Ann.
Acad.
Rom.
Sci.
Ser.
Math.
Appl.
ISSN
2066-6594
Vol.
18,
No.
3/2026
APPLICATIONS
OF
THE
DUALITY:
DETECTABILITY
VERSUS
STABILIZABILITY
IN
THE
STOCHASTIC
FRAMEWORK
∗
Vasile
Dr˘
agan
†
Ioan-Lucian
Popa
‡
Dedicated
to
the
memory
of
Professor
Mihail
Megan
DOI
10.56082/annalsarscimath.2026.3.77
Abstract
The
aim
of
this
paper
is
to
exploit
the
property
of
duality
between
detectability
and
stabilizability
from
the
stochastic
framework,
to
de-
rive
some
results
from
the
domain
of
control
of
systems
with
random
parameters.
First,
we
shall
derive
a
dual
Barbashin-Krasovski
type
criterion
for
asymptotic
stability
of
the
zero
solution
of
an
Itˆ
o
differ-
ential
equation
under
the
conditions
when
there
exists
a
Lyapunov
function
with
negative
semidefinite
derivative.
Then,
we
provide
a
set
of
necessary
and
sufficient
conditions
that
guaranty
the
existence
of
the
bounded
on
R
and
stabilizing
solution
of
a
forward
matrix
Riccati
dif-
ferential
equation
with
periodic
coefficients
arising
in
concordance
with
the
problem
of
filtering
of
a
signal
generated
by
a
dynamical
system
modeled
by
stochastic
differential
equations
of
Itˆ
o
type.
Keywords:
stochastic
detectability,
stochastic
stabilizability,
Barbashin-
Krasovski
type
criterion,
Riccati
differential
equations
of
filtering.
MSC:
93E15,
93B05,
93B07,
93D15,
47B65.
∗
Accepted
for
publication
on
March
14,
2026
†
vasile.dragan@imar.ro
,
Institute
of
Mathematics
Simion
Stoilow
of
the
Romanian
Academy,
P.O.
Box
1-764,
014700
Bucharest,
Romania
and
Academy
of
Romanian
Scien-
tists,
3
Ilfov,
050044
Bucharest,
Romania
‡
lucian.popa@uab.ro
,
Department
of
Computing,
Mathematics
and
Electronics,
1
Decembrie
1918
University
of
Alba
Iulia,
Gabriel
Bethlen
5,
Alba
Iulia,
510009,
Romania
and
Faculty
of
Mathematics
and
Computer
Science,
Transilvania
University
of
Bra¸
sov,
Iuliu
Maniu
Street
50,
Bra¸
sov,
500091,
Romania
77
Applications
of
the
duality
78
1
Introduction
Controlability
and
observability,
as
well
as
stabilizability
and
detectability
are
structural
properties
of
controlled
systems
which
are
involved
in
the
solu-
tion
of
various
problems
of
optimal
control
in
both
deterministic
framework
and
stochastic
framework
(see
[
20
],
[
2
],
[
23
],
[
4
],
[
5
],
[
6
],
[
12
]).
In
the
deterministic
framework
it
is
known
a
duality
relation
between
controlability
and
observability
properties
as
well
as
between
stabilizability
and
detectability
properties
of
a
linear
system
(see
[
20
],
[
2
],
[
23
],
[
4
]).
In
[
8
]
the
duality
between
stochastic
stabilizability
and
stochastic
detectability
was
analyzed
in
the
case
of
linear
systems
modeled
by
stochastic
linear
differential
equations
of
Itˆ
o
type
with
periodic
coefficients
possibly
subject
to
random
switching
modeled
by
a
Markov
process.
One
of
the
aim
of
the
present
paper
is
to
use
the
duality
between
stochas-
tic
detectability
and
stochastic
stabilizability
of
a
linear
stochastic
system
modeled
by
stochastic
differential
equations
of
Itˆ
o
type
to
derive
a
dual
Barbashin-Krasovski
type
criterion
for
exponential
stability
in
mean
square
of
the
zero
solution
of
a
stochastic
linear
differential
equation
under
the
con-
ditions
when
only
a
quadratic
Lyapunov
function
with
negative
semidefinite
derivative
is
available.
In
many
applications
of
the
Lyapunov
function
method
for
the
study
of
the
asymptotic
stability
of
some
systems
from
real
life,
were
found
some
Lyapunov
functions
attached
in
a
natural
way
to
the
system
under
con-
sideration,
as,
for
example,
the
case
of
the
energy
function
for
mechanical
systems.
In
most
cases,
these
functions
are
positive
definite,
but
their
deriva-
tive
is
only
negative
semidefinite.
In
those
cases,
the
original
method
of
the
Lyapunov
functions
cannot
be
used
to
obtain
information
about
the
asymp-
totic
stability
of
the
considered
systems.
In
[
3
],
Barbashin
and
Krasovski
proposed
a
set
of
conditions
which
guaranty
the
asymptotic
stability
also
in
the
case
when
the
derivative
of
the
Lyapunov
function
is
only
negative
semidefinite.
In
the
particular
case
of
a
linear
time-invariant
deterministic
system,
a
sufficient
condition
which
guaranty
that
a
quadratic
Lyapunov
function
with
negative
semidefinite
derivative
allows
us
to
conclude
if
the
system
under
consideration
is
or
not
exponentially
stable.
That
condition
is
expressed
by
the
detectability
of
a
pair
of
matrices
adequately
chosen
(see
[
7
]).
This
result
was
then
extended
in
the
case
of
some
linear
time-
varying
systems
both
in
continuous-time
and
discrete-time
in
the
determin-
istic
framework
and
stochastic
framework.
For
the
reader
convenience
we
refer
to
[
14
],
[
24
],
[
10
],
[
11
]
and
their
references.
In
the
present
paper,
we
derive
a
dual
Barbashin-Krasovski
type
criterion
where
the
detectability
V.
Dr˘
agan,
I.-L.
Popa
79
condition
involved
in
the
afore
mentioned
references
is
replaced
by
a
stabi-
lizability
condition.
Another
aim
of
this
work
is
to
provide
a
set
of
necessary
and
sufficient
conditions
that
guaranty
the
existence
of
a
bounded
and
stabilizing
solution
of
a
matrix
Riccati
differential
equation
with
periodic
coefficients
arising
in
connection
with
the
problem
of
filtering
of
some
signals
generated
by
dynamical
systems
modeled
by
stochastic
differential
equations
with
state
multiplicative
white
noise
perturbations.
This
kind
of
Riccati
equations
are
forward
matrix
Riccati
differential
equations
less
studied
in
the
time-varying
case.
in
the
existing
literature,
a
greater
attention
was
payed
to
backward
matrix
Riccati
differential
equations
[
1
],
[
12
].
In
this
work,
we
shall
use
the
duality
between
stochastic
detectability
and
stochastic
stabilizability
to
obtain
conditions
for
the
existence
of
a
bounded
and
stabilizing
solution
of
a
forward
matrix
Riccati
differential
equation.
Also,
we
provide
an
iterative
procedure
for
the
numerical
computation
of
this
kind
of
solutions.
2
The
problem
setting
2.1
The
model
description
We
consider
the
stochastic
linear
system
G
having
the
state
space
represen-
tation
described
by
the
following
stochastic
linear
differential
equations
of
Itˆ
o
type:
dx
(
t
)
=
(
A
0
(
t
)
x
(
t
)
+
B
0
(
t
)
u
(
t
))
+
r
k
=1
(
A
k
(
t
)
x
(
t
)
+
B
k
(
t
)
u
(
t
))
dw
k
(
t
)
(1a)
dy
(
t
)
=
C
0
(
t
)
x
(
t
)
dt
+
r
k
=1
C
k
(
t
)
x
(
t
)
dw
k
(
t
)
(1b)
t
∈
R
+
=
[0
,
∞
)
,
where
x
(
t
)
∈
R
n
is
the
vector
of
the
state
variables,
u
(
t
)
∈
R
m
is
the
vector
of
inputs
of
the
system
and
y
(
t
)
∈
R
q
is
the
vector
of
an
output
of
the
system.
The
vector
u
(
t
)
includes
control
parameters
as
well
as
and/or
some
exogenous
disturbances
which
affect
the
behavior
of
the
system.
The
vectors
y
(
t
)
include
signals
generated
by
the
system
G
which
can
be
measured
and
used
to
elaborate
a
control
law
with
given
performances
specifications.
The
output
y
(
t
)
may
include
also
some
signals
generated
by
system
under
consideration
named
quality
outputs
whose
evolution
in
time
can
be
changed
using
the
adopted
control
law.
Applications
of
the
duality
80
In
(
1
),
{
w
k
(
t
)
}
t
≥
0
are
processes
of
the
Brownian
motions
defined
on
a
given
probability
space
(Ω
,
F
,
P
)
and
are
satisfying
the
assumption:
(
H
1
)
{
w
(
t
)
}
t
≥
0
,
(
w
(
t
)
=
(
w
1
(
t
)
w
2
(
t
)
.
.
.
w
n
(
t
))
)
is
an
r-dimensional
standard
Wiener
process
i.e.
w
(0)
=
0
,
E
[
w
(
t
)]
=
0
,
E
[(
w
(
s
)
−
w
(
t
))(
w
(
s
)
−
w
(
t
))
]
=
I
r
(
s
−
t
)
,
for
all
s
>
t
≥
0
,
I
r
being
the
identity
matrix
of
size
r
×
r.
Here
and
in
the
sequel
E
[
·
]
stands
for
the
mathematical
expectation.
Regarding
the
coefficients
of
the
system
(
1
)
we
make
the
assumption:
(
H
2
)
(
A
k
(
·
)
,
B
k
(
·
)
,
C
k
(
·
))
:
R
→
R
n
×
n
×
R
n
×
m
×
R
q
×
n
,
0
≤
k
≤
r,
are
continuous
matrix
valued
functions
which
are
periodic
with
common
period
θ
>
0
.
Remark
1.
Even
if
the
system
(
1
)
evolves
for
t
∈
R
+
,
the
functions
A
k
(
·
)
,
B
k
(
·
)
,
C
k
(
·
)
can
be
extended
in
a
natural
way
by
periodicity
to
the
whole
real
axis.
For
each
t
>
0
,
F
t
⊂
F
denotes
the
smallest
σ
−
algebra
with
respect
to
which
the
random
vector
w
(
s
)
,
0
≤
s
≤
t
are
measurable.
For
t
=
0
,
F
0
is
the
trivial
σ
−
algebra
(
∅
,
Ω)
.
For
each
t
≥
0
,
the
σ
−
algebra
F
t
is
augmented
with
all
subsets
A
∈
F
which
is
satisfying
P
(
A
)
=
0
.
If
J
⊂
R
+
,
L
2
F
{
J,
R
d
}
stands
for
the
linear
space
of
the
measurable
stochastic
processes
v
:
J
×
Ω
→
R
d
adapted
to
the
filtration
F
=
{F
t
}
t
≥
0
and
have
the
property
E
J
|
v
(
t
)
|
2
dt
<
∞
.
Based
on
Theorem
1.1
Chapter
5
from
[
13
]
(see
also
Theorem
5.2.1
from
[
22
])
we
obtain:
Proposition
1.
For
any
pair
(
t
0
,
x
0
)
∈
R
+
×
R
n
and
any
u
(
·
)
∈
L
2
F
{
[
t
0
,
∞
);
R
m
}
the
stochastic
linear
differential
equation
(SLDE)
(
1a
)
has
a
unique
solution
x
(
·
;
t
0
,
x
0
,
u
(
·
))
:
[
t
0
,
∞
)
→
R
n
with
the
properties:
(a)
{
x
(
t
;
t
0
,
x
0
,
u
(
·
))
}
t
≥
t
0
is
a
continuous
stochastic
process;
(b)
x
(
·
;
t
0
,
x
0
,
u
(
·
))
∈
L
2
F
{
[
t
0
,
T
];
R
n
}
,
for
all
T
>
t
0
;
(c)
x
(
t
0
;
t
0
,
x
0
,
u
(
·
))
=
x
0
.
V.
Dr˘
agan,
I.-L.
Popa
81
To
ease
the
presentation
of
the
results
derived
in
the
next
sections
we
shall
say
sometimes
that
the
system
G
is
associated
to
the
triple
(
C
(
·
)
,
A
(
·
)
,
B
(
·
))
where
A
(
·
)
=
(
A
0
(
·
)
,
A
1
(
·
)
,
.
.
.
,
A
r
(
·
))
(2a)
B
(
·
)
=
(
B
0
(
·
)
,
B
1
(
·
)
,
.
.
.
,
B
r
(
·
))
(2b)
C
(
·
)
=
(
C
0
(
·
)
,
C
1
(
·
)
,
.
.
.
,
C
r
(
·
))
(2c)
A
k
(
·
)
,
B
k
(
·
)
,
C
k
(
·
)
,
0
≤
k
≤
r,
are
the
matrix
valued
functions
which
are
coefficients
of
(
1
)
and
are
satisfying
the
condition
from
assumption
(
H
2
)
.
The
system
(
1
)
will
be
named
the
state-space
representation
of
the
triple
(
C
(
·
)
,
A
(
·
)
,
B
(
·
))
.
2.2
Basic
definitions
Definition
1.
(a)
We
say
that
the
system
described
by
SLDE
(
1a
)
is
stochastic
stabilizable
by
linear
state
feedback,
or
equivalently
the
pair
(
A
(
·
)
,
B
(
·
))
is
stabilizable
by
linear
state
feedback
if
there
exist
continu-
ous
and
θ
−
periodic
matrix
valued
functions
F
(
·
)
:
R
→
R
m
×
n
with
the
property
that
the
zero
solution
of
the
corresponding
closed-loop
SLDE:
dx
(
t
)
=
(
A
0
(
t
)
+
B
0
(
t
)
F
(
t
))
x
(
t
)
dt
+
r
k
=1
(
A
k
(
t
)
+
B
k
(
t
)
F
(
t
))
x
(
t
)
dw
k
(
t
)
(3)
is
exponentially
stable
in
mean
square
(ESMS);
(b)
We
say
that
the
linear
system
dx
(
t
)
=
A
0
(
t
)
x
(
t
)
dt
+
r
k
=1
A
k
(
t
)
x
(
t
)
dw
k
(
t
)
(4a)
dy
(
t
)
=
C
0
(
t
)
x
(
t
)
dt
+
r
k
=1
C
k
(
t
)
x
(
t
)
dw
k
(
t
)
(4b)
(obtained
from
(
1
)
taking
u
(
t
)
=
0
)
is
named
stochastic
detectable,
or
equivalently,
the
pair
(
C
(
·
)
,
A
(
·
))
is
detectable
if
there
exist
continuous
and
θ
−
periodic
matrix
valued
functions
K
(
·
)
:
R
→
R
n
×
q
with
the
property
that
the
zero
solution
of
the
corresponding
resulting
system:
dx
(
t
)
=
(
A
0
(
t
)
+
K
(
t
)
C
0
(
t
))
x
(
t
)
dt
+
r
k
=1
(
A
k
(
t
)
+
K
(
t
)
C
k
(
t
))
x
(
t
)
dw
k
(
t
)
(5)
Applications
of
the
duality
82
is
ESMS.
Remark
2.
We
assume
that
the
definition
of
the
exponential
stability
in
mean
square
(ESMS)
of
the
zero
solution
of
a
stochastic
linear
differential
equation
as
(
3
)
or
(
5
),
respectively,
is
known.
For
the
reader
convenience
we
refer
to
[
18
]
or
[
12
].
In
Proposition
3.3
in
[
8
]
may
be
found
a
list
of
conditions
which
are
equivalent
to
the
ESMS
property
of
the
zero
solution
of
a
SLDE.
2.3
The
problem
setting
In
[
8
],
we
have
shown
that
if
(
C
(
·
)
,
A
(
·
)
,
B
(
·
))
is
as
in
(
2
)
one
may
construct
a
new
triple
(
C
#
(
·
)
,
A
#
(
·
)
,
B
#
(
·
))
named
dual
triple
of
the
given
triple,
having
the
properties:
(a)
the
pair
(
A
(
·
)
,
B
(
·
))
is
stabilizable
if
and
only
if
the
dual
pair
(
C
#
(
·
)
,
A
#
(
·
))
is
detectable;
(b)
the
pair
(
C
(
·
)
,
A
(
·
))
is
detectable
if
and
only
if
the
dual
pair
(
A
#
(
·
)
,
B
#
(
·
))
is
stabilizable.
In
the
present
work,
we
shall
use
this
duality
in
order
to
derive
a
dual
Barbasin-Krasovski
type
criterion
for
exponential
stability
in
mean
square
of
the
zero
solution
of
a
SLDE.
A
more
detailed
setting
of
this
issue
will
be
given
in
Section
3
.
Another
ap-
plication
of
the
duality
between
the
stabilizability
property
and
detectability
property
in
the
stochastic
context
consists
of
the
derivation
of
sufficient
con-
ditions
which
guaranty
the
existence
of
the
stabilizing
and
bounded
on
R
so-
lution
of
the
following
forward
matrix
Riccati
differential
equation
(FRDE):
˙
Y
(
t
)
=
A
0
(
t
)
Y
(
t
)
+
Y
(
t
)
A
0
(
t
)
+
r
k
=1
A
k
(
t
)
Y
(
t
)
A
k
(
t
)
−
(
Y
(
t
)
C
0
(
t
)
+
r
k
=1
A
k
(
t
)
Y
(
t
)
C
k
(
t
)
+
L
(
t
))(
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
(
t
)
C
k
(
t
))
−
1
·
(
C
0
(
t
)
Y
(
t
)
+
r
k
=1
C
k
(
t
)
Y
(
t
)
A
k
(
t
)
+
L
(
t
))
+
M
(
t
)
(6)
associated
to
the
pair
(
C
(
·
)
,
A
(
·
))
of
the
system
(
2a
),
(
2c
)
and
(
M
(
·
)
,
L
(
·
)
,
R
(
·
))
:
R
→
S
n
×
R
n
×
q
×S
m
which
are
continuous
and
periodic
matrix
valued
func-
tions
with
period
θ
(the
same
as
the
period
from
(
H
2
))
.
Throughout
this
work
S
d
⊂
R
d
×
d
denotes
the
linear
space
of
symmetric
matrices
of
size
d
×
d.
V.
Dr˘
agan,
I.-L.
Popa
83
Definition
2.
A
globally
defined
solution
Y
s
(
·
)
:
R
→
S
n
of
the
FMRDE
(
6
)
is
named
stabilizing
solution
if
the
zero
solution
of
the
following
closed-loop
SLDE:
dx
(
t
)
=
(
A
0
(
t
)+
K
s
(
t
)
C
0
(
t
))
x
(
t
)
dt
+
r
k
=1
(
A
k
(
t
)+
K
s
(
t
)
C
k
(
t
))
x
(
t
)
dw
k
(
t
)
,
(7)
t
≥
0
,
is
ESMS,
where
K
s
(
t
)
:=
−
(
Y
s
(
t
)
C
0
(
t
)
+
r
k
=1
A
k
(
t
)
Y
s
(
t
)
C
k
(
t
)
+
L
(
t
))
·
(
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
s
(
t
)
C
k
(
t
))
−
1
,
t
∈
R
.
(8)
Our
aim
is
to
provide
a
set
of
conditions
which
guaranty
the
existence
and
uniqueness
of
the
stabilizing
and
bounded
on
R
solution
of
FMRDE
(
6
).
We
shall
provide
also
an
iterative
procedure
for
the
numerical
computation
of
the
bounded
and
stabilizing
solution
of
the
equation
(
6
).
Remark
3.
(a)
It
is
worth
mentioning
that
we
do
not
know
neither
an
initial
value,
nor
a
boundary
condition
of
the
stabilizing
solution
of
the
equation
(
6
).
The
only
available
information
is
that
the
zero
solution
of
the
corresponding
resulting
SLDE
(
7
)-(
8
)
is
ESMS
and
additionally
it
is
the
unique
globally
defined
solution
having
the
property
that
the
zero
solution
of
the
resulting
SLDE
is
ESMS.
That
is
why
the
known
methods
for
numerical
computation
of
the
solution
with
given
initial
value
or
a
bilocal
problem
are
not
applicable
for
the
numerical
compu-
tation
of
the
bounded
and
stabilizing
solution
of
the
FMRDE
(
6
).
(b)
Starting
with
the
classic
works
of
Kalman
(see
[
16
],
[
17
])
the
stabi-
lizing
solution
of
some
particular
forms
of
the
Riccati
equation
of
the
form
(
6
)
played
a
central
role
in
the
construction
of
the
filter
with
ap-
plications
in
various
research
domains.
See
for
example
[
9
],
[
21
],
[
19
]
and
[
25
]
for
a
detailed
discussion.
3
A
dual
Barbasin-Krasovski
criterion
for
exponential
stability
in
mean-square
Let
us
consider
the
stochastic
linear
differential
equation
(SLDE):
dx
(
t
)
=
A
0
(
t
)
x
(
t
)
dt
+
r
k
=1
A
k
(
t
)
x
(
t
)
dw
k
(
t
)
,
t
≥
0
.
(9)
Applications
of
the
duality
84
Our
aim
is
to
extend
the
result
stated
in
Proposition
4.5
b
)
from
[
15
]
to
the
case
of
stochastic
time-varying
system
(
9
).
According
to
the
equivalence
(
i
)
⇔
(
vi
)
from
Proposition
3.3
in
[
8
]
with
the
updates
M
k
(
·
)
←
A
k
(
·
)
we
deduce
that
the
zero
solution
of
the
SLDE
(
9
)
is
ESMS
if
and
only
if
there
exists
a
matrix
valued
function
H
(
·
)
:
R
→
S
n
continuous
and
θ
−
periodic,
H
(
t
)
>
0
,
for
all
t
∈
R
with
the
property
that
the
non-homogeneous
forward
Lyapunov-type
matrix
linear
differential
equation:
˙
X
(
t
)
=
A
0
(
t
)
X
(
t
)
+
X
(
t
)
A
0
(
t
)
+
r
k
=1
A
k
(
t
)
X
(
t
)
A
k
(
t
)
+
H
(
t
)
,
(10)
t
∈
R
has
a
unique
solution
X
(
·
)
:
R
→
S
n
which
is
a
θ
−
periodic
function
and
satisfies
X
(
t
)
>
0
,
t
∈
R
.
In
this
section
we
show
that
under
a
suitable
stabilizability
assumption,
the
fact
that
the
non-homogeneous
Lyapunov
type
linear
differential
equa-
tion
(
10
)
with
the
free
term
H
(
t
)
≥
0
,
t
∈
R
(instead
of
H
(
t
)
>
0
,
t
∈
R
)
has
a
solution
X
(
t
)
≥
0
(instead
of
X
(
t
)
>
0
,
)
t
∈
R
,
guaranties
the
ESMS
property
of
the
zero
solution
of
the
SLDE
(
9
).
Let
B
0
(
t
)
∈
R
m
×
m
be
obtained
from
the
factorization
H
(
t
)
=
B
0
(
t
)
B
0
(
t
)
,
t
∈
R
.
(11)
If
H
(
·
)
is
a
θ
−
periodic
function,
then
B
0
(
·
)
from
(
11
)
may
be
chosen
to
be
a
θ
−
periodic
function
(with
the
same
period).
We
set
q
B
(
·
)
:=
(
B
0
(
·
)
,
0
,
.
.
.
,
0)
.
(12)
If
A
(
·
)
is
that
described
in
(
2a
)
and
q
B
(
·
)
is
introduced
in
(
12
),
then,
Defini-
tion
1
(
a
)
becomes:
Definition
3.
We
say
that
the
pair
(
A
(
·
)
,
q
B
(
·
))
is
stabilizable
if
there
exists
a
continuous
and
θ
−
periodic
matrix
valued
function
F
(
·
)
:
R
→
R
m
×
n
with
the
property
that
the
zero
solution
of
the
corresponding
closed-loop
system:
dx
(
t
)
=
(
A
0
(
t
)
+
B
0
(
t
)
F
(
t
))
x
(
t
)
dt
+
r
k
=1
A
k
(
t
)
x
(
t
)
dw
k
(
t
)
,
t
≥
0
(13)
is
ESMS.
The
main
result
of
this
section
is:
V.
Dr˘
agan,
I.-L.
Popa
85
Theorem
1.
Assume:
(a)
the
assumptions
(
H
1
)
,
(
H
2
)
hold;
(b)
(
A
(
·
)
,
q
B
(
·
))
is
stabilizable.
Under
these
conditions,
the
following
are
equivalent:
(i)
the
zero
solution
of
the
SLDE
(
9
)
is
ESMS;
(ii)
the
non-homogeneous
forward
Lyapunov
type
matrix
linear
differential
equation:
˙
X
(
t
)
=
A
0
(
t
)
X
(
t
)
+
X
(
t
)
A
0
(
t
)
+
r
k
=1
A
k
(
t
)
X
(
t
)
A
k
(
t
)
+
B
0
(
t
)
B
0
(
t
)
,
(14)
t
∈
R
has
a
unique
solution
X
(
t
)
:
R
→
S
n
which
is
θ
−
periodic
and
satisfying
X
(
t
)
≥
0
,
for
all
t
∈
R
.
Proof.
Let
T
(
t,
t
0
)
:
S
n
→
S
n
be
the
linear
evolution
operator
in
S
n
defined
by
the
linear
differential
equation:
˙
X
(
t
)
=
A
0
(
t
)
X
(
t
)
+
X
(
t
)
A
0
(
t
)
+
r
k
=1
A
k
(
t
)
X
(
t
)
A
k
(
t
)
.
(15)
Employing
Theorem
3.2.2
from
[
12
]
in
the
particular
case
D
=
{
1
}
,
we
may
infer
that
the
statement
(
i
)
holds
if
and
only
if
T
(
t,
t
0
)
≤
βe
−
α
(
t
−
t
0
)
,
for
all
t
≥
t
0
∈
R
,
where
β
≥
1
,
α
>
0
do
not
depend
upon
t,
t
0
.
We
set
X
(
t
)
:=
t
−∞
T
(
t,
s
)[
B
0
(
s
)
B
0
(
s
)]
ds.
By
direct
calculation,
we
show
that
X
(
·
)
defined
as
before,
is
the
unique
solution
of
the
equation
(
14
)
which
is
θ
−
periodic
and
satisfies
X
(
t
)
≥
0
,
t
∈
R
.
So,
the
validity
of
the
implication
(
i
)
⇒
(
ii
)
is
confirmed.
Let
us
now
show
that
the
implication
(
ii
)
⇒
(
i
)
holds
too.
If
X
(
·
)
:
R
→
S
n
is
the
unique
θ
−
periodic
and
positive
semidefinite
solution
of
the
equation
(
14
),
we
define
Y
(
t
)
:=
X
(
−
t
)
,
t
∈
R
.
Applications
of
the
duality
86
By
direct
calculation
one
obtains
that
Y
(
·
)
solves
the
non-homogeneous
backward
Lyapunov
type
matrix
linear
differential
equation:
−
˙
Y
(
t
)
=
A
0
(
t
)
Y
(
t
)
+
Y
(
t
)
A
0
(
t
)
+
r
k
=1
A
k
(
t
)
Y
(
t
)
A
k
(
t
)
+
C
0
(
t
)
C
0
(
t
)
,
(16)
t
∈
R
,
where
A
k
(
t
)
:=
A
k
(
−
t
)
,
0
≤
k
≤
r
(17a)
C
0
(
t
)
:=
B
0
(
−
t
)
,
t
∈
R
.
(17b)
The
non-homogeneous
backward
Lyapunov
type
matrix
linear
differential
equation
(
16
)
is
associated
to
the
system:
d
x
(
t
)
=
A
0
(
t
)
x
(
t
)
dt
+
r
k
=1
A
k
(
t
)
x
(
t
)
dw
k
(
t
)
(18a)
y
(
t
)
=
C
0
(
t
)
x
(
t
)
,
t
∈
R
+
.
(18b)
We
show
that
the
system
(
18
)
is
stochastic
detectable
in
the
sense
of
Defi-
nition
4.1.1
from
[
12
]
if
the
pair
(
A
(
·
)
,
q
B
(
·
))
is
stabilizable.
According
to
Definition
3
,
(
A
(
·
)
,
q
B
(
·
))
is
stabilizable
if
and
only
if
there
exists
a
continuous
and
θ
−
periodic
matrix
valued
function
F
(
·
)
:
R
→
R
m
×
n
with
the
property
that
the
zero
solution
of
the
SLDE
(
13
)
is
ESMS.
Em-
ploying
the
equivalence
(
i
)
⇔
(
v
)
from
Proposition
3.3
[
8
]
with
the
updates
M
0
(
t
)
←
A
0
(
t
)
+
B
0
(
t
)
F
(
t
)
,
M
k
(
t
)
←
A
k
(
t
)
,
1
≤
k
≤
r,
we
deduce
that
the
zero
solution
of
the
SLDE
(
13
)
is
ESMS
if
and
only
if
the
following
non-
homogeneous
forward
Lyapunov
type
matrix
linear
differential
equation:
˙
Y
(
t
)
=(
A
0
(
t
)
+
B
0
(
t
)
F
(
t
))
Y
(
t
)
+
Y
(
t
)(
A
0
(
t
)
+
B
0
(
t
)
F
(
t
))
+
r
k
=1
A
k
(
t
)
Y
(
t
)
A
k
(
t
)
+
I
n
,
t
∈
R
(19)
has
a
unique
solution
Y
(
·
)
:
R
→
S
n
which
is
θ
−
periodic
and
satisfies
Y
(
t
)
>
0
,
t
∈
R
.
Let
Z
(
t
)
:=
Y
(
−
t
)
,
t
∈
R
.
By
direct
calculation,
involving
(
19
),
we
obtain
that
Z
(
·
)
is
the
unique
θ
−
periodic
and
uniform
positive
defined
solution
V.
Dr˘
agan,
I.-L.
Popa
87
of
the
following
non-homogeneous
backward
Lyapunov
type
matrix
linear
differential
equation:
−
˙
Z
(
t
)
=(
A
0
(
t
)
+
K
(
t
)
C
0
(
t
))
Z
(
t
)
+
Z
(
t
)(
A
0
(
t
)
+
K
(
t
)
C
0
(
t
))
+
r
k
=1
A
k
(
t
)
Z
(
t
)
A
k
(
t
)
+
I
n
,
(20)
t
∈
R
,
where
A
k
(
t
)
,
0
≤
k
≤
r,
and
C
0
(
t
)
were
introduced
by
(
17
)
and
K
(
t
)
:=
F
(
−
t
)
,
t
∈
R
.
Invoking
the
implication
(
iv
)
⇒
(
i
)
from
Proposition
3.3
in
[
8
]
with
the
updates
M
0
(
t
)
←
A
0
(
t
)
+
K
(
t
)
C
0
(
t
)
and
M
k
(
t
)
←
A
k
(
t
)
,
1
≤
k
≤
r,
we
may
infer
that
the
existence
of
a
θ
−
periodic
and
uniform
positive
definite
solution
of
the
non-homogeneous
backward
Lyapunov
type
matrix
linear
differential
equation
(
20
)
guaranties
the
ESMS
property
of
the
zero
solution
of
the
following
SLDE:
d
x
(
t
)
=
(
A
0
(
t
)
+
K
(
t
)
C
0
(
t
))
x
(
t
)
dt
+
r
k
=1
A
k
(
t
)
x
(
t
)
dw
k
(
t
)
,
t
∈
R
+
.
(21)
The
exponential
stability
in
mean-square
of
the
zero
solution
of
the
SLDE
(
21
)
is
equivalent
to
the
stochastic
detectability
of
the
system
(
18
)
in
the
sense
of
the
Definition
4.1.1
(
b
)
from
[
12
].
Hence,
we
can
apply
Theorem
4.1.7
from
[
12
]
in
the
case
of
the
pair
formed
by
the
system
(
18
)
and
the
non-
homogeneous
backward
Lyapunov
type
matrix
linear
differential
equation
(
16
),
in
order
to
conclude
that
the
zero
solution
of
the
SLDE:
d
x
(
t
)
=
A
0
(
t
)
x
(
t
)
dt
+
r
k
=1
A
k
(
t
)
x
(
t
)
dw
k
(
t
)
,
t
∈
R
+
(22)
is
ESMS.
Let
T
(
t,
t
0
)
:
S
n
→
S
n
,
t,
t
0
∈
R
be
the
linear
evolution
operator
defined
by
the
linear
differential
equation
on
S
n
:
˙
X
(
t
)
=
A
0
(
t
)
X
(
t
)
+
X
(
t
)
A
0
(
t
)
+
r
k
=1
A
k
(
t
)
X
(
t
)
A
k
(
t
)
,
t
∈
R
.
(23)
According
to
Theorem
3.2.2
(
i
)
from
[
12
]
in
the
particular
case
D
=
{
1
}
we
may
infer
that
the
zero
solution
of
the
SLDE
(
22
)
is
ESMS
if
and
only
if
T
(
t,
t
0
)
≤
βe
−
α
(
t
−
t
0
)
,
(24)
for
all
t
≥
t
0
,
t,
t
0
∈
R
,
where
β
≥
1
,
α
>
0
,
do
not
depend
upon
t
and
t
0
.
Applications
of
the
duality
88
Let
L
(
t
)
:
S
n
→
S
n
be
the
linear
operator
defined
by:
L
(
t
)[
X
]
=
A
0
(
t
)
X
+
XA
0
(
t
)
+
r
k
=1
A
k
(
t
)
XA
k
(
t
)
,
for
all
X
∈
S
n
.
With
this
notation
the
linear
differential
equation
(
15
)
may
be
written
as:
˙
X
(
t
)
=
L
(
t
)[
X
(
t
)]
.
(25)
Bearing
in
mind
(
17a
)
we
may
rewrite
the
differential
equation
(
23
)
as:
˙
X
(
t
)
=
L
∗
(
−
t
)[
X
(
t
)]
,
(26)
L
∗
(
t
)
being
the
adjoint
operator
of
L
(
t
)
with
respect
to
the
inner
product
X,
Y
=
Tr[
XY
]
,
for
all
X,
Y
∈
S
n
,
Tr[
X
]
being
the
trace
operator.
Recalling
that
T
(
t,
s
)
is
the
linear
evolution
operator
of
the
linear
differ-
ential
equation
(
25
)
and
T
(
t,
s
)
is
the
linear
evolution
operator
of
the
linear
differential
equation
(
26
)
we
obtain
that
T
(
s,
t
)
=
T
∗
(
−
t,
−
s
)
,
for
all
s
≥
t,
t,
s
∈
R
.
Employing
(
24
)
we
get:
T
∗
(
−
t,
−
s
)
≤
βe
−
α
(
s
−
t
)
,
(27)
for
all
s
≥
t,
or
equivalently,
for
any
−
t
≥
−
s,
t,
s
∈
R
.
This
allows
us
to
reply
t
by
−
t,
s
by
−
s
in
(
27
)
to
obtain
T
∗
(
t,
s
)
≤
βe
−
α
(
t
−
s
)
,
(28)
for
all
t
≥
s,
t,
s
∈
R
.
Taking
into
account
that
c
1
T
(
t,
s
)
≤
T
∗
(
t,
s
)
≤
c
2
T
(
t,
s
)
,
for
all
t,
s
∈
R
(where
c
j
>
0
,
j
=
1
,
2
,
do
not
depend
on
t,
s
)
we
obtain
from
(
28
)
that
T
(
t,
s
)
≤
βc
−
1
1
e
−
α
(
t
−
s
)
,
(29)
for
all
t
≥
s,
t,
s
∈
R
,
which
means
that
the
zero
solution
of
the
linear
differential
equation
(
25
)
is
exponentially
stable.
Invoking
again
Theorem
3.2.2
from
[
12
]
in
the
particular
case
D
=
{
1
}
we
find
that
(
29
)
is
equivalent
to
the
ESMS
property
of
the
zero
solution
of
the
SLDE
(
9
).
Thus,
the
proof
ends.
V.
Dr˘
agan,
I.-L.
Popa
89
4
On
the
bounded
and
stabilizing
solution
of
a
forward
matrix
Riccati
differential
equation
4.1
A
dual
backward
matrix
Riccati
differential
equation
as-
sociated
to
a
forward
matrix
Riccati
differential
equation
Let
Y
(
·
)
:
R
→
S
n
be
a
bounded
on
R
solution
of
the
FMRDE
(
6
).
We
set
X
(
t
)
:=
Y
(
−
t
)
,
t
∈
R
.
By
direct
calculations
one
obtains
that
X
(
·
)
is
a
bounded
on
R
solution
of
the
following
backward
matrix
Riccati
equation
(BMRDE):
−
˙
X
(
t
)
=
p
A
0
(
t
)
X
(
t
)
+
X
(
t
)
p
A
0
(
t
)
+
r
k
=1
p
A
k
(
t
)
X
(
t
)
p
A
k
(
t
)
−
X
(
t
)
p
B
0
(
t
)
+
r
k
=1
p
A
k
(
t
)
X
(
t
)
p
B
k
(
t
)
+
p
L
(
t
)
·
p
R
(
t
)
+
r
k
=1
p
B
k
(
t
)
X
(
t
)
p
B
k
(
t
)
−
1
·
p
B
0
(
t
)
X
(
t
)
+
r
k
=1
p
B
k
(
t
)
X
(
t
)
p
A
k
(
t
)
+
p
L
(
t
)
(30)
where
p
A
k
(
t
)
:=
A
k
(
−
t
)
,
(31a)
p
B
k
(
t
)
:=
C
k
(
−
t
)
,
0
≤
k
≤
r
(31b)
x
M
(
t
)
:=
M
(
−
t
)
(31c)
p
L
(
t
)
:=
L
(
−
t
)
(31d)
p
R
(
t
)
:=
R
(
−
t
)
,
t
∈
R
.
(31e)
The
backward
matrix
Riccati
type
differential
equation
(
30
)-(
31
)
will
be
named
the
dual
BMRDE
associated
to
the
FMRDE
(
6
).
The
BMRDE
(
30
)
is
a
kind
of
matrix
Riccati
differential
equation
involved
in
the
solution
of
a
problem
of
stochastic
linear
quadratic
optimal
control.
For
the
reader
convenience
we
recall:
Applications
of
the
duality
90
Definition
4.
A
solution
X
s
(
·
)
:
R
→
S
n
of
BMRDE
(
30
)
is
named
stabi-
lizing
solution
if
the
zero
solution
of
the
corresponding
closed-loop
stochastic
linear
differential
equation:
dx
(
t
)
=
(
p
A
0
(
t
)
+
p
B
0
(
t
)
F
s
(
t
))
x
(
t
)
dt
+
r
k
=1
(
p
A
k
(
t
)
+
p
B
k
(
t
)
F
s
(
t
))
x
(
t
)
dw
k
(
t
)
,
(32)
t
∈
R
+
,
is
ESMS,
where
F
s
(
t
)
:=
−
p
R
(
t
)
+
r
k
=1
p
B
k
(
t
)
X
s
(
t
)
p
B
k
(
t
)
−
1
·
p
B
0
(
t
)
X
s
(
t
)
+
r
k
=1
p
B
k
(
t
)
X
s
(
t
)
p
A
k
(
t
)
+
p
L
(
t
)
,
t
∈
R
(33)
The
problem
of
the
existence
of
the
stabilizing
and
bounded
on
R
solution
for
a
BMRDE
of
type
(
30
)
was
studied
in
Chapter
5
from
[
12
].
4.2
An
auxiliary
result
Let
us
consider
the
following
two
SLDEs:
dx
(
t
)
=
M
0
(
t
)
x
(
t
)
dt
+
r
k
=1
M
k
(
t
)
x
(
t
)
dw
k
(
t
)
(34)
and
dx
(
t
)
=
x
M
0
(
t
)
x
(
t
)
dt
+
r
k
=1
x
M
k
(
t
)
x
(
t
)
dw
k
(
t
)
(35)
where
M
k
(
·
)
:
R
→
R
n
×
n
,
0
≤
k
≤
r
are
continuous
and
θ
−
periodic
matrix
valued
functions
and
x
M
k
(
t
)
:=
M
k
(
−
t
)
,
0
≤
k
≤
r.
(36)
We
have:
Lemma
1.
The
following
are
equivalent:
(i)
the
zero
solution
of
the
SLDE
(
34
)
is
ESMS;
(ii)
the
zero
solution
of
the
SLDE
(
35
)
is
ESMS.
V.
Dr˘
agan,
I.-L.
Popa
91
Proof.
If
the
zero
solution
of
the
SLDE
(
34
)
is
ESMS
then,
based
on
the
implication
(
i
)
⇒
(
v
)
from
Proposition
3.3
from
[
8
]
applied
in
the
case
of
the
equation
(
34
),
we
deduce
that
the
non-homogeneous
forward
Lyapunov
type
linear
differential
equation:
˙
X
(
t
)
=
M
0
(
t
)
X
(
t
)
+
X
(
t
)
M
0
(
t
)
+
r
k
=1
M
k
(
t
)
X
(
t
)
M
k
(
t
)
+
I
n
(37)
has
a
unique
solution
X
(
·
)
:
R
→
S
n
which
is
a
θ
−
periodic
function
and
is
uniform
positive
on
R
.
We
set
Y
(
t
)
∆
=
X
(
−
t
)
,
t
∈
R
.
By
direct
calculations
involving
(
36
)
and
(
37
)
we
obtain
that
Y
(
·
)
is
unique
uniform
positive
definite
on
R
and
θ
−
periodic
solution
of
the
following
non-
homogeneous
backward
Lyapunov
type
linear
differential
equation:
−
˙
Y
(
t
)
=
x
M
0
(
t
)
Y
(
t
)
+
Y
(
t
)
x
M
0
(
t
)
+
r
k
=1
x
M
k
(
t
)
Y
(
t
)
x
M
k
(
t
)
+
I
n
.
(38)
Invoking
the
implication
(
iv
)
⇒
(
i
)
from
Proposition
3.3
from
[
8
]
in
the
particular
case
of
the
SLDE
(
35
)
and
of
non-homogeneous
Lyapunov
type
differential
equation
(
38
)
we
may
conclude
that
the
zero
solution
of
the
SLDE
(
35
)
is
ESMS.
Thus
we
have
checked
the
validity
of
the
implication
(
i
)
⇒
(
ii
)
from
the
statement.
Conversely,
if
(
ii
)
holds,
then
the
implication
(
i
)
⇒
(
iii
)
from
Propo-
sition
3.3
in
[
8
]
applied
in
the
case
of
the
SLDE
(
35
),
we
deduce
that
the
non-homogeneous
backward
Lyapunov
type
matrix
linear
differential
equa-
tion
(
38
)
has
a
unique
solution
Y
(
·
)
:
R
→
S
n
which
is
a
θ
−
periodic
function
and
satisfies
Y
(
t
)
>
0
,
∀
t
∈
R
.
Setting
X
(
t
)
=
Y
(
−
t
)
,
t
∈
R
,
we
obtain
via
(
36
)
and
(
38
)
that
X
(
·
)
is
the
unique
uniform
positive
and
θ
−
periodic
solution
of
the
non-homogeneous
forward
Lyapunov
type
matrix
linear
differential
equation
(
37
).
Based
on
the
implication
(
vi
)
⇒
(
i
)
from
Proposition
3.3
in
[
8
]
we
deduce
that
the
zero
solution
of
the
SLDE
(
34
)
is
ESMS.
Thus
the
proof
is
complete.
The
equivalence
from
Lemma
1
applied
in
the
case
of
the
pair
of
SLDE
(
7
)-(
8
)
and
(
32
)-(
33
),
respectively,
gives:
Applications
of
the
duality
92
Corollary
1.
Under
the
assumptions
(
H
1
)
and
(
H
2
)
,
Y
s
(
·
)
:
R
→
S
n
is
the
bounded
on
R
and
stabilizing
solution
of
the
FMRDE
(
6
)
if
and
only
if
X
s
(
·
)
defined
by
X
s
(
t
)
=
Y
s
(
−
t
)
,
t
∈
R
is
bounded
on
R
and
stabilizing
solution
of
its
dual
BMRDE
(
30
)-(
31
).
Proof.
Taking
M
k
(
t
)
∆
=
A
k
(
t
)
+
K
s
(
t
)
C
k
(
t
)
,
0
≤
k
≤
r,
K
s
(
t
)
being
defined
in
(
8
)
and
x
M
k
(
t
)
∆
=
p
A
k
(
t
)
+
p
B
k
(
t
)
F
s
(
t
)
,
0
≤
k
≤
r,
with
F
s
(
t
)
defined
by
(
33
)
we
obtain
via
(
31
)
that
x
M
k
(
t
)
=
M
k
(
−
t
)
,
0
≤
k
≤
r,
t
∈
R
.
The
conclusion
is
obtained
from
Lemma
1
applied
to
the
SLDEs
(
7
)
and
(
32
),
respectively.
So,
conditions
which
guaranty
the
existence
and
some
useful
properties
of
the
bounded
and
stabilizing
solution
of
the
FMRDE
(
6
)
can
be
obtained
applying
to
the
dual
FMRDE
(
30
)-(
31
)
some
results
from
Chapter
5
in
[
12
].
4.3
On
the
bounded
and
maximal
solution
of
the
FMRDE
Let
us
remark
that
a
FMRDE
of
type
(
6
)
is
defined
by
the
following
triple
Σ
=
{A
(
·
)
,
C
(
·
)
,
Q
(
·
)
}
where
A
(
·
)
,
C
(
·
)
are
those
from
(
2a
),
(
2c
)
and
Q
(
·
)
=
M
(
·
)
L
(
·
)
L
(
·
)
R
(
·
)
=
Q
(
·
)
.
(39)
To
the
triple
Σ
we
associate
the
set
of
functions
Γ
Σ
formed
by
all
func-
tions
Z
(
·
)
:
R
→
S
n
continuously
differentiable,
θ
−
periodic
satisfying
the
following
conditions:
−
˙
Z
(
t
)
+
A
0
(
t
)
Z
(
t
)
+
Z
(
t
)
A
0
(
t
)
+
M
(
t
)
Z
(
t
)
C
0
(
t
)
+
L
(
t
)
C
0
(
t
)
Z
(
t
)
+
L
(
t
)
R
(
t
)
+
r
k
=1
A
k
(
t
)
C
k
(
t
)
Z
(
t
)
A
k
(
t
)
C
k
(
t
)
≥
0
(40)
V.
Dr˘
agan,
I.-L.
Popa
93
and
R
(
t
)
+
r
k
=1
C
k
(
t
)
Z
(
t
)
C
k
(
t
)
>
0
,
t
∈
R
.
(41)
A
crucial
role
in
the
development
from
this
section
will
be
played
by
the
subset
Γ
Σ
⊂
Γ
Σ
formed
by
all
functions
Z
(
·
)
∈
Γ
Σ
which
are
satisfying
−
˙
Z
(
t
)
+
A
0
(
t
)
Z
(
t
)
+
Z
(
t
)
A
0
(
t
)
+
M
(
t
)
Z
(
t
)
C
0
(
t
)
+
L
(
t
)
C
0
(
t
)
Z
(
t
)
+
L
(
t
)
R
(
t
)
+
r
k
=1
A
k
(
t
)
C
k
(
t
)
Z
(
t
)
A
k
(
t
)
C
k
(
t
)
>
0
,
t
∈
R
.
(42)
Remark
4.
(a)
The
inequalities
from
(
41
)
and
(
42
)
are
uniform
with
re-
spect
to
t
∈
R
due
to
the
periodicity
property
of
the
matrix
valued
functions
arising
in
these
inequalities.
(b)
Employing
the
Schur
complement
technique,
one
may
shown
that
the
set
Γ
Σ
contains
all
θ
−
periodic
solutions
Y
(
·
)
:
R
→
S
n
of
the
FMRDE
(
6
)
which
are
satisfying
the
condition:
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
(
t
)
C
k
(
t
)
>
0
,
t
∈
R
.
(43)
(c)
From
(
40
),
(
41
)
we
obtain
via
the
Schur
complement
technique
that
Z
(
t
)
≡
0
∈
Γ
Σ
if
and
only
if
R
(
t
)
>
0
,
(44)
M
(
t
)
−
L
(
t
)
R
−
1
(
t
)
L
(
t
)
≥
0
,
t
∈
R
.
(45)
The
dual
BMRDE
(
30
)
is
associated
to
the
dual
triple
Σ
#
=
(
A
#
(
·
)
,
B
#
(
·
)
,
Q
#
(
·
))
,
(46)
where
A
#
(
·
)
=
(
p
A
0
(
·
)
,
p
A
1
(
·
)
,
.
.
.
,
p
A
r
(
·
))
,
B
#
(
·
)
=
(
p
B
0
(
·
)
,
y
AB
1
(
·
)
,
.
.
.
,
p
B
r
(
·
))
Q
#
(
·
)
=
x
M
(
·
)
p
L
(
·
)
p
L
(
·
)
p
R
(
·
)
(47)
Applications
of
the
duality
94
where
p
A
k
(
·
)
,
p
B
k
(
·
)
,
0
≤
k
≤
r,
and
x
M
(
·
)
,
p
L
(
·
)
,
p
R
(
·
)
were
defined
in
(
31
).
To
the
triple
Σ
#
we
associate
the
set
Γ
Σ
#
formed
by
all
continuously
differ-
entiable
and
θ
−
periodic
matrix
valued
functions
p
Z
(
·
)
:
R
→
S
n
which
are
satisfying:
˙
p
Z
(
·
)
+
p
A
0
(
t
)
p
Z
(
t
)
+
p
Z
(
t
)
+
p
A
0
(
t
)
+
x
M
(
t
)
p
Z
(
t
)
p
B
0
(
t
)
+
p
L
(
t
)
p
B
0
(
t
)
p
Z
(
t
)
+
p
L
(
t
)
p
R
(
t
)
+
r
k
=1
(
p
A
k
(
t
)
p
B
k
(
t
))
p
Z
(
t
)(
p
A
k
(
t
)
p
B
k
(
t
))
≥
0
(48)
p
R
(
t
)
+
r
k
=1
p
B
k
(
t
)
p
Z
(
t
)
p
B
k
(
t
)
>
0
,
(49)
t
∈
R
.
The
set
of
functions
Γ
Σ
#
is
the
analogous
of
the
set
Γ
Σ
introduced
in
Section
5.1
in
[
12
]
being
associated
to
the
particular
case
of
the
BMRDE
(
30
).
Remark
5.
From
(
40
),
(
41
)
on
one
hand
and
(
48
),
(
49
)
on
the
other
hand,
one
obtains
via
(
31
)
that
a
function
Z
(
·
)
:
R
→
S
n
lies
in
Γ
Σ
if
and
only
if
the
function
p
Z
(
·
)
defined
by
p
Z
(
t
)
=
Z
(
−
t
)
,
t
∈
R
lies
in
Γ
Σ
#
.
Definition
5.
(a)
A
solution
Y
max
(
·
)
:
R
→
S
n
of
the
FMRDE
(
6
)
is
named
maximal
solution
with
respect
to
the
set
Γ
Σ
or
maximal
solution
(for
shortness)
of
(
6
)
if
Y
max
(
t
)
≥
Z
(
t
)
,
t
∈
R
,
for
any
Z
(
·
)
∈
Γ
Σ
.
(b)
A
solution
X
max
(
·
)
:
R
→
S
n
of
the
BMRDE
(
30
)
is
named
maximal
solution
with
respect
to
the
set
Γ
Σ
#
or
maximal
solution
(for
shortness)
of
(
30
)
if
X
max
(
t
)
≥
p
Z
(
t
)
,
t
∈
R
,
for
any
p
Z
(
·
)
∈
Γ
Σ
#
.
Remark
6.
According
to
Remark
5
and
Definition
5
it
follows
that
Y
max
(
·
)
is
the
maximal
solution
of
the
FMRDE
(
35
)
if
and
only
if
X
max
(
·
)
defined
by
X
max
(
t
)
=
Y
max
(
−
t
)
,
t
∈
R
,
is
the
maximal
solution
of
its
dual
BMRDE
(
30
)-(
31
).
Moreover,
if
it
exists,
the
maximal
solution
Y
max
(
·
)
of
(
6
)
is
unique.
V.
Dr˘
agan,
I.-L.
Popa
95
Theorem
2.
Assume:
(a)
the
assumptions
(
H
1
)
,
(
H
2
)
hold;
(b)
the
pair
(
C
(
·
)
,
A
(
·
))
is
detectable.
Under
these
conditions,
the
following
are
equivalent:
(i)
the
set
Γ
Σ
is
not
empty;
(ii)
the
FMRDE
(
6
)
has
a
bounded
and
maximal
solution
Y
max
:
R
→
S
n
which
is
satisfying
the
sign
condition:
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
max
(
t
)
C
k
(
t
)
>
0
,
t
∈
R
.
(50)
Moreover
Y
max
(
·
)
is
a
periodic
function
with
period
θ.
Proof.
The
implication
(
ii
)
⇒
(
i
)
is
obvious
because
according
to
Remark
4
(
d
)
,
the
maximal
solution
Y
max
(
·
)
,
if
any,
lies
in
Γ
Σ
.
To
check
the
validity
of
the
implication
(
i
)
⇒
(
ii
)
we
invoke
the
implication
(
i
)
⇒
(
ii
)
from
Theorem
5.3.5
in
[
12
]
applied
in
the
case
of
the
dual
BMRDE
(
30
)-(
31
).
First,
let
us
remark
that
according
to
Proposition
4.2
from
[
8
],
the
pair
(
C
(
·
)
,
A
(
·
))
is
detectable
if
and
only
if
the
dual
pair
(
A
#
(
·
)
,
B
#
(
·
))
is
stabi-
lizable.
On
the
other
hand
based
on
Remark
5
we
may
infer
that
the
set
Γ
Σ
#
is
nonempty
if
the
set
Γ
Σ
is
not
empty.
So,
we
can
apply
(
i
)
⇒
(
ii
)
of
Theorem
5.3.5
[
12
]
in
the
case
of
BMRDE
(
30
)-(
31
),
to
deduce
that
this
equation
has
a
maximal
solution
X
max
:
R
→
S
n
which
is
a
θ
−
periodic
function
and
satisfies
the
sign
condition:
p
R
(
t
)
+
r
k
=1
p
B
k
(
t
)
X
max
(
t
)
p
B
k
(
t
)
>
0
,
t
∈
R
.
(51)
Then,
invoking
Remark
6
,
we
may
conclude
that
Y
max
(
·
)
defined
by
Y
max
(
t
)
∆
=
X
max
(
−
t
)
,
t
∈
R
is
the
maximal
solution
of
the
FMRDE
(
6
).
Replacing
t
by
−
t
in
(
51
)
we
obtain
that
Y
max
(
·
)
is
satisfying
(
50
)
too.
Thus,
the
proof
ends.
Applications
of
the
duality
96
4.4
The
bounded
and
stabilizing
solution
of
the
FMRDE
(
6
)
The
next
result
establishes
the
relation
between
the
maximal
solution
and
the
stabilizing
solution
of
a
FMRDE
of
type
(
6
).
Proposition
2.
Assume:
(a)
the
assumptions
(
H
1
)
,
(
H
2
)
hold;
(b)
the
set
Γ
Σ
is
not
empty.
Under
these
conditions,
the
following
are
true:
(i)
the
stabilizing
solution
Y
s
(
·
)
of
the
FMRDE
(
6
)
if
any
coincides
to
its
maximal
solution
Y
max
(
·
);
(ii)
the
bounded
on
R
and
stabilizing
solution
of
the
FMRDE
(
6
)
if
any
is
unique
and
it
is
θ
−
periodic
function.
Proof.
Let
Y
s
(
·
)
be
the
bounded
and
stabilizing
solution
of
the
FMRDE
(
6
).
According
to
Corollary
1
,
X
s
(
·
)
defined
by
X
s
(
t
)
∆
=
Y
s
(
−
t
)
,
t
∈
R
is
the
bounded
and
stabilizing
solution
of
the
dual
BMRDE
(
30
)-(
31
).
Let
Z
(
·
)
∈
Γ
Σ
.
Based
on
Remark
5
,
the
function
p
Z
(
·
)
defined
by
p
Z
(
t
)
∆
=
Z
(
−
t
)
,
t
∈
R
,
lies
in
Γ
Σ
.
Employing
Theorem
5.4.1
from
[
12
]
in
the
case
of
the
pair
formed
by
the
BMRDE
(
30
)-(
31
)
and
the
se
Γ
Σ
,
we
obtain
that
X
s
(
·
)
coincides
to
the
maximal
solution
X
max
(
·
)
of
(
30
)-(
31
).
Finally,
employing
Corollary
1
and
Remark
6
,
we
conclude
that
Y
s
(
t
)
=
Y
max
(
t
)
,
t
∈
R
,
which
confirms
(
i
)
from
the
statement.
The
validity
of
the
assertion
(
ii
)
is
obtained
invoking
(
ii
)
from
Theorem
2
together
with
the
previous
equality.
The
next
result
provides
a
set
of
necessary
and
sufficient
conditions
which
are
guarantying
the
existence
of
the
bounded
on
R
and
stabilizing
solution
of
the
FMRDE
(
6
).
Theorem
3.
Under
the
assumptions
(
H
1
)
,
(
H
2
)
the
following
are
equiva-
lent:
(i)
the
pair
(
C
(
·
)
,
A
(
·
))
is
detectable
and
the
set
Γ
Σ
is
not
empty;
V.
Dr˘
agan,
I.-L.
Popa
97
(ii)
the
FMRDE
(
6
)
has
a
bounded
on
R
and
stabilizing
solution
Y
s
(
·
)
:
R
→
S
n
which
is
satisfying
the
sign
condition:
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
s
(
t
)
C
k
(
t
)
>
0
,
t
∈
R
.
(52)
Proof.
To
show
that
the
implication
(
i
)
⇒
(
ii
)
holds,
we
apply
the
impli-
cation
(
i
)
⇒
(
ii
)
from
Theorem
5.4.6
in
[
12
],
in
the
particular
case
of
the
BMRDE
(
30
)-(
31
).
First,
let
us
remark
that
if
(
C
(
·
)
,
A
(
·
))
is
detectable,
then
the
dual
pair
(
A
#
(
·
)
,
B
#
(
·
))
is
stabilizable,
A
#
(
·
)
,
B
#
(
·
)
being
introduced
in
(
46
).
On
the
other
hand,
if
Z
(
·
)
∈
Γ
Σ
one
obtains
directly
from
(
31
)
and
(
42
)
that
p
Z
(
·
)
defined
by
p
Z
(
t
)
∆
=
Z
(
−
t
)
,
t
∈
R
,
lies
in
Γ
Σ
#
.
So,
the
condi-
tions
from
the
statement
(
i
)
from
Theorem
5.4.6
[
12
]
particularized
to
the
case
of
the
BMRDE
(
30
)-(
31
)
are
fulfilled.
Hence,
the
implication
(
i
)
⇒
(
ii
)
from
Theorem
5.4.6
from
the
afore
mentioned
reference
allows
us
to
deduce
that
under
the
considered
assumption
the
BMRDE
(
30
)-(
31
)
has
a
unique
bounded
and
stabilizing
solution
X
s
(
·
)
:
R
→
S
n
which
is
satisfying
the
sign
condition
p
R
(
t
)
+
r
k
=1
p
B
k
(
t
)
X
s
(
t
)
p
B
k
(
t
)
>
0
,
t
∈
R
.
(53)
Employing
Corollary
1
we
may
infer
that
Y
s
(
·
)
defined
by
Y
s
(
·
)
∆
=
X
s
(
−
t
)
,
t
∈
R
is
just
the
bounded
and
stabilizing
solution
of
the
FMRDE
(
6
).
Writing
(
53
)
for
t
replaced
by
−
t,
we
obtain
that
Y
s
(
·
)
is
satisfying
the
sign
condition
(
52
)
too.
Thus,
the
validity
of
the
implication
(
i
)
⇒
(
ii
)
is
confirmed.
Let
us
show
that
the
implication
(
ii
)
⇒
(
i
)
holds
too.
If
Y
s
(
·
)
is
the
bounded
and
stabilizing
solution
of
the
FMRDE
(
6
)
which
is
satisfying
the
sign
condition
(
52
),
it
follows
that
X
s
(
·
)
defined
by
X
s
(
t
)
=
Y
s
(
−
t
)
,
t
∈
R
,
is
the
bounded
and
stabilizing
solution
of
the
BMRDE
(
30
)-(
31
)
which
is
satisfying
the
sign
condition
(
53
).
Based
on
the
implication
(
ii
)
⇒
(
i
)
from
Theorem
5.4.6
in
[
12
],
applied
in
the
case
of
the
BMRDE
(
30
)-(
31
)
we
deduce
that
the
pair
(
A
#
(
·
)
,
B
#
(
·
))
is
stabilizable
and
the
set
Γ
Σ
#
is
not
empty.
Based
on
Proposition
4.2
in
[
8
]
we
may
infer
that
the
pair
(
C
(
·
)
,
A
(
·
))
is
detectable
if
(
A
#
(
·
)
,
B
#
(
·
))
is
stabilizable.
On
the
other
hand,
if
p
Z
(
·
)
∈
Γ
Σ
#
one
obtains
that
the
function
Z
(
·
)
defined
by
Z
(
t
)
:=
p
Z
(
−
t
)
,
t
∈
R
lies
in
Γ
Σ
.
Thus
we
have
shown
that
the
statement
(
i
)
holds
if
(
ii
)
is
true.
The
proof
ends.
Applications
of
the
duality
98
4.5
An
iterative
procedure
for
the
numerical
computation
of
the
maximal
solution
and
of
the
stabilizing
solution
of
the
FMRDE
(
6
).
As
it
can
be
seen
from
the
previous
developments,
we
do
not
know
apriori
neither
its
initial
value
nor
a
boundary
value
of
the
maximal
solution
and/or
the
stabilizing
solution
of
a
FMRDE.
That
is
why,
the
existing
procedures
for
the
numerical
computation
of
the
solution
of
Cauchy
problem
or
a
solution
of
a
boundary
value
problem
cannot
be
used
to
compute
the
maximal
solution
or
the
stabilizing
solution
of
a
FMRDE.
In
this
subsection,
we
propose
an
iterative
procedure
for
numerical
com-
putation
of
the
maximal
solution
and
of
the
stabilizing
solution,
respectively,
of
the
FMRDE
(
6
).
The
first
step
in
the
construction
of
the
iterative
pro-
cedure
is
described
in
the
following
lemma:
Lemma
2.
Assume
that
(
C
(
·
)
,
A
(
·
))
is
detectable.
Let
K
0
(
·
)
:
R
→
R
n
×
q
be
a
continuous
and
θ
−
periodic
matrix
valued
function
with
the
property
that
the
zero
solution
of
the
corresponding
resulting
system:
dx
(
t
)
=
(
A
0
(
t
)
+
K
0
(
t
)
C
0
(
t
))
x
(
t
)
dt
+
r
k
=1
(
A
k
(
t
)
+
K
0
(
t
)
C
k
(
t
))
x
(
t
)
dw
k
(
t
)
(54)
is
ESMS.
Let
Y
0
(
·
)
:
R
→
S
n
be
a
continuous
differential
matrix
valued
func-
tion
which
is
θ
−
periodic
and
solves
the
following
linear
matrix
differential
inequality
(LMDI):
−
˙
Y
0
(
t
)
+
(
A
0
(
t
)
+
K
0
(
t
)
C
0
(
t
))
Y
0
(
t
)
+
Y
0
(
t
)(
A
0
(
t
)
+
K
0
(
t
)
C
0
(
t
))
+
r
k
=1
(
A
k
(
t
)
+
K
0
(
t
)
C
k
(
t
))
Y
0
(
t
)(
A
k
(
t
)
+
K
0
(
t
)
C
0
(
t
))
+
M
0
(
t
)
≤
0
(55)
where
M
0
(
t
)
:=
M
(
t
)
+
εI
n
+
L
(
t
)
K
0
(
t
)
+
K
0
(
t
)
L
(
t
)
+
K
0
(
t
)
R
(
t
)
K
0
(
t
)
,
t
∈
R
.
Under
these
conditions
we
have:
Y
0
(
t
)
−
Z
(
t
)
≥
γI
n
(56)
for
all
t
∈
R
,
and
any
Z
(
·
)
∈
Γ
Σ
.
V.
Dr˘
agan,
I.-L.
Popa
99
Proof.
Let
F
0
(
t
)
:=
K
0
(
−
t
)
,
t
∈
R
.
Applying
Lemma
1
with
the
updates
M
k
(
t
)
←
A
k
(
t
)
+
K
0
(
t
)
C
k
(
t
)
,
0
≤
k
≤
r,
we
obtain
that
the
zero
solution
of
the
SLDE
dx
(
t
)
=
(
p
A
0
(
t
)
+
p
B
0
(
t
)
F
0
(
t
))
x
(
t
)
dt
+
r
k
=1
(
p
A
k
(
t
)
+
p
B
k
(
t
)
F
0
(
t
))
x
(
t
)
dw
k
(
t
)
,
is
ESMS
if
and
only
if
the
zero
solution
of
(
54
)
is
ESMS,
p
A
k
(
·
)
,
p
B
k
(
·
)
being
defined
in
(
31a
),
(
31b
),
respectively.
If
we
set
X
0
(
t
)
:=
Y
0
(
−
t
)
,
we
obtain
that
X
(
·
)
satisfies
˙
X
0
(
t
)
+
(
p
A
0
(
t
)
+
p
B
0
(
t
)
F
0
(
t
))
X
0
(
t
)
+
X
0
(
t
)(
p
A
0
(
t
)
+
p
B
0
(
t
)
F
0
(
t
))
(57)
+
r
k
=1
(
p
A
k
(
t
)
+
p
B
k
(
t
)
F
0
(
t
))
X
0
(
t
)(
p
A
k
(
t
)
+
p
B
k
(
t
)
F
0
(
t
))
+
x
M
0
(
t
)
≤
0
,
t
∈
R
,
where
x
M
0
(
t
)
:=
x
M
(
t
)
+
εI
n
+
p
L
(
t
)
F
0
(
t
)
+
F
0
(
t
)
p
L
(
t
)
+
F
0
(
t
)
p
R
(
t
)
F
0
(
t
)
,
x
M
(
·
)
,
p
L
(
·
)
,
p
R
(
·
)
being
defined
in
(
31c
)-(
31e
).
Applying
Lemma
5.8.1
from
[
12
]
in
the
case
of
(
57
),
we
obtain
via
Remark
5
that
X
0
(
t
)
−
Z
(
−
t
)
≥
γI
n
,
for
all
t
∈
R
and
for
any
Z
(
·
)
∈
Γ
Σ
.
The
conclusion
is
obtained
setting
−
t
instead
of
t
in
the
last
inequality.
From
(
56
)
we
obtain
via
(
41
)
that
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
0
(
t
)
C
k
(
t
)
>
0
,
t
∈
R
.
Hence,
we
may
define:
K
0
(
t
)
:=
−
Y
0
(
t
)
C
0
(
t
)
+
r
k
=1
A
k
(
t
)
Y
0
(
t
)
C
k
(
t
)
+
L
(
t
)
·
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
0
(
t
)
C
k
(
t
)
−
1
,
t
∈
R
.
(58)
Employing
again
(
56
)
together
with
(
40
),
(
41
),
(
55
),
(
58
),
we
obtain:
Applications
of
the
duality
100
Corollary
2.
If
K
0
(
t
)
is
computed
via
(
58
),
based
on
a
θ
−
periodic
solution
of
the
LMDI
(
55
),
then
the
zero
solution
of
the
SLDE
dx
(
t
)
=
(
A
0
(
t
)
+
K
0
(
t
)
C
0
(
t
))
x
(
t
)
dt
+
r
k
=1
(
A
k
(
t
)
+
K
0
(
t
)
C
k
(
t
))
x
(
t
)
dw
k
(
t
)
is
ESMS.
Particularly,
the
zero
solution
of
the
linear
differential
equation
on
R
n
˙
x
(
t
)
=
(
A
0
(
t
)
+
K
0
(
t
)
C
0
(
t
))
x
(
t
)
,
is
exponentially
stable.
Taking
Y
0
(
·
)
,
K
0
(
·
)
as
first
terms,
we
construct
the
sequences
{
Y
p
(
t
)
}
p
≥
1
,
{
K
p
(
t
)
}
p
≥
1
as:
(a)
Y
p
(
·
)
is
the
unique
θ
−
periodic
solution
of
the
following
Lyapunov
dif-
ferential
equation:
˙
Y
p
(
t
)
=(
A
0
(
t
)
+
K
p
−
1
(
t
)
C
0
(
t
))
Y
p
(
t
)
+
Y
p
(
t
)(
A
0
(
t
)
+
K
p
−
1
(
t
)
C
0
(
t
))
+
M
p
(
t
)
(59)
where
M
p
(
t
)
=
M
(
t
)
+
ε
p
+
1
I
n
+
L
(
t
)
K
p
−
1
(
t
)
+
K
p
−
1
(
t
)
L
(
t
)
+
K
p
−
1
(
t
)
R
(
t
)
K
p
−
1
(
t
)
(60)
+
r
k
=1
(
A
k
(
t
)
+
K
p
−
1
(
t
)
C
k
(
t
))
Y
p
−
1
(
t
)(
A
k
(
t
)
+
K
p
−
1
(
t
)
C
k
(
t
))
.
(b)
the
gain
matrices
K
p
(
t
)
are
defined
by:
K
p
(
t
)
:=
−
Y
p
(
t
)
C
0
(
t
)
+
r
k
=1
A
k
(
t
)
Y
p
−
1
(
t
)
C
k
(
t
)
+
L
(
t
)
·
R
(
t
)
+
r
k
=1
C
k
(
t
)
Y
p
−
1
(
t
)
C
k
[(
t
)
−
1
,
t
∈
R
,
p
≥
1
.
(61)
Theorem
4.
Assume:
(a)
the
assumptions
(
H
1
)
,
(
H
2
)
hold;
(b)
(
C
(
·
)
,
A
(
·
))
is
detectable;
V.
Dr˘
agan,
I.-L.
Popa
101
(c)
the
set
Γ
Σ
is
not
empty.
Under
these
conditions,
for
any
continuous
and
θ
−
periodic
matrix
valued
functions
K
0
(
·
)
:
R
→
R
n
×
q
with
the
property
that
the
zero
solution
of
the
corresponding
resulting
SLDE
(
54
),
is
ESMS,
the
sequences
{
Y
p
(
t
)
}
p
≥
1
,
{
K
p
(
t
)
}
p
≥
1
,
t
∈
R
constructed
via
(
59
)-(
60
)
and
(
61
),
respectively,
starting
from
an
arbitrary
θ
−
periodic
solution
of
the
LMDI
(
55
),
are
convergent.
If
Y
(
t
)
:=
lim
t
→∞
Y
p
(
t
)
,
t
∈
R
,
(62)
then
Y
(
·
)
is
the
maximal
and
bounded
on
R
solution
of
the
FMRDE
(
6
)
which
is
satisfying
the
sign
condition
(
50
).
If
the
set
Γ
Σ
is
not
empty,
then
Y
(
·
)
defined
in
(
62
)
is
just
the
bounded
on
R
and
stabilizing
solution
of
the
FMRDE
(
6
)
which
is
satisfying
the
sign
condition
(
52
).
Proof.
Hint.
One
shows
inductively,
with
respect
to
p,
that
the
terms
of
the
sequence
{
Y
p
(
t
)
}
p
≥
1
have
the
properties:
(
a
p
)
Y
p
(
t
)
−
Z
(
t
)
≥
γI
n
.
t
∈
R
,
(63)
for
any
Z
(
·
)
∈
Γ
Σ
,
where
γ
>
0
does
not
depend
upon
p,
but
it
may
depends
upon
the
function
Z
(
·
);
(
b
p
)
the
zero
solution
of
the
linear
differential
equation
on
R
n
:
˙
x
(
t
)
=
(
A
0
(
t
)
+
K
0
(
t
)
C
0
(
t
))
x
(
t
)
(64)
is
exponentially
stable;
(
c
p
)
Y
p
(
t
)
≥
Y
p
+1
(
t
)
,
t
∈
R
.
(65)
The
exponential
stability
of
the
zero
solution
of
the
equation
(
64
)
guaranties
the
existence
of
the
unique
θ
−
periodic
solution
Y
p
(
·
)
of
the
Lyapunov
differ-
ential
equation
(
59
)-(
60
).
The
condition
(
63
)
together
with
(
41
)
guaranty
that
K
p
(
·
)
are
well
defined
via
(
61
).
Bearing
in
mind
that
(
63
)
and
(
65
)
hold
for
any
p
≥
1
and
t
∈
R
,
we
may
infer
that
{
Y
p
(
t
)
}
p
≥
1
is
point-wise
convergent.
Employing
dominant
convergence
theorem
one
shows
that
Y
(
·
)
defined
by
(
62
)
is
a
solution
of
the
FMRDE
(
6
).
Letting
p
→
∞
in
(
63
)
we
deduce
that
Y
(
t
)
≥
Z
(
t
)
,
Applications
of
the
duality
102
for
all
t
∈
R
,
and
any
Z
(
·
)
∈
Γ
Σ
.
This
means
that
Y
(
·
)
defined
by
(
62
)
coincides
to
the
maximal
solution
of
the
FMRDE
(
6
).
If
Γ
Σ
is
not
empty,
then,
from
Proposition
2
and
Theorem
3
,
we
deduce
that
Y
(
·
)
is
just
the
stabilizing
solution
of
(
6
).
Thus
the
proof
is
complete.
Remark
7.
Since
all
functions
Y
p
(
·
)
,
K
p
(
·
)
,
p
≥
0
are
θ
−
periodic
functions,
it
is
sufficient
to
compute
them
on
an
interval
[
t
0
,
t
0
+
θ
]
.
References
[1]
H.
Abou-Kandil,
G.
Freiling,
V.
Ionescu
and
G.
Jank,
Matrix
Riccati
Equations
in
Control
Systems
Theory
,
Birkhauser,
Basel,
2003.
[2]
B.D.O.
Anderson
and
J.B.
Moore,
Optimal
Control:
Linear
Quadratic
Methods
,
Prentice-Hall,
Englewood
Cliffs,
1989.
[3]
E.M.
Barbashin
and
N.N.
Krasovskii,
Stability
of
motion
in
the
large,
Dokl.
Akad.
Nauk.
86
(1952),
453-456.
[4]
S.
Bittanti
and
P.
Colaneri,
Periodic
Systems,
Filtering
and
Control
,
Springer-Verlag,
London,
2009.
[5]
O.L.V.
Costa,
M.D.
Fragoso
and
R.P.
Marques,
Discrete-Time
Markov
Jump
Linear
Systems
,
Springer,
2005.
[6]
O.L.V.
Costa,
M.D.
Fragoso
and
M.G.
Todorov,
Continuous-Time
Markov
Jump
Linear
Systems
,
Springer-Verlag
Berlin
Heidelberg,
2013.
[7]
J.C.
Doyle,
K.
Glover,
P.P.
Khargonekaar
and
B.A.
Francis,
State
space
solutions
to
standard
H
2
and
H
∞
control
problems,
IEEE
Trans.
Au-
tomatic
Control
34
(1989),
831-847.
[8]
V.
Dr˘
agan,
Duality:
detectability
versus
stabilizability
in
the
stochastic
context,
Ann.
Acad.
Rom.
Sci.,
Serie
Math.
Appl.
16
(2024),
287-310.
[9]
V.
Dr˘
agan,
Optimal
filtering
for
discrete-time
linear
systems
with
mul-
tiplicative
white
noise
perturbations
and
periodic
coefficients,
IEEE
Trans.
Autom.
Control.
58
(2012),
1029-1034.
[10]
V.
Dr˘
agan
and
T.
Morozan,
Global
solutions
to
a
game
theoretic
Riccati
equation
of
stochastic
control,
J.
Differ.
Equations
138
(1997),
328-350.
V.
Dr˘
agan,
I.-L.
Popa
103
[11]
V.
Dr˘
agan,
G.
Freiling,
A.
Hochhaus
and
T.
Morozan,
A
class
of
nonlin-
ear
differential
equations
on
the
space
of
symmetric
matrices,
Electron.
J.
Differ.
Equations
96
(2004),
1-48.
[12]
V.
Dr˘
agan,
T.
Morozan
and
A.M.
Stoica,
Mathematical
Methods
in
Robust
Control
of
Linear
Stochastic
Systems
,
Second
edition,
Springer,
New
York,
2013.
[13]
A.
Friedman,
Stochastic
Differential
Equations
and
Applications
,
vol.
I,
Academic
Press,
New
York,
1975.
[14]
A.
Halanay
and
V.
Ionescu,
Time
Varying
Discrete
Linear
Systems
,
Berlin,
Birkhauser,
1994.
[15]
A.
Ichikawa
and
H.
Katayama,
Linear
Time-Varying
Systems
and
Sampled-Data
,
Springer,
2001.
[16]
R.E.
Kalman,
Contributions
to
the
theory
of
optimal
control,
Bol.
Soc.
Aht.
Mexicana
5
(1960),
102-119.
[17]
R.E.
Kalman
and
R.S.
Bucy,
New
results
in
linear
filtering
and
predic-
tion
theory,
ASME
Trans.
Part
D,
J.
Basic
Eng,
83
(1961),
95-108.
[18]
R.
Khasminskii,
Stochastic
Stability
of
Differential
Equations
,
Sythoff
and
Noordhoff,
Alpen
aan
den
Ryn,
1980.
[19]
A.
Kumar
Roy
and
S.
Kannan,
Event-based
optimal
filter
for
a
net-
worked
system
with
multiplicative
and
auto/cross-correlated
process
and
measurement
noise,
Int.
J.
Adapt.
Control
Signal
Process.
36
(2022),
2453-2478.
[20]
H.
Kwakernaak
and
R.
Sivan,
Linear
Optimal
Control
Systems
,
Wiley-
Intersicience,
New
York,
1972.
[21]
W.
Liu,
H.
Zhang,
K.
Yu
and
X.
Tan,
Optimal
linear
filtering
for
networked
systems
with
communication
constraints,
fading
measure-
ments,
and
multiplicative
noises,
Int.
J.
Adapt.
Control
Signal
Process.
31
(2017),
1019-1039.
[22]
R.
Oxendal,
Stochastic
Differential
Equations
,
Springer,
Berlin,
1998.
[23]
V.M.
Popov,
Hyperstability
of
Control
Systems
,
Editura
Academiei-
Springer
Verlag,
Berlin,
1973.
Applications
of
the
duality
104
[24]
R.
Ravi,
K.M.
Nagpal
and
P.P.
Khargonekar,
H
∞
control
for
linear
time-varying
systems:
a
state
space
approach,
SIAM
J.
Control
Optim.
29
(1991),
1394-1413.
[25]
J.
Zhang,
X.
He
and
D.
Zhou,
Robust
optimal
filtering
for
linear
time-varying
systems
with
stochastic
uncertainties,
IET
Control
Theory
Appl.
11
(2017),
3297-3304.