
(54b)
for
any
initial
time
t
0
∈
R
n
and
any
initial
state
x
0
∈
X
t
0
.
To
this
end,
we
took
into
account
that
based
of
Lemma
1
applied
in
the
case
of
the
MFSLS
(
14
),
we
may
deduce
that
the
random
vectors
x
(
t
)
and
w
(
t
)
are
mutually
independent.
Then,
subtracting
(
54
)
from
(
14
)
we
obtain:
x
(
t
+
1)
−
E
[
x
(
t
+
1)]
=
(
A
0
(
t
)
+
B
0
(
t
)
F
(
t
))(
x
(
t
)
−
E
[
x
(
t
)])+
r
k
=1
[(
A
k
(
t
)
+
B
k
(
t
)
F
(
t
))(
x
(
t
)
−
E
[
x
(
t
)])
+
(
A
k
(
t
)
+
A
k
(
t
)+
(
B
k
(
t
)
+
B
k
(
t
))
F
(
t
))
E
[
x
(
t
)]]
w
k
(
t
)
,
t
≥
t
0
(55a)
x
(
t
0
)
−
E
[
x
(
t
0
)]
=
x
0
−
E
[
x
0
]
.
(55b)
Employing
the
notation
(
4
),
we
obtain
from
(
54
)
and
(
55
),
the
following
On
mean-square
stability
and
stabilizability
358
IVP:
x
1
(
t
+
1)
x
2
(
t
+
1)
=
(
A
0
(
t
)
+
B
0
(
t
)
F
(
t
))
x
1
(
t
)
x
2
(
t
)
+
r
k
=1
(
A
k
(
t
)
+
B
k
(
t
)
F
(
t
))
x
1
(
t
)
x
2
(
t
)
w
k
(
t
)
,
t
≥
t
0
,
(56a)
x
1
(
t
0
)
x
2
(
t
0
)
=
x
0
−
E
[
x
0
]
E
[
x
0
]
(56b)
for
all
t
0
∈
Z
+
and
x
0
∈
X
t
0
,
where
A
k
(
t
)
,
B
k
(
t
),
0
≤
k
≤
r
were
defined
in
(
7
),
(
8
)
and
F
(
t
)
=
F
(
t
)
0
0
F
(
t
)
(57)
t
∈
Z
.
Applying
Proposition
2
in
the
case
of
the
pair
formed
by
the
IVP
(
14
)
and
the
IVP
(
56
),
we
obtain
via
Definition
2
,
the
following
equivalence
whose
proof
is
omitted.
Corollary
1.
Under
the
assumption
H1
)
the
following
are
equivalent:
(i)
the
MFCSLS
(
1
)
is
MF
stabilizable
in
mean-square;
(ii)
there
exist
R
m
×
n
valued
θ
-periodic
matrix
sequences
{
F
(
t
)
}
t
∈
Z
,
{
˜
F
(
t
)
}
t
∈
Z
with
the
properties
that
for
each
initial
time
t
0
∈
Z
+
and
any
initial
state
x
0
∈
X
t
0
,
the
solutions
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
0
,
x
0
)
of
the
corresponding
IVP
(
56
)
have
a
decay
of
the
form:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
β
1
ˆ
δ
t
−
t
0
(
E
[
|
x
0
−
E
[
x
0
]
|
2
]+
|
E
[
x
0
]
|
2
)
(58)
for
all
t
≥
t
0
,
where
ˆ
β
≥
1
,
ˆ
δ
∈
(0
,
1)
are
constants
which
are
not
depending
upon
t,
t
0
,
x
0
.
In
order
to
compare
the
mean-field
stabilizability
in
mean-square
prop-
erty
with
a
kind
of
stochastic
stabilizability
property
of
the
DTCSLS
(
9
)
we
introduce
the
following
definition:
Definition
5.
We
say
that
the
DTCSLS
(
9
)
is
stochastic
stabilizable
in
mean-square
by
a
decentralized
linear
state
feedback
if
there
exists
a
control
of
the
form
u
(
t
)
=
F
(
t
)
x
(
t
)
=
F
(
t
)
x
1
(
t
)
0
0
F
(
t
)
x
2
(
t
)
(59)
V.
Dr˘
agan
359
with
the
properties:
a)
{
F
(
t
)
}
t
∈
Z
,
{
F
(
t
)
}
t
∈
Z
are
θ
-periodic
R
m
×
n
matrix
valued
sequences;
b)
the
closed-loop
system
x
(
t
+
1)=(
A
0
(
t
)+
B
0
(
t
)
F
(
t
))
x
(
t
)+
r
k
=1
(
A
k
(
t
)+
B
k
(
t
)
F
(
t
))
x
(
t
)
w
k
(
t
)
,
(60a)
x
(
t
0
)
=
x
1
0
x
2
0
∈
X
t
0
×
X
t
0
(60b)
(obtained
plugging
the
control
(
59
)
to
the
DTCSLS
(
9
))
is
ESMS.
Remark
8.
For
a
given
F
(
t
)
as
in
(
57
),
the
IVP
(
56
)
is
the
special
case
of
the
IVP
(
60
)
for
x
1
0
←
x
0
−
E
[
x
0
]
and
x
2
0
←
E
[
x
0
]
.
Hence,
if
the
DTCSLS
(
9
)
is
stochastic
stabilizable
in
mean-square
by
a
decentralized
linear
state
feedback
of
the
form
(
59
),
then
the
solutions
of
the
IVP
(
56
)
obtained
when
in
(
6
)
is
used
the
control
(
59
),
have
a
decay
of
the
form
(
58
).
Corollary
2.
Under
the
assumption
H1
)
the
MFCSLS
(
1
)
is
MF
stabiliz-
able
in
mean-square
if
its
companion
DTCSLS
(
9
)
is
stochastic
stabilizable
in
mean-square
by
a
decentralized
linear
state
feedback.
Proof
follows
immediately
combining
Remark
8
with
the
implication
(
ii
)
⇒
(
i
)
from
Corollary
1
.
The
next
example
shows
that
the
converse
implication
of
the
one
from
Corollary
2
is
not
always
true.
Example
2.
We
consider
the
1-dimensional
discrete-time
time-invariant
MFCSLS:
x
(
t
+
1)
=
a
0
x
(
t
)
+
a
0
E
[
x
(
t
)]
+
b
0
u
(
t
)
+
b
0
E
[
u
(
t
)]+
+
(
a
1
x
(
t
)
+
a
1
E
[
x
(
t
)]
+
b
1
u
(
t
)
+
b
1
E
[
u
(
t
)])
w
1
(
t
)
,
t
≥
t
0
(61a)
x
(
t
0
)
=
x
0
∈
R
(61b)
where
a
k
,
a
k
,
b
k
,
b
k
are
real
numbers
subject
to
the
constraints:
c
1
)
a
1
+
a
1
=
b
1
+
b
1
=
0
,
b
0
+
b
0
=
0
;
c
2
)
a
0
b
0
+
a
1
b
1
=
0
;
c
3
)
a
2
0
+
a
2
1
>
1
.
In
(
61
),
{
w
1
(
t
)
}
t
≥
0
is
a
sequence
of
independent
random
variables
with
zero
mean
and
variance
1.
We
show
that
under
these
conditions,
the
MFC-
SLS
(
61
)
is
MF
stabilizable
in
mean-square
but
its
companion
DTCSLS
if
type
(
9
)
is
not
stochastic
stabilizable
in
mean-square
by
a
decentralized
linear
On
mean-square
stability
and
stabilizability
360
state
feedback.
Based
on
Remark
7
,
we
may
infer
that,
without
loss
of
gener-
ality,
we
can
assume
that
the
initial
state
x
0
is
deterministic.
Let
δ
∈
(0
,
1)
fixed
and
f
defined
from:
a
0
+
a
0
+
(
b
0
+
b
0
)
f
=
δ.
(62)
Let
f
∈
R
be
arbitrary
but
fixed.
We
show
that
the
closed-loop
system
obtained
by
coupling
the
control
u
(
t
)
=
fx
(
t
)
+
(
f
−
f
)
E
[
x
(
t
)]
(63)
t
≥
t
0
,
to
the
MFCSLS
(
61
)
is
MF-ESMS,
which
means
that
the
MFCSLS
(
61
)
is
MF
stabilizable
in
mean-square.
The
IVP
of
the
form
(
56
)
obtained
by
coupling
the
control
(
63
)
to
the
MFC-
SLS
(
61
)
subject
to
the
constraints
c
1
)
is
x
1
(
t
+
1)
=
(
a
0
+
b
0
f
)
x
1
(
t
)
+
(
a
1
+
b
1
f
)
x
1
(
t
)
w
1
(
t
)
,
t
≥
t
0
(64)
x
1
(
t
0
)
=
0
,
x
2
(
t
+
1)
=
(
a
0
+
a
0
+
(
b
0
+
b
0
)
f
)
x
2
(
t
)
,
t
≥
t
0
(65)
x
2
(
t
0
)
=
x
0
.
From
the
unicity
of
the
solution
of
an
IVP,
we
have
that
the
unique
solution
of
(
64
)
is
x
1
(
t
)
=
0
a.s.
t
≥
t
0
≥
0
.
From
(
62
)
and
(
65
)
we
obtain
that
x
2
(
t
)
=
δ
t
−
t
0
x
0
,
t
≥
t
0
≥
0
.
(66)
Hence,
(
66
)
gives
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
=
δ
t
−
t
0
|
x
0
|
2
,
t
≥
t
0
≥
0
,
x
0
∈
R
.
Bearing
in
mind
that
δ
∈
(0
,
1)
we
may
conclude
via
(
ii
)
⇒
(
i
)
from
Corollary
1
applied
in
the
case
of
the
MFCSLS
(
61
)
that
this
system
is
MF
stabilizable
in
mean-square.
Further,
we
show
that
the
corresponding
companion
of
the
form
(
9
)
associated
to
(
61
)
is
not
stochastic
stabilizable
in
mean-square
by
a
decentralized
linear
state
feedback,
in
the
sense
of
the
Definition
5
.
Let
u
1
(
t
)
u
2
(
t
)
=
f
0
0
f
x
1
(
t
)
x
2
(
t
)
(67)
V.
Dr˘
agan
361
be
an
arbitrary
control
of
the
form
(
59
)
where
f,
f
are
unknown
real
numbers
which
should
be
determined
such
that
the
corresponding
closed-loop
system
to
be
ESMS.
The
IVP
of
the
form
(
60
)
associated
to
the
MFCSLS
(
61
)
and
to
the
control
(
67
)
is:
x
1
(
t
+
1)
=
(
a
0
+
b
0
f
)
x
1
(
t
)
+
(
a
1
+
b
1
f
)
x
1
(
t
)
w
1
(
t
)
,
t
≥
t
0
(68)
x
1
(
t
0
)
=
x
1
0
,
x
2
(
t
+
1)
=
(
a
0
+
a
0
+
(
b
0
+
b
0
)
f
)
x
2
(
t
)
,
t
≥
t
0
(69)
x
2
(
t
0
)
=
x
2
0
.
According
to
Definition
5
,
the
DTCSLS
of
type
(2.9)
associated
to
the
MFC-
SLS
(
61
)
is
stochastic
stabilizable
in
mean-square
by
a
decentralized
lin-
ear
state
feedback
if
the
closed-loop
DTCSLSs
(
68
),
(
69
)
are
simultaneously
ESMS.
Employing
a
Lyapunov
type
criteria
we
may
infer
that
the
DTCSLS
(
68
)
is
ESMS
if
and
only
if
(
a
0
+
b
0
f
)
2
+
(
a
1
+
b
1
f
)
2
−
1
<
0
.
(70)
Bearing
in
mind
the
constraints
c
2
)
we
deduce
that
(
70
)
is
equivalent
to
a
2
0
+
a
2
1
−
1
+
(
b
2
0
+
b
2
1
)
f
2
<
0
.
(71)
Employing
c
3
)
we
may
conclude
that
does
not
exist
f
∈
R
for
which
(
71
)
to
be
satisfied.
Thus,
we
have
shown
that
the
companion
DTCSLS
of
the
form
(
9
)
associated
to
the
MFCSLS
(
61
)
is
not
stochastic
stabilizable
in
mean-square
by
a
decentralized
linear
state
feedback.
Motivated
by
the
implication
stated
in
Corollary
2
,
we
introduce
a
new
kind
of
stabilizability
in
mean-square
of
the
MFCSLS
(
1
).
Definition
6.
We
say
that
the
MFCSLS
(
1
)
is
MF
strong
stabilizable
in
mean-square
if
its
companion
DTCSLS
(
9
)
is
stochastic
stabilizable
in
mean-
square
by
a
decentralized
linear
state
feedback.
Remark
9.
From
Corollary
2
and
Definition
6
,
we
may
infer
that
the
MF
strong
stabilizability
in
mean-square
of
the
MFCSLS
(
1
)
implies
its
MF
sta-
bilizability
in
mean-square.
From
Example
2
and
Definition
6
it
follows
that
the
converse
implication
is
not
always
true.
Hence,
the
necessary
and
suffi-
cient
conditions
for
MF
strong
stabilizability
in
mean-square
of
the
MFCSLS
(
54
)
serves
as
sufficient
conditions
for
the
MF
stabilizability
in
mean-square
of
this
system.
On
mean-square
stability
and
stabilizability
362
Theorem
2.
Under
the
assumption
H1
)
,
the
following
are
equivalent:
(i)
the
MFCSLS
(
1
)
is
MF
strong
stabilizable
in
mean-square;
(ii)
the
discrete-time
controlled
stochastic
linear
system
(DTCSLS):
x
(
t
+
1)
=
A
0
(
t
)
x
(
t
)
+
B
0
(
t
)
u
(
t
)
+
r
k
=1
(
A
k
(
t
)
x
(
t
)
+
B
k
(
t
)
u
(
t
))
w
k
(
t
)
,
t
≥
0
(72)
(obtained
from
(
1
)
ignoring
the
mean-field
part)
is
stochastic
stabilizable
in
mean-square
and
the
discrete-time
controlled
deterministic
linear
system
(DTCDLS):
x
(
t
+
1)
=
(
A
0
(
t
)
+
A
0
(
t
))
x
(
t
)
+
(
B
0
(
t
)
+
B
0
(
t
))
u
(
t
)
(73)
(obtained
from
(
1
)
taking
A
k
(
t
)
=
A
k
(
t
)
=
0
,
B
k
(
t
)
=
B
k
(
t
)
=
0
,
1
≤
k
≤
r
,
t
∈
Z
)
is
stabilizable
by
linear
state
feedback.
Proof.
To
prove
the
implication
(
i
)
⇒
(
ii
),
let
us
remark
that
if
(
i
)
holds,
there
exist
a
control
law
of
the
form
(
59
)
with
the
property
that
the
solutions
x
1
(
·
;
t
0
,
x
0
)
x
2
(
·
;
t
0
,
x
0
)
of
the
corresponding
closed-loop
system
(
60
)
have
a
decay
of
the
form:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
+
E
[
|
x
2
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
(
E
[
|
x
1
0
|
2
]
+
E
[
|
x
2
0
|
2
])
(74)
for
all
t
≥
t
0
≥
0,
x
0
=
(
x
1
0
)
(
x
2
0
)
∈
X
t
0
×
X
t
0
,
where
β
≥
1,
δ
∈
(0
,
1)
are
constants
not
depending
upon
t,
t
0
,
x
k
0
.
Based
on
the
unicity
of
the
solution
of
an
IVP
we
may
infer
that
if
x
0
←
ˆ
x
0
=
x
0
0
where
x
0
∈
R
n
,
we
have
x
1
(
t
;
t
0
,
x
0
)
x
2
(
t
;
t
0
,
x
0
=
x
1
(
t
;
t
0
,
x
0
)
0
where
t
→
x
1
(
t
;
t
0
,
x
0
)
is
the
solution
of
the
closed-loop
DTSLS:
x
1
(
t
+1)
=
(
A
0
(
t
)+
B
0
(
t
)
F
(
t
))
x
1
(
t
)+
r
k
=1
(
A
k
(
t
)+
B
k
(
t
)
F
(
t
))
x
1
(
t
)
w
k
(
t
)
(75)
corresponding
to
the
initial
pair
(
t
0
,
x
0
)
∈
Z
+
×
R
n
.
In
this
case
(
74
)
be-
comes:
E
[
|
x
1
(
t
;
t
0
,
x
0
)
|
2
]
≤
βδ
t
−
t
0
|
x
0
|
2
.
This
means
that
the
DTSLS
(
75
)
is
ESMS,
or
equivalently,
the
DTCSLS
(
72
)
is
stochastic
stabilizable
in
the
sense
of
Definition
??
from
[
6
].
On
the
other
hand,
if
x
0
←
ˇ
x
0
=
0
x
0
where
x
0
∈
R
n
,
we
have
that
V.
Dr˘
agan
363
x
1
(
t
;
t
0
,
x
0
)
x
2
(
t
;
t
0
,
x
0
)
=
x
1
(
t
;
t
0
,
ˇ
x
0
)
x
2
(
t
;
t
0
,
x
0
)
where
t
→
x
2
(
t
;
t
0
,
x
0
)
is
the
solution
of
the
closed-loop
deterministic
linear
equation:
x
2
(
t
+
1)
=
[
A
0
(
t
)
+
A
0
(
t
)
+
(
B
0
(
t
)
+
B
0
(
t
))
F
(
t
)]
x
2
(
t
)
(76)
t
≥
t
0
corresponding
to
the
initial
pair
(
t
0
,
x
0
)
∈
Z
+
×
R
n
.
Employing
(
74
)
we
obtain
that
in
this
case
we
have:
|
x
(
t
;
t
0
,
x
0
)
|
2
≤
βδ
t
−
t
0
|
x
0
|
2
for
all
t
≥
t
0
≥
0,
x
0
∈
R
n
.
This
means
that
the
discrete-time
linear
equation
(
76
)
is
exponentially
stable,
or
equivalent,
the
DTCDLE
(
73
)
is
stabilizable.
Thus
we
have
shown
that
(
ii
)
holds
if
(
i
)
is
true.
The
implication
(
ii
)
⇒
(
i
)
may
be
proved
in
a
similar
way
to
the
proof
of
the
implication
(
ii
)
⇒
(
i
)
from
Theorem
1
replacing
the
DTSLS
(
37
)
by
the
DTSLS
(
75
)
and
the
DTDLE
(
38
)
by
the
closed-loop
discrete-time
equation
(
76
).
The
details
are
omitted.
Thus
the
proof
ends.
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