TELK
OMNIKA,
V
ol.16,
No
.6,
December
2018,
pp
.
3024–3033
ISSN:
1693-6930,
accredited
First
Gr
ade
b
y
K
emenr
istekdikti,
Decree
No:
21/E/KPT/2018
DOI:
10.12928/TELK
OMNIKA.v16i6.10112
3024
T
ransf
er
Function,
Stabilizability
,
and
Detectability
of
Non-A
utonomous
Riesz-spectral
Systems
Sutrima
*1
,
Christiana
Rini
Indrati
2
,
and
Lina
Ar
y
ati
3
1
Univ
ersitas
Sebelas
Maret,
Ir
.Sutami
St
no
.36
A
K
entingan
Sur
akar
ta,
Ind
onesia
1,2,3
Univ
ersitas
Gadjah
Mada,
Sekip
Utar
a
K
otak
P
os:
BLS
21,
Y
ogy
akar
ta,
Indo
nesia
*
Corresponding
author
,
e-mail:
sutr
ima@mipa.uns
.ac.id,
zutr
ima@y
ahoo
.co
.id
Abstract
Stability
of
a
state
linear
system
can
be
identified
b
y
controllability
,
obser
v
ability
,
stabilizability
,
detectability
,
and
tr
ansf
er
function.
The
appro
ximate
controllability
and
obser
v
ability
of
non-autonomous
Riesz-spectr
al
systems
ha
v
e
been
estab
lished
as
w
ell
as
non-autonomous
Stur
m-Liouville
systems
.
As
a
contin
uation
of
the
estab
lishments
,
this
paper
concer
n
on
the
analysis
of
the
tr
ansf
er
function,
stabilizability
,
and
detectability
of
the
non-autonomous
Riesz-spectr
al
systems
.
A
str
ongly
contin
uous
quasi
semig
roup
approach
is
implemented.
The
results
sho
w
that
the
tr
ansf
er
function,
stabilizability
,
and
detectability
can
be
estab
lished
comprehen
siv
ely
in
the
non-autonomous
Rie
sz-spectr
al
systems
.
In
par
ticular
,
sufficient
and
necessar
y
cond
itions
f
or
the
stabilizability
and
detectability
can
be
constr
ucted.
These
results
are
par
allel
with
infinite
dimensional
of
autonomous
systems
.
K
e
yw
or
ds:
detectability
,
non-autonomous
Riesz-spectr
al
system,
stabilizability
,
tr
ansf
er
funct
ion
Cop
yright
c
2018
Univer
sitas
Ahmad
Dahlan.
All
rights
reser
ved.
1.
Intr
oduction
Let
X
,
U
,
and
Y
be
comple
x
Hilber
t
spaces
.
This
paper
concer
ns
on
the
linear
non-
autonomous
control
systems
with
state
x
,
input
u
,
and
output
y
:
_
x
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
;
x
(0)
=
x
0
;
y
(
t
)
=
C
(
t
)
x
(
t
)
;
t
0
;
(1)
where
A
(
t
)
is
a
linear
closed
oper
ator
in
X
with
domain
D
(
A
(
t
))
=
D
,
independent
of
t
and
dense
in
X
;
B
(
t
)
:
U
!
X
and
C
(
t
)
:
X
!
Y
are
bounded
oper
ato
rs
such
that
B
(
)
2
L
1
(
R
+
;
L
s
(
U
;
X
))
and
C
(
)
2
L
1
(
R
+
;
L
s
(
X
;
Y
))
,
where
L
s
(
V
;
W
)
and
L
1
(
;
W
)
denote
the
space
of
bounded
oper
ators
from
V
to
W
equipped
with
strong
oper
ator
topology
and
the
space
of
bounded
measurab
le
functions
from
to
W
pro
vided
with
essential
suprem
um
nor
m,
respectiv
ely
.
The
state
linear
system
(1)
is
denoted
b
y
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
.
Symbol
(
A
(
t
)
;
B
(
t
)
;
)
and
(
A
(
t
)
;
;
C
(
t
))
denote
the
state
linear
system
(1)
if
C
(
t
)
=
0
and
B
(
t
)
=
0
,
respectiv
ely
.
In
par
ticular
,
this
paper
shall
in
v
estigate
stability
of
the
state
linear
system
(1)
where
each
A
(
t
)
is
a
gener
aliz
ed
Riesz-spectr
al
oper
ator
[1]
using
tr
ansf
er
function,
stabilizability
,
and
detectability
.
The
f
ollo
wing
e
xplanations
giv
e
some
reasons
wh
y
these
in
v
estigations
are
impor
tant
f
or
the
systems
.
In
the
autonomous
control
system
(
A;
B
;
C
)
,
where
A
is
an
infinitesimal
gener
ator
of
a
C
0
-semig
roup
T
(
t
)
,
there
e
xist
special
relationships
among
input,
state
,
and
output
[2].
Controlla-
bility
map
specifies
the
relationship
betw
een
the
input
and
the
state
,
obser
v
ability
map
specifies
the
relationship
betw
een
an
init
ial
state
and
the
output.
The
relationship
betw
een
the
input
and
the
output
can
be
char
acter
iz
ed
b
y
tr
ansf
er
function,
a
linear
map
that
specifies
relationship
betw
een
the
Laplace
tr
ansf
or
m
of
the
inputs
and
outputs
.
The
f
ollo
wing
studies
sho
w
ho
w
urgency
of
the
tr
ansf
er
fu
nction
is
in
the
line
ar
system.
In
the
infinite-dimen
sional
autonomous
system,
e
xter
nal
stability
is
indicated
b
y
boundedness
of
its
tr
ansf
er
function
[3].
W
eiss
[4]
ha
v
e
pro
v
ed
a
f
or
m
ula
f
or
the
tr
ansf
er
function
of
a
regular
linear
system,
which
is
similar
to
the
f
or
m
ula
in
the
finite
di-
mensions
.
As
an
e
xtension
of
results
of
[4],
Staff
ans
and
W
eiss
[5]
ha
v
e
gener
aliz
ed
the
results
Receiv
ed
Ma
y
31,
2018;
Re
vised
October
5,
2018;
Accepted
No
v
ember
3,
2018
Evaluation Warning : The document was created with Spire.PDF for Python.
3025
ISSN:
1693-6930
of
regular
linear
systems
to
w
ell-posed
linear
systems
.
The
y
ha
v
e
also
introduced
the
Lax-Phillips
semig
roup
induced
b
y
a
w
ell-posed
linear
system.
W
eiss
[6]
ha
v
e
studied
f
ou
r
tr
ansf
or
mations
which
lead
from
one
w
ell-posed
linear
system
to
another
:
time-in
v
ersion,
flo
w-in
v
ersion,
time-
flo
w-in
v
ersion,
and
duality
.
In
par
ticular
,
a
w
ell-posed
linear
system
is
flo
w-in
v
er
tib
le
if
and
only
if
the
tr
ansf
er
function
of
the
system
has
a
unif
or
mly
bounded
in
v
erse
on
some
r
ight
half-plane
.
Finally
,
P
ar
tington
[7]
ha
v
e
giv
en
simple
sufficient
condit
ions
f
or
a
space
of
f
unctions
on
(0
;
1
)
such
that
all
shift-in
v
ar
iant
oper
ators
defined
on
the
space
are
represented
b
y
tr
ansf
er
functions
.
Controllability
and
stability
are
qualitativ
e
control
prob
lems
which
are
impor
tant
aspects
of
the
theor
y
of
control
systems
.
Kalman
et
al.
[8]
had
initiated
the
theor
y
f
or
the
finite
dimensional
of
autonomous
systems
.
Recently
,
the
theor
y
w
as
gener
aliz
ed
into
controllability
and
stabilizability
of
the
non-autonomous
control
systems
of
v
ar
ious
applications
,
see;
e
.g.
[9,
10,
11],
and
the
ref
erences
t
herein.
The
concept
of
the
stabilizability
is
to
find
an
admissib
le
cont
rol
u
(
t
)
such
that
the
corresponding
solution
x
(
t
)
of
the
system
has
some
required
proper
ties
.
If
the
stabilizabil-
ity
is
identified
b
y
n
ull
controllability
,
system
(1)
is
said
to
be
stabilizab
le
if
there
e
xists
a
control
u
(
t
)
=
F
(
t
)
x
(
t
)
such
that
the
z
ero
solution
of
the
closed-loop
system
_
x
(
t
)
=
[
A
(
t
)
+
B
(
t
)
F
(
t
)]
x
(
t
)
;
t
0
;
is
asymptotically
stab
le
in
the
L
y
apuno
v
sense
.
In
this
case
,
u
(
t
)
=
F
(
t
)
x
(
t
)
is
called
the
stabili-
zing
f
eedbac
k
control.
In
par
ticular
f
or
autonomous
system
(
A;
B
;
)
,
where
A
is
an
infinitesimal
gener
ator
of
an
e
xponentially
stab
le
C
0
-semig
roup
T
(
t
)
,
(
A;
B
;
)
is
stabilizab
le
if
oper
ator
A
+
B
F
is
an
infinitesimal
gener
ator
of
an
e
xponentially
stab
le
C
0
-semig
roup
T
B
F
(
t
)
[2].
In
the
finite-
dimensional
autonomous
control
system,
Kalman
et
al.
[8]
and
W
onham
[12]
had
sho
wn
that
the
system
is
stabilizab
le
if
it
is
n
ull
controllab
le
in
a
finite
time
.
Bu
t,
it
does
not
hold
f
or
the
con
v
erse
.
Fur
ther
more
,
if
the
system
is
completely
stabilizab
le
,
then
it
is
n
ull
controllab
le
in
a
finite
time
.
F
or
finite-dimensional
non-autonomous
control
systems
,
Ik
eda
et
al.
[13]
pro
v
ed
that
the
system
is
completely
stabilizab
le
whence
it
is
n
ull
controllab
le
.
Gener
alizations
of
the
results
of
t
he
stabilizability
f
or
the
finite-dimensional
systems
into
infinite-dimensional
systems
ha
v
e
been
successfully
done
.
Extending
the
L
y
apuno
v
equation
in
Banach
spaces
,
Phat
and
Kiet
[14]
specified
the
relationship
betw
een
stability
and
e
xact
n
ull
controllability
in
the
autonomous
systems
.
Guo
et
al.
[15]
pro
v
ed
the
e
xistence
of
the
infinitesimal
gener
ator
of
the
per
turbation
semig
roup
.
F
or
neutr
al
type
linear
systems
in
Hilber
t
spaces
,
Rabah
et
al.
[16]
pro
v
ed
that
e
xact
n
ull
controllability
implies
the
complete
stabilizability
.
In
the
paper
,
unbounded
f
eedbac
k
is
also
in
v
estigated.
In
the
non-autonomous
systems
,
Hinr
ichsen
and
Pr
itchard
[17]
in
v
estigat
ed
r
adius
stability
f
or
the
systems
under
str
uctured
non-autonomous
per
turbations
.
Niamsup
and
Phat
[18]
pro
v
ed
that
e
xact
n
ull
controllability
implies
the
complete
stabilizability
f
or
linear
non-autonomous
systems
in
Hilber
t
spaces
.
Leiv
a
and
Barcenas
[19]
ha
v
e
introduced
a
C
0
-quasi
semig
roup
as
a
ne
w
approach
to
in
v
estigate
the
non-autonomous
systems
.
In
this
conte
xt,
A
(
t
)
is
an
infinitesimal
gener
ator
of
a
C
0
-quasi
semig
rou
p
on
a
Banach
space
.
Sutr
ima
et
al.
[20]
and
Sutr
ima
et
al.
[21]
in
v
estigated
the
adv
anced
proper
ties
and
some
types
of
stabilities
of
the
C
0
-quasi
semig
roups
in
Banach
spaces
,
respectiv
ely
.
Ev
en
Barcenas
et
al.
[22]
ha
v
e
char
acter
iz
ed
the
con
trollability
of
the
non-autonomous
control
systems
using
the
quasi
semig
roup
approach,
although
it
is
still
limited
to
the
autonomous
controls
.
In
par
ticular
,
Sutr
ima
et
al.
[1]
char
acter
iz
ed
the
controllability
and
obser
v
ability
of
non-autonomous
Riesz-spectr
al
systems
.
The
ref
erences
e
xplain
that
the
tr
ansf
er
function,
stabilizability
,
and
detectability
of
the
control
systems
are
impor
tant
indicators
f
or
the
stability
.
Unf
or
tunately
,
there
are
no
studies
of
these
concepts
in
the
non-autonomous
Riesz-spectr
al
systems
.
Theref
ore
,
this
paper
concer
ns
on
in
v
estigations
of
the
tr
ansf
er
function,
stabilizability
,
and
detectability
of
the
non-autonomous
Riesz-spectr
al
systems
that
ha
v
e
not
been
in
v
estgated
at
this
time
y
et.
These
in
v
estigations
use
the
C
0
-quasi
semig
roup
approach.
2.
Pr
oposed
Methods
and
Discussion
2.1.
T
ransf
er
Function
W
e
recall
the
de
finition
of
a
non-autonomous
Riesz-spectr
al
system
that
ref
ers
to
Sutr
ima
et
al.
[1].
The
definition
of
a
Riesz-spectr
al
oper
ator
f
ollo
ws
[2].
TELK
OMNIKA
V
ol.
16,
No
.
6,
December
2018
:
3024
3033
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
3026
Definition
1.
The
linear
non-autonomous
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
is
called
a
non-autonomous
Riesz-spectr
al
system
if
A
(
t
)
is
the
infinitesimal
gener
ator
of
a
C
0
-quasi
semig
roup
which
has
an
e
xpression
A
(
t
)
=
a
(
t
)
A
(2)
where
A
is
a
Riesz-spectr
al
oper
ator
on
X
and
a
is
a
bounded
contin
uous
function
such
that
a
(
t
)
>
0
f
or
all
t
0
.
By
e
xpression
(2),
f
or
e
v
er
y
t
0
,
w
e
see
that
A
(
t
)
and
A
ha
v
e
th
e
common
domain
and
eigen
v
ectors
.
In
f
act,
if
n
,
n
2
N
,
is
an
eigen
v
alue
of
A
,
then
a
(
t
)
n
is
the
eigen
v
alue
of
A
(
t
)
of
(2).
Hence
,
in
gener
al
A
(
t
)
ma
y
ha
v
e
the
non-simple
eigen
v
alues
.
In
the
non-autonomous
Riesz-spectr
al
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
,
where
A
(
t
)
is
an
infinitesimal
gener
ator
of
a
C
0
-quasi
semig
roup
R
(
t;
s
)
,
relationships
among
input,
state
,
and
output
are
deter
mined
b
y
R
(
t;
s
)
.
The
relationships
of
state-input
and
state-
output
of
the
system
ha
v
e
been
discussed
b
y
Sutr
ima
et
al.
[1].
In
this
subsection,
w
e
f
ocus
on
char
acter
izing
the
relationship
betw
een
the
input
and
output
of
the
system
using
the
tr
ansf
er
function,
where
B
(
t
)
2
L
2
(
U
;
X
)
and
C
(
t
)
2
L
2
(
X
;
Y
)
f
or
all
t
0
.
W
e
define
m
ultiplicativ
e
oper
ators
B
:
L
2
(
R
+
;
U
)
!
L
2
(
R
+
;
X
)
and
C
:
L
2
(
R
+
;
X
)
!
L
2
(
R
+
;
Y
)
b
y:
(
B
u
)(
t
)
=
B
(
t
)
u
(
t
)
and
(
C
x
)(
t
)
=
C
(
t
)
x
(
t
)
;
t
0
;
respectiv
ely
.
W
e
see
that
the
oper
ators
B
and
C
are
bounded.
The
definition
of
the
tr
ansf
er
function
of
the
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
f
ollo
ws
the
definition
f
or
the
autonomous
systems
of
[2].
Definition
2.
Let
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
be
the
non-autonomous
Riesz-spectr
al
system
with
z
ero
initial
state
.
If
there
e
xists
a
real
such
that
^
y
(
s
)
=
G
(
s
)
^
u
(
s
)
f
or
Re
(
s
)
>
,
where
^
u
(
s
)
and
^
y
(
s
)
denote
the
Laplace
tr
ansf
or
ms
of
u
and
y
,
respectiv
ely
,
and
G
(
s
)
is
a
L
(
U
;
Y
)
-v
alued
function
of
comple
x
v
ar
iab
le
defined
f
or
Re
(
s
)
>
,
then
G
is
called
tr
ansf
er
function
of
the
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
.
The
impulse
response
h
of
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
is
defined
as
the
Laplace
in
v
erse
tr
ansf
or
m
of
G
.
The
tr
ansf
er
fun
ction
of
the
non-autonomous
Riesz-spectr
al
systems
with
finite-r
ank
in-
puts
and
outputs
can
be
stated
in
eigen
v
alues
and
eigen
v
ectors
of
the
Riesz-spectr
al
oper
ator
.
Theorem
3.
The
tr
ansf
er
function
G
and
impulse
re
sponse
h
of
the
non-autonomous
Riesz-
spectr
al
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
e
xist
and
are
giv
en
b
y
G
(
s
)
=
C
a
(
)
(
sI
a
(
)
A
)
1
B
on
(
A
(
))
;
and
h
(
t
)
=
C
T
(
t
)
B
;
t
0
0
;
t
<
0
;
where
T
(
t
)
is
a
C
0
-semig
roup
with
the
infinitesimal
gener
ator
A
.
Proof
.
By
definition
of
Riesz-spectr
al
oper
ator
and
Theorem
3
of
[1],
w
e
ha
v
e
that
the
resolv
ent
set
of
A
is
connected,
so
1
(
A
)
=
(
A
)
.
W
e
shall
v
er
ify
the
e
xistences
of
the
tr
ansf
er
function
and
impulse
response
.
Let
R
(
t;
s
)
be
a
C
0
-quasi
semig
roup
with
infinitesimal
gener
ator
A
(
t
)
of
the
f
or
m
(2).
F
or
Re
(
s
)
>
!
0
a
1
,
where
!
0
is
the
g
ro
wth
bound
of
a
C
0
-semig
roup
T
(
t
)
with
infinitesimal
gener
ator
A
and
a
1
:=
sup
t
0
a
(
t
)
.
By
Lemma
2.1.11
of
[2],
w
e
ha
v
e
C
a
(
)
(
sI
a
(
)
A
)
1
B
u
=
C
Z
1
0
e
s
a
(
)
T
(
)
B
ud
=
Z
1
0
e
s
a
(
)
C
T
(
)
B
ud
f
or
all
u
2
L
2
(
R
+
;
U
)
and
Re
(
s
)
>
!
0
a
1
.
As
in
the
proof
of
Lemm
a
2.5.6
of
[2],
these
s
can
be
e
xtended
to
all
s
2
(
a
(
)
A
)
.
Corollar
y
4.
Let
A
(
t
)
be
an
oper
ator
of
the
f
or
m
(2),
where
A
is
a
Riesz-spectr
al
oper
ator
with
eigen
v
alues
f
n
2
C
:
n
2
N
g
.
If
B
:
R
+
!
L
(
C
m
;
X
)
and
C
:
R
+
!
L
(
X
;
C
k
)
such
that
T
r
ansf
er
Function,
Stabilizability
,
and
Detectability
of
Non-A
utonomous
...
(Sutr
ima)
Evaluation Warning : The document was created with Spire.PDF for Python.
3027
ISSN:
1693-6930
B
(
t
)
2
L
(
C
m
;
X
)
and
C
(
t
)
2
L
(
X
;
C
k
)
,
then
the
tr
ansf
er
function
and
impulse
response
of
the
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
are
giv
en
b
y
G
(
s
)
=
1
X
n
=1
a
(
)
s
a
(
)
n
C
n
B
n
tr
f
or
s
2
(
A
(
))
(3)
h
(
t
)
=
8
<
:
1
P
n
=1
e
n
a
(
)
t
C
n
B
n
tr
;
t
0
0
;
t
<
0
;
(4)
where
n
and
n
are
the
corresponding
eigen
v
ectors
of
A
and
A
,
respectiv
ely
.
Symbol
W
tr
denotes
tr
anspose
of
W
.
Proof
.
By
representation
of
(
sI
a
(
)
A
)
1
of
Theorem
3
of
[1]
and
f
acts
that
oper
ators
B
(
t
)
and
C
(
t
)
are
bounded
f
or
all
t
0
,
then
from
Theorem
3
f
or
s
2
(
A
(
))
w
e
ha
v
e
G
(
s
)
u
=
C
a
(
)
(
sI
a
(
)
A
)
1
B
u
=
C
"
lim
N
!1
N
X
n
=1
a
(
)
s
a
(
)
n
hB
u;
n
i
n
#
=
1
X
n
=1
a
(
)
s
a
(
)
n
C
n
B
n
tr
:
Here
,
w
e
use
the
proper
ty
h
v
;
w
i
C
m
=
w
tr
v
.
Thus
,
e
xpression
(3)
is
pro
v
ed.
By
a
similar
argument
and
condition
(c)
of
Theorem
3
of
[1]
w
e
ha
v
e
e
xpression
(4)
f
or
h
(
t
)
.
The
f
ollo
wing
e
xample
illus-
tr
ates
the
tr
ansf
er
function
of
a
non
-autonomous
Riesz-spectr
al
system.
The
e
xample
is
modified
from
Example
4.3.11
of
[2].
Example
5.
Consider
the
controlled
non-autonomous
heat
equation
on
the
inter
v
al
[0
;
1]
,
@
x
@
t
(
t;
)
=
a
(
t
)
@
2
x
@
2
(
t;
)
+
2
b
(
t
)
u
(
t
)
[
1
2
;
1]
(
)
;
0
<
<
1
;
t
0
;
@
x
@
(
t;
0)
=
@
x
@
(
t;
1)
=
0
;
y
(
t
)
=
2
c
(
t
)
Z
1
=
2
0
x
(
t;
)
d
;
(5)
where
a
:
R
+
!
R
is
a
boundedly
unif
or
mly
contin
uous
positiv
e
function
and
b
:
R
+
!
C
is
a
bounded
contin
uous
function.
W
e
v
er
ify
the
tr
ansf
er
function
and
impulse
response
of
the
go
v
er
ned
system.
Setting
X
=
L
2
[0
;
1]
and
U
=
Y
=
C
,
the
prob
lem
(5)
is
a
non-autonomous
Riesz-spectr
al
system:
_
x
(
t
)
=
A
(
t
)
x
(
t
)
+
B
(
t
)
u
(
t
)
;
y
(
t
)
=
C
(
t
)
x
(
t
)
;
t
0
;
(6)
where
A
(
t
)
=
a
(
t
)
A
with
Ax
=
d
2
x
d
2
on
domain
D
=
D
(
A
)
=
f
x
2
X
:
x;
dx
d
are
absolutely
contin
uous
;
d
2
x
d
2
2
X
;
dx
d
(0)
=
dx
d
x
(1)
=
0
g
;
B
(
t
)
u
(
t
)
=
2
b
(
t
)
u
(
t
)
[
1
2
;
1]
(
)
;
and
C
(
t
)
x
(
t
)
=
2
c
(
t
)
Z
1
=
2
0
x
(
t;
)
d
:
The
eigen
v
alues
and
eigen
v
ectors
of
A
are
f
0
;
n
2
2
:
n
2
N
g
and
f
1
;
p
2
cos(
n
)
:
n
2
N
g
,
respectiv
ely
.
It
is
easy
to
sho
w
that
A
is
a
self-adjoint
Riesz-spectr
al
oper
ator
with
its
Riesz
basis
f
1
;
p
2
cos(
n
)
:
n
2
N
g
.
In
this
case
w
e
ha
v
e
B
(
t
)
x
=
Z
1
0
x
[
1
2
;
1]
(
)
d
;
TELK
OMNIKA
V
ol.
16,
No
.
6,
December
2018
:
3024
3033
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
3028
f
or
all
2
X
and
t
0
.
In
vir
tue
of
(3)
e
v
aluating
at
t
,
the
tr
ansf
er
function
of
the
system
(6)
is
giv
en
b
y
G
(
s
)(
t
)
=
1
s
+
1
X
n
=1
2
a
(
t
)[cos(
n
)
1]
[
s
+
a
(
t
)
n
2
2
](
n
)
2
;
and
e
v
aluating
at
s
in
(4),
w
e
ha
v
e
the
impulse
response
h
(
t
)(
s
)
=
1
+
1
X
n
=1
2[cos(
n
)
1]
(
n
)
2
e
(
n
)
2
a
(
s
)
t
;
on
t
0
and
z
ero
else
where
,
respectiv
ely
.
2.2.
Stabilizability
W
e
first
recall
the
definition
of
unif
or
mly
e
xponentially
stab
le
C
0
-quasi
semig
roup
that
ref
ers
to
[21,
23].
The
concept
pla
y
an
impor
tant
role
in
char
acter
izing
stabilizability
of
the
non-
autonomous
Riesz-spectr
al
systems
.
Definition
6.
A
C
0
-quasi
semig
roup
R
(
t;
s
)
is
said
to
be:
(a)
unif
or
mly
e
xponentially
stab
le
on
a
Banach
space
X
if
there
e
xist
constants
>
0
and
N
1
such
that
k
R
(
t;
s
)
x
k
N
e
s
k
x
k
;
(7)
f
or
all
t;
s
0
and
x
2
X
;
(b)
-unif
or
mly
e
xponentially
stab
le
on
a
Banach
space
X
if
(7)
holds
f
or
<
.
A
constant
is
called
deca
y
r
ate
and
the
suprem
um
o
v
er
all
possib
le
v
alues
of
is
called
stability
margin
of
R
(
t;
s
)
.
Indeed,
the
stability
margin
is
min
us
of
the
unif
or
m
g
ro
wth
bound
!
0
(
R
)
defined
!
0
(
R
)
=
inf
t
0
!
0
(
t
)
;
where
!
0
(
t
)
=
inf
s>
0
1
s
log
k
R
(
t;
s
)
k
.
W
e
see
that
R
(
t;
s
)
is
-unif
or
mly
e
xponentially
stab
le
if
its
stability
margin
is
at
least
.
W
e
giv
e
tw
o
preliminar
y
results
which
are
urgent
in
discussing
the
stabilizability
.
Theorem
7.
Let
R
(
t;
s
)
be
a
C
0
-quasi
semig
roup
on
a
Banach
space
X
.
The
R
(
t;
s
)
is
unif
or
mly
e
xponentially
stab
le
on
X
if
and
only
if
!
0
(
R
)
<
0
.
Proof
.
By
taking
log
on
(7),
w
e
ha
v
e
the
asser
tion.
Theorem
8.
Let
A
(
t
)
be
an
infinitesimal
gener
ator
of
C
0
-quasi
semig
roup
R
(
t;
s
)
on
a
Banach
space
X
.
If
B
(
)
2
L
1
(
R
+
;
L
s
(
X
))
,
then
there
e
xists
a
uniquely
C
0
-quasi
semig
roup
R
B
(
t;
s
)
with
its
infinitesimal
gener
ator
A
(
t
)
+
B
(
t
)
such
that
R
B
(
r
;
t
)
x
=
R
(
r
;
t
)
x
+
Z
t
0
R
(
r
+
s;
t
s
)
B
(
r
+
s
)
R
B
(
r
;
s
)
xds;
(8)
f
or
all
t;
r
;
s
0
with
t
s
and
x
2
X
.
Moreo
v
er
,
if
k
R
(
r
;
t
)
k
M
(
t
)
,
then
k
R
B
(
r
;
t
)
k
M
(
t
)
e
k
B
k
M
(
t
)
t
:
Proof
.
W
e
define
R
0
(
r
;
t
)
x
=
R
(
r
;
t
)
x;
R
n
(
r
;
t
)
x
=
Z
t
0
R
(
r
+
s;
t
s
)
B
(
r
+
s
)
R
n
1
(
r
;
s
)
xds;
(9)
T
r
ansf
er
Function,
Stabilizability
,
and
Detectability
of
Non-A
utonomous
...
(Sutr
ima)
Evaluation Warning : The document was created with Spire.PDF for Python.
3029
ISSN:
1693-6930
f
or
all
t;
r
;
s
0
with
t
s
,
x
2
X
,
and
n
2
N
,
and
R
B
(
r
;
t
)
=
1
X
n
=0
R
n
(
r
;
t
)
;
(10)
F
ollo
wing
the
proof
of
Theorem
2.4
of
[19],
w
e
obtain
the
asser
tions
.
The
in
v
estigations
of
stabiliz-
ability
and
detectability
of
the
non-autonomous
Riesz-spectr
al
systems
are
gener
alizations
of
the
concepts
of
stabilizability
and
detectability
f
or
the
autonomous
systems
that
had
been
de
v
eloped
b
y
Cur
tain
and
Zw
ar
t
[2].
Definition
9.
The
non-autonomous
Riesz-spectr
al
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
is
said
to
be:
(a)
stabilizab
le
if
there
e
xsits
an
oper
ator
F
2
L
1
(
R
+
;
L
s
(
X
;
U
))
such
that
A
(
t
)
+
B
(
t
)
F
(
t
)
,
t
0
,
is
an
infinitesima
l
gener
ator
of
a
unif
or
mly
e
xponentially
stab
le
C
0
-quasi
semig
roup
R
B
F
(
t;
s
)
.
The
oper
ator
F
is
called
a
stabilizing
f
eedbac
k
oper
ator
;
(b)
detectab
le
if
there
e
xists
an
oper
ator
K
2
L
1
(
R
+
;
L
s
(
Y
;
X
))
such
that
A
(
t
)
+
K
(
t
)
C
(
t
)
,
t
0
,
is
an
infinitesimal
gener
ator
of
a
unif
or
mly
e
xponentially
stab
le
C
0
-quasi
semig
roup
R
K
C
(
t;
s
)
.
The
oper
ator
K
is
called
an
output
injection
oper
ator
.
If
the
quasi
semig
roup
R
B
F
(
t;
s
)
is
-unif
or
mly
e
xponentially
stab
le
,
w
e
sa
y
that
the
system
(
A
(
t
)
;
B
(
t
)
;
)
is
-stab
lizab
le
.
If
R
K
C
(
t;
s
)
is
-unif
or
mly
e
xponentially
stab
le
,
w
e
sa
y
that
the
system
(
A
(
t
)
;
;
C
(
t
))
is
-detectab
le
.
F
or
2
R
,
w
e
can
decompose
the
spectr
um
of
A
in
comple
x
plane
into
tw
o
distinct
par
ts:
+
(
A
)
:=
(
A
)
\
C
+
;
C
+
=
f
2
C
:
Re
(
)
>
g
(
A
)
:=
(
A
)
\
C
;
C
=
f
2
C
:
Re
(
)
<
g
:
In
the
autonomous
case
,
if
B
has
finite-r
ank
and
the
system
(
A;
B
;
)
is
stabilizab
le
,
then
w
e
can
decompose
the
spectr
um
of
A
into
a
-stab
le
par
t
and
a
-unstab
le
par
t
which
compr
ises
eigen
v
alues
with
finite
m
ultiplicity
.
In
other
w
ord,
A
has
at
most
finitely
man
y
eigen
v
alues
in
C
+
.
W
e
shall
apply
the
decom
position
of
the
spectr
um
to
the
non-autonomous
Riesz-spectr
al
systems
.
Definition
10.
An
oper
ator
A
is
said
to
be
satisfying
the
spectr
um
decomposition
assumption
at
if
+
(
A
)
is
bounded
and
separ
ated
from
(
A
)
in
such
w
a
y
that
a
rectifiab
le
,
simple
,
closed
cur
v
e
,
,
can
be
dr
a
wn
so
as
to
enclose
an
open
set
containing
+
(
A
)
in
its
inter
ior
and
(
A
)
in
its
e
xter
ior
.
According
to
Definition
10,
w
e
ha
v
e
that
classes
of
Riesz-spectr
al
o
per
ators
with
a
pure
point
spectr
um
and
only
finitely
man
y
eigen
v
alues
in
+
(
A
)
satisfy
the
spectr
um
decomposition
assumption.
F
or
e
v
er
y
t
0
w
e
define
the
spectr
al
projection
P
(
t
)
on
X
b
y
P
(
t
)
x
=
1
2
i
Z
(
I
A
(
t
))
1
xd;
(11)
f
or
all
x
2
X
,
where
is
tr
a
v
ersed
once
in
the
positiv
e
direction
(countercloc
kwise).
By
this
oper
ator
,
w
e
can
decompose
an
y
Hilber
t
space
X
to
be:
X
=
X
+
X
;
where
X
+
:=
P
(
t
)
X
and
X
:=
(
I
P
(
t
))
X
:
(12)
By
this
decomposition,
w
e
denote
A
(
t
)
=
A
+
(
t
)
0
0
A
(
t
)
;
R
(
t;
s
)
=
R
+
(
t;
s
)
0
0
R
(
t;
s
)
(13)
B
(
t
)
=
B
+
(
t
)
B
(
t
)
;
C
(
t
)
=
C
+
(
t
)
C
(
t
)
;
(14)
TELK
OMNIKA
V
ol.
16,
No
.
6,
December
2018
:
3024
3033
Evaluation Warning : The document was created with Spire.PDF for Python.
TELK
OMNIKA
ISSN:
1693-6930
3030
where
B
+
(
t
)
=
P
(
t
)
B
(
t
)
2
L
(
U
;
X
+
)
,
B
(
t
)
=
(
I
P
(
t
))
B
(
t
)
2
L
(
U
;
X
)
,
C
+
(
t
)
=
C
(
t
)
P
(
t
)
2
L
(
X
+
;
Y
)
,
and
C
(
t
)
=
C
(
t
)(
I
P
(
t
))
2
L
(
X
;
Y
)
.
In
vir
tue
of
the
decomposition,
w
e
can
e
x-
press
the
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
as
the
v
ector
sum
of
the
tw
o
subsystems:
(
A
+
(
t
)
;
B
+
(
t
)
;
C
+
(
t
))
on
X
+
and
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
on
X
.
In
par
ticular
,
if
the
input
oper
ator
B
(
)
has
finite-r
ank,
then
the
subsystem
(
A
+
(
t
)
;
B
+
(
t
)
;
)
has
a
finite
dimension.
Theorem
11.
Let
(
A
(
t
)
;
B
(
t
)
;
)
be
a
non-autonomous
Riesz-spectr
al
system
on
the
state
space
X
where
B
(
)
has
a
finite
r
ank.
If
A
satisfies
the
spectr
um
decomp
osition
assumption
at
,
X
+
has
a
finite
dimension,
R
(
t;
s
)
is
-unif
or
mly
e
xponent
ially
stab
le
,
and
subsystem
(
A
+
(
t
)
;
B
+
(
t
)
;
)
is
controllab
le
,
then
the
system
(
A
(
t
)
;
B
(
t
)
;
)
-stabilizab
le
.
In
this
case
,
a
-stabilizing
f
eed-
bac
k
oper
ator
is
giv
en
b
y
F
(
t
)
=
F
0
(
t
)
P
(
t
)
,
where
F
0
is
a
-stabilizing
f
eedbac
k
oper
ator
f
or
(
A
+
(
t
)
;
B
+
(
t
)
;
)
.
Proof
.
Since
(
A
+
(
t
)
;
B
+
(
t
)
;
)
is
controllab
le
,
there
e
xists
a
f
eedbac
k
op
er
ator
balik
F
0
(
)
2
L
1
(
R
+
;
L
s
(
X
+
;
U
)
such
that
the
spectr
um
of
A
+
(
t
)
+
B
+
(
t
)
F
0
(
t
)
in
C
f
or
a
ll
t
0
.
W
e
choose
a
f
eedbac
k
oper
ator
F
(
)
=
(
F
0
(
)
;
0)
2
L
1
(
R
+
;
L
s
(
X
;
U
))
f
or
the
system
(
A
(
t
)
;
B
(
t
)
;
)
.
According
to
Theorem
8,
the
per
turbed
oper
ator
A
(
t
)
+
B
(
t
)
F
(
t
)
=
A
+
(
t
)
+
B
+
(
t
)
F
0
(
t
)
0
B
(
t
)
F
0
(
t
)
A
(
t
)
is
the
infinitesimal
gener
ator
of
a
C
0
-quasi
semig
roup
R
(
t;
s
)
.
Moreo
v
er
,
if
1
is
the
g
ro
wth
bound
of
R
(
t;
s
)
,
then
1
is
the
maxim
um
of
that
of
the
quasi
semig
roups
gener
ated
b
y
A
+
(
t
)
+
B
+
(
t
)
F
0
(
t
)
and
A
(
t
)
.
Theref
ore
,
1
<
,
i.e
.
the
system
(
A
(
t
)
;
B
(
t
)
;
)
is
-stabilizab
le
.
By
the
dual
proper
ty
betw
een
the
stabilizability
and
detectability
,
Theorem
11
pro
vides
the
sufficiency
f
or
the
-detectability
.
Theorem
12.
Let
(
A
(
t
)
;
;
C
(
t
))
be
the
non-autonomous
Riesz-spectr
al
system
on
the
state
space
X
where
C
(
)
has
a
finite
r
ank.
If
A
satisfies
the
spectr
um
decomposition
assumption
at
,
X
+
has
a
finite
dimension,
R
(
t;
s
)
is
-unif
or
mly
e
xponentially
stab
le
,
and
(
A
+
(
t
)
;
;
C
+
(
t
))
is
obser
v
ab
le
,
then
(
A
(
t
)
;
;
C
(
t
))
is
-detectab
le
.
In
this
case
,
-stabilizing
output
injection
oper
ator
is
giv
en
b
y
K
(
t
)
=
i
(
t
)
K
0
(
t
)
,
where
K
0
is
an
oper
ator
such
that
A
+
(
t
)
+
K
0
(
t
)
C
+
(
t
)
is
the
infinitesimal
gener
ator
of
the
-unif
or
mly
e
xponentially
stab
le
C
0
-quasi
semig
roup
and
i
(
t
)
is
an
injection
oper
ator
from
X
+
to
X
.
Proof
.
By
the
dual
concept,
w
e
ha
v
e
that
the
system
(
A
(
t
)
;
;
C
(
t
))
is
-detectability
if
and
only
if
(
A
(
t
)
;
C
(
t
)
;
)
is
-stabilizability
.
F
rom
Theorem
11,
A
satisfies
the
spectr
um
decomposition
assumption
at
.
The
corresponding
spectr
al
projection
is
giv
en
b
y
P
(
t
)
x
=
1
2
i
Z
(
I
A
(
t
))
1
xd;
f
or
all
x
2
X
.
W
e
can
choose
such
that
is
symmetr
ic
with
respect
to
the
real
axis
.
Hence
,
the
decomposition
of
(
A
(
t
)
;
C
(
t
)
;
)
is
the
adjoint
of
the
decomposition
of
(
A
(
t
)
;
;
C
(
t
))
.
By
the
dual
argument,
w
e
ha
v
e
the
required
results
.
Example
13.
Consider
the
non-autonomous
Riesz-spectr
al
system
in
Example
5.
Using
all
of
the
ag
reement
there
,
w
e
sho
w
that
the
system
is
stabilizab
le
and
detectab
le
.
W
e
ha
v
e
the
spectr
al
decomposition
of
A
(
t
)
:
A
(
t
)
x
=
a
(
t
)
1
X
n
=1
(
n
)
2
h
x;
n
i
n
f
or
x
2
D
;
where
n
(
)
=
p
2
cos(
n
)
,
and
the
f
amily
of
the
oper
ators
gener
ates
a
C
0
-quasi
semig
roup
R
(
t;
s
)
giv
en
b
y
R
(
t;
s
)
x
=
h
x;
1
i
1
+
1
X
n
=1
e
(
n
)
2
(
g
(
t
+
s
)
g
(
t
))
h
x;
n
i
n
;
T
r
ansf
er
Function,
Stabilizability
,
and
Detectability
of
Non-A
utonomous
...
(Sutr
ima)
Evaluation Warning : The document was created with Spire.PDF for Python.
3031
ISSN:
1693-6930
where
g
(
t
)
=
R
t
0
a
(
s
)
ds
.
Since
the
set
of
eigen
v
alues
of
A
has
an
upper
bound,
then
A
satisfies
the
spectr
um
assumption
at
an
y
real
.
Suppose
w
e
choose
=
2
.
In
this
case
w
e
ha
v
e
+
2
(
A
)
=
f
0
g
and
so
there
e
xists
a
closed
simple
cur
v
e
2
that
encloses
the
eigen
v
alue
0
.
F
or
an
y
x
2
X
and
t
0
,
Cauch
y’
s
Theorem
giv
es
P
2
(
t
)
x
=
1
2
i
Z
2
(
I
a
(
t
)
A
)
1
xd
=
1
2
i
Z
2
1
h
x;
1
i
d
=
h
x;
1
i
:
In
vir
tue
of
(12),
w
e
ha
v
e
that
X
+
2
has
dimension
1
.
Moreo
v
er
,
in
the
subspace
w
e
ha
v
e
(
A
+
2
(
t
)
;
B
+
2
(
t
)
;
C
+
2
(
t
))
=
(0
;
I
;
I
)
which
are
controllab
le
and
obser
v
ab
le
.
There
e
xists
F
0
(
)
2
L
1
(
R
+
;
L
s
(
X
+
2
;
U
))
such
that
the
spectr
um
of
A
+
2
(
t
)
+
B
+
2
(
t
)
F
0
(
t
)
in
C
2
f
or
all
t
0
.
W
e
can
choose
F
0
(
t
)
x
=
3
h
x;
1
i
f
or
all
x
2
X
+
2
.
Theorem
11
concludes
that
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
is
(
2)
-
stabilizab
le
with
the
f
eedbac
k
oper
ator
u
=
F
(
t
)
z
,
where
F
(
t
)
=
F
0
(
t
)
0
=
3
I
0
.
By
the
duality
,
Theorem
12
sho
ws
that
the
system
(
A
(
t
)
;
;
C
(
t
))
is
(
2)
-detectab
le
wit
h
output
injection
oper
ator
K
(
t
)
=
3
I
0
such
tha
t
K
(
t
)
y
=
3
y
1
.
The
deca
y
constants
of
the
quasi
semig
roup
gener
ated
b
y
A
(
t
)
+
B
(
t
)
F
(
t
)
and
A
(
t
)
+
L
(
t
)
C
(
t
)
are
3
.
The
f
ollo
wing
theorem
is
a
similar
result
with
Theorem
13
of
[1]
f
or
controllability
and
obser
v
ability
of
the
non-autonomous
Riesz-spectr
al
systems
on
Hilber
t
spaces
.
Theorem
14.
Let
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
be
a
non-autonomous
Riesz-spectr
al
system.
Necessar
y
and
sufficient
conditions
f
or
(
A
(
t
)
;
B
(
t
)
;
)
is
-stabilizab
le
are
that
ther
e
e
xists
an
>
0
such
that
+
(
A
)
compr
ises
at
most
finitely
man
y
eigen
v
alues
and
rank
(
h
b
1
(
t
)
;
'
n
i
;
:
:
:
;
h
b
m
(
t
)
;
'
n
i
)
=
1
;
(15)
f
or
all
n
such
that
n
2
+
(
A
)
and
t
0
.
Necessar
y
and
sufficient
conditions
f
or
(
A
(
t
)
;
;
C
(
t
))
is
-detectab
le
are
that
there
e
xists
an
>
0
such
that
+
(
A
)
compr
ises
at
most
finitely
man
y
eigen
v
alues
and
rank
(
h
n
;
c
1
(
t
)
i
;
:
:
:
;
h
n
;
c
k
(
t
)
i
)
=
1
;
(16)
f
or
all
n
such
that
n
2
+
(
A
)
and
t
0
.
Proof
.
W
e
only
need
to
pro
v
e
necessar
y
and
sufficient
conditions
f
or
-stabilizability
.
F
or
-
detectability
f
ollo
ws
the
dual
argument
of
the
system.
Sufficiency
f
or
-stabilizability
.
Since
+
(
A
)
only
contains
at
most
finitely
man
y
eigen
v
alues
of
A
,
then
X
+
has
a
finite
dimension
and
A
satisfies
the
spectr
um
decomposition
assumption
at
.
By
condition
(a)
of
Theorem
3
of
[1]
and
Cauch
y’
s
Theorem,
f
or
an
y
t
0
w
e
ha
v
e
P
(
t
)
x
=
1
a
(
t
)
X
n
2
+
h
x;
n
i
n
:
Since
A
(
t
)
and
A
ha
v
e
common
eigen
v
ectors
f
or
e
v
er
y
t
0
,
again
Theorem
3
of
[1]
giv
es
X
+
=
span
n
2
+
f
n
g
;
X
=
span
n
2
f
n
g
;
R
(
t;
s
)
=
X
n
2
e
n
(
g
(
t
+
s
)
g
(
t
)
h
x;
n
i
n
;
A
+
(
t
)
x
=
a
(
t
)
X
n
2
+
n
h
x;
n
i
n
;
and
B
+
(
t
)
u
=
X
n
2
+
h
B
(
t
)
u;
n
i
n
;
(17)
where
g
(
t
)
=
R
t
0
a
(
s
)
ds
.
This
result
sho
ws
that
R
(
t;
s
)
is
a
C
0
-quasi
semig
roup
correspond-
ing
to
the
Riesz-spectr
al
oper
ator
A
on
X
.
Consequently
,
!
0
(
R
)
<
0
and
Theorem
7
states
that
R
(
t;
s
)
is
-unif
or
mly
e
xponentially
stab
le
.
W
e
need
pro
v
e
that
(
A
+
(
t
)
;
B
+
(
t
)
;
)
is
controllab
le
.
The
reachibility
subspace
of
(
A
+
(
t
)
;
B
+
(
t
)
;
)
is
the
smallest
A
+
-in
v
ar
iant
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TELK
OMNIKA
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3032
subspace
of
X
+
which
contains
ran
B
+
(
t
)
.
In
vir
tue
of
Lemma
2.5.6
of
[2],
this
subspace
is
spanned
b
y
the
eigen
v
ectors
of
A
+
.
Hence
,
if
this
subspace
does
not
equal
with
the
state
space
X
+
,
then
there
e
xists
a
j
2
+
(
A
)
such
that
j
=
2
X
+
.
Theref
ore
,
its
bior
thogonal
element
j
is
or
thogonal
to
the
reachibility
subspace
,
in
par
ticular
h
B
(
t
)
u;
j
i
=
0
f
or
e
v
er
y
u
2
C
m
and
t
0
.
This
contr
adicts
to
(15),
and
so
(
A
+
(
t
)
;
B
+
(
t
)
;
)
is
controllab
le
.
Thus
,
Theorem
11
sho
ws
that
the
system
(
A
(
t
)
;
B
(
t
)
;
)
is
-stabilizab
le
.
Necessity
f
or
-stabilizability
.
F
rom
Def-
inition
6
and
Definition
9,
if
(
A
(
t
)
;
B
(
t
)
;
)
is
-stabilizab
le
,
then
there
e
xists
>
0
such
that
the
system
is
also
(
)
-stabilizab
le
.
In
vir
tue
of
Theorem
11,
A
satisfies
the
spectr
um
decompo-
sition
assumption
at
.
Moreo
v
er
,
the
subspace
X
+
is
R
(
t;
s
)
-in
v
ar
iant.
Lemma
2.5.8
of
[2]
giv
es
X
+
=
span
n
2
J
f
n
g
:
Since
X
+
has
a
finite
dimension,
then
J
contains
at
most
finitely
man
y
elements
.
So
the
spec-
tr
um
of
A
+
=
A
j
X
+
is
contained
in
C
+
and
the
spectr
um
of
A
=
A
j
X
is
contained
in
C
.
This
concludes
that
the
inde
x
set
J
equals
with
the
set
f
n
2
N
:
n
2
+
(
A
)
g
.
Thus
,
+
(
A
)
g
compr
ises
at
most
finitely
man
y
eigen
v
alues
.
By
(17)
and
Theorem
11,
w
e
conclude
that
(
A
+
(
t
)
;
B
+
(
t
)
;
)
is
controllab
le
.
Suppose
that
the
condition
(15)
does
not
hold.
There
e
xists
n
2
+
(
A
)
such
tha
t
h
B
(
t
)
u;
n
i
=
0
f
or
all
u
2
C
m
.
This
states
that
the
reachibility
subspace
of
(
A
+
(
t
)
;
B
+
(
t
)
;
)
does
not
equal
to
X
+
,
that
is
(
A
+
(
t
)
;
B
+
(
t
)
;
)
is
not
con-
trollab
le
.
This
giv
es
a
contr
adiction.
In
pr
actice
,
Theorem
14
is
more
applicab
le
than
Theorem
11
and
12
in
char
acter
izing
the
stabilizability
and
detectability
of
the
non-autonomous
Riesz-spectr
al
systems
.
W
e
retur
n
to
Example
13.
F
or
=
2
,
w
e
can
choose
=
1
such
that
+
3
(
A
)
=
f
0
g
and
the
corresponding
eigen
v
ector
is
0
(
)
=
1
.
In
this
case
,
it
is
ob
vious
that
Z
1
0
b
(
t
)
0
(
)
d
6
=
0
and
Z
1
0
c
(
t
)
0
(
)
d
6
=
0
;
f
or
all
t
0
,
i.e
.
the
conditions
(15)
and
(16)
are
confir
med.
Hence
,
the
system
(
A
(
t
)
;
B
(
t
)
;
C
(
t
))
are
stabilizab
le
and
detectab
le
.
3.
Conc
lusion
The
concepts
of
the
tr
ansf
er
function,
stabilizability
,
and
detectability
of
the
autonomous
systems
can
be
gener
aliz
ed
to
the
non-autonomous
Riesz-spectr
al
systems
.
The
results
are
alter
nativ
e
consider
ations
in
analyzing
the
related
control
prob
lems
.
There
are
oppor
tunities
to
gener
aliz
e
these
results
to
the
gener
ally
non-autonomous
systems
inclu
ding
the
time-dependent
domain.
Ac
kno
wledg
ement
The
authors
are
g
r
ateful
to
Research
Institution
and
Comm
unity
Ser
vice
of
Univ
ersitas
Sebelas
Maret
Sur
akar
ta
f
or
funding
and
to
the
re
vie
w
ers
f
or
helpful
comments
.
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TELK
OMNIKA
V
ol.
16,
No
.
6,
December
2018
:
3024
3033
Evaluation Warning : The document was created with Spire.PDF for Python.