Research Article Group Synchronization of Nonlinear Complex Dynamics Networks with Sampled Data

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1 Matematical Problems i Egieerig, Article ID 14261, 8 pages ttp://d.doi.org/1.1155/214/14261 Researc Article Group Scroizatio of Noliear Comple Damics Networks wit Sampled Data Ma Li, 1 Bo Liu, 2 Yuqig Zu, 1 Liju Wag, 3 ad Mei Zou 2 1 Matematics ad Psics, Naag Istitute of Tecolog, Naag, Hea 4734, Cia 2 College of Sciece, Nort Cia Uiversit of Tecolog, Beijig 1144, Cia 3 Scool of Ecoomics ad Maagemet, Beijig Uiversit of Tecolog, Beijig 122, Cia Correspodece sould be addressed to Bo Liu; boliu@cut.edu.c Received 6 December 213; Accepted 5 Februar 214; Publised 24 Marc 214 Academic Editor: Huaiceg Ya Coprigt 214 Ma Li et al. Tis is a ope access article distributed uder te Creative Commos Attributio Licese, wic permits urestricted use, distributio, ad reproductio i a medium, provided te origial work is properl cited. Based o a oliear cosesus protocol, tis paper cosiders te group scroizatio of comple damical etworks wit sampled data. Usig te Lapuov metod, te group scroizatio of te oliear comple etworks is aalzed. All te odes i eac group ca coverge to teir ow scroous state asmptoticall, if te sampled period satisfies some matri iequalit coditios. Furtermore, te teoretical results are verified b some simulatios. 1. Itroductio Because of te wide applicatio of te comple damical etworks 1 26, te scroizatio problem of te comple etworks as become a ot topic recetl. A lot of researces cocetrate o te cotiuous iformatio trasmissio; tat is, ever ode ca receive or detect eigbor iformatio of te etwork all te time 1 3, 5 7, 1. I 1, te autors cosidered te group scroizatio of te oliear comple damical etwork via piig cotrol. A ew criterio for te cluster scroizatio of couplig etworks wit time-varig delas was establised i cotiuous time i 2. Moreover, Liu et al. 3 cosidered te adaptive scroizatio of comple damical etworks wit switcig topolog b local Lipscitz oliearit. I 5, group cosesus of multiaget sstems wit switcig topologies ad commuicatio delas was studied i te cotiuous-time case. However, i ma egieerig practices, te iformatio commuicatio betwee odes could be iterrupted at a time due to some factors, suc as te ureliabilit of commuicatio caels ad te limitatios of te ode detectio abilit. Hece, it is ecessar to cosider te iformatio trasmissio uder te discrete state 9, 13 16, 18 21, 23, 26. Iformatio data o te discrete time ca also be regarded as sampled data 11, 25. I 11, te group cosesus of liear multiaget sstems wit sampled-data was cosidered. Xiao ad Wag furter cosidered a discrete-time model wit time delas i 13 ad aalzed a cosesus problem i te eistece of te time dela we agets ecaged iformatio betwee eac oter. Gao ad Wag 16 studied te cotiuous-time cosesus of multiaget sstems wit sampled data b time-varig topolog. I 18, cosesus of multiple damic agets wit sampled iformatio was cosidered. I 2, average cosesus cotrol of etworks wit sampled data ad measuremet oises was studied ocotiuoustime.itispaper,wecosidertegroup scroizatio of oliear comple damical etwork wit sampled data. Te rest of tis paper is orgaized as follows. I Sectio 2, we cosider te group scroizatio of a comple damicaletworkwitsampleddataadgivesomeassumptios ad lemmas. I Sectio3, we aalze group scroizatio ofteproposedetworkadgivetescroizatiocoditio based o liear matri iequalit (LMI). I Sectio 4, we give te scroizatio coditio for a special case. I Sectio 5, te simulatios verif te teoretical results. Coclusio is fiall summarized i Sectio 6.

2 2 Matematical Problems i Egieerig 2. Prelimiaries ad Problem Statemet Cosider a comple damical etwork of +modes wit sampled data as follows: ca 11 =ca ij R ad ca 22 =ca ij R m m represet te couplig cofiguratio of te subgroups, respectivel. Network (1) is said to group scroizatio if wit u i (t) = i (t) =u i (t), (1) f( i (t k )) + c a ij Γ( j (t k ) i (t k )) +c b ij Γ j (t k )+μ i (t k ) i l 1, t t k,t k+1 ; f( i (t k )) + c a ij Γ( j (t k ) i (t k )) +c b ij Γ j (t k )+μ i (t k ) i l 2, t t k,t k+1, (2) lim t ( i (t) j (t)) =, i,j l 1, lim t ( i (t) j (t)) =, i,j l 2, were te oliear fuctio satisfies f( 1 (t),t)= f( 2 (t),t)= 1 (t), 2 (t), (6) (7) were i = ( 1 i (t), 2 i (t),..., i (t))t R represets te state vectors of te ode i at time t, f( ) R is cotiuousl differetiable, c is couplig stregt, ad Γ = diag(γ 1,γ 2,...,γ ) R is a ier-couplig matri, if te ode liks troug its it state wit its eigbors γ i > ; oterwise γ i =. I tis protocol, let X 1 = 1, 2,..., }, ad let X 2 = +1, +2,..., +m },werex = X 1 X 2. l 1 =1,2,...,, l 2 =+1, +2,...,+m,witl=l 1 l 2. N i is te eigbors of te ode i, N i N 1i N 2i,were N 1i = j X 1 : a ij >,i,j l 1 }, N 2i = j X 2 : a ij >,i,j l 2 }. I protocol (2), μ i (t) is te cotrol iput: μ i (t) = c iγ( i (t k ) 1 (t k )), i l 1 c i Γ( i (t k ) 2 (t k )), i l 2, were i is a o-off cotrol; if te sstem is samplig te data, te i = 1;oterwise i =.Ifodei ca get iformatio from ode j itesamegroup,tea ij > ; oterwise a ij =;ifodei ca get iformatio from ode j betwee differet groups, te b ij =;oterwiseb ij =.Tus te couplig cofiguratio matri A R (+m) (+m) ca be writte as were A= ca 11 cb m 12 cb m 21 ca m m 22 (3), (4) ad 1 (t), 2 (t) R are te scroous states. Give a positive real umber α ad a sampled period T, we suppose tat t i+1 t i =αt i, i =, 1, 2,..., (8) were t <t 1 < sows te discrete time tat te ode j ca obtai iformatio from its eigbors ad positive iteger T i is a sampled time about te it time ( i =, 1, 2,...), ad it satisfies T i T. Uder tis coditio, a liear cosesus protocol based o a liear estimatio-based samplig period is desiged as follows 11: u i (t k +α)=u i (t k ) 1 T u i (t k )=(1 1 T )u i (t k ), u i (t k +2α)=u i (t k +α)+(u i (t k +α) u i (t k )). =(1 2 T )u i (t k ), u i (t k+1 α)=u i (t k+1 2α)+(u i (t k 2α) u i (t k+1 3α)) ca ii = ca ii = +m j=+1,j =i j=1,j =i ca ij, ca ij,,2,...,, i=+1,+2,...,+m. (5) =(1 t k+1 t k 1 )u T i (t k ) =(1 T k 1 T )u i (t k ). (9)

3 Matematical Problems i Egieerig 3 Tus, we ave +c b ij Γ j (t k )+μ i (t k ) i l u i (t) = 1, t t k,t k+1 ; (1 T k 1 T ) (1 T k 1 T ) f( i (t k )) + c a ij Γ( j (t k ) i (t k )) j N 1,i f( i (t k )) + c a ij Γ( j (t k ) i (t k )) j N 2,i +c b ij Γ j (t k )+μ i (t k ) i l 2, t t k,t k+1. (1) Te te comple damical etwork (1) i ever sampled period becomes wit (1 T ) i (t) =u i (t), (11) f( i (t k )) + c a ij Γ( j (t k ) i (t k )) j N 1,i +c b ij Γ j (t k )+μ i (t k ) i l 1, t t k +α,t k + (+1) α, =,1,...,T u i (t) = k 1; (1 T ) f( i (t k )) + c a ij Γ( j (t k ) i (t k )) j N 2,i +c b ij Γ j (t k )+μ i (t k ) i l 2, t t k +α,t k + (+1) α, =,1,...,T k 1. Some assumptios ad lemmas are eeded. (12) Assumptio 1 (see 3). If eac f i of te oliear fuctio f( i (t)) = (f 1 ( i (t)), f 2 ( i (t)),...,f ( i (t))) T i etwork (1) satisfies te local Lipscitz coditio, for a compact set S R m, tere eists a positive costat matri η(s),suctat ( ) T f () f() ( ) T KΓ ( ) η (s) ( ) T Γ( ),, S, η (s) = K 1. (13) Assumptio 2 (see 1). Assume tat protocol (1) satisfies te balace of effectiveess betwee te subgroups +m b ij =, i l 1, j=+1 j=1 b ij =, i l 2. (14) Lemma 3 (see 1). Defie d = (1/2) N ( i() ) 2,ad costruct a closed space B(σd, )= R N 1 2 N i() 2 σd }, (15) were =( T 1,T 2,...,T N )T ad σ>1is a costat. For B(σd, ), tere eists a costat η(σ, ) suc tat N ( i )(f( i ) f( )) η (σ, ) i 2. (16) Lemma 4 (see 1). Suppose tat a R ad b R m are vectors, ad i matri M,tefollowigiequalitolds: 2 T T M + T M 1. (17) I tis paper, A > (, <, ) meas tat A is a positive (or semipositive, egative, or semiegative) defiite matri; A 1 = ma 1 j a ij, A = ma 1<i< j=1 a ij. 3. Group Scroizatio Aalsis i Comple Network wit Sampled Data I tis sectio, we cosider group scroizatio problem of te comple etworks wit sampled data. We ave te followig teorem. Teorem 5. For etwork (1) wit protocol (12) of +modes, uder Assumptios 1-2 ad Lemmas 3-4, if, Tsatisf 1 T η(σ, 1)I +c(a 11 H 1 ) Γ + c 2 ( B 12 1 I + B 21 I ) Γ}<, =,1,...,T k 1;

4 4 Matematical Problems i Egieerig 1 T η(σ, 2)I m +c(a 22 H 2 ) Γ + c 2 ( B 21 1 I m + B 12 I m) Γ}<, =,1,...,T k 1, (18) +c a ij Γ(( j (t k ) 1 (t k )) ( i (t k ) 1 (t k )) ) +c b ij Γ( j (t k ) 2 (t k )) teallteodesieacgroupcacovergetoteirow scroous state asmptoticall, were H 1 = diag 1, 2,..., l,,...,}, H 2 = diag +1, +2,..., +l,,...,}. Proof. Costruct a Lapuov fuctio as follows: were V 1 (t k )= 1 2 V 2 (t k )= 1 2 Defie +m i=+1 (19) V (t k ) =V 1 (t k ) +V 2 (t k ), (2) ( i (t k ) 1 (t k )) T ( i (t k ) 1 (t k )), ( i (t k ) 2 (t k )) T ( i (t k ) 2 (t k )). e i,1 (t k )= i (t k ) 1 (t k ), e i,2 (t k )= i (t k ) 2 (t k ),,2,...,, i=+1,+2,...,+m, e 1 (t k )=(e 1,1 (t k ),e 2,1 (t k ),...,e,1 (t k )) T, e 2 (t k )=(e +1,2 (t k ),e +2,2 (t k ),...,e +m,2 (t k )) T. Te, (21) (22) +c b ij 2 (t k ) c i Γ( i (t k ) 1 (t k )) = e T i,1 1 T (t k) f( i (t k )) f ( 1 (t k )) +c a ij Γ(e j,1 (t k ) e i,1 (t k )) +c b ij Γe j,2 (t k ) c i Γe i,1 (t k ). (24) Uder Assumptio 1 ad Lemma 3,wecaave V 1 (t k ) 1 T e T i,1 (t k) η(σ, 1 )Γe i,1 (t k ) +c a ij Γ(e j,1 (t k ) e i,1 (t k )) +c b ij Γe j,2 (t k ) c i Γe i,1 (t k ) V 1 (t k )= 1 2 V 2 (t k )= 1 2 +m i=+1 e i,1 (t k ) T e i,1 (t k ), e i,2 (t k ) T e i,2 (t k ). Uder Assumptio 2,we ca kow V 1 (t k )= 1 T ( i (t k ) 1 (t k )) T (23) = 1 T e T 1 (t k)(η(σ, 1 )I Γ)e 1 (t k ) Te, we ca get V 2 (t k ) 1 T +e T 1 (t k)(c(a 11 H 1 ) Γ)e 1 (t k ) +c e T i,1 (t k) b ij Γe j,2 (t k ). (25) f( i (t k )) f ( 1 (t k )) +m i=+1 e T i,2 (t k) η(σ, 2 )Γe i,2 (t k )

5 Matematical Problems i Egieerig 5 +c a ij Γ(e j,2 (t k ) e i,2 (t k )) +c b ij Γe j,1 (t k ) c i Γe i,2 (t k ) = 1 T e T 2 (t k)(η(σ, 2 )I m Γ)e 2 (t k ) +e T 2 (t k)(c(a 22 H 2 ) Γ)e 2 (t k ) +m +c i=+1 e T i,2 (t k) b ij Γe j,1 (t k ). (26) Tus, we ca obtai V(t k )= V 1 (t k )+ V 2 (t k ) 1 T et 1 (t k)(η(σ, 1 )I Γ)e 1 (t k ) +e T 1 (t k)(c(a 11 H 1 ) Γ)e 1 (t k ) +e T 2 (t k)(η(σ, 2 )I m Γ)e 2 (t k ) +e T 2 (t k)(c(a 22 H 2 ) Γ)e 2 (t k ) +e T i,1 (t k) c 2 B 12 1 I Γe i,1 (t k ) +e T i,2 (t k) c 2 B 12 I m Γe i,2 (t k ) c Usig Lemma 4,wecaave e T i,1 (t k) b ij Γe j,2 (t k ) c b 2 ij (e T i,1 (t k)γe i,1 (t k )+e T j,2 (t k)γe j,2 (t k )) +m c i=+1 c 2 B 12 1 e T i,1 (t k)γe i,1 (t k ) + c 2 B 12 e T j,2 (t k)γe j,2 (t k ) =e T i,1 (t k) c 2 B 12 1 I Γe i,1 (t k ) +e T i,2 (t k) c 2 B 12 I m Γe i,2 (t k ), e T i,2 (t k) b ij Γe j,1 (t k ) If +e T i,2 (t k) c 2 B 21 1 I m Γe i,2 (t k ) +e T i,1 (t k) c 2 B 21 I Γe i,1 (t k )} =e T 1 (t k) 1 T η (σ, 1)I +c(a 11 H 1 ) Γ + c 2 ( B 12 1 I + B 21 I ) Γ}e 1 (t k ) +e T 2 (t k) 1 T η (σ, 2)I m +c(a 22 H 2 ) Γ + c 2 ( B 21 1 I m + B 12 I m) Γ}e 2 (t k ). (28) 1 T η(σ, 1)I +c(a 11 H 1 ) Γ + c 2 ( B 12 1 I + B 21 I ) Γ}<, 1 T η(σ, 2)I m +c(a 22 H 2 ) Γ (29) c 2 +m i=+1 c 2 B 21 1 b ij (e T i,2 (t k)γe i,2 (t k )+e T j,1 (t k)γe j,1 (t k )) +m i=+1 e T i,2 (t k)γe i,2 (t k ) + c 2 B 21 e T j,1 (t k)γe j,1 (t k ) =e T i,2 (t k) c 2 B 21 1 I m Γe i,2 (t k ) +e T i,1 (t k) c 2 B 21 I Γe i,1 (t k ). (27) te + c 2 ( B 21 1 I m + B 12 I m) Γ}<, V(t k )<. (3) Terefore, V(t k ) decreasesoaiterval t t k +α,t k + ( + 1)α, =,1,...,T k 1,adV(t k ) V(t ).Fromte above discussio, we ave lim t ( i (t) j (t)) =, i,j l 1, lim t ( i (t) j (t)) =, i,j l 2. (31) Te, etwork (1) wit sampled data is groupscroized uder protocol (12).

6 6 Matematical Problems i Egieerig 4. A Special Case I tis sectio, we cosider a special case about etwork (1). For coveiece, we let A 11 =A 11 H 1 ad A 22 =A 22 H 2. Lemma 6 (see 1). If A = (a ij ) R N N is a smmetric irreducible matri wit a ii = N j=1,j =i a ij ad a ij = a ji (i =j),teforamatrie=diag(e,,...,)wit e all eigevalues of te matri (A E) are egative. I etworks, we te iformatio trasmissio is te same betwee te odes i ad j, te couplig cofiguratio matri A is smmetric. From Lemma 6, wea is smmetric, A 11 ad A 22 are egative. Hece, all of teir eigevalues are strictl egative; oe deotes tem as λ (A 11 ) λ 2 (A 11 ) λ 1 (A 11 )<, λ m (A 22 ) λ 2 (A 22 ) λ 1 (A 22 )<. (32) Te, usig Lemma 6,wecaobtaitefollowigresultwe te couplig cofiguratio matri is smmetric. Teorem 7. For etwork (1) wit protocol (12) of +modes, we te couplig cofiguratio matri is smmetric, uder Assumptios 1-2 ad Lemmas 3 6,if, Tsatisf Figure 1: Te positios of te odes of etwork (1) oitervalt, T η(σ, 1)I +cλ 1 (A 11 ) Γ + c 2 ( B 12 1 I + B 21 I ) Γ}<, =,1,...,T k 1; 1 T η(σ, 2)I m +cλ 1 (A 22 ) Γ + c 2 ( B 21 1 I m + B 12 I m) Γ}<, (33) =,1,...,T k 1, teallteodesieacgroupcacovergetoteirow scroous state asmptoticall, were A 11 =A 11 H 1 ad A 22 =A 22 H 2. Te proof of Teorem 7 is similar to tat of Teorem 5 ad ere is omitted. 5. Simulatios I tis sectio, we give some simulatio results of te above discussios. For coveiece, let =5, m=4, c=12, a ij =a ji for all i, j, adh 1 = diag,1,1,1,}, H 2 = diag,,,1}. We cosider group scroizatio of comple damical etwork wit sampled data i te time iterval, 6. Te curves i te graps are te locatios of 5+4odes ad te scroous targets i te etwork. From Figures 1, 2, ad3, tescroousstatesare 1 () =, cos, 2 () = 2, cos(2). From tese figures, we ca see tat all te odes i eac subgroupcacovergetoteirowscrooustargets.but, Figure 2: Te positios of te odes of etwork (1) oitervalt, 6,sampledperiodT=1. i Figures 4, 5, ad6, all of te odes ca asmptoticall coverge to a scroous state, we te scroous state is te ol oe 1 () = 2 () =, cos(2). I Figures 1 ad 4, wesimulatescroizatioofte etwork icludig two subgroups witout sampled data. I Figures 2 ad 5,wecoseT=1as te sampled period. Tat is, te comple etwork is samplig at te momet t = k, for k =,1,...IFigures3 ad 6, wecoset=2.5as te sampled period; tat is, te comple etwork is samplig at te momet t = 2.5k,fork=,1,... I tese simulatio results, we ca fid tat te odes of te sstem keep teir state value i te first T time util before te momet ( + 1)T, keepig teir state value i ( + 1)T time util te momet ( + 2)T ad so o. B comparig, we coosig te sampled data o ever period, te rate of covergece of te odes i te comple damics etwork is slow. Moreover, te bigger te sampled period is,

7 Matematical Problems i Egieerig Figure 3: Te positios of te odes of etwork (1) oitervalt, 6,sampledperiodT = 2.5. Figure 5: Te odes of etwork (1) o iterval t,6coverge to te same target, sampled period T= Figure 4: Te odes of etwork (1) o iterval t,6coverge to te same target. Figure 6: Te odes of etwork (1) o iterval t,6coverge to te same target, sampled period T = 2.5. te slower te rate of covergece of te odes i te comple damics etwork is. 6. Coclusio I tis paper, we ave ivestigated te group scroizatio problem of a comple damical etwork wit sampled data. We prove tat te odes of te etwork arrive at scroizatio i two subgroups if te samplig period satisfies te coditio based o te liear matri iequalit (LMI). I additio, we ave give some simulatio results about te proposed comple etwork. Coflict of Iterests Te autors declare tat tere is o coflict of iterests regardig te publicatio of tis paper. Ackowledgmets Tis work was supported b te Natioal Natural Sciece Foudatio of Cia uder Grat o , Fudig Project for Academic Huma Resources Developmet i Istitutios of Higer Learig uder te Jurisdictio of Beijig Muicipalit (PHR211855), ad Sciece ad Tecolog Developmet Pla Project of Beijig Educatio Commissio (os. KM ad KM ). Refereces 1 B. Liu, P. F. Wei, X. F. Wag et al., Group scroizatio of comple etworks wit oliear damics via piig cotrol, i Proceedig of te 32d Ciese Cotrol Coferece, pp , S. Wag, H. Yao, S. Zeg, ad Y. Xie, A ovel criterio for cluster scroizatio of comple damical etworks wit

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