1. Introduction
WepresentavariationalframeworkbasedonsesquilinearformsforGalerkin
approxima-tiontechniques forstatefeedback controlinproblemsgoverned byinnitedimensional
dynamicalsystems. Bothparab olicandsecondorderintimehyp erb olicpartial
dieren-tialequationswithunb oundedinputandunb oundedobservationop eratorsareincluded
as sp ecialcases ofourtreatment.
In thispap er we discuss the linear quadratic regulator(LQR) problemfor a class
of (essentially parab olic) unb ounded input or b oundary control problems. A
varia-tionalframeworkusing sesquilinear formsisdevelop edto treat Dirichlet andNeuman
b oundary control problems for parab olic equations and strongly damp ed elastic
sys-tems. Using such a framework,convergence of Galerkinapproximations to solutions
of Riccati equations is also established. The b oundary control problemfor parab olic
systems hasb eenstudied extensivelyoverthelast twodecades, inspired by the
mono-graph of J.L.Lions [Li] (e.g., see [Ba], [CP],[Fl1],[PS], [IP] and the references cited
there). Inaseries ofpap ers [La1],[LT1,LT2],LasieckaandTriggianiobtainexistence
andregularityresultsforsolutionstotheop eratorRiccatiequationsthatapp earinthe
linear quadraticregulator problem. Themainto olintheir treatment isthe theory of
analyticsemigroups.Theyhaverecentlydevelop edanapproximationtheoryforRiccati
solutionsinasimilarspirit[LT3].
Our approach is very close in spirit to the studies by Lions [Li] and Sorrine [So]
3 3
0
2
V H H V
V: V
H :
U V U
V:
( = )ofellipticop erators[Li],[Ta],[Sh],[W]usingco ercivesesquilinear
formsdenedon Itfollowsfrom[Ta]thatthe co ercive formdenesageneratorof
ananalyticsemigroupon Thecontrolforcesthroughb oundaryorp ointwiseactuators
are represe nted byan input sesquilinear formdened on where is theinput
space. Then theapproximation metho ds are derived by restricting these sesquilinear
formsontonitedimensionalsubspacesof
Theobjectiveofthispap eristoexploretowhatextentonecandevelopthe
sesquilin-ear form formulation to study unb ounded input control problems and to establish a
computationallyfeasibleapproximationtheoryforlinearquadraticregulatorproblems.
These investigationsweremotivatedby[BK]whereourtreatmentoriginated,by [BI1]
inwhichasimilartreatmenthasb eendevelop edwithinthecontext ofparameter
esti-mationproblems,andbythefactthatmanyb oundarycontrolproblemscanb etreated
usingthesesquilinearforms.Someofourresultscanb efoundintheexistingliterature,
buttheapproachhereisnewand,insomecases, greatlysimpliesthetechnicaldetails
required toestablishtheresults.
The results given here extend to the unb ounded case the approximation theory
develop edin[Gi],[BK],[It]inwhichtheinputandoutputop eratorsareassumedtob e
b ounded. Thecase ofunb oundedinput op eratorhas alsob een discussed in[La2]and
[IT].
Theoutlineofthe pap eris asfollows. InSection 2we discuss thebasic resultsfor
abstract parab oliccontrol systemsdescrib ed bysesquilinear formsandstate thelinear
quadraticregulatorproblemweconsiderinthispap er. Also,severalexamplesthatcan
H : mationstotheLQRproblemareformulatedandaconvergencetheoryisestablishedfor
the case where theoutputop erator app earing inthequadratic costis b oundedon
InSection 5we dealwiththecasewhen theoutputop erator isnotb ounded. Damp ed
second ordersystemsaretreated inSection6.
AfewoftheTheorems andLemmaarestatedherewithoutpro ofs. Detailedpro ofs
3
3
3
3 3
3
2 2
1
1
0
1
0
0
1
0 3
3 3
3
3
0 3
0
0 3
3 3
3
3
0
V ;V H
V V
V H
V ;V
V
V
H H
V V
V;V
V V
! !
h i 2 h i
2 2 !
j j jj jj jj jj 2
jj jj 0 jj jj 2
A2L
h0A i 2
6
jj 0A jj
jj jj
2
jj 0A jj
j 0 j
jj jj 2
jj 0A jj
j 0 j
jj jj 2
A 2L
h 0A i 2
A
6
jj 0A jj
j 0 j
jj jj 2
V H V H
V V H
V , H H , V
; V V ;
H H V V V V
: ; C ; V
: Re ; ! V
!> V;V
: ; ; ; V;
; ;
: I
!
V
: I
M
H
: I
M
V
M C=! V;V
: ; ; ; V
Re
; ;
: I
M
V:
Assume and are complex Hilb ert spaces and with continuous dense
injection. Let denote thetop ological(conjugate) dual space of . We identify
withitsdual,sothat = inaGelfandtriple[W].Thedualitypro duct
on istheuniqueextensionbycontinuityofthescalarpro duct
of denedon . Considerasesquilinearform on (i.e., : Cl )such
that
(21) ( ) for
(22) ( ) for
where 0. Itthenfollowsfrom[Ta,Lemma3.6.1]thatif ( )isdened by
(23) ( )= forall
then forRe =
(24) ( ) for
(25) ( ) for
(26) ( ) for
where =1+ . Thedualoradjointop erator ( )dened by
(27) ( )= forall
alsosatisestheestimates(2.4) (2.6). Moreover,(2.6)for impliesthatfor
=
Z
1 1
0
0
3 3
3
3
3
3
V;V V;V
V V
H
H
V
V
V
V
V
t
0 3 0
3
3
3
3
3
3 3
3 3
3
3 3 3
3
3
3
jh 0A i j jh 0A i j
jj jj
j 0 j
jj jj 2
A A f 2
j j k k 2 g
A
A
A f 2 A 2 g
A
j
j
! 2
2 A !
0
2
A
I ; ; I
M
V :
S t H ;V V
V ; k V
V;H ;V
S t V ;
: V V V:
S t H
S t V S t
S t V
: S t V S t :
t f t V
V t t z t
: z t S t S t sf s ds
z t V; V
:
d
dt
z t z t f t V :
( ) = ( )
forall
This implies (2.8). It thus follows from [Pa, Chap. 2, Theorem 5.2] (see also [Ta])
that generates ananalyticsemigroup ()on and where dom ( )=
: ( ) for all . Following conventional notation (see [Ta])
wewillnotdistinguishb etween thesemigroupsin since eachinvolveseither a
restriction orextension ofoneoftheothers. From(2.1)one canshowthat dom ( ),
thedomainoftheinnitesimalgenerator ofthesemigroup ( )denedon satises
(29) dom ( )= : =
Inaddition,wehave that generatestheadjointsemigroup ()on [Ta,Pa]and
moreoverthedual of thesemigroup ()dened on equals ( ) , therestriction
of ()on ;i.e.,
(210) ( () on ) = ( )
It follows from[Ka, Theorem IX.1.19] that if () is continuously
dif-ferentiable and = dom ( ), then for 0 the function () dened
by
(211) ()= () + ( ) ( )
satises () iscontinuouslydierentiablein ,andsatises
(212) ()= ()+ () in
3
3
3
3
3
3
3
3
3
Z
Z Z
R
Z Z
Z Z
Theorem 2.1
Theorem2.2
H H
t
V
T
V H
T
V
t
t
H
t
V
t
H
t
V
V ;V
H H
2 1
1
2
1 2
0
2
0
2
1 2
0
2
2
0
0
2
1
0
2
0
2
1
0
2
0 1
0
1
0 1 0 1 0
1
1 1
1 1
2 2 2
\
jj jj jj jj jj jj 2
jj jj jj jj jj k
! 2
2 ! 0 2
jj 0 jj jj jj
jj 0 jj jj jj
0
2
2 2L
h i 2
f 2 j j k k 2 g
H f L ;T V z
L ;T V H ;T V M
: z t M f s ds ; t ;T
: z s ds M f s ds :
t z t H
f L ;T V t S t s f s ds H
: S t s f s ds M f s ds
: S t sf sds M f s ds:
V
;
; ;
;
; ;
; V:
; Re ;
V A V;V
A ; ; ; V
A V ; k V :
For and (0 ; ), (2.12) has a unique solution
(0 ; ) (0 ; )givenby(2.11)and forsomep ositiveconstant
(213) ( ) ( + ( ) ) [0 ]
(214) ( ) ( + ( ) )
From Thm. I I I.1.2 in[Li],we have that () is continuous. Thus from
(2.11)and (2.13),for all (0 ; ), thefunction ( ) ( ) is
continuousand
(215) ( ) ( ) ( )
(216) ( ) ( ) ( )
Moreover,wehave (againfromstandardresults andarguments)eg. see[BI2]
Letusdenethesesquilinear forms and on by
( ) =
( )+ ( )
2
( ) =
( ) ( )
2
for
Then = + ,with symmetricand skew-symmetric,and ( )= ( )
for . Dene ( )by
= ( ) forall
and
Z Z
Z
3 3
3
H H
t
H V
t
H
V H
H
T
Y U
1
2 1
2
0
2 2
2
0
2
1
1
0
0
2 2
2
0
0
2
0
A
2 ! 2 \
jj jj jj jj jj jj 2
2
j j k k k k
jj jj jj jj h i
2
A
2
2L
2L 2L
2 2
: A :
f L ;T H t z t H ;T H C ;T V
M >
: z s ds z t M f s ds; t ;T :
; V
: a ; K
K > A V
U Y U U Y Y
: J ;T u;z Cz t ut dt z T ;Gz T
u L ;T U
:
d
dt
z t z t Bu t
z z H :
ut U B U;V
C V;Y G H
z H u L ;T U J u;z
(217) dom ( ) dom ( )
Thenif =0and (0 ; )in(2.11),wehave () (0 ; ) (0 ; )
and thereexistsaconstant 0such that
(218) _( ) + () ( ) [0 ]
Thecondition(2.17)issatised ifweassumethat forall
(218 ) ( )
forsome 0since inthis case dom ( )= (itisalsosatised under themilder
condition(5.9)giveninSection5).
Assume and are Hilb ert spaces ( = and = ). Consider the nite
horizon optimalcontrol problem: Minimizethequadraticfunctional
(219) (0 ; )= ( ( ) + () ) + ( ) ( )
over (0 ; )subject to
(220)
() = ()+ ()
(0) =
Here we assume () isa -valuedcontrolfunction, theinputmap ( ), the
observation map ( ), and ( )isself-adjoint,nonnegative denite. By
Theorem2.1,theminimizationproblemfor(2.19)iswell-p osed;i.e.,giveninitialdatum
and (0 ; ),thecost ( )isnite. We willanalyzethe solutionof
2 Z
1
3 Example1
n n
@
@
L
H
H 2
2
0
2 2
2 1 2
0 (0)
1
< <
5 5 5
2
j
2 2
5 5 0 5 0
2
!
h i h j i 2
A2L h0A i A f 2
jA 2 g A
; n ; n
C
:
@
@t
z a z b z cz ;
z L
a @
@
z u t
z z t L ;u u t L :
H L ; V H ;U L
V
: ; ax b x c x dx
a;b;c L a x ! >
B U H
: Bu; u; H :
V;V ; ;
V H :
Section 4.
Letusconcludethissectionbydiscussingsomemotivatingexamplesofsomeimp
or-tanceinapplications.
(Parab olicsystems withb oundarycontrol)
Letb eeitherap olygonaldomainin 3orab oundeddomainof 3
with -b oundary0. Weconsiderthecontrolproblemfor(2.19)withNeumanb oundary
control system:
(221)
= ( )+ + in
(0)= ()
= ()
where = () () = () (0) and is the normal derivative on 0 To
write this in abstract form (2.20), without loss of generality we use the real Hilb ert
spaces = () = () = (0)and deneareal valuedsesquilinear form
on (the complexicationof spacesand formsisreadilydoneintheusualway)by
(222) ( )= ( ( ) ( ) ( ) )
where () and ( ) 0 a.e. in . We consider the input map
: givenby
(223) = forall ()
As ab ove we dene ( )via = ( ) andtake dom ( )=
3
0
0
1
1 2
H H
H
V ;V H
H
H L
3
0 3
3 0 3
3
0 3 0
0 0
0 1
2
2
0
2
2 1 1
0
1
2
()
1 2 1
1 1
0
1
2
()
1
()
1
2
A A f 2 j j g
2 A A
5 5 5
1 4 1
4
!
!
h1 1i h1 1i h1 1i
2 2 2 04
2
2
h i h 04 i
b
b a
a C
: H
@
@
:
A A
: A a b c:
@
@t
z t; z t;
z t ut
U L :
V L ;H H H V ,
H, V H H H
V L V :
; ; ;
H V H L V V: R
H H H
L
: ; ; :
can readilyargue that(2.18a)and hence (2.17),holds inthecase that ^=0on 0.
Moreover, in the sp ecial case that = 0 and p ossesse s sucient smo othness (e.g.,
())itcan b eshownthat
(224) dom ( )= dom ( )= () =0
Inanycase,wehave for dom ( )that = where isformallygivenby
(225) = ( )+ +
We next consider the Dirichletb oundary control problemconsisting ofminimizing
(2.19)subjectto
( )= ( )in
(2.26)
()= ( )on0
where denotestheLaplacianand = (0) Tocastthisprobleminthecontext of
(2.20), we take = () = () = () inaGelfand triple setting
with ()thepivotspace. Wethusidentify and throughtheRiesz
mapandbutofcoursedonotidentify = ()and Asusual,thedualitypairing
is theextension bycontinuityofthe inner pro duct = from
= () ()to We recallthat =( ) denesanisometric
isomorphismfrom () onto () (see [W],[Gr]) andmoreover for ()
and () we have
Z
3
3 3
0
3 3
0
0
0
0 0
2
2
1
2
2 2 2
2 2 2
2 2 2
1
2
L
V ;V L
@
@
V ;V V ;V
L L L
L L L
L L L
H L V ;V
()
1
0 (0)
2 2 1 2
1
0
2 1 2
2 1
0
() () 0 (0)
() ()
0
(0)
1 2
1
() ()
1
0 (0)
() ()
h i
2L
h i h0 04 j i
2 2 04 \
2
h i h i
2 \
2 \
h4 i h 4 i 0h j i
h i h 4 i 0h j i
04
h 04 i h0 i h0 04 j i
h i h i h i
V L
; dx ;
B U;V
: Bu; u;
@
@
V L : L H
H H
:
@z t
@t
; z t ; But ;
z C ;T H C ;T L :
H H
z; z; z;
@
@
:
@z
@t
; z; u;
@
@
:
L
@z
@t
; z; u;
@
@
;
@z
@t
; z; Bu;
We dene on = () by
( )= =
and ( )by
(228) = ( )
forany = () Notethatfor ()we have =( ) is in ()
() so that (0) (see [W, p. 127]). Thus we may consider the abstract
system
(229)
()
+ ( ( ) )= ()
which is in theform (2.20). To see that this is an abstract formulationof (2.26),we
consider(2.26)inthecasewhere (0 ; ()) (0 ; ()) ThenbyGreen's
formula([Gr,p. 53][W])forall () ()we have
=
Using (2.26)inthisrelationshipwend
=
If we cho ose =( ) for arbitraryin (),this lastequation canb e written
as
( ) = + ( )
or,inviewof(2.27)and(2.28),
R
Example2
@
@x
@
@x
=
= d
dt
1 2 R
f 2 g
R R R
2 2 2
2
2
1
1
2
2
1 2 1 2
0
2
2
1 2
2
2
0 0
1 2
0
0 :
@
@t z t;x
@
@x
z t;x ;
z t; u t ; z t; : y t; H =
H dx y t;x z t;x:
@
@t y t;x
@
@x y t;x
@
@x
y t; u t
@
@x
y t; :
V H ; = ; H L ; = ;U ;Y L ; ; C ;
w H
:
d
dt
w t A
d
dt
w t Aw t But
:
d
dt
w t A
d
dt
w t Aw t But
wt H w V A w H :
A H >
As a sp ecial case of the system in (2.21), for the one dimensional case one can
also formulate the Dirichlet b oundary control problem as a Neumann problem. For
simplicityof discussion,we considertheheatequation
(230) ( )= ( ) on=(0 1)
with b oundary condition ( 0) = () ( 1) = 0 We dene ( ) () =
(): =0 satisfying ( )= ( ) Then(2.30)can b ewrittenas
( )= ( )
with
( 0)= ()and ( 1)=0
If we cho ose = (0 1) = (0 1) = = (0 1) and =
then this examplecan b e cast into anabstract Neumanb oundary control problemas
ab ove.
(Secondordersystems withdamping)
We considersecondorderequationsfor intheHilb ert space
(231) ( )+2 ()+ ( )= ()
(232) ()+2 ( )+ ()= ()
for () with initialdata (0) = dom( ) and (0) =
Here is a p ositive denite self-adjoint op erator on , and 0 is the damping
3
0
0 0
00 00
00 00
3 =
=
V ;V
= =
H
V V
V H
4
4
2
1 2
4
0
2 1
0
1 2
2
2
0
0 0 0 0
1 2 1 2 1 2 2 1 2 2
0 0
0
1 2 1 2
0
1 2 1 2
2 2
f 2 j g
\
0 2
2 2
h i0hh ii h i
hh ii L
h i h i h i 2
j j k k k k 2
k k 0 k k
A
: A
d
dx
L ;
A
: ;
A H ;
V H ; H ;
: A
d
dx
V :
H V H V V V
V
: ; ; ; A ; ; A ; ;
; V A V ;V
A; A; A ;A ; V
; ; ; ; V;
Re ; ;
BI1]andnumerousotherplaces, andisknownas amo delwithKelvin-Voigtdamping.
Sp ecial cases of these equationsinclude anEuler-Bernoullib eamequation inwhich
is denedby
(233) = in (0 1)
with theappropriateb oundary conditions. Ingeneral, isnotnecessarily a
dier-ential op erator. But if we consider the b oundary condition (corresp onding to hinged
ends)
(234) (0)= (1)= (0)= (1)=0
then dom( ) = (0 1) (0) = (1) = (0) = (1) = 0 and, moreover,
= (0 1) (0 1)with
(235) = for
Firstweconsiderthesystemdenedby(2.32). Let = , = ,and
dene asesquilinearform on by
(236) (( ) ( ))= +2
where istheinnerpro ductin and istheusualextensionin ( )given
by = = for . Then
( ) 2(1+ ) for =( ) =( )
and
= 3
2
4
3
5
R
Z
0 0
0 0
0
0 0
1 2
0 00 000
0
00 00
00
V V V
H V H
H
H
H
=
V ;V
A
= 1 2
2
1 2
2 2
2
1 2
2 2
0 0
0
0 2
0
1 2 2
1
0
0
2
( ) 1
0 2
1 2
k k k k k k
k k k k k k
A
0 0
A f 2 2 2 g
A
f 2 2 2 g
f 2 g
h i 2
k k j j
;
:
:
I
A A
H
: ; H V A A H :
; H V A :
A H L ;
:
V A H ; ;
A; dx ; V
dx;
A
= +
and
= +
Hence (2.1)and(2.2)aresatised forthesesquilinearform dened by(2.36),and
(237) =
0
2
generates ananalyticsemigroupon ,where
(238) dom ( )= ( ) : and +2
Notethat dom ( )canb e equivalentlydenedby
( ) : and +2 dom ( )
Wemayalsoconsiderthedierentialop erator of(2.33)on = (0 1)withthe
b oundary conditions(corresp onding toxed-free endsor acantilevered b eam)
(0)= (0)= (1)= (1)=0
Then,
= dom( ) = (0 1): (0)= (0)=0
= for
with
dom =
2
2
3
2 00
00 00 0
0 X Z p 8 < : p p p p
h i 2<
2
2
0
0
2 2
2 \ \
0 2
2
0
0 0
2 2
h i0 0 h i
0 h i h i
m
i
i i m
m i @ @x @ @t@x = = = d dt = = d dt = = d dt = = = = = = = = =1 1 0 1 2 0 0 + 0 0 + 0 0 1 2 2 0
1 1 2
1 2 1 2 2 1 2 1 2 1 1 2 1 2 1 2 2 2 1 2 2 2 1 2 1 0 0
1 4 1 4
1 2 1 2
1 2 1 1 2 1 2 2 1 2 1 2 1 2 1 2 2 2
: Bu; u x xdx u u ;:::;u U
L ; i
x
;x ;x ; ;
: w t;x w t;x ut
M t;x M t;x ut
M t;x w t;x w t;x
w A w A u
w C ;T H C ;T A C ;T A z t
A w t z t w t A w t w A
w A
:
z A z A z
z A z A z :
H H H V A A
V
:
; ; ; A ; A ;
A ; A ; :
(239) = ( ) ( ) for =col( ) =
where (0 1)represe ntsthesupp ortofan thmomentcontrol(suchcontrollersare
realizedusingpiezo ceramicpatchactuators). Forexample,if ( )isthecharacteristic
functionoftheinterval[0 ] (0 1) thenthecontrolinputwillhave theform
(240) ( ) ( )= ()
foranundamp edsystem,ormoregenerally,
( ) ( )= ()
whenthemomentisgivenby ( )= ( )+2 ( )foraKelvin-Voigt
damp edb eam.
Next we consider the system dened by (2.31). It is not dicult to show that
if (0) dom( ) and _(0) dom( ) and 0, then (2.31)has the solution
(0 ; ) (0 ; dom( )) (0 ; dom( ))(e.g.,see[CR]).Let ()=
1 () and ( ) = ()+ (). Then for (0) dom( ) and
_(0) dom( )
(241)
+ = 1
+ = 1
Let = and =dom( ) dom( ),anddene asesquilinear form
on by
(242)
(( ) ))= 1
= Z
Z
14 =
=
=
A
i 0 0
0
000 000 0
f 2 g
h i 2
k k j j
2
0 0
0
1 4 1
1 2
1
0
1 4
2
( ) 1
0 2
1
0 0
+
0 0
A
A H ;
A ; dx ; A ;
dx
H ; x x x ;x
w t;x w t;x u t ;
isthedierentialop eratorgivenby(2.33)withtheb oundaryconditions(2.34),then
dom( ) = (0 1): (0)= (1)=0
= for dom( )
and
dom
=
For this system we mayalso consider the input op erator dened by (2.39)with
(0 1). If ( )= ( )on[0 ]andzerootherwise,thenthecontrolinputis of
theform
( ) ( )= ()
2 2 Z
Z 3
3 3 3 0 3 3
Theorem 3.1
t
T
T
T
= T
T
=
L ;TY L ;TU
T T
H
T T
T
T T
2
2
0
2
2
1 2
0
1 2
0 0
2
(0 ; ) 2
(0 ; )
0 2
0
1
0 L
L 0 2
M
M 2
L M
2 L 2 L
L !
M !
L 0
M
kL M k k k kL M k
2
0 L L L L L M L M
L ;T U
L ;T Y
: u t C S t s Bu sds; t ;T ;
H L ;T Y
: t CS t ; t ;T :
B U;V C V;Y S t
T > L ;T U H
H H
: u G S T s Bu sds
: G S T :
J u;z u z u u z
z H
: u I z
FirstweformulatetheLQRproblemfor(2.19)usinganop eratortheoreticframework
as in[Ba], [Gi],[IT].Let b e the b ounded linearop eratormappingfrom (0 ; )
into (0 ; )denedby
(31) ( )()= ( ) ( ) (0 )
and b e theb ounded linearop eratormappingfrom into (0 ; )denedby
(32) ( )()= () (0 )
Boundedness of the op erators and followsfrom Theorem 2.1, the b oundedness
assumptions ( ) and ( ), and prop erties of the semigroup ().
For a given 0, dene the b ounded linear op erators : (0 ; ) and
: by
(33) = ( ) ( )
and
(34) = ( )
Thenthecostfunctional(2.19)can b ewrittenas
( )= + + + +
and usingstandard arguments(e.g.,see[PS],[IT])we have
Assume the sesquilinear form satises (2.1) and (2.2). Then for
,theoptimalcontrolthatminimizes(2.19)isgivenby
3
2
2 Remark 3.2
1
0 0 0 0
(0 ; )
2
0
(0 ; )
3 3
3 3 0
3
3
3
3
3 3
0
3 3 3
3
3
2
4 2
4
3
5 2
4 3
5
3
5 0
@
1
A
Z T
T
T
T
T
T
T
u
T
T
T T t
T T T T T
L ;TY V
t
H L ;TU
M M
L
L
L L
M
M
h i
2 L 2 L
2L
0
A A0
2
k k k k 2
M
!
k 0 k k k
: ;
I
I
: z ;z J ;T u ;z J u;z :
C H ;Y G V ;V Y
V ;V
u
: u t B t z t
t
:
d
dt
t t t t BB t C C
V z t
: CS t c H ;
W V H V
V L ;T Y
: S t sBus ds b u ;
(36) 5 =( )
0
0
+ [ ]
satises
(37) 5 = (0 ; )=min ( )
Moreover, if we assume ( ) and ( ), and either (i) is nite
dimensionalor(ii) satises(2.17),inadditionto(2.1)-(2.2),then5 ( )and
theoptimalcontrol isgivenby
(38) ()= 5 () ()
where 5 () 5 satisesthedierentialRiccatiequation
(39) 5 ()+ 5 ()+5 ( ) 5 ( ) 5 ()+ =0
forall and ()isthecorresp ondingoptimaltrajectory.
(1) The condition(i) or (ii) app earing in Theorem 3.1 is used only to
argue(e.g.,[BI2])
(310) () forall
which is (H2) of [PS] if we identify and of [PS] with the spaces and ,
resp ectively, of our notation here. The condition (H3) of [PS] followsfrom (2.4)and
(2.9). We notethat(3.10)impliestheop erator of(3.2)can readilyb e continuously
extended to (0 ; ).
From(2.15)wehave
Z
Z
Z
Z Z
3
3 3 3
3
0
3
3 3
1
3
3
0 Lemma3.3
Pro of:
0 0
0
2 2
0 0 0
0 0
2
0
2
0
2
0 2 T
T T
T
t
T
T
T
t
s
T s
T
T T
Y U
H
t
H
H
t
V
t
H H
2L 2L 2L
!
0 0 2
0 0 0 2
0 2
jj jj jj jj
k k k k 2
2
k k k k k k k k
B U;V ;C V;Y G H
t t
: t x S T t GU T;t x S s t C CU s;t xds x H ;
U s;t
: U t;s x S t s x S t Bu s xd x H ;
u
: u t B t U t; z z H :
: J u z Cz t u t dt
S t
H S t S t
H V V
S t H
S t z Me z > ; M z H
z t S t z ; z V
z t ! z s ds z s ds z :
treated usingthePritchard-Salamonframework[PS](e.g.,seeThm2.2of[IT]).
(2)(See [Ba])Inthegeneral case (i.e., ( ) ( )and ( )),
thefunction 5 ()satisestheimplicitintegralRiccatiequation
(312) 5 ( ) = ( ) ( ) + ( ) ( ) for
where theevolutionop erator ( )isdenedby
(313) ( ) = ( ) + ( ) ( ; ) for
and theoptimalcontrol isgivenby
(314) ()= 5 () ( 0) for
Nextweconsider theproblemofminimizingthequadraticcostfunctional
(315) ( ; )= ( ( ) + () )
subject to (2.20). The following lemmaplays an imp ortant role in the innite time
intervalproblemandhenceweshallsketch apro of.
Asemigroup ( )generatedviaasesquilinearform satisfying(2.1),(2.2)
is(uniformly)exp onentiallystableon ifandonlyif ()and ()areexp onentially
stable on , and .
We rst assume that () is exp onentially stable on . Then we have
() forsome 0 1and any . Using (2.2)and
(2.12)for ()= () , we obtain
1
2
() + ( ) ( ) +
1
3 3 Z
Theorem 3.4
0
2
2
0 2
( ) ( )
( )
V H
V;V V ;V
H H
H V
V
V 1
L
3
L
3 3
3
3 3
3
L
3
3
3 3 3
3 3
3
k k k k
k k k k
2 2 A A
2
k k k k k 0 k k k
2L
2L
A
2L A0
A 2L A 0
A A0 2
2L A0
z s ds
!
M
z
S t V
S t S t S t V
S t H
S t H S t V S t
V
S t
V H t> ; S t
V H t>
S t k S t k S t S
S t V H
C C H ;Y
C H ;Y Y
;B
K H ;U BK
H ;C F Y;V FC
V
BB C C x x V
V ;V BB
U t H V V
( )
1
2 +
1
2
andanapplicationofDatko'slemma[D]yieldsexp onentialstabilityof ( )on . Since
() = () ,wealsoobtainexp onentialstabilityof ()on .
Since () is also exp onentially stable on , similar arguments b eginning with
the b ound for () on yield that () is exp onentially stable on and () is
exp onentiallystableon .
Toargue theconverse statement inthelemma,supp ose that ( )is exp onentially
stable on andrecall thatfor and 0 ( ) dom ( )where dom ( )
is denseandcontinuouslyemb edded in . For and 1wehave
() () ( 1) (1)
so thatexp onentialstabilityof ()on impliesexp onentialstabilityon .
For thecase when isb ounded(i.e., ( )),wehave
Assume ( )and either isnite dimensional,or satises
(2.17), in addition to (2.1) and (2.2). If ( ) is stabilizable, i.e., there exists an
op erator ( )such that generates anexp onentiallystablesemigroup
on and( )isdetectable,i.e.,thereexists ( )suchthat generates
anexp onentiallystable semigroupon ,thenthealgebraicRiccatiequation
( 5+5 5 5+ ) =0 forall
hasauniquenonnegativesolution5 ( ), 5generatesanexp onentially
3
3
3 Pro of
0
2 2
( )
2
( ) 2
2 2
3 3
3
L L
3 3
3
3
3
3 3
3 3
3
K U
K
V
U;V H;U
H
V K
H
V
V 0
2L
h i 2
jj jj 0 jj jj jj jj jj jj
jj jj 0 jj jj
A 2
L A0
A 2 L
A0
A
A
2L A0
A0
2L
u t B U t z :
K H ;U
; ; K;B ; V
V
Re ; !
!
B K
! :
;B K
V ;U BK V
;C F Y;V
FC V
V H V
Z Z V
B H ;U BB
U t H V V BB V
U t V H V
C C V;Y
()= 5 ( )
: Notethat for ( )
( )= ( )+ forall
denesacontinuoussesquilinearformon where
( ) 1
2
( + 1
2
)
= 1
2
Thus, by Lemma 3.3, the stabilizability of ( ) implies that there exists an
( ) such that generates an exp onentially stable semigroupon , and
similarly, the detectability of ( ) implies that there exists an ( ) such
that generates an exp onentially stable semigroup on . The theorem now
followsfrom[PS,Theorems3.3and3.4andRemark3.5]. RecallthatthespacesWand
of[PS]are chosen as and , resp ectively, inourframeworkhere. Wenote that
the space of (H3) in [PS] is here taken to b e = dom ( ) = endowed with
thegraph normcorresp onding to ;thecondition(H3) followsthenfrom(2.4). Since
5 ( )itfollowsfromthediscussions inSection 2that 5 generates
an analytic semigroup () on , and . Since dom ( 5) = , the
exp onentialstabilityof ( )on impliesthaton and .
Z
3
3
3
3 2
0
0
0
0
2
N N N N
N N
N N N
N N
N N
N N
N N
N
V
N N
N N
V ;V
N
N N
N N
N
H
t
N
V
!
h0 i 2
2 2L
2L
h i h i 2
2
k 0 k ! !1
x
2 2 2
2
h i h i 2
0
k k ! k k ! !1
4.1 ConvergenceofGalerkinapproximations
Lemma 4.1
V V H A V V
: A ; ; ; V ;
A V V B U;V
B U;V
: B u; u;B V ;
C C V
H
V V
N
f L ;T V z H z t V
z z t V ;t
:
d
dt
z t ; z t ; f t ; V
z P z ;
e t z t z t
e t e s ds N
Weturnnowtotheapproximationresultsthatarethemainfo cusofthispap er. Let
b e asequence ofnite dimensionalsubspaces of . Dene :
by
(41) = ( ) forall
that is, is dened viarestriction of to . For given ( ), we
dene ( )by
(42) = forall
and denotestherestrictionof onto . Weassumetheapproximationcondition:
( 1)
Forany ,thereexists asequence in such that
0as .
Thefollowinglemmaisstandardintheliterature(e.g.,seeChapterI I IinLions[Li]).
Supp ose(H1)issatisedandlet (0 ; )and . If ()
is denedby(2.11)with = and () 0satises
(43) ( ) + ( () )= ( ) forall
(0)=
then theerror function ()= () ()satises
Z
N N
3
3
3 1
2
1
2 3
3
0
3
0
0
0
3
0 0
3
0 Lemma 4.2
Pro of
0
0 ( )
1
1
0
1
1
0 1
1 1
1
N N
N N N N
N N
V ;V
N
N tA N
t
t sA N
N
N N
V
V
N N
H H
N N
V H
N N
N
N N
N N N N
N N
V ;V
N 2
2 h i h i 2
2L h i
h i 2
! 2
2
k 0 k
k k
k 0 k
j 0 j k k
k 0 k
j 0 j k k
2
0 ! 0
2 2
0 2
h i h i 2
P H V
H
P V P ; ; V :
P V ;V ;
; V z t
z t e P z e P f s ds:
P H H
Re V
: I A P
!
: I A P
M
: I A P
M
M M=! M M! =!
V
I A P I A V:
V
V u V V
Re ; u I A P V
u ; u ; ; V :
Inthis lemma denotesthe usualorthogonal projectionof onto ; i.e.,for
and = forall
This projectioncanreadilyb eextendedto ( )byreplacing inthe
denitionby forall . Thenthesolution ()of(4.3)canb ewritten
as
( )= + ( )
Itfollowsfrom(H1)that in forall . Inaddition,wehave
For and
(44) ( )
(45) ( )
(46) ( )
where =1+ and =((1+ ) ) .
Moreover,forevery
( ) ( ) in
: First we observe that thesesquilinear form restricted to denes a
con-tinuoussesquilinear formon satisfying(2.2)forall . Hencefor and
=( ) existsand satises
1
3 0
3
3 1
2
1
2 3
3
3
3 N
N
N N
N
N Lemma 4.3
8
>
<
>
:
8
>
<
>
:
V ;V
N N N
N N N
V
N N N N
N N
N N N
N N
V
N
V
tA N
H
tA N
H
tA N
V H
tA N
V H
tA N
V
tA N
V
2 0A 2
h i h i 2
h 0 i 0 2
2 k 0 k ! !1
h 0 i 0
h 0 i 0
0 2
k 0 k k 0 k
2
k 0 k ! !1
k 0 k ! !1
2
k k k k k k k k
k 0 k ! !1
k 0 k ! !1
V ; u I V
u; u; ; V;
u u ; u u ; V :
u V u u N
u u ; u u ;
u u; u u; :
u u V Re
u u
c
! u u
H
i
e P S t N
e P S t N
t H
ii e P
c
t
; e P
c
t
t>
iii
e P S t N ;
e P S t N :
3.6.1of[Ta]. Sinceforall =( ) satises
+ ( )= forall
weobtain
+ ( )=0 forall
By (H1),there existsasequence ^ such that ^ 0as ,and
^ + (^ )
= ^ + (^ )
Cho osing =^ ,we thenndfrom(2.1)and (2.2)that for ,
^ ^
forsomep ositiveconstantc. Fromthisthedesired convergencefollowsimmediately.
Using the Trotter-Kato theorem [Pa] and the estimates (4.5) and (4.6) one can
readilyestablishthefollowingresults (see [BI1]forpro ofs).
Forall
()
( ) 0 as
() 0 as
where theconvergenceisuniformonb ounded -intervals. Moreover,forall
( )
and for 0
( )
() 0 as
R N 3
3
3
3 3
3
3
3
2
2
2
0
2 2
0 N
N
N
N tA N
N N
N
N
t
N
Y U
N N
N N N N
N N
2L
M !M 2
M !M 2
2 M
M 2L
M 2
M M
x
k k k k
h i
Lemma 4.4
4.2 Convergence ofRiccati Solutions
C H ;Y
L ;T Y V
y y V y L ;T Y
V ; V
V ;L ;T Y
t Ce P V :
N V
:
J ;T u Cz t u t dt
Gz T ;z T
:
d
dt
z t A z t B u t ; t>
z P z :
corresp onding toTheorem2.2.
If ( )andthesesquilinearform satises(2.17),inadditionto
(2.1)and(2.2),then
in (0 ; ) forall
and
in forall (0 ; )
where for isgivenbythe continuousextensionto (see (3.10))of the
op erator in(3.2)and ( (0 ; ))isdenedby
( )()= for
Theargumentsb ehindtheseresultsaretediousbutstraightforwardonceoneusesthe
estimatesof Theorems2.1and2.2(which requires(2.17))to establishuniformb ounds
for and . Thedetails canb e foundin[BI2].
We rstconsider thenitehorizon optimalcontrol problemfor(2.19)-(2.20).
Cor-resp onding to agivenapproximationscheme as dened via (4.1), (4.2), we formulate
the thapproximateproblemin : Minimize
(48)
(0 ; ) = ( ( ) + () )
+ ( ) ( )
subject to
(49)
() = ( )+ () 0
2 0 3
3
3 3 3
3
3 A
2L 2L 2L
2
0
2
2
0
L M L M
2L Theorem4.5
Pro of:
Theorem4.6
N
T
T
N
T
N
N
T
N N
T
N
T
N N
T
N N N
T
N N
N
T
N N N
T N
T
N
T
N
T
N N
T N
N N N
T N
T N
T N
T
N
T
B U;V ;C V;Y G H
u N
u
L ;T U z H t ;t T
V
:
d
dt
t A t t A t B B t C C
T P GP H t P t
H t ;T u t ;t T
: u t B t z t :
; ; ; ;u ;
C H ;Y Y
4.1-4.3inthegeneralcase ofunb oundedoutput.
Supp osethesesquilinearform satises(2.1)and(2.2)and isdened
by(2.3), ( ) ( )and ( )isnonnegative. Supp osethe
approx-imationscheme satises (H1). Then theoptimalcontrol to the th approximate
problemfor (4.8)-(4.9)converges strongly to , the optimalcontrol for (2.19)-(2.20)
in (0 ; )for all . Moreover,if 5 () isthe solutionto the Riccati
equation in :
(410) 5 ()+ 5 ()+5 () 5 () 5 ()+ =0
and5 ( )= ,thenforall ,5 () convergesstronglyto5 () in
,uniformlyin [0 ],wheretheoptimalcontrol () is givenby
(411) ()= 5 () ()
Theargumentsare quitestraightforwardandare similartothose givenin
[IT].Wedeneforthenitedimensionalproblemanalogues 5
oftheop eratorsin(3.1)-(3.6). UseofLemmas4.1and4.3thenyieldatoncethatthese
op eratorsconvergeintheappropriatetop ologiestothecorresp ondingop eratorsin
(3.1)-(3.6).
Under the stronger assumptions (including b ounded output) in Theorem 3.1, we
obtain stronger convergence prop erties for 5 . The arguments are quite similar to
those just outlined, except we use the stronger results for the extended op erators in
Lemma4.4inthiscase.
1
3 3
0 0
3
3
3 N N N
N
K
N
K
1 1
0 0
( )
1
2 2
2 2
Lemma4.7
Pro of: N
T
T
N N
tA B K N
H
! t
H
N
N
K
K
K
K H V
V K
H
K
N
K
N N N
N
K
K
N
tA N
tA N L
!
A
2L
k k k k
A 2L
A0
h i 2
2
0k kk kk k k k
k k 0 k k
k k k k 0
h0 i 2
!
!
t t V ;V ;T
i V H
;B
K V ;U M ! N
N N N
e P M e ; t> :
;B K H ;U
BK H K
K V
V
: ; ; K;B ; V
V
:
Re ; Re ; K B
!
B K = ! A A B K
A ; ; ; V
e P T t H
e P T t H
Then5 ()converges to5 ()in ( ),uniformlyon[0 ].
We turnnext totheinnitehorizonprobleminvolving(3.15). Weassume
(H2) theinjection : iscompact.
Thenwe havetheuniformstabilizabilityresult whichisthe unb oundedinput
ana-logueof thefundamentalresultsof[BK].
Supp ose( )isstabilizableand(H2)holds.Thenthereexistsasequence
of op erators ( )andp ositiveconstants 1and indep endentof
and ap ositiveinteger such that forall
0
Since ( ) is stabilizable, there exists an op erator ( )such that
generates auniformlyexp onentially stable semigroupon . Let b e the
restriction of onto . Consider,asinthepro ofofTheorem3.4,thecorresp onding
sesquilinear form on
(412) ( )= ( )+ for
Then is continuousandfor
(413)
( ) ( )
2
where = + 2 . Thus,Lemma4.3appliesto = where
= ( )for all ,sothat
() in
3 3 3
3
1
2
0
2
L
L
3
3
0 0
Theorem 4.8
N
K
N
K
N
K
N
K
N N N N 1
( )
0
0 1
2
( )
2 2
( 5 )
2 tA N
V H
N
tA N
tA N
H
t A N
H
N
N
N N N N N N N N N N
tA B B N
H
!t
H
k k k k
A0
[ 0
k 0 k ! !1
k k k k
2L
2L
A A
2 L
0
k k k k
e P
M
t
; t> ;
T t ; t BK i
t> S e P T t B
H B H
e P T t N :
T t ;t t >
T t N e P <
C H ;Y
B U;V
;B ;C N
V ;V
V
: A A B B C C ;
M ! >
: e P M e ;
~
0
where ( ) 0 is thesemigroup generated by . Since the emb edding is
compact,foraxed 0theset = ( ()) isrelativelycompactin
,where istheunitsphere of . ItthenfollowsfromProp osition3.7in[Ch,p.126]
that
( ) 0as
Since () 0 is uniformly exp onentially stable, there exists a 0 such that
( ) . Hencefor sucientlylarge 1fromwhichthedesired
resultsfollowsusingstandardsemigrouparguments(e.g.,seethepro ofofTheorem3.1.1
in[T]).
CombiningLemma4.7withargumentsin[BK]and[IT],oneobtainsthemainresult
ofthissectionconcerningconvergencefortheb oundedoutput ( ),unb ounded
input, ( ),case.
Assume the conditions of Theorem 4.6 hold and (H1) and (H2) are
satised. Supp ose( )isstabilizableand( )isdetectable. Thenfor suciently
large, there exists a unique nonnegative self-adjoint solution 5 ( ) to the
algebraicRiccati equationin
(414) 5 +5 5 5 + =0
there existsconstants 1and 0such that
1
1 Pro of:
3
3
3
3
3 N
K
N
K
N N N
3
3
L
3
0
0
3
0 ( )
0 0
2
1 2
1 2
2
0
( )
! 2 k 0
k ! !1 A0
2
0 1 0 k k
! !1
k k k k
k k k k
2
k k
N N
s
N N N
H;U
N N
N N
H
K
N
tA N
V
= w t
H
tA N
H
= w t
V
N N
V
tA G C N
P V V B P
B N BB
H H
J B z z J u z " N
"N N
K
: e P
M
t
e
: e P
M
t
e
V
J u
N N > N
N
e P
and 5 5 in for every . Moreover, we have 5
5 0as . The feedback systemop erator 5 (i.e.,the
approximatefeedbackcontrolsintheoriginalinnitedimensionalsystem)generates an
exp onentiallystableanalyticsemigroupon andforevery
( 5 (); ) ( ; ) ( )
where ( ) 0as .
Theargumentsare essentially a rep eat of those forTheorems 2.2 and 3.1
of [BK] and Theorem 2.6 of [IT]where we use theappropriate norms and top ologies
at each step. Forthesake ofcompleteness we giveabriefoutline,referringthe reader
to [BI2]fordetails. Werstconsider and asinLemma4.7. Usingtheinverse
Laplace transform(see [T] or [Sh])to obtainarepresentation foranalytic semigroups
intermsofresolventsand standardcontour arguments(againsee[Sh])alongwith the
b ounds(4.6),one readilyobtainsb ounds
(416)
(417)
for . These canthenb eusedtoargue
( ; )
for and 0indep endentof . This isthe result of Step1inthe pro of of
Theorem2.6intheApp endixof[IT]andguarantees (Theorem3.3of[PS])existence of
thedesired solutionsto (4.14)for sucientlylarge.
Detectabilityand argumentssimilartothoseinLemma4.7 yieldb oundssimilarto
w t
2
2
2
2 3
3
3
3
3
0
L
0
L
0
L
3 3
3
L
3
L
3
L 1 2
( )
2
( )
2
( )
( ) ( )
( ) M
t
e :
P P
M
T e
M
T
e ;
P i H V
B P B B P
B P
N
k k
k 0 k
k 0 k
k 0 k ! !
k 0 k k 0 k
k kk 0 k !
!1
=
V
N N N
T N
V ;V
wT
T
V ;V
w T
N N
V ;V
N N N
H;U
N N
H;U
N N
H;V
Young'sinequalityandDatko'slemma),one canobtaintheb ound(4.15). Infact,one
alsoobtainsab oundinwhich therightsideof(4.15)isreplacedby
Thenwithstandard estimates(see theApp endix in[BK]and(A.6),(A.7)of[IT])
5 5 (0)
5 5 (0)
theconvergenceofLemma4.6andthetriangleinequality,weareabletoconcludethat
5 5 0. Moreover, since theinjection : is compact,we
nd
5 5 (5 5)
5 5 0
Z
3
3 3
3
2 2
2 2 0 0
3 0 3 0 0 0 0 3 0 L 0 1 0 0 0 0 0 0 0 0 2 1 2 0 1 1 2 1 1 1 1 0 2 0 2 0 + 0 1 ( ) + 0 1 u H H H V = = V H
V V V
V V V V V U V V
2L 2L
f 2 j j k k 2 g
h i h i 2
h i k k 2
k k k k k k k k k k k k
k k k k k k k k
k k k k
2 A
A f j 0 j g[
k 0A k
j 0 j
2
0A
C V;Y B U;V A
H
:
A u V u;v k v v V ;
A u;v u;v u;v u;v V;
u;u u;u ! u u V:
A V ;V
V V;H V;V H
:
c c
c c ;
B b < b> :
sp > ;M > U
< U I M I
t S t dt
In thissection we consider the case ( ) and ( ). Let b e a
self-adjointop erator on denedby
(51)
dom( ) = : ( ) forall and
= ( )+ forall
where isthesymmetric partof (see Theorem2.2)and
( )+ for
Let3= withdom(3)= andfor0 1 denote(see[LM])theintermediate
space: =[ ] =dom(3 ). Thenwe have[ ] = and theinterp olation
inequality
(52)
=
where = . Weassumethroughoutthissection
(H3) with 1forsome 0andlet =
Since Cl isnotin ( ),thereexist 0 0and ,aneighb orho o dof such
that
( ) 6 = :0 arg( )
2 + and ( ) 1+ for 6
(e.g.,see[Ta]). Itthenfollowsfrom[Pa,p.69-75]that
( ) =
1
0( )
R R R 1 0 0 1 0 3 3 3 3 3 3 3 Theorem 5.1 1
( ) 1
( )
1
1
0(1 ) 0
0(1 ) 0 1 0(1 ) 0 + 1 0 1 1 1 ( ) ( ) L 0 L 0 3 3 0 0 1 0 3 0 1
0 3 3
0
0 1
0 0 0
3 0
3 3 3 0
3 3 0 3
L
0
3
1 2 A
k k
k k
2 A
k 0A k k k
k k k k
k k
k 0A k k k 2
0A
0A 2L
1
1
k k
k 0A k k k
V t V ;V M t t V V V c V V cM t V
V V
T T T T V t s
T V V
S t e S t ;t
> M
:
S t M e
S t e :
x V
I x t S t x dt
t S t x S t x
t e x dt:
<< M >
I x M x x V:
V V V V I V
: B I V ;U :
G U t;s
T > M >
U t;s Me
< M
: I t x M x t T:
where 0()denotes thegammafunctionand ()= ( ) 0. Since ( ),
there existconstants 0and 1suchthat
(53)
()
()
Thenfor =dom ( )
( ) ()
() ( )
Hence if0 ,thenthereexists 0suchthat
( ) forall
Since is dense in and is closed in , this implies dom(( ) )
and therefore
(54) ( ) ( )
Under the assumption(H3) using the samearguments as in Theorem 1 [Fl2],we
have thefollowingresultsonthesolution5 ()totheRiccatiequation (3.12).
Assume(H3) holds. Let5 ()b e the solutionto theintegral Riccati
equation (3.12) with =0 and ( ) b e the evolution op erator dened by (3.13).
Thenforeach 0,thereexists aconstant 0such that
( )
and for 1,thereexistsaconstant suchthat
Z
2
2
Theorem5.2
3 3 3
3 3
3 3
1
3 3 1 3
1
1 1
3 1 3 1
T T T
T
T
T
t
T T
H
T
d
dt
T T T
T T
H T
T
T w
H
2 2 1 1
2L
2
0 0 0
2 A ! 2
h i hA i h A i
0h i h i
2 A
0 !1 L
2
0A ! 0A
h A i hA i
0h i h i
2 A
H z H u L ; U J u z <
M;M T
: B t V;H B t t B:
x V; t x;t T
: a t x S s t C C sBB s S s t xds:
x t S t x V L ;T V
S t Ax t x;t T
t x;y x; t y t x; y
B t x;B t y Cx;Cy
x;y T
s T H
V <
I I V :
:
x; y x; y
B x;B y Cx;Cy
x;y
( 4) forany thereexistsacontrol (0 ; )such that ( ; ) ,
then theconstants inab oveareuniformin .
ByTheorem5.1 and(5.4),
(56) 5 ( ) ( ) and( 5 ()) =5 ()
Nowitisnotdiculttoargue(e.g.,see[Fl1])thatfor 5 ( ) isgivenby
(57 ) 5 () = ( )( 5 ( ) 5 ( )) ( )
Since for dom ( ), () is strongly dierentiable in (0 ; ) with
derivative () (see Theorem 2.1), 5 () satises the dierential Riccati
equation
5 () + 5 () + 5 ( )
5 () 5 () + =0
forall dom ( )and 5 ( )=0. Thefollowingtheoremfollowsfrom(5.5).
If(H3),(H4)holdandlet5 = lim5 (0)as in ( ),then
for and 1
( ) 5 (0) ( ) 5 in
Moreover,if(H2)issatised,theconvergenceofab oveb ecomesstrongand5 satises
thealgebraicRiccatiequation
(58)
5 + 5
5 5 + =0
1
1 1
3
3 3
1
1 1 3 1 3 1
3
3
3
3 3
1
V V
T
T
d
dt
T T T
T T
T
N N N N
N Theorem5.3
Theorem 5.4 2L
j j k k k k
2
2 2 \
h i
h i0h i
2
0 2L
h i
0h i 2
A A
2
L A0
0 2
2L V
: ; M
M > < ; V;
t ;t T
G x V t x C ;T V C ;T V
:
t x;y x; t y y ; t x
B t x;B t y Cx;Cy
x;y V:
s V
:
x; y y ; x B x;B y
Cx;Cy x;y V:
;B ;C
V BB T t ;t H
u
u t B T t z; t z H :
V V ;A ;B ;C
P and 5 ( );i.e.,
Assume thatthesesquilinearform satises
(59) ( )
~
forsome ~
0and0
1forall inadditionto (2.1)and(2.2)andthat
(H2) and (H3)hold. Let5 () b e thesolutionto theintegral Riccatiequation
(3.12)with =0. Then, foreach , 5 () ([0 ]; ) ([0 ]; )and
satisestheRiccatiequation
(510)
5 () = ( 5 ( ) )+ ( 5 () )
+ 5 () 5 ()
forall
Moreover,if(H4) holds,then 5 (= lim5 (0)) ( )and satisesthealgebraic
Riccatiequation
(511)
( 5 ) + ( 5 )+ 5 5
=0 forall
Thenwe havetheoptimalfeedbacksolution:
Assumethe sesquilinearform satises(5.9)inaddition to(2.1)and
(2.2),and (H2)and(H4)hold. Then,if( )isstabilizableand( )isdetectable,
thenthealgebraicRiccatiequation(5.11)hasauniquenon-negativesolution5=5
( ) and 5 generates a stable analytic semigroup () 0 on . The
optimalcontrol for(3.15)isgivenby
()= 5 () 0 for
NextweconsidertheGalerkinapproximationof5 ( ). Let
N N N N
3 3
3
3
3
3 1
0
1
0
( 5 ) ~
Theorem 5.5
3 3
0
3 0
3 3
A 0 0
N
V V
N
N
N N
V V
N N N
N N N
N
V V
N
V V
V V
N
N N
N
N N N N
N
t B B N
H
! t
H
k k k k 2
! 2
! 2
k 0 k
j 0 j k k
2 0
h i h i 2
j jkh ik k k k k k k k k
k k k k
2
2 L \L
h i
0h i 2
k k k k
H c> P c V:
:
P V V
P V V ;
Re
: I A P
cM
:
V u I A P
u ; u ; ; V :
u ; c u
M
P V
N
V H
V
:
x; y y ; x B x;B y
Cx;Cy x;y V ;
M !>
e P Me ; t :
( 5) thereexistsaconstant~ 0suchthat ~ forall
Whiletheresults b elowcanb e shown bysomeelab orateargumentswithoutassuming
(H5), the condition (H5) is a standard assumptionin nite element approximations
(e.g.,see[BA]).Thenif(H1)and(H5)aresatised,then
(512)
in for and
in for
and moreoverfor
(513) ( )
~
Infact,iffor , =( ) ,then
+ ( )= forall
Thus,
+
and cho osing = forall ,thisinequalityyields(5.13).
Supp ose theassumptionsinTheorem 5.4hold andthe approximation
conditions (H1) and (H5) are satised. Then, for suciently large, there exists
a unique non-negative solution 5 = 5 ( ) ( ) to the algebraic Riccati
equation in
(514)
( 5 ) + ( 5 )+ 5 5
=0 forall
and thereexistconstants ~
1and ~ 0such that
~
3 1
1
3 3 1 1
! 2
! 2
! L
A0 N N
s
N N w
N N s
N N
P H H
P V V:
B P B V;U
N BB
H
5 5 in foreach
5 5 in foreach
and 5 5 in ( ) where 5 is the unique non-negative solution
to (5.11). Finally, for suciently large, 5 generates astable analytic
3
3
p
3
3
3 3
0 0
0 0
0
V ;V
V ;V
V
! !
2L 2L
2 2
k k 2 2
2
0
A
A
0 0
A f 2 2 2 g
0 0
0
2
2
0 0 0
0
0
0
0 0 0
0
0
0
0 0 0
0
0
0 0 0 0
1 2 1 2 2 1 1 2 2 2
0 0
1 2 2 0 0 1 0 2 0
V , H , V
:
d
dt
w t D d
dt
w t A w t B ut V
B U;V A ;D V ;V
: a ; <A ; >
: b ; <D ; >
; V a b V V :
a V
! b V
a ;: V V V H V H
V V
: ; ; ; a ; a ; b ; :
H
:
I
A D
; H V A D H :
Let b e a Gelfand triple as discussed inSection 2. We consider
general secondordersystemsinthecontextofsesquilinearformsusingtheapproachin
Section 3of[BI1]. Considerthenthesecondordersystem:
(61) ()+ ( )+ ( )= () in
where ( )and ( )aredened by
(62) ( )=
(63) ( )=
for and and arecontinuoussymmetricsesquilinear formson We
assumethat is -co ercive(i.e. satisesanequalityoftheform(2.2)whereinwithout
loss of generality we take = 1 and = 0) and is nonnegative. Thus can b e
equipp edwith theequivalentnorm = ( ) Let = , =
and denethesesquilinear form on by
(64) (( ) ( ))= ( )+ ( )+ ( )
We again have asolution semigroupon for thesystems (6.1)written in rstorder
form(e.g.,see[BI1]). Thegenerator inthiscase isgivenby
(65) =
0
withdom( )= ( ) : and + Thesystemdescrib ed
by(6.1)-(6.5)requiressomeseparateanalysissince someofconditionsassumedin
2
2
3 3
Z
Z t
C ;TV L ;TH
t
V L ;tY
0
0
0
0
0
(0 ; ) (0 ; )
0
(0 ; )
2
0
A
x
A
A
k 0 k k k
2L
k 0 k k k
2 2
6.1 Case ofBounded
Lemma 6.1
B ;B :
C
C H b
V S t ;t
b V M
T
S t s f s ds M f :
C H ;Y
c>
: S t s C y sds c y
t ;T y L ;T Y
b
V
conditionson and =col[0 ]
It hasb een shown in[BI1]that generates a -semigroupon and that if is
-co ercive,then generates ananalyticsemigroup () 0. Infact,thelattercan
b eshownreadilybyobservingthatthesesquilinearform denedby(6.4)satisesthe
conditions (2.1)-(2.2). Moreover, we actually have from standard estimates (e.g., see
[BIW])
Assume that is -co ercive. Then there exists a p ositive constant
(whichdep ends onlyon )such that
( ) ( )
ItfollowsdirectlyfromLemma6.1thatforany ( ),thefollowingcondition
holds: there exists 0suchthat
(66) ( ) ( )
forall [0 ]and (0 ; ). Fromduality(see[PS])itfollowsthenthat(3.10)
holds and indeedthe conditionsof[PS] aresatised (see Remark3.2(1)). Thusif is
-co ercive,wehave that theconclusions ofTheorems3.1 and3.4hold forthesecond
ordersystemproblemgovernedby(6.1)-(6.5). Moreover,itisnotdiculttoshowthat
theconclusionsofLemma4.4fortheGalerkinapproximationholdsforthiscase. Thus
the conclusions of Theorem4.6 appliesto thiscase for approximationsof solutionsto
3
Z
N
N
N 0
3
L
L
0
0 0
0 0
0 0
Lemma 6.2
Pro of:
0 0
0 0
2 2
0
1
0
0 0 0 0
0
0
( )
( )
( + )
1 2
2
2 2
2
0
1
2 2R
0 k k 0 k k
2 2L
2R
k k
k k
2
k k 0 k k
0
V H
V ;V
H
t
" "
N N
H
"t
N A t N
V H
tA N
tA N t N N
V
H
V H b a b;
> b V V
b ;
V A D V D V ;V
<D ; > b ; : M
S t Me ; t
"> N N N
S t P Me ; t
M > N S t e A
z ; V
z;z
:
! e V
e P
i
e I A P d
to obtainTheorem 4.8 and Lemma4.7 we assumed (H2): is compactly emb edded
into . Thiscondition(H2)isnottrueforthesecondordersystemproblemformulated
here. Butwehave aresult corresp ondingtoLemma4.7for theseproblems.
Assumethat iscompactlyemb eddedinto and = +
forsome
0where thecontinuoussesquilinearform
on satisesforsome
Re
( )
2
forall ,and
isacompactop eratoron with
( )dened by
=
( ) Thenifforsome and 1
() 0
then forany 0thereexists aninteger such thatfor
()
~
0
for someconstant ~
0indep endent of , where ( )= with dened as
in(4.1).
From(6.4)we havefor =( )
Re ( )
2
Thus satises (2.1)-(2.2)with = and thesemigroup on is represente d
by
= 1
2
2
0 0
0 0 0 0
k 0 k f g
0 2 2 0 0 0 2 0 3 0 0 0 0 0 0 0 0 0 0 0 N N
N N N N
V N
H
N N N N
N N
N N N N
N N N N N
V N N H N V N H
N N N
N
N
N N 1
1 2 0
2 0 2 0 1
2 1
2
0 0 1 0
0
0 0 0
2 1 0 1 0 0 1 1 0 0 +1 1 0 +1 1 0 0 1 1 2 2 0 1 0 1 0 1 0 1 0 2 1 0 0 2 0 1 0 1 0 1
I A P "
N:
: z z f V
: z D z A z g H :
z z f:
D A z g f D f V ;
D A D :
I A
A D z
f
A
g f D f :
>; I A A D
I A z ;z P f;P g
: I A
A D z
f
A
g f D f
z z f f P f;g P g P P
V V H ; z V
I A A D z
itsucestoarguethat ( ) isuniformlyb oundedon Re + and
in Considertheresolventequation
(67) =
(68) + + =
From(6.7), = Itthusfollowsfrom(6.8)that
( + + ) = + + in
where bytheassumption = + Equivalently, ( + +1 + +1 ) = +1 + +1 ( + + )
Thus,ifRe then( + +
) existsandisb ounded. Similarly
wehave
( )( )=( )
is equivalentto
(69) ( + +1 ( ) + +1 ( ) ) = +1 + ( ) +1 ( + + )
and = where = = and and aretheorthogonal
projectionsonto of and resp ectively. Herefor
0 0 0 0 0 2 0 0 6.2 Cases ofUnb ounded
0 0 0 0 0 0 0 0 0 L 0 0 L
0 0 0
0 3 N N N H N N N
N N N
V N N H H N N H N
N N N
V " " N N U V T 0 0
k 0 k k 0 k !
!1
f g
0
f g
A
2L
x
2L
k k k k 2
2
2 L \L !
2 2 1 0 0 0 1 2 1 0 0 1 1 1 0 0 0 1 0 1 0 1 0 1 0
0 0 0
0 1 1 0 ( ) 0 1 0 1 0 0 ( ) +1 0 1 +1 0 1 0 1 1 0 0 0 0 0 0 0 I A A D z
A A P z
A D A D z :
A D P A D V H
A P A A D A D
N : N I A A D P
> :
N N N I A P
" ;
B ;B ;
C H ;Y :
C
C V;Y <
: B b V
V V ;H ;
V V;H V V ;
H V;V < V
V =( + +1 + +1 ) + +1 ( ( ) ) + +1 ( ( ) )
Since( )
=
and iscompactlyemb eddedinto
( ) and ( )
0
as Hence for sucientlylarge( + ( ) + ( )
)
exists and is uniformly b ounded in any compact set in Re It thus follows
from(6.9)that thereexists aninteger suchthat for ( ) is
uniformlyb ounded on Re + whichcompletesthepro of.
It thus follows from Lemma 6.2 that under the assumptions in Lemma 6.2, the
assumptions(H1) andstabilizabilityand detectabilityof Theorem4.8,theconclusions
of Theorem 4.8 hold for the case when is dened by (6.5), = col[0 ] and
( )
For thecase when ( )weassumethatthere exists0 1such that
(69) for
where istheintermediatespace=[ ] 0 1denedasinSection5. Since
= [ ] = 0 1 the assumption(H3) is satised. Assuming the
nite cost condition (H4), it followsfrom Theorem 5.2 that there exists an op erator
5 ( ) ( ) (for all0 1) such that 5 (0) 5 weakly in for
0 T
H
q
3 3
3 3 3
3
3
3
3
3 3
3 3
3
f g2
! 2
A A0
0
2
A A
A
0
0
2
0 0 0
0 0 0
0 0 0
2 2
h i j j 2
P V V P H H B P B
P H H
B B U V
: BB C C
; a ; a ;
; ; ; V
I
A D
;
V H
A B B Q
: A D A B B Q
A D D B B Q
Q C C
Q Q
Q Q
V H H V H
H A ; ; ; H
2 0 2 0 2
2 0
1 2 1 1 2
1 2 1 2
0
0
11 12
21 22
0 0
21 12 0 12 0
0
21 11
0 11
0
21 22 0 22 0
0
21 21
0 12 21
0
22 22 0 22 0
0
22 22
11 12
21 22
0 0 0 0
0 1 1 2
2
1 2
notvalid. However, = iscompactlyemb edded into = and =
where is theprojectionof onto thesubspace 0 . Thisfact canb e usedto
showthat 5 (0) 5 strongly in foreach and that 5 satises the
algebraicRiccati equation
(610) 5+5 5 5+ =0
inthesenseofTheorem5.2(i.e.,see(5.8)).
Thecondition(5.9)isnotsatisedforthesecondordersystemdenedby(6.4)-(6.5)
since
( )= ( )+ ( )
for =( ) =( ) . Hence,Theorems5.3-5.5cannotb eapplieddirectly
to the second order system (6.1)-(6.3). In what follows we employ the structure of
solutionsto theRiccatiequation toestablishthe results thatcorresp ond toTheorems
5.3-5.5forthesecondordersystem. Sincetheadjointop erator of isgivenby
= 0
theRiccatiequation(6.10)canb ewrittenintermsoftheop eratormatrix5=
5 5
5 5
on as
5 5 5 5 + =0
(611) 5 5 5 5 5 + =0
5 +5 5 5 5 5 + =0
where = = on . Since5isself-adjointon =
V V
3 3
0
0
3 3
0
0
0
3 3
0 0
0 0
1
2
1
2
1
2 1
2
1
2
1
2
1
2
1
2
1
2
1
2
1
2
2L 2L 2L
2L 0 2L
j j k k k k
2 2 L
0 0 0
0 2L 2L
2 L 2 L \L
21 0 0
21
0 12 0 0
11
0 11 0
0
22 0 21 0 0 22 0 12 21 22 22 0
0
0 0
0 0 0
0 22 0 0
0
0
0 0
22 0
0 0 0
21
0 11
0 21
0 0
22 0
0 0
0
0 11
0
21 0 0
0
21 0 0
V ;H ; A H ;V A V ;V :
H V ;H Q A BB V ;V
V H
b b a b D A D
b
: b ; M ; <
; V : V ;V :
A A
I A D B B A A
A A Q
A D B B A H A A Q V ;H
A V ;H H V
that 5 ( ) 5 = 5 ( ) and5 = 5 ( ) Note
that thethirdequation of(6.11)is theLyapunovequationfortheself adjointop erator
5 on ,given5 ( )andthat + 5 + 5 5 5 ( )
is b oundedand symmetric.
AsinLemma6.2,wehaveassumedthat iscompactlyemb eddedinto andthat
the sesquilinear form in(6.3)is given by = +
(equivalently, = +
)
where
nowsatises
(612)
( )
0
1
for all It then followsfrom Theorem 5.3 that 5 ( ) Moreover,
premultiplyingthesecondequationof(6.11)by (recallthat isself-adjoint)we
obtain
( (
5 ) ) 5
= 5 +
where (
5 ) ( )iscompactand 5 + ( ).
Thus 5 ( ) and therefore 5 ( ) ( ). Furthermore, a similar
argument using the partitioning of Riccati solutions can b e applied to establish the
corresp onding resulttoTheorem 5.5.
Thecase ofthestructural damping(2.31)can b etreated directly using theresults
4
8
22
8
39
The Mathematical Foundations of the Finite Element
Methodwith ApplicationstoPartial Dierential Equations,
Spectral Approximation of Linear Operators,
[Ba] A.V. Balakrishnan, Boundary control of parab olic equations: L-Q-R theory, in
nonlinear op erators, Pro c. 5th InternationalSummer Scho ol, Akademie-Berlin,
1978.
[BA] I. Babuska and A. Aziz,
NewYork (1972).
[BI1] H.T.Banks andK. Ito,A unied frameworkfor approximationininverse
prob-lemsfor distributed parametersystems, Control-Theory andAdvanced T
echnol-ogy, (1988),73-90.
[BI2] H.T. Banks and K. Ito, A variational approach to a class of b oundary control
problems: numericalanalysis,BrownUniversity(1989),preprint.
[BIW] H.T.Banks,K.Ito,andY.Wang,Wellp osednessfordamp edsecondordersystems
withunb ounded inputop erators, CRSC TR93-10,N.C.StateUniv., June1993;
DierentialandIntegralEquations, (1995),587-606.
[BK] H.T.BanksandK.Kunisch,Thelinearregulator problemfor parab olicsystems,
SIAMJ.ControlandOptimization, (1984),684-698.
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