APPROXIMATE CONTROLLABILITY OF DELAYED SEMILINEAR CONTROL SYSTEMS

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1 APPROXIMATE CONTROLLABILITY OF DELAYED SEMILINEAR CONTROL SYSTEMS LIANWEN WANG Received 22 Jnury 2004 nd in revised form 9 July 2004 We del with the pproximte controllbility of control systems governed by delyed semiliner differentil equtions ẏ(t) = Ay(t)+A 1 y(t )+F(t, y(t), y t )+(Bu)(t). Vrious sufficient conditions for pproximte controllbility hve been obtined; these results usully require some complicted nd limited ssumptions. Results in this pper provide sufficient conditions for the pproximte controllbility of clss of delyed semiliner control systems under nturl ssumptions. 1. Introduction The min concern in this pper is the pproximte controllbility of the following delyed semiliner control system: ẏ(t) = Ay(t)+A 1 y(t )+F ( ) t, y(t), y t +(Bu)(t), t, y = ξ, (1.1) in rel Hilbert spce X with the norm. The mening of ll nottions is listed s follows: 0 is system dely; y( ):[,b] X is the stte function; ξ C([,0];X), the Bnch spce of ll continuous functions ψ :[,0] X endowed with the norm ψ =sup{ ψ(θ) : θ 0}; A is the genertor of C 0 semigroup T(t) inx;a 1 is boundedlineropertorfromx to X; F :[,b] X C([,0];X) X is nonliner opertor; u( ) L 2 (,b;u) is control function; U is Hilbert spce; B is bounded liner opertor from L 2 (,b;u) tol 2 (,b;x). In ddition, for ny y C([,b];X) nd t [,b], define y t C([,0];X)byy t (θ) = y(t + θ)forθ [,0]. Denote the stte function of (1.1) corresponding to control u( ) by y( ;u). Then y(b;u) is the stte vlue t terminl time b. Introduce the set R b (F) = { y(b;u):u( ) L 2 (,b;u) }, (1.2) which is clled the rechble set of system (1.1) tterminltimeb, its closure in X is denoted R b (F). Copyright 2005 Hindwi Publishing Corportion Journl of Applied Mthemtics nd Stochstic Anlysis 2005:1 (2005) DOI: /JAMSA

2 68 Approximte controllbility of delyed systems Definition 1.1. The system (1.1) is sid to be pproximtely controllble on[,b] if R b (F) = X. The following system is clled the corresponding liner system of (1.1): ẏ(t) = Ay(t)+A 1 y(t )+(Bu)(t), t, y = ξ. (1.3) This is specil cse of (1.1) withf 0. The rechble set of system (1.3) tterminl time b is denoted R b (0). Similrly, system (1.3) is sid to be pproximtely controllble on [,b]ifr b (0) = X. For semiliner control systems without delys, pproximte controllbility hs been extensively studied in the literture. We list only few of them. Zhou [10] studied the pproximte controllbility for clss of semiliner bstrct equtions. Nito [6] estblished the pproximte controllbility for semiliner control systems under the ssumption tht the nonliner term is bounded. Approximte controllbility for semiliner control systems lso cn be found in Choi et l. [1], Fernndez nd Zuzu [2], Li nd Yong [4], Mhmudov [5], nd mny other ppers. Most of them concentrte on finding conditions of F, A, nd B such tht semiliner systems re pproximtely controllble on [,b]if the corresponding liner systems re pproximtely controllble on [,b]. For semiliner delyed control systems, some ppers re devoted to the pproximte controllbility. For exmple, Klmk [3] provided some pproximte controllbility results. Nito nd Prk [7] delt with pproximte controllbility for delyed Volterr systems. In [9] Ryu et l. studied pproximte controllbility for delyed Volterr control systems. The purpose of this pper is to study the pproximte controllbility of control system (1.1). We obtin the pproximte controllbility of system (1.1)if the corresponding liner system is pproximte controllble nd other nturl ssumptions such s the locl Lipschitz continuity for F nd the compctness of opertor W re stisfied. 2. Bsic ssumptions We strt this section by introducing the fundmentl solution S(t) ofthefollowing system: ẏ(t) = Ay(t)+A 1 y(t ), t, y = ξ. (2.1) We lredy know tht (2.1) hs unique solution, denoted by y ξ (t), for ech ξ C([, 0];X). Hence, we cn define n opertor S(t)inX by y ξ (t + ), t 0, S(t)ξ(0) = 0, t<0. (2.2)

3 Linwen Wng 69 S(t) is clled the fundmentl solution of (2.1). It is esy to check tht S(t) is the unique solution of the following opertor eqution: S(t) = T(t)+ T(t s)a 1 S(s )ds. (2.3) 0 Let K := mx{ T(t) :0 t b}.by(2.3)wehve S(t) K + K A 1 S(s ) ds K + K t A 1 S(s) ds. 0 (2.4) Gronwll s inequlity implies tht S(t) ( ) K exp K A (b ) := M1, 0 t b. (2.5) Throughout the pper we impose the following condition on F. (H1) F :[,b] X C([,0];X) X is loclly Lipschitz continuous in y, η uniformly in t [,b]; tht is, for ny r>0, there is constnt L(r)suchtht F ( ) ( ) ( t, y 1,η 1 F t, y2,η 2 L(r) y 1 y 2 + ) η 1 η 2 (2.6) for ny t [,b], y 1 r, y 2 r, η 1 r,nd η 2 r. With minor modifiction of [8], we cn prove tht system (1.1) hsuniquemild solution y( ;u) C([,b];X) for ny control u( ) L 2 (,b;u) underssumption (H1). This mild solution is defined s solution of the integrl eqution: y(t;u) = S(t )ξ(0) + S(t s) [ F ( ) ] s, y(s;u), y s +(Bu)(s) ds, t, (2.7) y = ξ. Similrly, for ny z( ) L 2 (,b;x), the following integrl eqution: x(t;z) = S(t )ξ(0) + S(t s) [ F ( ) ] s,x(s;z),x s + z(s) ds, t, x = ξ, (2.8) hs unique mild solution x( ;z). Therefore, we cn define n opertor W from L 2 (,b;x) to C([,b];X)by Regrding the opertor W, we ssume tht (H2) W is compct opertor. (Wz)( ) = x( ;z). (2.9) Remrk 2.1. (H2) is the cse if, for instnce, T(t), the semigroup generted by A, is compct semigroup. The following ssumption (H3) ws introduced by Nito in [6]. Define liner opertor ϕ from L 2 (,b;x)tox by ϕp = S(b s)p(s)ds for p( ) L 2 (,b;x). (2.10)

4 70 Approximte controllbility of delyed systems Let the kernel of the opertor ϕ be N; tht is, N ={p : ϕp = 0}. ThenN is closed subspce of L 2 (,b;x). Denote its orthogonl spce in L 2 (,b;x)byn.letg be the projection opertor from L 2 (,b;x)inton nd let R[B]betherngeofB. We ssume tht (H3) for ny p( ) L 2 (,b;x), there is function q( ) R[B]suchthtϕp = ϕq. Remrk 2.2. (H3)is vlid for mny control systems,see [6] for detiled discussion. It follows from ssumption (H3) tht {x + N} R[B] for ny x N. Therefore, the opertor P from N to R[B]definedby Px = x, (2.11) where x {x + N} R[B] nd x =min{ y : y {x + N} R[B]}, iswelldefined. It is proved in [6]thtP is bounded. 3. Lemms This section provides two lemms tht will be used to prove the min theorem. Lemm 3.1. Assume tht (t) is continuous on [,b], b(t) is nonnegtive nd integrble on [,b],nd x(t) is nonnegtive continuous function stisfying the following inequlity: If the eqution x(t) (t)+ b(s)x α (s)ds, 0 α<1, t [,b]. (3.1) y(t) = (t)+ b(s)y α (s)ds (3.2) hs unique solution ȳ(t) on [,b], then x(t) ȳ(t), t [,b]. (3.3) Proof. Let C[,b] be thebnchspceofllcontinuousfunctionson [,b] endowedwith the mximum norm. Define n opertor E from C[,b] toc[,b] by (Ey)(t) = (t)+ b(s)y α (s)ds. (3.4) Construct sequence {y n } s follows: We hve Note tht y 0 (t) = x(t), y n+1 (t) = ( Ey n ) (t), n = 0,1,... (3.5) x(t) = y 0 (t) y 1 (t), x = y 0 y 1. (3.6) Ey + y α 0 b(s)ds, (3.7)

5 Linwen Wng 71 then we cn find number d>0suchtht Ey y for y d. (3.8) If y n d holds for ny integer n = 0,1,...,then y n is bounded. Otherwise, it follows from (3.6)tht sufficiently lrge integer N exists such tht Thus Consequently, Therefore, y 0 y N d, y n >d for n>n. (3.9) mx { y n : n 0 } mx { d, y N+1 } := m. (3.10) 0 x(t) = y 0 (t) y 1 (t) y n (t) m. (3.11) lim n y n(y) = ȳ(t). (3.12) Note tht ȳ(t) is the unique solution of (3.2). The conclusion of Lemm 3.1 follows from (3.11). Lemm 3.2. Assume tht (H1) is fulfilled. Furthermore, for ny y X nd η C([,0];X) F(t, y,η) M ( 1+ y α + η α), 0 α<1, t [,b]. (3.13) Then the mild solution x(t;z) of (2.8) hs the estimte x t H ( z ), (3.14) where H(r) is n incresing function nd H(r) = O(r) s r. Proof. Recll tht It follows from x(s) x s nd (2.8)tht M 1 = mx { S(t) :0 t b }. (3.15) x(t) ( M1 ξ + M 1 M 1+ x(s) α + t x s α) ds+ M1 z(s) ds (3.16) M 1 ξ + M 1 M(b )+M 1 b z +2M1 M x s α ds. For ny θ [,0], we hve x(t + θ) M1 ξ + M 1 M(b )+M 1 b z +2M1 M +θ x s α ds. (3.17)

6 72 Approximte controllbility of delyed systems Hence x t M1 ξ + M 1 M(b )+M 1 b z +2M1 M Note tht for ny two constnts V 1 nd V 2, the following eqution x s α ds. (3.18) hs unique solution y(t) = Applying Lemm 3.1 to (3.18), we obtin y(t) = V 1 + V 2 y α (s)ds (3.19) 0 [ ] 1/(1 α). (1 α)v2 t + V1 1 α (3.20) x t [ 2(1 α)mm1 (b )+ ( M 1 ξ + M 1 b z + MM1 (b ) ) 1 α ] 1/(1 α) := H ( z ). (3.21) Clerly, the function H(r) stisfies ll requirements of Lemm 3.2 nd the proof of the lemm is complete. 4. Approximte controllbility The following theorem is the min result of this pper. Theorem 4.1. Assume tht liner system (1.3) is pproximtely controllble on [,b]. If (H1), (H2), (H3), nd (3.13) re fulfilled, then system (1.1) is pproximtely controllble on [,b]. Proof. Note tht system (1.3) is pproximtely controllble on [,b] by the ssumption, then R b (0) = X. To prove the pproximte controllbility of (1.1); tht is, R b (F) = X,itis sufficient to show tht R b (0) R b (F). (4.1) Tht mens for ny ɛ > 0ndx b R b (0), there exists ν R b (F)suchtht ν x b < ɛ. By the definition of rechble set R b (0)ofsystem(1.3), there is control u( ) L 2 (,b;u)suchtht x b = S(b )ξ(0) + S(b s)(bu)(s)ds. (4.2) Let z 0 = Bu, z 0 = G z 0.Thenz 0 N. Define n opertor J from N to N by Jv = z 0 GΓPv, v N, (4.3)

7 Linwen Wng 73 where Γ is the opertor from L 2 (,b;x)tol 2 (,b;x)definedby (Γz)(t):= F ( t,(wz)(t),(wz) t ) = F ( t,x(t;z),xt ). (4.4) For ny v N,wehvePv L 2 (,b;x), ΓPv L 2 (,b;x), nd GΓPv N. Therefore, J is well defined. Since W is compct by ssumption (H2), for ny bounded sequence z n ( ) L 2 (,b;x); tht is, z n r 1 for some r 1 > 0, there is subsequence z nk ( )ofz n ( )suchtht(wz nk )( ) converges to x( ) inc([,b];x) sk.so,wz nk is bounded in C([,b];X); tht is, Wz nk r 2 for some constnt r 2 > 0. (H1) implies tht constnt L(r) > 0 exists such tht F ( t, ( ) ( ) ) ( ) Wz nk (t), Wznk t F t, x(t), xt L(r) ( ( ) Wz nk (t) x(t) + ( Wz nk )t x ) t, where r = mx(r 1,r 2 ). Hence, we hve Γz nk F (, x( ), x ) 2 = F (, ( ) ( ) ) ) Wz nk ( ), Wznk F (, x( ), x 2 ( L 2 (r)(b ) sup t b ( Wz nk ) (t) x(t) +sup ( ) Wz nk t x 2 ) t 0 t b s k. Therefore, Γ is compct nd J is compct s well. From Lemm 3.2,fornyz( ) L 2 (,b;x), we hve (4.5) (4.6) F ( t,x(t;z),x t ) M ( 1+ x(t;z) α + x t α) M ( 1+2H α ( z )). (4.7) Note tht H(r)isincresingndP is bounded opertor, then z 0 GΓPv z 0 + GΓPv z 0 + M b +2M b H α( P v ). (4.8) Tking into ccount lim z 0 + M b +2M b H α( P v ) = 0, (4.9) v v then lim z 0 GΓPv = 0. (4.10) v v Therefore, we cn find sufficiently lrge number r such tht z 0 GΓPv r for v r. (4.11)

8 74 Approximte controllbility of delyed systems This mens tht J mps the bounded closed set D( r) ={v : v r,v N } of N into itself. Consequently, fixed point of opertor J exists due to the Schuder fixed point theorem; tht is, there is v D( r)suchtht On ccount of we hve Jv = z 0 GΓPv = v. (4.12) Pv ( v + N ) R[B], (4.13) S(b s)(pv )(s)ds = S(b s)v (s)ds. (4.14) Note tht G is the projection opertor from L 2 (0,T;X)intoN,thenwehve S(b s)gp(s)ds = S(b s)p(s)ds S(b s)(bu)(s)ds = = for p( ) L 2 (,b;x), S(b s) [ F ( s,x ( s;pv ),x s ) + v (s) ] ds S(b s) [ F ( s,x ( s;pv ),x s )+ ( Pv ) (s) ] ds. (4.15) Finlly, x b = S(b )ξ(0) + S(b s) [ F ( s,x ( s;pv ) ) (,x ) s + Pv (s) ] ds = x ( b;pv ). (4.16) Observe tht Pv R[B], then there is sequence u n ( ) L 2 (,b;u) suchthtbu n Pv s n. W is continuous due to its compctness, then This implies WBu n WPv in C ( [,b];x ). (4.17) x ( b;bu n ) x ( b;pv ) = x b (4.18) s n.sincex(b;bu n ) = y(b;u n ) R T (F), we obtin x b R T (F) nd complete the proofofthetheorem. Remrk 4.2. If A 1 = 0, (H3) implies the pproximte controllbility of (1.3)on[,b] (see [6]). Therefore, Nito s result in [6] isspecilcseoftheorem 4.1 when A 1 = 0, = 0, nd F(t,x(t),x t ) = F(x(t)). In prticulr, we improve Nito s result by wekening the uniform Lipschitz continuity nd the uniform boundedness imposed on the nonliner term.

9 5. Exmple Linwen Wng 75 Let X = L 2 (0,π)nde n (x) = sin(nx)forn = 1,...Then {e n : n = 1,2,...} is n orthogonl bse for X.DefineA : X X by Ay = y with domin D(A) = { y X : y nd y re bsolutely continuous, y X, y(0) = y(π) = 0 }. (5.1) Then Ay = n 2 y,e n en, y D(A). (5.2) n=1 It is well known tht A is the infinitesiml genertor of n nlytic group T(t), t 0, in X nd is given by T(t)y = e n2 t y,e n en, y X. (5.3) n=1 T(t) is compct becuse it is n nlytic semigroup. Define n infinite dimensionl spce U by { U = u : u = u n e n, n=2 n=2 } u 2 n < (5.4) with the norm defined by ( 1/2 u U = un) 2. (5.5) n=2 Define mpping B from U to X s follows: Bu = 2u 2 e 1 + u n e n. (5.6) n=2 Consider the following delyed semiliner het eqution: y(t,x) t = 2 y(t,x) x 2 + y(t,x)+f ( y(t,x), y(t,x) ) + Bu(t,x), 0 <t<b,0<x<π, y(t,0)= y(t,π) = 0, 0 t b, y(t,x) = ξ(x), t 0, 0 x π. (5.7) Then system (5.7) cn be written to the bstrct form (1.1). (H2) holds becuse T(t) is compct semigroup. Following the sme rguments s in [6]wecnprovetht(H3)is vlid nd tht the corresponding liner system is pproximtely controllble on [0, b]. By Theorem 4.1, system(5.7) ispproximtelycontrollbleon[0,b] iff is loclly Lipschitz continuous nd condition (3.13) is stisfied.

10 76 Approximte controllbility of delyed systems References [1] J. R. Choi, Y. C. Kwun, nd Y. K. Sung, Approximte controllbility for nonliner integrodifferentil equtions, J. Kore Soc. Mth. Educ. Ser. B Pure Appl. Mth. 2 (1995), no. 2, [2] L. A. Fernández nd E. Zuzu, Approximte controllbility for the semiliner het eqution involving grdient terms, J. Optim. Theory Appl. 101 (1999), no. 2, [3] J. Klmk, Controllbility of Dynmicl Systems, Mthemtics nd its Applictions (Est Europen Series), vol. 48, Kluwer Acdemic Publishers Group, Dordrecht, [4] X. Li nd J. Yong, Optiml Control Theory for Infinite Dimensionl Systems, Birkhuser, Boston, Msschusettes, USA, [5] N.I.Mhmudov,Approximte controllbility of semiliner deterministic nd stochstic evolution equtions in bstrct spces,siamj.controloptim.42 (2003), no. 5, [6] K. Nito, Controllbility of semiliner control systems dominted by the liner prt, SIAMJ. Control Optim.25 (1987), no. 3, [7] K. Nito nd J. Y. Prk, Approximte controllbility for trjectories of dely Volterr control system, J. Optim. TheoryAppl.61 (1989), no. 2, [8] A. Pzy, Semigroups of Liner Opertors nd Applictions to Prtil Differentil Equtions, Applied Mthemticl Sciences, vol. 44, Springer-Verlg, New York, [9] J.W.Ryu,J.Y.Prk,ndY.C.Kwun,Approximte controllbility of dely Volterr control system, Bull. Koren Mth. Soc. 30 (1993), no. 2, [10] H. X. Zhou, Approximte controllbility for clss of semiliner bstrct equtions, SIAMJ. Control Optim.21 (1983), no. 4, Linwen Wng: Deprtment of Mthemtics nd Computer Science, Centrl Missouri Stte University, Wrrensburg, MO 64093, USA E-mil ddress: lwng@cmsu1.cmsu.edu

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