DSpace at VNU: The controllability of degenerate system described by invertible operator tài liệu, giáo án, bài giảng ,...
VNU 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 IL IT Y O F D E G E N E R A T E S Y S T E M D E S C R IB E D B Y R IG H T IN V E R T IB 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 anoi U niversity o f P edagogy A b s tr a c t T he controllability o f a linear sy ste m described by light, invertible opera tors was studied by m a n y autho rs H ow ever, fo r the degenerate system , the problem has n o t been so fa r considered In this paper, the controllability o f these system s is studied I n tr o d u c tio n The theory of right invertible operators was started in 1972 with 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 mixed boundary value problems have been considered Since 1977- 1978, Nguyen Dinh Quyet, in series of articles, has introduced the controllability of linear systems 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 stud 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 S o m e fu 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 to r D £ L ( X ) is said to be a rig h t in v ertib le operator i f there is an o p era to r R € L q( X ) su ch th a t I m R c d o m D a n d DR = I , (1.1) where I is id e n tity operator In th is ease , R is called a rig h t inverse o p c tã lo r o f D T h e set o f all rig h t in v ertib le o p e to rs b elo 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 0*7 N g u y e n D in h Q u y e t, H o a n g 38 Van T hi P r o p o s i t i o n [7] I f D € R { X ) , th e n fo r e v e ry R £ 7Z p we h a v e d om D = R X © ker D D e f in it i o n (1-2) [7] All operator F € L q ( X ) is said to be an initial operator for D corresponding to R 7ZI) i f F == jP, F X = ker D a n d F R = OĨ1 doinH The set o f all initial operators for D will be denoted by Tj j D e f in it i o n [7]Suppose th a i D £ R ( X ) and R € TZd - A n operator A E L o ( X ) is said to be statio n a ry if D A — A D = on cIomD and R A — A R = T h e o r e m 1 [7] Suppose th a t D € R ( X ) A necessary and sufficient condition for an o p e to r F € L o (X ) to b e an 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 F = I - RD (1.3) D e f in it i o n [7] An operator V £ L o ( X ) is said to be a left invertible operator i f there is an o p e r a to r L € L { X ) su c h th a t I m V c d o m L } L V = I W e d e n o te 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 veA (x) D e f in it io n |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 = Von 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 to rs in L ( X ) w ill b e d e n o te d b y W ( X ) I f V € W ( X ) , we denote by w v the set o f all generalized inverses o f V P r o p o s i t i o n (5j S u p p o s e th a t V Ç W ( X ) a n d w G vvv , th e n dom K = M/ y ( d o in K ) © k e r V (1*4) T h e o r e m [5] S u p p o s e th 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 rtib 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 denote by R a b { L a b >w a b ) & r ig h t inverse ( left inverse , generalized inverse ) o f I — A B , then there exists R b a € ^ i - i ì a { ^ b a € £ j - B A ) W b a € W /- /M , r e s p e c tiv e ly su c h th a t: (i) R a b — I + A R ba R ba = / + B R a b A , (iij L j\ h = J + A L b a B y L b A = I + B L a b A , (in ) ( I - A B ) - = I + A(I - B A ) - ' B } fivj VVUb = / + j W î m B , ^>1 = I + ( I -B A )-1= I + tìự - AB)~lA , T h e th eory o f right in vertib le o p era to rs and th ier a p p lica tio n s ca n be seen in [5,7] T h e c o n tr o lla b ility o f d e g e n e r a te s y s te m d e s c r i b e d b y 39 D e g e n e r a te s y s te m s D e f i n i t i o n Suppose th a t D € R ( X ) }di mkeT D ý- and A, B € L o ( X ) , with A ^ non-invertiblc T hen a linear sy ste m o f the form A Dx = Bx + y , y € X ( 1) is said to be a degenerate system P r o p o s i t i o n Suppose th a i I) € R ( X ) , d i m ker D 7^ 0; F is an initial operator for D corresponding to R € TZjj an d A y B € L q( X ), with A Ỷ non-invertibie Then the following id entities hold on d o m ü : ( 2 ) A D - t ì = D ự - R [ ự - A )D + £ ] } , (i4 D - t ì ) R = A - B R (2.3) A D - B = ( A - B R )D - B F F roof, (2.4) (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 proofs of (2.3) a n d (2.4) a re com pletely sim ilar P r o p o s i t i o n 2 Suppose th a t a 11 th e assum ptions o f P roposition 2.1 arc satisfied I f A — H R is right invertible ( leit in vertib le , invertible, generalized invertible), then so is I — R[(I — A ) D + tì] M oreover , i f R a b (J'AJJ) {A — B R ) ~ l , w A b ) is right inverse ( left inverse, in v e rse , g e n e r a liz e d in v e rs e ) o f A — B R , th e n th e re e x is ts R q € 'R 'Ị-R \ụ -A )D + iì\Ụ J0 £ £ i - R [ ( i - A ) n + B } , W Q € V\>Ị-R\ ụ - A)D+B] respectively , such that: (i) Ro = I + R R a b HI - A ) p + B] , (ii) Lo = / + ỈL ^ Ị(/-i4 )D + fi) , (iỉi) { I - R [ ự - A ) ỏ + ổ ] } “ = s / + « ( i - B / ĩ ) - l ị ( / - i ) 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 have / — [ ( / — i4)/J) -f /? ] /ỉ = y4 — B R S uppose th at /1 - B R € W (X ) and € W /I-B * Then / - /ĩ[(/ - i4)D + | G W (X ) (by T h eorem 1.2 ) M oreover, there e x is ts Wq = I + RW A b \ { I “ >4)/) + # ] is a generalized inverse o f I — R[ ( I — >4)D + # ] A n o p era to r - B R is sa id to b e a resolving operator for th e sy ste m (2 ), if >4 — R R is in vertib le, th en th e s y ste m (2 ) is said to b e w ell-d eterm in ed O therw ise, it is ill-d eterm in ed T h e o r e m Suppose th a t all a ssum ptions o f Proposition 2.1 arc satisfied Then, we have: (i) I f A — B R Ç Jl {X) a n d /Ỉ, /J € T^A-BR Ì then all solutions o f th e system (2.1) are g iv e n b y X = = {/ + RR a b \Ự — A )D + B } } ( R y + z) + z, (2.5) N g u y e n D in h Q u y e t, H o a n g 40 Van T hi where z € ker D } z € ker{ / — / 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 ) a re given by X = { / 4- /Ỉ L v4/ j [ ( / — /1 )D + £?]}(/£?/ + ) , G k erD , (2 ) (Hi) I f A — B R is invertible, then a 11 solutions o f th e system (2.1 ) are given by X = { / + / Ỉ( i4 - f í / ỉ ) “ l í - i4 )D + £ ] } ( i ỉ y + í ) , € k e rD , (2 ) (iv) I f A — B R € IV' ( X ) rUìd W A Ịì £ VV/t - Ịĩ 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 er{J — /-?.[(/ — j4 )D + Ổ ]} P ro o f S ince b o th o n e-sid ed in v e rtib le a n d in v ertib le o p e to rs axe g eneralized in v ertib le, it is sufficien t to sid er th e ca se ( 2?;) A ccording to eq u a lity the (2 ) ill P ro p o sitio n 2.1, we see th 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 , e ker £>, (2.9) B y th e a ssu m p tio n , /1 — B R € i y ( X ) an d W/ \ n € VVU-/3/* T h u s, P ro p o sitio n 2.2 im p lies th a t I — /{ [ ( / - / ) D + f í| G VV'(X) and th ere e x ists a gen eralized in vertib le operator Wq = I + /?[(/ — i4)D + B\ Therefore, (2.9) obtaines that all solutions of (2 ) axe g iv e n b y X = { J + /ỈM^ì4, b [ ( / - -4)-D + f t Ị } ( /t y 4- z ) + T h e in itia l v a lu e p r o b le m S u p p o se th a t 1) £ I Ỉ ( X ) , d im k e rD ^ 0; -F is a n in itia l o p e to r for D c o rresp o n d in g to I t £ 1Z ị)\ a n d A , B £ L o ( X ) , w ith A Ỷ n o n -in v e rtib le In th is sectio n , w e consider th e in itia l valu e p ro b le m fo r d e g e n e te sy ste m ( D S ) o f th e form : AD x= Bx + y F x = To T heorem J , (3.1) y € X Æ0 € k er D (3.2) Suppose th a t oil th e assum ptions o f Proposition 2.1 are satisfied and R y + X'o E { / - /? [ ( / — (i) I f A — B R y4)D + B ]} d o m D Then, we have: GI Ì { X ) a n d R a b € TZa - b r y th e n a ll s o lu tio n s o f th e p r o b le m (3 )-(3 ) axe given by x = { ỉ + H R a b ỉ - A ) D + B } } ( R y + x 0) + z , (3.3) where z € k e r { / — / ? [ ( / —A ) D + ] } (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 )-(3 ) are given by X = { / + /ỈL ^ B ÍƠ - i4 )D + « ị } ( i ỉ y + x o ) (3.4) (Hi) I f A — H R is invertible, then solution o f the problem (3.1)-(3.2) arc given by X = { I + R(A - i4)/J + B ]}(/ỉy + x 0) (3.5) T h e c o n tr o lla b ility o f d e g e n e r a te s y s te m 41 d e s c r i b e d b y (iv) I f A — B li (E W( X ) and wy\ ỊỊ £ y^A -B R (3.1)-(3.2) cue given by Ĩ then allsolutions o f the problem X = { / + Ỉ I W a j ỉ [Ự - A ) D 4- B ] } ( R y 4* Xo) 4- z , (3.6) Ì where z € kcr{7 - / ỉ [ ( / — )D + B]} Proof According; to the p ro o f o f T h eo rem 2.1, from (3 ) w e h a v e { / — lì.ịự - A )D + B ] } x = R y + z , z G k e r /J (3 ) Thus, acting on b oth sides of t his equality by operator 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 = 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 and there e x ists i t ’s g en era lised inverse 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 ave all so lu tio n s o f th e p ro b lem (3 )-(3 ) 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 S u p p o s e t h a t A B a re 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 rtib le T h e n , th e in itia l value p r o b le m (3 )-(3 ) 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 roof By th e a ss u m p tio n /1, ft a re s ta tio n a ry o p e to rs , A D — B = D ( A — f t/i) a n d (3.1) becomes D(A — B R )x = ty , this implies th at (/1 — H I Ỉ ) x = /fy -f co n d itio n (3.2) finds th a t = XQ M oreover, /t — £ € k e r/J The is in v e rtib le T h u s , th e so lu tio n of 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 le Suppose th a t X is the space («) o f all real sequences { x n }, n — , , , ■• ■ with addition and multiplication by scalars defined as follows: I f X = { i n } , y = {j/n }> A + B]}{Ry + x0) = {^ , ^0 - yo> Zo - Vo - Vì , ^0 - 2/0 - 2/1 - 2/2 , • • • }• N g u y e n D in h Q u y e ty H o a n g 42 Van T hi E x a m p le Suppose th a t X , Dy R and F are defined as in E xam ple 3.1; W rite i4 { x n } = {2 zo + x i , , X2 + X3 }X3 + X >• • • }, B { x n } = { x - Xo, 0, X4 - X2 , S - x ) • • • }• Clear, A ^ o and is non-invertible , sin ce ker i4 = {x o , —2xo, x ) —X2 , X2 , —£2, • • • } 7^ { } a n d ,4 X Ỷ X L et y = {?/„} € X and XQ = { x o } € ker D N ow we consider the degenerate system ( D S ) o f the form: (3 ) ADx = Bx + y F x = Xo , Xo € ker D (3 ) It is easy to verify th a t the resolving operator A — B R is generalized invertible, in d eed , (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 , X'2 , X3 , x , • • • }, x n € R , n = 2, 3, 4, • • • } B y (3.Ổ) , the solution o f the problem (D S ) is g ive n by X = { / + / ĩ [ ( / - A ) D + ổ ] } ( i ? y + x o ) 4- , Ỉ k e r { / - f i[ ( J - i4 ) D + B ] } = {xo, x + yo, 20 + yo + 2yi + x2,x0 + yo + 2t/i +2y2 + đ3ằ"* } C o n tro lla b ility o f th e d e g e n e te s y s te m Let X and u be linear spaces over the same field T [ T = R or C ) Suppose th at D € R ( X ) } d im k e r ữ ^ ] F e T o is an in itia l op erator for D co rresp o n d in g to R € 7Zd\ and A, R € L ()(X ), w ith A Ỷ n oil-in vertib le a n d c € Lo( U} X ) W e co n sid er the prob lem (£>S)o: f ADx = B x + Cỵz , w ith c o n d itio n JĨC Ơ © {xo} c { / — / ỉ [ ( / — >4)D + B )} d o m D (4 ) \ Fx = Xo , x € ker D (4 ) T he spaces X and [/ are called the space of states and the space of 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 sta te A pair ( x o , u) G (ker D) X u is called an in put In sectio n 3, w e have proved th a t th e p rob lem (4 )-(4 ) is eq u iv a len t to th e eq 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 sufficient co n d itio n for th e p rob lem (4 )-(4 ) have so lu tio n for ev ery u G V D en ote by i) = k er D (4.11) (Hi) T h e sy ste m ( D S ) is said to be Ỉ ) - controllable to XI € ker D i f Xi F i(R a n g UtXo + /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 ) is i s - c o n tro lla b le i f a n d o n ly i f ker C * R * T * F * = {()} (4.19) Proof It is sufficient, to sid er th e ca se = N o te th a t in all th e ca ses 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 ) is eq u iv a le n t t o F iT J iC U = k e rD (4.20) T h e a s s u m p tio n R C U © {xo} c { / — / ỉ [ ( / — i4 )D + /^1} d o in , im p lies th a t F aT4RCƯ = F M R C U © {x*o}) - {F T4 X0} c F4T4{/ - /i[(/ - /1)0 + B]}domD ~ {F4T4®o} c F {T4 {7 - /ỈỊ(/ - /i)D + B]}dọm D © k e r{ / - /ỉ[(J - i4)jD + B]}} - {F 4T4X0} - F4 ker{/ - /?[(/ - A)D + tì}} = / ‘4 d o m /) ~ { ^ 4^ } - F k e r { / - / i [ ( / - / ) / ; + /ỉ]} c k e r D N g u y e n D in h Q u y e tf H o a n g V a n T h i 46 B y (4 ), w e h ave F 4T 4R C U = r ^ d o m D —ịr^T /iX o }—F k e r { —/ ? [ ( / —A ) D + B]} = k e r D T h u s, /V A /ĩC Ơ - { K t T 4x o H F k e r { / - i l [ ( / - i ) j D + B ] } = A d o m D = ker D T h is m ean s th a t for e v e ry X\ € ker D i th e re e x is t s V G d o m D , u £ Ư a n d € k e r { — / ? [ ( / — i ) D 4- B ] } su ch th a t Xi = F V = F4 T RC U 4- F T XQ 4- -£4*0 = 4- Xo) + 20], i.e X\ is F 4- reachab le from Xo T h e a rb itra rin ess o f Xo, X\ € ker D im p lies th a t F 4( R angt/,Xo$ 4) = ker£> C onversely, su p p o se th a t F 4( Range; Xo^ 4) = k e r D C h o o sin g £0 = ,z o = 0, we g et th a t F 4T 4R C U = ker D T h e p ro o f is com pleted C o r o ll a r y S uppose th a t A , 2? are stationary operators T hen th e system ( D S )0 is F - controllable if and only if ker c * R * ( A - B R ) * ~ ' = { } (4.21) T h e o r e m S uppose th a t th e sy ste m ( D ) o is F i- controllable T h en , it is jF/- con- trollable for every initial o perator F[ G T o Proof L et F i b e an in itia l o p era to r for D correspon d in g to jR € 7^ 0, i-e - Ỉ*\ỈU = O n th e o th er h a n d , for ev ery £1 € k e r D an d V £ X , th ere e x ists X2 £ ker D su ch th at x x = T'2 B y our a ssu m p tio n th e sy ste m ( D S )0 is F t- co n trollab le T hus, for every x 0i %2 € ker D ị th ere e x ists a co n tro l U G Í / and Zo € k er{7 — jR [(/ — A ) D + J5]} su ch th at F i[T i(R C u + xq) + Zo] = x , or Fi[T t ( / ỉ u + æ0) + Zo] = F t ( x + Rị v) , for som e V € X Hence, F '[T i(R C u + xo) + zq] = F '( x + jRiv) = £2 + F 'R iV = X\ T he arbitrariness of 3?0, Xi € ker Đ , th e p ro o f is co m p lete d T h e o r e m 4 L et he given a degenerate system ( D S ) a n d an initial operator \ £ T p T h e n , the sy ste m ( D S )0 is F t- controllable i f and only i f it is Fi- controllable to every element v' e F jTiR X Proof F ir st, w e p rove th e eq u a lity : F {T 4( i ỉ X © ker D ) + k e r { I - / ì ị ( / - A )D + £f]>> = ker D (4.22) In d eed , sin ce { / - /? [ ( / — Ẩ ) D + ổ ] } d o m D c d o m = R X © ker D , th ere E c X an d ex ists c ker D su ch th a t f t # © z ' = { / - /?[(/ - ,4 )D + B] } d omD T h is im plies th a t Ta( R E ® z ' ) + kerự - R [ ự - A )D + B}} = \ { I - / i [ ( / - A ) D + B ]} d o m D © k e r { / - R [ { I - /1)1) + B } } = dom D H ence, ker D = F td o m D = F 4{T 4( / Ỉ E © z ' ) + k er{7 - / ĩ [ ( / - >1)D + B ]} } c F {T4(flX © ker D) + k er{ / - /ỉ[ ( / - Ấ )D + B]}} c ker Ơ, i.e th e form ula (4 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 F -c o n tro lla b le to e v e ry elem ent v' G F T RV, V £ X , i.e th e re e x is t s a c o n tro l UQ € u a n d 20 € k e r{I — /ỉ[ ( / — A ) D + J5]} su ch th a t i*4[ T ^ R O uq 4* xq ) 4* zq] = F 4T 4/ỴV T h e c o n tr o lla b ility o f d e g e n e r a te s y s te m d e s c r ib e d b y 47 T h i s im p lie s th a t f 4A{ A [noUo X0 + X2Ị 4- ZQ 4" Z\} = { / 4( ỈỈV 4* X ) 4- w here G k e r { / — /i![(/ — /4 )/J 4- ft]} a n d X2 € ker D are }, (4 ) arbitrary B y form ula (4 2 ), for ev ery X\ G ker D , there e x ists j € k e r { - / < [ ( / - / l ) D + / i ] } , v' G X a n d x*2 € ker D such th a t X\ = /^4 (7 ^4 + x f2 ) 4- z[] T h is e q u a lity and (4 ) I‘\[T/[(RC a '0 + XQ + x l2) + Zo + z[} = X\ (4 ) O n the oth er hand, th e co n d itio n € F 4T R X an d our a ssu m p tio n im p ly th at ( l ) S )0 is F 4- trollab le to zero, i.e € Fii( R a n g (;To0 ) for ev ery XQ £ ker D T h ere fore, th ere ex ists U\ € u and Z2 £ k o r { / — / ỉ [ ( / — i4 )D + ft]} su ch th a t F ị7 ,4 ( / Ỉ C u - ) + * 2} = 0- ( / 5) I f w e a d d (4 ) and (4 ), th e n F^[T^(RCu + Xo) 4* } = X\ , w h e re ,3 = ZQ -f z[ -f 22 r u — W() 4- U i T h e a rb it rin e s s o f Xo, X j g ive s Jp4 ( R a n g í/ xo$ ) = ker /J Suppose that X is the space (s) o f all real sequences {.Tn }, n = , ,2 , • • • E x a m p l e W rite D { x n } = {•/'T1+1 #n }, / ỉ { x n } = { y n } , yo = 0, y n = n r =0 ^ ajld /•’{.7V,} = { X ) C o n sid e r th e degenerate system ( DS) o o f the form: A D x = B x 4- C u F x = Zo , , u € Ỉ/ x € ker D (4 ) (4 ) We have A - H R € i y ( x ) , / € W ^ f l / 2, sin ce (A - # / ỉ ) i ( / l - £ / ỉ ) { x n } = (A - B R ) { x n } Moreover , k e r { / - / ỉ [ ( / - i4 )D + £ ] } = { { , ,a;2 ,a:3 ,X ,* • • },X n € R ,n = ,3 ,4 , • • • } • Therefore, by the form ula (4.7), th e solution o f the problem ( D S ) is g iv en by (2 , w) = { / + / ỉ ị ( / — / i ) D + B ] } ( R C u + Xo) + i , £ € k e r { / - / ỉ [ ( / - / l ) D + ỡ ] } = { Xo, Xo + Uo,Xo + TX0 + tii + X2 ,XQ + u0 4- 2ìXi + Ỉ3 > • • • }• (4.31) Let í be defined by F 4{ x n } = {a?i} T hen for every X() = { x o } € k e r D , there exists ũ = { - X o , U ị , 0 ,0 , • • • } € such th a t F /ị^ (x Q }ĩio) = { ,0 , 0, • • • } T h u s , the system ( D S )0 is /' 4- controllable to zero M oreover , / ' (T ker I) + k c r { - « [ ( / - i ) ü + £ ] } ) = ke r D , w it h r = / + iỊ( - ) D + fí) H y the Theorem 4.1.the sy ste m ( f ) S )0 is i*4- controllable R eferen c es N gu yen D in h Q u yet, O n linear sy ste m s d escrib ed by right in v ertib le op era to rs a ctin g in a linear space, Control and Cybernetics (1 ), 33 - 45 N gu yen D in h Q-uyet, C on tro lla b ility an d ob servab ility o f linear s y ste m s d escrib ed by th e right in vertib le op erators in lin ear sp ace, P rep rin t N o 113, In stitu te o f M athem atics, Polish Acad Sci y Warszawa, 1977 N g u y en D in h Q u y et, On the F \- controllability of the system described by the right invertible operators in linear spaces , M eth o d s o f M a th em a tic a l P rogram m in g, S ystem R esearch I n s titu te , P o lish A cad Sci., P W N - P o lish S cientific P u b lish e r, W arszaw a 1981, 223- 226 N guyeii Van M ail, C on tro lla b ility o f gen eral linear sy ste m s w ith right in vertib le op erators, p re p rin t N o 472 Institute o f M athem atics y P o lis h Acad S c i.j W arszawa, 1990 N g u y en Van M au, B ou n d ary valu e p rob lem s and co n tro lla b ility o f linear sy ste m s w ith rig h t invertib le o p e to rs, D issertationes Math., C C C X V I j Warszawa, 1992 A.Pogorzelec, ” Solvability and controllability of ill-determined system s w ith right invertible operators” , Ph.D.Diss., Institute of Mathematics, Technical University of Warsaw, Warszawa , 1983 D P rezew orsk a - R olew icz, Algebraic A nalysis , P W N an d R eid el, W arszaw a- D or drecht, 1988 ... issertationes Math., C C C X V I j Warszawa, 1992 A.Pogorzelec, ” Solvability and controllability of ill-determined system s w ith right invertible operators” , Ph.D.Diss., Institute of Mathematics,... Suppose that- nil a ssu m p tio n s o f L em m a 4/2 arc satisfied ircncrnt.fi system ( D S )0 is Then the d e- - controllable Proof Suppose; that /1 — B Ji € li7(X ) By our assum ption there exists... Warszawa, 1977 N g u y en D in h Q u y et, On the F - controllability of the system described by the right invertible operators in linear spaces , M eth o d s o f M a th em a tic a l P rogram