DSpace at VNU: The controllability of degenerate system described by invertible operator

DSpace at VNU: The controllability of degenerate system described by invertible operator tài liệu, giáo án, bài giảng ,...

V N U JOURNAL OF SCIENCE M athem atics - Physics T X V IIỊ N()3 - 2002

T H E C O N T R O L L A B I L I T Y O F D E G E N E R A T E S Y S T E M

D E S C R I B E D B Y R I G H T I N V E R T I B L E O P E R A T O R S

N g u y e n D in h Q u y e t, H o a n g V an T h i

H a n o i U niversity o f P edagogy

A b s tr a c t T h e co n tro lla b ility o f a linear s y s te m described by light, invertible opera-

to rs was stu d ie d by m a n y a u th o rs H ow ever, fo r the degenerate syste m , the problem

ha s n o t been so fa r considered In this paper, the co n tro llability o f these syste m s is studied.

0 I n t r o d u c t io n

T he theory of right invertible operators was sta rte d in 1972 w ith works of I) Przeworska- Rolewicz and then has been developed by M Tasche, II veil Trotha, z Binderm an and m any other M athem aticians (see [7]) W ith the appearing of this theory, the initial, boundary arid m ixed boundary value problems have been considered Since 1977- 1978, Nguyen Dinh Q uyet, in series of articles, has introduced the controllability of linear system s described by right invertible operators in the case of a resolving operator being invertible (see[2, 3]) T he results related to the controllability of linear systems were generalized by Pogorzelec for the case of one-sized invertible resolving operarors In 1992, Nguyen Van Man, in his study, introduced the controllability of general system and stu d ­ ied the controllability of linear system s described by generalized invertible operators (see [5]) However, for the degenerate systems, the problem has not been so far investigated

In this paper, the controllability of the degenerate system described by right, invertible operators is studied.

1 S o m e f u n d a m e n ta l n o tio n s

L et X be a linear space over a field T of scalars [ T = M or C).

Denote by l j ( X) the set of all linear operators w ith dom ains and ranges contained

in X and L 0( X ) = {A € L ( X ) : donii4 = X }

D e fin itio n 1.1 [7] A n o p e ra to r D £ L ( X ) is said to be a rig h t in v e r tib le o p era to r i f there is an o p e ra to r R € L q ( X ) su ch th a t I m R c d o m D a n d

w here I is id e n tity o p era to r In th is ease, R is called a rig h t in verse o p c tã lo r o f D T h e se t

o f all rig h t in v e r tib le o p e ra to rs b e lo n g in g to L { X ) will be d e n o te d b y R ( X ) I f D £ R { X ) ,

we d e n o te 7Z p = { I i £ L q { X ) : D R = /}

'Fypeset by

38 N g u y e n D i n h Q u y e t , H o a n g V a n T h i

P r o p o s i t i o n 1 2 [7] I f D € R { X ) , th e n fo r e v e r y R £ 7Z p we h a v e

D e f i n i t i o n 1 2 [7] All operator F € L q ( X ) is said to be an in itia l operator for D corresponding to R 6 7ZI) i f F2 == jP, F X = ker D a n d F R = 0 OĨ1 doinH T h e set o f all initial operators for D will be den o ted by Tj j

D e f i n i t i o n 1 3 [7] Suppose th a i D £ R ( X ) and R € TZd- A n operator A E L o ( X ) is said to be sta tio n a ry if D A — A D = 0 on cIomD and R A — A R = 0.

T h e o r e m 1 1 [7] Suppose th a t D € R ( X ) A necessary and sufficient condition for an

o p e r a to r F € Lo (X ) to b e a n in itia l o p e r a to r for D c o r r e s p o n d in g to R € is th a t

D e f i n i t i o n 1 4 [7] An operator V £ L o ( X ) is said to be a left in vertible operator i f there

is an o p e r a to r L € L { X ) s u c h th a t I m V c d o m L } L V = I W e d e n o t e A ( X ) th e s e t o f

all left invertible operators belonging to L ( X ) and by C y the set o f all left inverses o f

v e A ( x )

D e f i n i t i o n 1 5 |5j All operator V € L ( X ) is said to be generalized invertible i f there is

an operator w £ L ( X ) ( called a generalized inverse o f V ) such that:

ImV c domW y Im W c domV and v w v = V on domV.

T h e s e t o f all g e n e r a liz e d in v e r tib le o p e r a to r s in L ( X ) w ill b e d e n o t e d b y W ( X ) I f

V € W ( X ) , we den ote by w v th e set o f all generalized inverses o f V.

P r o p o s i t i o n 1.2 (5j S u p p o s e t h a t V Ç W ( X ) a n d w G vvv , th e n

T h e o r e m 1 2 [5] S u p p o s e t h a t A , B € L ( X ) , I m A c d o m B a n d I m B c ( lo m A, th e n

I - A B is r ig h t in v e r tib le ( le ft in v e r tib le ' in v e r tib le , g e n e r a liz e d in v e r tib le ) i f cind o n ly i f

so is I — B A M oreover, i f we d en ote by Ra b{ La b> w a b) & r ig h t inverse ( left inverse,

generalized inverse ) o f I — A B , then there exists Rb a € ^ i-i ì a{ ^b a € £ j - B A ) Wb a €

W /- /M , r e s p e c tiv e ly s u c h th a t:

(i) Ra b — I + A Rb a Rb a = / + B R a b A ,

(iij Lj\h = J + A Lb aB y Lb A = I + B La bA ,

(in ) ( I - A B ) - 1 = I + A ( I - B A ) - ' B } ( I - B A ) - 1 = I + t ì ự - A B ) ~ l A ,

fiv j VVUb = / + j W î m B , ^ > 1 = I + .

T h e th eo ry o f right in v ertib le o p era to r s and th ie r a p p lic a tio n s ca n b e seen in [5,7].

T h e c o n t r o l l a b i l i t y o f d e g e n e r a t e s y s t e m d e s c r i b e d b y 39

2 D e g e n e r a t e s y s t e m s

D e f i n i t i o n 2 1 Suppose th a t D € R ( X ) }d i mk e T D ý- 0 and A, B € L o ( X ) , with A ^ 0

non-invertiblc T hen a linear s y s te m o f the form

is said to be a degenerate system

P r o p o s i t i o n 2 1 Suppose th a i I) € R ( X ) , d i m k er D 7^ 0; F is an initial operator for

D corresponding to R € TZjj a n d A y B € Lq( X), with A Ỷ 0 non-invertibie Then the following id en tities hold on d o m ü :

F roof, (i) O n (loin D we have

D { I - R [ ( I - A ) D + B ] } = D - D R [ Ự - A ) D + B} = D - Ự - A ) D - tì

= I ) - D + AD - H = AD - t ì

T h e p ro o fs o f (2.3) a n d (2 4 ) a re co m p letely sim ilar

P r o p o s i t i o n 2 2 Sup p ose th a t a11 th e assum ptions o f P roposition 2.1 arc satisfied I f

A — H R is righ t invertible ( leit in v ertib le, invertible, generalized invertible), then so is I — R[(I — A ) D + tì] M oreover, i f Ra b(J'AJJ) {A — B R ) ~ l , w A b) is right inverse ( left inverse,

in v e rs e, g e n e r a liz e d in v e r s e ) o f A — B R , th e n th e r e e x is ts R q € 'R 'Ị - R \ụ - A ) D + iì\ỤJ0 £

£ i - R [ ( i - A ) n + B } , W Q € V\>Ị-R \ ụ - A)D+B] respectively, su ch that:

(i) Ro = I + R Ra bHI - A ) p + B] ,

(ii) Lo = / + 7 ỈL ^ Ị(/-i4 )D + fi) ,

(iỉi) { I - R [ ự - A ) ỏ + ổ ] } “ 1 = s / + « ( i 4 - B / ĩ ) - l ị ( / - i 4 ) D + B l ,

(ỉv) IVo = I + R W a M V - A ) D + fl] •

Proof. W e w ill prove th e ca se ( iv ) W e h ave / — [ ( / — i4)/J) -f /? ] /ỉ = y4 — B R S u p p ose

th a t /1 - B R € W (X ) and € W /I-B * T hen / - /ĩ[ ( / - i4)D + 0 | G W (X ) (by

T h eo rem 1.2 ) M oreover, th ere e x is t s Wq = I + R W A b \ { I “ >4)/) + # ] is a generalized inverse o f I — R [ ( I — >4)D + # ] .

A n o p e r a to r - B R is sa id to b e a resolving operator for th e s y ste m (2 1 ), if

>4 — R R is in v er tib le, th en th e s y s te m (2 1 ) is said to b e w ell-d e te rm in e d O th erw ise, it is

ill-d ete rm in ed

T h e o r e m 2 1 Supp ose th a t all a ssu m p tio n s o f Proposition 2.1 arc satisfied Then, we

have:

(i) I f A — B R Ç Jl { X) a n d /Ỉ,4/J € T^A- BR Ì then all solutions o f th e s y ste m (2.1) are

A D x = B x + y , y € X (2.1)

A D - t ì = D ự - R [ ự - A )D + £ ] } , (i4 D - t ì ) R = A - B R

A D - B = ( A - B R )D - B F

(2.2)

(2.3) (2.4)

g i v e n b y

40 N g u y e n D i n h Q u y e t, H o a n g V a n T h i

where z € kerD } z € k er{ / — / ỉ [ ( / — v4)D 4- # ] } ,

(ii) I f A — # / ? € A ( X ) a n d Zvy\# € C a - b h , th e n a ll s o lu tio n s o f th e s y s te m (2 1 ) a re

given by

X = { / 4- /Ỉ L v4/ j [ ( / — /1 )D + £?]}(/£?/ + 2) , 2 G k erD , (2 6 )

(Hi) I f A — B R is invertible, th en a11 solutions o f th e s y ste m (2.1 ) are given by

X = { / + / Ỉ( i4 - f í / ỉ ) “ l í ơ - i4 )D + £ ] } ( i ỉ y + í ) , 2 € k e rD , (2 7 )

(iv) I f A — B R € IV' ( X ) rUìd W A Ịì £ VV/t - Ị ĩ Ị{ , then all solutions o f the system (2.1)

a re g i veil b y

X = { I + R W AtB[(I - A ) D + tì]}ự ỉy + z) + z, (2.8)

where z € ker I) , £ € k e r {J — /-?.[(/ — j4 ) D + Ổ ]}.

P ro o f. S in ce b o th o n e -sid e d in v e rtib le a n d in v e rtib le o p e r a to r s axe g en eralized in v e rtib le ,

it is su fficien t t o co n sid er th e ca se (2?;) A ccord in g to e q u a lity th e (2 2 ) ill P ro p o sitio n 2.1,

w e see t h a t (2 1 ) is e q u iv a le n t to D { I — / ĩ [ ( / — A )D + B ) } x = y H ence,

{ / - I i [ { I - A ) D + £ ] } * = « y + ỉ , 2 e k er £>, (2.9)

B y th e a s s u m p tio n , /1 — B R € i y ( X ) a n d W/ \ n € VVU-/3/* T h u s , P ro p o s itio n

2.2 im p lies th a t I — / { [ ( / - / 4 ) D + f í | G VV'(X) an d th ere e x is ts a gen era lized in v er tib le

operator Wq = I + /?[(/ — i4)D + B\ Therefore, (2.9) obtaines that all solutions of

(2 1 ) axe g iv e n b y X = { J + /ỈM^ì4, b [ ( / - -4)-D + f t Ị } ( / t y 4- z ) + 5.

3 T h e i n i t i a l v a l u e p r o b l e m

S u p p o s e t h a t 1) £ I Ỉ ( X ) , d im k e rD ^ 0; -F is a n in itia l o p e r a to r for D c o rre sp o n d in g

to I t £ 1Z ị)\ a n d A , B £ L o ( X ) , w ith A Ỷ 0 n o n -in v e rtib le In th is sec tio n , w e co n sid er

th e in itia l v a lu e p ro b le m fo r d e g e n e ra te s y s te m ( D S) 0 o f th e form :

T h e o r e m 3 1 S u p p o se th a t oil th e assum p tio ns o f P roposition 2.1 are satisfied and

R y + X'o E { / - / ? [ ( / — y4)D + B ] } d o m D T hen, we have:

(i) I f A — B R G I Ì { X ) a n d R a b € TZ a - b r y th e n a ll s o lu tio n s o f t h e p r o b le m (3 1 )-(3 2 )

axe given by

x = { ỉ + H R a b ỉ ơ - A ) D + B } } ( R y + x 0 ) + z , (3.3)

where z € k e r { / — / ? [ ( / — A ) D + 5 ] }

(ii) I f A - B R € /1(X ) a n d L a b € C a - b r , th e n a /i s o lu tio n s o f th e p r o b le m (3 1 )-(3 2 )

are given by

X = { / + /Ỉ L ^ B Í Ơ - i4 )D + « ị } ( i ỉ y + x o ) (3.4)

(Hi) I f A — H R is invertible, th en solution o f th e p ro b lem (3.1)-(3.2) arc given by

X = { I + R(A - i4)/J + B ]}(/ỉy + x 0) (3.5)

(iv) I f A — B l i (E W ( X ) and wy\ ỊỊ £ y^A -B R Ĩ then all solutions o f the problem

(3.1)-(3.2) cue given by

X = { / + Ỉ I Wa j ỉ[Ự - A ) D 4- B ] } ( R y 4* Xo) 4- z , (3 6 )

Ì where z € kcr{7 - / ỉ [ ( / — 4 )D + B]}.

Proof. According; to the p r o o f o f T h e o r e m 2.1, fro m (3 1 ) w e h a v e

{ / — lì.ịự - A )D + B ] } x = R y + z , z G k e r /J (3 7 )

Thus, acting on b o th sides of t his equality by o p erator F , we find th a t F x — F H [ ( I —

,4 )/J + Bịa: = F R y + F z Ile n c c XQ = F x = /*2 = 2 T h e re fo re ,

{ / - / ỉ | ( / - A ) D + B ) } x = R y + *0 (3.8)

By our assum ption, A - H R € implies th a t I — /£[(/ — A )D + B] € w p o

an d th ere e x is ts i t ’s g en e r a lise d in verse Wo = I + R W A /* [ ( / — A ) D + B] , w ith co n d itio n

R y -f Xo € { I — R[ ( I — A) I) + B ] } d o m D , w e h a v e a ll s o lu tio n s o f th e p r o b lem (3 1 )-(3 2 )

is given by

X = { I + R W AtR[{I - A ) D + B \ } ( R y + Xo) + z , z € k e r { / - / ỉ [ ( / - / ! ) / ; + / * ] }

T h e o r e m 3 2 S u p p o s e t h a t A B a r e s ta tio n a r y o p e r a to r s a n d A — f t / ỉ is in v e r tib le

T h e n , th e in itia l va lu e p r o b le m ( 3 1 )-(3 2 ) h a s a u n iq u e s o lu tio n

x = ( A - B R ) - \ R y + x 0) (3.9)

P ro of. By th e a s s u m p tio n /1, ft a re s ta tio n a r y o p e ra to rs , A D — B = D ( A — f t / i ) a n d (3.1) becomes D(A — B R )x = ty , this implies that (/1 — H I Ỉ ) x = /fy -f £ € k e r/J The

co n d itio n (3.2) fin d s th a t 2 = XQ M oreover, /t — is in v e rtib le T h u s , th e s o lu tio n o f the problem (3.1)-(3.2) is unique and given by

X = {A — B R ) ~ l (R y + z 0)

E x a m p l e 3 1 Suppo se th a t X is the space («) o f all real sequences { x n } , n — 0 , 1 , 2 , ■ • ■

with addition and m ultiplication by scalars defined as follows: I f X = { i n } , y = {j/n }> A <E

R then X + y — { x n 4- y n } , Ằ x = {Ằ.T„},71 = 0 , 1 , 2 , W rite D { x n } = { i n+1 - £ « } ,

R { x n} = { y n } , y0 = 0, y „ = Y,kZo n ^ 1 HI1(I F {x n} = { ^ n } , z n - XQ, n = 0 , 1 , 2 , —

It is easy to verify th a t D € R ( X ) , R € 7Zp, F is an in itia l op erato r for Ỉ) corresponding

to R and ker D = {z = {Zn}, ZTX = c , n € N ,c G R } C onsider th e degenerate sy ste m ( D S )0 o f th e form :

where A ) B are defined b y A { x n } — {&’n + i } , B { x n } = {a:n + 2 — ^ n } nn d y = {?/„} G X ,

x 0 = { x 0 } G ker D arc given Wc conclude th a t 4 ^ 0 is Đon-invertible, the resolving operator A - B R is invertible, A - B l l = (A — = —I B y T h eo rem 3.1, the solution o f th e s y ste m ( D S ) 0 is o f the form :

X = { / + fì(i4 - B /Ĩ) “ l [(/ - / 1 )D + « ] } ( % + 5?o)

= {/ - -Rí(/ — i4)£> + B]}{Ry + x0)

= {^ 0 , ^0 - yo> Zo - Vo - Vì, ^0 - 2/0 - 2/1 - 2 / 2 , • • • }•

T h e c o n t r o l l a b i l i t y o f d e g e n e r a t e s y s t e m d e s c r i b e d b y 41

42 N g u y e n D i n h Q u y e t y H o a n g V a n T h i

E x a m p l e 3 2 Suppose th a t X , Dy R an d F are defined as in E xam ple 3.1; W rite i4 { x n } = {2 z o + x i , 0 , X2 + X3}X3+ X4> • • • }, B { x n } = { x2 - Xo, 0, X4 - X2, S5 - x3) • • • }• Clear, A ^ o and is non-invertible, sin c e ker i4 = {x o , —2 x o , x2) —X2, X2, —£2, • • • } 7^ { 0 } a n d ,4 X Ỷ X

L et y = {?/„} € X and XQ = { x o } € ker D

N ow we consider th e degenerate sy ste m ( D S ) 0 o f th e form:

It is easy to verify th a t th e resolving operator A — B R is generalized invertible, in d e e d ,

(A - B R ) I ( A - B R ) { x n } = (i4 - ổ / ỉ ) { x n } , i.e A - B R £ W { X ) a n d I € W a - b k

Moreover, ker{ / — -R [(/ — j4 )D + f i] } = { { 0 , 0 , X'2, X3, x 4 , • • • }, x n € R , n = 2, 3, 4, • • • } B y

(3.Ổ) , the solution o f the problem ( D S) 0 is g ive n by

X = { / + / ĩ [ ( / - A ) D + ổ ] } ( i ? y + x o ) 4- 2 , Ỉ 6 k e r { / - f i[ ( J - i4 ) D + B ] }

= {xo , x 0 + yo , 20 + yo + 2yi + x2,x0 + yo + 2t/i +2y2 + ®3»"* }•

4 C o n tr o lla b ility o f t h e d e g e n e r a te s y s te m

Let X and u be linear spaces over the same field T [ T = R or C) Suppose that

D € R ( X ) } d im k e r ữ ^ 0] F e T o is an in itia l o p era to r for D c o r r esp o n d in g to R € 7Zd\

and A, R € L ()(X ), w ith A Ỷ 0 n o il-in v ertib le a n d c € Lo( U} X ) W e c o n sid er th e p ro b lem (£>S)o:

f A D x = B x + Cîz , w ith c o n d itio n JĨC Ơ © {xo} c { / — / ỉ [ ( / — >4)D + B )} d o m D (4 1 )

T he spaces X and [/ are called the space of states and the space o f controls, re­ spectively Elements X £ X and u £ u are called states and controls, respectively The

elem en t Xo G ker D is called an in itia l s ta te A pair ( x o , u ) G (ker D) X u is ca lled an

in p u t.

In se c tio n 3, w e have proved th a t th e p ro b lem (4 1 )-(4 2 ) is e q u iv a le n t to th e e q u a ­ tion

{ / - R[ ( I - A ) D + B ] } x = R C u + xo (4 3 )

Hence, the inclution R C U © {xo} c { / — R [ (I — A )D + B ]} dom D is a necessary

and su fficien t co n d itio n for th e p ro b lem (4 1 )-(4 2 ) h ave s o lu tio n for ev ery u G V . D en o te

by <px(i = 1 , 2 , 3 , 4 ) th e fo llo w in g se ts, d efin ed for ev ery Xo € ker Dy u £ Ư:

(i) If A — B R G R (X ) and Ra s g T^a- b r } then:

# i(x o , u) = {x = T \(R C u + Xo) + z , z € ker{ I — R [ (I — A )D + B]} ,

Tx = I + R Ra b[{I - A )D + B] } (4.4)

(ii) If A — H R G Ả ( X ) a n d La b ^ £a- b r ) then:

0 2(xo,îi) = {x = T2( R C u + x q ) , T2 = / + /Ỉ La b[(/ - + 5 ) } (4.5)

(iii) If A — t ìỉỉ is invertible, then:

* 3 (x o ,u ) = {x = 73(R C u + xo) ,7 3 = / + R (A - B R ) - ' [ { I - A )D + H\ } (4.6)

T h e c o n t r o l l a b i l i t y o f d e g e n e r a t e s y s t e m d e s c r i b e d b y 43

(iv) If A - B R e w ( x ) and W^,B e , then:

i>4(xo, u) = {x = T4( R Cu + Xo) + z , z € ker{7 — /ỉ[ ( / - A )D + Ổ]} ,

n = I + R W A,B[ ( I - A ) D + H \} (4.7)

N o t e that = 1 , 2 , 3 , 4 ) are s e ts o f all so lu tio n s o f th e p ro b lem (4 1 )-(4 2 ) in th e

c o rr esp o n d in g ca se.

D e f i n i t i o n 4 1 S up pose that wc are given a s y ste m ( D S ) o a n d file s e ts <pị(.'Co, u) o f the form s (4.4)-(4.7) A sta te X € X is said to be (i)- reachable (i = 1 , 2 , 3 4 ) from an initial

s ta te XQ € kcr D i f for every TìỰTị = I + R R AB[(I - A) D + B ] , T2 = J + / Ỉ Í M B | ( / - / 1 ) 0 + Ã ]

73 = 7 + / ỉ ( / l - B f í ) - l [(J - A ) D + B), T a = I + - i4 ) D + e ] ) there e x ists a

control u € u such that X G '■/•>, (xn , u )

W rite Rangi/.xo^i = ( J u) J ^0 € ker D , (2 = 1 , 2 ,3 ,4 ).

u€ ơ

It is easy to see th a t Rangy is ( 2 )- reachable from x *0 6 k e rD by m eans of controls

u £ IJ , and th ese sets are co n ta in ed in dom D .

L e m m a 4 1 S u p p o se th a t T i(i = 1 ,2 , 3 ,4 ) are defined as in (4.4)-(4.7) , then:

r t ( / ì ơ ơ © {z0} )+ k e r{ / - /ỉ[ ( / - 4 ) /J + £]}

= TaR C U © {TiXo} © ker{/ - /ỉ( ( / - ,4)1) + B]} (4.8)

Proof. It is su fficien t to p rove th e ca se 2 = 4.

By our assum ption, I — R [(I — A)D + B] € W (X ) and T 4 G

Therefore, Proposition 1.2 implies th a t X = 7 4 { / — jR((J — A )D 4 B ] } x © k er{ I — R [(I

-A )D + B)} On the other hand R C U © {xq} c { / — R [ (I - A )D + B}} dom D, there exists

E C dom D such th a t R C U © {x 0} = { / - R [(I - A)D + B ] } E c { / - /ỉị ( / - i4)D +

# ]} đom /) Hence,

r 4(/?ct/ © {.T0})+ ker{/ - R[(I - yl)D + B]}

= T4{ I - R [(I - i4)D + JB]}E + ker{J - /ỉ[ ( / - A )D + B}}

= T4{7 - R [(I - ^ ) D + B ] } E e ker{/ - R [ (I - yl)D + B]}

= r A{R C U © {a?Q}) © ker{7 - /ỉ[(/ - i4)D + /?]}

W e w ill p rove th e e q u a lity T ^ (R C U © { ^0} ) = T4R C U @ {T4X0} Indeed, let

æ € (T4R C U ) n {T4X0}, i.e th ere e x is ts u € U , t € T su ch th a t X = T ^ R C u = T Ạ xo , or

T n (R C u — £xo) = 0 B y our a ssu m p tio n /ỈC Ỉ7 © { z o } c { / — / ỉ [ ( / — ^4)D + B ]} d o m D ,

th ere e x is t s V € đ o m D s u c h th a t R C u — t x 0 = { I — R [ ( I — A ) D + B] } v , th is im p lies th a t

0 = Ta( R Cu - t xo) = r 4{ / - R [ ( I - A ) D + B) } v , or

0 = { / - R[ { I - A ) Ü + B \} T a{ I - R [ ( I - A ) D + B ) } v

= { I - R [ ( I - A ) D + B }}v = R C u - t x 0

Ilence, 0 = Dtx 0 = D R C u = Cu, tx 0 = 0 and X = T4R C u — T 4 Í 10 = 0

R e m a r k If eith er A — B R Ç: A ( x ) or A — B R is an invertible operator, then

k e r { / - R{ ( I — A ) D - f # ] } = { 0 } T h u s th e fo rm u la (4 8 ) ta k es th e form:

TịỤlCU © {xo}) = TtRCƯ ® {TiX 0 } , (t = 1,2,3,4) (4.9)

C o r o l l a r y 4 1

Rangự)Xo<Pl = Tị R C U © {TịX 0 } © k e r{ / - R[ự - A) D + BỊ} (4.10)

C o r o l l a r y 4 2 A s ta te X is (i)- reachable from a given initial sta te XQ € ker D if and

only if X e T iR C U © {Ttxo} ® ker{7 - R [(I - yl)D + B}} , (i = 1 ,2 ,3 ,4 ).

D e f i n i t i o n 4 2 L et be given a degenerate sy ste m ( D S ) o o f th e form (4.1)-(4.2) and let

F, G T u ụ — 1,2, 3 , 4 ) be arbitrary initial operators (not necessarily different).

(i) A s ta te XI € ker D is s a id to b e F t- reachable from an initial s ta te XQ € ker D i f

th e re e x i s t s a c o n tr o l u € u su ch t h a t Xị £ F ị $ i ( x o , u ) T h e s t a t e Xi is c a lle d a

final sta te.

(ii) T h e s y s te m ( D S ) o is s a id t o b e F ị- c o n tr o lla b le i f fo r e v e r y in itia l s t a t e x o 6 k er D,

F i( RangU 'X04>i) = k e r D (4.11)

(Hi) T h e s y ste m ( D S ) 0 is said to be Ỉ ) - controllable to XI € ker D i f

Xi 6 F i(R a n g UtXo<Pi) (4 1 2 )

for every initial s ta te XQ E k e r D.

L e m m a 4 2 L et be given a degenerate s y ste m ( O S )0 a n d an initial operator 1\ £ J 7/;

S u ppo se th a t th e s y ste m { O S )0 is F i- controllable to zero and th a t

Fi{Ti k e r D + k e r { / - / ỉ [ ( / - y l) D + £ ] } ) = ke r D (4 1 3 )

T h en every s ta le Xị € ker D is F t- controllable to zero.

Proof It is sufficient to prove th e case 2 = 4 Let Xi € ker D , since the assum ption ( DS)o is

F4- c o n tro lla b le to zero T h u s , 0 € F t ( R a n g ỉ /x0^4) for ev ery Xo € ker D , i.e th e re e x ists

a c o n tro l Uo € u a n d 20 ^ k e r { / — /Ỉ[ (Z — A ) D 4- B ]} su c h t h a t F ^ [ T ^ { R C u q + Xo) + 20] = 0,

or

F 4(TaR Cuo + 20 ) = - f \ T AXQ (4.14)

44 N g u y e n D i n k Q u y e t , H o a n g V a n T h i

T h e c o n d itio n (4 1 3 ) im p lies t hat for ev ery X\ £ ker Dy th e r e e x is t s X2 € ker D and

Z\ £ k e r { / — ft[(I — A) I) -f H]} s u c h th a t

F a ( T * x2 + Z\ ) = X ] (415)

O n th e o th e r h a n d , by form u la (4 1 4 ) for ev ery X'2 £ ker Ü , th e r e e x is t s uQ € u an d

2q € k(*r{/ — / ỉ [ ( / — 4 )/J 4- H]} su c h that

P<i( 1 \ R Cuq 4- Zq 4* 2 i) = Kị ( A x 2 + 2 i) (4 1 6 )

S o th a t (4 1 5 ) and (4 1 6 ) im p ly F'k( 7 \ R C u'0 + z[) = X\ , w ith = 2q -f- 2i €

k er{J - /£ [ ( / — / l ) / J + ft]} • This p roves th a t every s t a t e X\ G ker I) is i*4- rea ch a b le from

/,oro

T h e o r e m 4 1 Suppose that- nil a ssu m p tio n s o f L em m a 4/2 arc satisfied Then th e d e-

ircncrnt.fi s y ste m ( D S )0 is - controllable.

Proof Suppose; that /1 — B Ji € li7(X ) By our assum ption there exists Uo € u and

2<) £ k e r { / — /? [ ( / — A )D + w ]} su c h th a t

^ [ ' A Í / Ỉ C i i o + .To) + *o] = 0 (4 1 7 )

On th e o th e r h a n d , by Lerm na 4 2 , for ev ery X\ € kcr D th e r e e x is t s Uq € u and

Z\ € k e r { / — I i \ ( I — y t)/} + # ] } su c h th a t

T h e re fo re , (4 1 7 ) a n d (4 1 8 ) im p ly th a t Fa{T/[[RC(uo + Uq) + Xo) + (zo + Z\ ) } = X\y

i.(\ the sta te X\ is Fi\- re ach ab le fro m in it ia l state XQ. T h e th e o re m h a s b e en proved

T h e o r e m 4 2 L e t b e g iv e n a d e g e n e r a te s y s te m ( D S ) o o f th e fo r m (4 1 )-(4 2 ) a n d ail

in itia l o p e r a to r F t € T o (i = 1 ,2 ,3 ,4 ) a n d le t T \ = / 4- R I Ỉ A t ì [ ( ỉ — A ) D + B \ i f A — B R €

ï l { X) 9 7 2 = i f A - B R € y l(X ) , 7 3 = J + / ỉ ( / i - B / ỉ ) - l [ ( / - y l ) D + « l

i/\4 - / Í / Ỉ is in v e r tib le a n d 1 \ = 7 + /ỈW 0\,£*[(J - / I ) /-> + /í) i / '/ i - H R € W ( X ) S u p p o s e that c e Lq(U X , X ' -> u % D € X')> and A, B , R € L0( X , X ' ) Then , the

s y s te m ( P S) 0 is i s - c o n tr o lla b le i f a n d o n ly i f

Proof. It is sufficient, to co n sid er th e c a se 2 = 4 N o te th a t in a ll th e c a s e s con sid er,

F / l ) l i e m a p s Ơ in to k erD T h e c o n d itio n (4 1 9 ) is e q u iv a le n t t o

T h e a s s u m p tio n R C U © {xo} c { / — / ỉ [ ( / — i4 )D + /^1} d o i n ơ , im p lie s th a t

FaT4RCƯ = F M R C U © {x*o}) - {F 4 T 4 X 0 }

c F4T4{/ - /i[(/ - /1)0 + B]}domD ~ {F4T4®o}

c F 4 {T 4 {7 - /ỈỊ(/ - /i)D + B]}dọm D © k e r{ / - /ỉ[(J - i4)jD + B]}}

- {F 4 T 4 X 0 } - F 4 ker{/ - /?[(/ - A )D + tì}}

= / ‘4d o m /) ~ { ^474^0} - F4 k e r { / - / i [ ( / - /1) / ; + /ỉ]} c k e r D

T h e c o n t r o l l a b i l i t y o f d e g e n e r a t e s y s t e m d e s c r i b e d b y 45

46 N g u y e n D i n h Q u y e t f H o a n g V a n T h i

B y (4 2 0 ), w e h a v e F4T4R C U = r ^ d o m D —ịr^ T /iX o } —F4 k e r { 7 —/ ? [ ( / —A ) D + B]} = k e r D

T h u s, /V A / ĩ C Ơ 4 - { K t T4x o H F4 k e r { / - i l [ ( / - i 4 ) j D + B ] } = A d o m D = ker D T h is m ea n s

th a t for e v e ry X\ € ker D i th e re e x is ts V G d o m D , u £ Ư a n d 2 0 € k e r { 7 — / ? [ ( / — i 4 ) D 4- B ] }

su ch th a t Xi = F4V = F4T4R C U 4- F4T4XQ 4- -£4*0 = 4- Xo) + 20], i.e X\ is F4 -rea ch a b le from Xo T h e a rb itra r in ess o f Xo, X\ € ker D im p lie s th a t

F 4( R an gt/,Xo$ 4) = ker£>.

C o n v ersely , su p p o se th a t F 4( Range; Xo^4) = k e r D C h o o sin g £0 = 0 ,z o = 0, we

g e t t h a t F4T4R C U = ker D T h e p ro o f is co m p leted

C o r o l l a r y 4 3 S u pp o se th a t A, 2? are sta tio nary operators T h en th e system ( D S )0 is

F 3 - controllable if and only if

ker c * R * ( A - B R) * ~ ' = { 0 } (4 2 1 )

T h e o r e m 4 3 S u p po se th a t th e s y s te m ( D 5 ) o is F i- controllable T h e n, it is jF/- con-

trollable for every initial o p era to r F[ G T o

-Proof. L et F i b e a n in itia l o p e r a to r for D co rresp o n d in g t o jR € 7^ 0 , i-e - Ỉ*\ỈU = 0

O n th e o th e r h a n d , for ev er y £1 € k e r D a n d V £ X , th e re e x is t s X2 £ ker D su ch th a t

x x = .T'2 B y our a s s u m p tio n th e s y ste m ( D S )0 is F t- co n tro lla b le T h u s, for every

x 0i %2 € ker D ị th e r e e x is ts a co n tro l U G Í / an d Zo € k e r { 7 — jR [(/ — A ) D + J5]} su c h th a t

F i[T i(R C u + xq) + Zo] = x2 , or Fi[T t ( / ỉ ơ u + æ0) + Zo] = F t ( x2 + Rị v ) , for so m e V € X

Hence, F '[T i(R C u + xo) + zq] = F '( x 2 + jRiv) = £2 + F 'R iV = X\ The arbitrariness of

3?0, Xi € ker Đ , th e p ro o f is c o m p le te d

T h e o r e m 4 4 L et he g iven a degenerate system ( D S ) 0 a n d an in itial operator 1 \ £ T p

T h e n, th e s y ste m ( D S )0 is F t- controllable i f and o nly i f it is Fi- controllable to every

element v' e F jT i R X

Proof. F ir s t, w e p rove th e eq u a lity :

F4{ T 4( i ỉ X © ker D ) + k e r { I - / ì ị ( / - A ) D + £f]>> = ker D (4 2 2 )

In d eed , sin c e { / - / ? [ ( / — Ẩ ) D + ổ ] } d o m D c d o m ơ = R X © ker D , th ere e x ists

E c X a n d c ker D su c h t h a t f t # © z ' = { / - ./?[(/ - ,4 ) D + B] } d o mD. T h is im p lies

th a t

Ta( R E ® z ' ) + ker ự - R [ ự - A ) D + B}}

= 1 \ { I - / i [ ( / - A ) D + B ]} d o m D © k e r { / - R [ { I - /1)1) + B } } = d o m D

H en ce,

ker D = F td o m D = F4{ T 4( / Ỉ E © z ' ) + k er{7 - / ĩ [ ( / - >1)D + B ] } }

c F 4 {T4(f lX © ker D) + k e r{ / - / ỉ[ ( / - Ấ )D + B]}} c ker Ơ,

i.e th e form u la (4 2 2 ) is h a s b e e n proved.

S u p p o se that, the s y ste m ( DS ) o is F4-c o n tro lla b le to e v e ry elem ent v' G F4T4RV, V £

X , i.e th e re e x is t s a co n tro l UQ € u a n d 2 0 € k e r { I — / ỉ [ ( / — A ) D + J5]} su ch th a t

i*4[ T ^ R Ouq 4* xq) 4* zq] = F4T4/ÎV.