DSpace at VNU: The Approximate Controllability for the Linear System Described by Generalized Invertible Operators

DSpace at VNU: The Approximate Controllability for the Linear System Described by Generalized Invertible Operators tài l...

V N U J O U R N A L O F S C IE N C E , M a t h e m a tic s - Physics T.xx, Nq3 - 2004 TH E A PPR O X IM A TE C O N T R O LLA B ILITY FO R T H E LIN EA R SY STEM D E S C R IB E D BY G EN ER A LIZED IN V E R T IB L E O P E R A T O R S H oang V an T hi Hong D ue University A b s t r a c t In this paper, we deal with the approximate controllability for a linear system described by generalized invertible operators in the infinite dimensional Hilbert spaces K e y w o r d s : Right invertible and generalized invertible operators, alm ost inverse operator, initial operator, right and left initial operators, initial value problem I n t r o d u c t i o n \ The theory of right invertible operators was s ta rte d w ith works of D PrzeworskaRolewicz and then has been developed by M Tasche, H von T ro th a , z Binder m an and many other M athematicians By the appearance of this theory, the initial, bo un dary and mixed boundary value problems for the linear system s described by right invertible op­ erators and generalized invertible operators were studied by m an y M athem aticians (see [4, 8]) Nguyen Dinh Quyet considered th e controllability of linear system described by right invertible operators in the case when the resolving o p erato r is invertible (see [10, 12, 13]) These results were generalized by A Pogorzelec in th e case of one-sized invert­ ible resolving operarors (see [6, 8]) and by Nguyen Van M au for the system described by generalized invertible operators (see [3, 4]) T h e above m entioned controllability is exactly controllable from one state to another However, in infinite dimensional space, the exact controllability is not always realized To overcome these restrictions, we define the so-called approximately controllable, in th e sense of: ” A system is approximately controllable if any state can be transferee! to th e neighbourhood of o th er sta te by an ad ­ missible control” In this paper, we consider the app ro xim ate controllability for th e system (LS)o of the form (2.1)-(2.2) in infinite dimensional H ilbert space, w ith dim (ker V) = 4- 0 The necessary and sufficient conditions for the linear system ( L S ) to be approximately reachable, approxim ately controllable and e x a c tly trollab le are also found P r e l i m i n a r i e s Let X be a linear space over a field of scalars T ( T = R or C) Denote by L ( X ) the set of all linear operators with domains and ranges belonging to X , and by L q( X ) the set of all operators of L ( X ) whose domain is X , i.e Lq(X) = { A e L ( X ) : domẨ = X } An operator D E L ( X ) is said to be right invertible if th ere exists an R € L q( X ) such th a t R X c dom D and D R — I on dom i? (where I is th e identity operator), in this T ypeset by 50 The a p p r o x i m a t e c o n t r o l l a b i l i t y f o r th e l i n e a r s y s t e m d e s c r i b e d by 51 case R is called a right inverse of D T h e set of all right invertible operators of L ( X ) will be denoted by R ( X ) For a given D e R { X ) , we will denote by 11D the set of all right inverses of D, i.e 7Z o — { R € L o ( X ) : D R = I} An operator F e L o ( X ) is said to be an initial operator for D corresponding to R e 1I d if F = F, F X = kei'jD and F R = on dom R The set of all initial operators for D will be denoted by T-D' T h e o r e m 1.1 [8] S uppose th a t D e condition for an operator F e L ( X ) to R ( X ) and R e 1ZD- A necessary and sufficient be an initial operator for D corresponding ỉo R is that F = I —RD on dom ữ (1.1) D e f in itio n 1.1 [4, 5] (i) An operator V G L ( X ) is said to be generalized invertible if there is an operator w G L ( X ) (called a generalized inverse of V ) such th a t Im V c dom w , Im w c d o m V and V w v = V on dom K The set of all generalized invertible operators of L ( X ) will be denoted by W ( X ) For a given V e W ( X ) , th e set of all generalized inverses of V is denoted by W y (ii) If V € W ( X ) , w G W v an d w v w = w on dom w , then w is called an almost inverse of V T h e set of all alm ost inverse operators of V will be denoted by W y D e f in itio n 1.2 [4] (i) An operator F (r) G L ( X ) is said to be a right initial operator of V e W ( X ) corresponding to w € vvịr i f ( F ( r) ) = F ( r ) , I m = kerV, d o m F ( r) = d o m y and F ^ W = on m W (ii) An operator e L o ( X ) is said to be a left initial operator of V € W ( X ) corre­ sponding to w W y if ( F ^ ) = f W , F W X = kerw and F ® v = on domV T he set of all right and left initial operators of V € W (X ) are denoted by and T y , respectively L e m m a 1.1 [4] Le t V G W ( X ) ãnd w W y Then d o m V = W V ( d o m V ) © kerV T h e o r e m 1.2 [4] Let V G W ( X ) and let w G W y (i) A necessary and sufficient condition for an operator F e operator o f V corresponding to w is th a t F — I —w v (ii) A necessary and sufficient condition for an operator € operator o f V corresponding to w is th a t = I —v w (1.2) L ( X ) to be a right initial on d o m V L q ( X ) to be a left initial on dom w 52 H o a n g Van Thi T h e o r e m 1.3 [14] L et X , Y , Z be the infinite dimensional H ilbert spaces Suppose that F £ L ( X , z ) and T £ L (Y , Z ) Then two following conditions are equivalent (i) ImF c Im T , (ii) There exists c > such that ||T * /|| ^ c\\F*f\\ for all f e z* (where z* is the conjugate space o f Z ) T h e o r e m 1.4 (The separation theorem) Suppose th a t M a n d N are convex sets in the Banach space X and M n N = (i) I f intM Ỷ then there exists a X * € X * , x * / 0, A R such th a t (ii) I f M is a compact set in X , N is a closed set then there exists X* G Ai, À2 g R such that (x*,x) ^ Ai < À2 ^ for every X Ỷ G M, y € N The theory of right invertible, generalized invertible o p erato rs and their applications can be seen in [4, 8] The proof of Theorems 1.3 and T heorem 1.4 can be found in [2, 14] A p p r o x i m a t e c o n t r o l l a b i l i t y Let X and u be infinite dimensional Hilbert spaces over th e same field of scalars T [ T — M or C) Suppose th a t V E W ( X ) i with d im (k e ry ) = +oo; and are right and left initial operators of V corresponding to w G W y , respectively; A € L o ( X ) i and B e L 0( U, X) Consider the linear system ( L S ) of the form: Vx = Ax + B u , u € F ^ x = Xo , XQ u , BU G ke rV c ( V - ) d o m ^ ( 1) ( 2 ) The spaces X and u are called the space o f states and th e space of controls, respec­ tively So th at, elements X G X and u € u are called states and controls, respectively The element £o € kerV" is said to be an initial state A pair ( x q , u ) G (kerV) X u is called an input If the system (2.1)-(2.2) has solution X = G( x o , u ) th en this solution is called output corresponding to input ( x o , u) Note that, the inclusion B U c (V —A ) d o m V holds If th e resolving operator I - W A is invertible then the initial value problem (2.1)-(2.2) is well-posed for an arbitrarily fixed pair (xo, u) G (kerV) X u , and its unique solution is given by (see [Mbou]) G ( x , u) = E a (W B u + Xo) , where E a = ( I — w A ) (2.3) Write Rangt/iIOG = Ị J G ( x 0, u) , u£U x € kerF ( ) 53 The a p p r o x i m a t e c o n t r o l l a b i l i t y f o r th e l i n e a r s y s t e m d e s c r i b e d by Clearly Range/.;,;0G is th e set of all solutions of (2.1)-(2.2) for arbitrarily fixed initial st ate /•() Gk c v V This is reachable set from the initial state X() by moans of controls //, e u D e f in itio n Let the*linear system ( L S ) of the form (2.1) - (2.2) ho given Suppose' that G{.V(),ti.) is (Ic'fiiK'd by (2.3) (i) A stat e r £ X is said to he approximately reachable from the initial state X'o £ kerV if for any £ > there exists a control a E u such that 11;r — G(j*o, u)|| < £ (ii) The linear systom ( L S ) is said to he approximately reachable from the initial state /•() G krvV if R a n g U, X0 G = X • T h e o r e m T h e linear system ( L S ) is approxim ately reachable from zero if and only if the identity B*W*E*Ah = 0, it implies h =0 (2.5) Proof By Definition 2.1 th e system (LS)o is approximately reachable from zero if EaWDU = X According to T h ro iriii 1.4 th e condition (2.G) is (/?,;!:) = , V.T G E a W B U , (2.6) equivalent to th e tiling t hat if h £ A* it follows h, = (2.7) SÌIKV E \ W B Ư is a subspace of X , (2.7) holds it and only if th at (//.,:/:) = , V;/: e E A W B U E A W V { d o m V ) Proof By f [ t ^Ea W V { dom V) c kerF , the necessary condition is easy to be obtained To prove the sufficient condition, we prove the equality f [ t)E A [ W V ( d o mV ) ® kerV] = kerV (2.12) Indeed, since ( / - W A ) { d o m V ) c d o m F = W V (d om V ) © k e ĩV (by Lem m a 1.1 and the properties of the generalized invertible operators [Mcon, Mbou, Mai]), there exists a set E c domV and z c kerV such th a t W V E © z = Ự - WA){domF) This implies E A { W V E © Z ) = E A ( I - Ịy A )(d o m F ) = domV Thus, we have kerF = f [t\ domV) = f [t)Ea ( WVE © z) c E a [ W V (domV) © kerF] c k e iV Therefore, the formula (2.12) holds H o a n g Van Thi 56 Suppose th a t the system ( L S ) is F ị T^-approximately controllable to y' — f [ ^ E a W V y G doniV, i.e for every y G dom V and a rb itra ry e > th ere exists a control Uo G u such th at \\f [t)E a ( W B u0 + xo) - Fị r)E AWVy\\ < I T h a t is ||F 1(r)£ 4(W/-£'U0 + xo + x 2) - F[ T)E A { W V y + x 2)|| < I where (2.13) X2 G kerV is arbitrary By the formula (2.12), for every X\ G ker V, there exists 2/1 £ dom V and xi) € kerV such that xx = f [ t)E A ( W V Vl + x ’2 ) This equality and (2.13) together imply IIF ị r)E A{WBu'ữ + x ữ + x'2) - X l \\ < ị (2.14) E A W V d o m V and th e assum ptions, it follows th at On the other hand, from G ( L S ) is F i 7^-approximately controllable to zero, i.e G Fị \ R m \ g u XoG ) , for arb itrary Xo G k e r V Thus, for the element x'2 € kerK there exists U\ G u such th a t -z'aJII < | (2.15) Using (2.14) and (2.15 then for X0 ,X \ G kerV" and Ổ > there exist u = u'Q+ U\ £ u so that + X0 ) -X ! || = \\F[r)E A [W B(u'0 + m ) + x 0] - Xi II = \\F[r)E A (W B u 'ữ + xo + x'2) - X! + F[ r)E A { W D u - x£)|| ^ + + ) - XIII + < - x'2)|| £ + = £ - Thus, F• h = 0, or equivalently (h, f [ t )E a W B u ) = , Vu e Ỉ/ =► h = It is satisfied if and only if (.B*W*E*A {F[r)) * h ,u ) = , Vu e t / =► /1 = (2.19) Hence, the condition (2.19) m eans t h a t B*W*E*A (F[r)Ỵ h = implies h = Conversely, if (2.16) is satisfied th en (2.19) holds This implies (2.17) and therefore we obtain (Range/,0G) = k e r T h e o r e m A necessary and sufficient condition for the linear system ( L S ) to be -controllable is th a t there exists a real num ber a > such th a t \\B*W*E*A{Fịr)y f \ \ > «11/11, for all f € (kerVO* ■ Proof Necessity Suppose th a t th e system ( L S ) is jF\(r) (R a n g y XoG) = kerK , ^-controllable, we (2-20) have for every x e k e i V It follows t h a t f [t )E a W B U = kerV By T heorem 1.3, there exists a real number a > Osuch th a t \\{Fịr)E A W B Ỵ Ị \ \ > ck||/|| , for all / € (kerF )*, i.e the condition (2.20) holds Sufficiency Suppose t h a t th e condition (2.20) is satisfied By using Theorem 1.3, we obtain F[ r)E A W B U D kerV" Moreover F 1(r)E A W B U c kerV Since we have f [ t)E a W f Ịt) is a right initial operator for V Consequently, B U = kerV T his implies f \ (r) (Range/,X0G) = k e ĩV , for x € kerV 58 H o a n g Van Thỉ T h e o r e m 2.6 The linear system ( L S ) is exists /3 > such that f [v)-controllable \\B*W*E*A (F[r))*f\\ > (3\\E*A ( F ịr)) * f \ \ , for every to zero if and only if there f e (kervy (2.21) Proof Suppose th a t the system (LS)o is F ^ -c o n tr o lla b le to zero We then ha,ve 0e (Rang{/iXoG ) , for all x k e rF Therefore, for arbitrary x € kerV, there exists u e u such th a t Fị r)E A{ WB u + x o) = It implies th a t for x ' qe Thus, F j 7' ^ ( k e r F ) c kerw , there exists v!