Research Article Wirtinger-Type Inequality and the Stability Analysis of Delayed Lur e System

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1 Hindwi Pulishing Corporion Discree Dynics in Nure nd Sociey Volue 13, Aricle ID , 9 pges hp://dx.doi.org/1.1155/13/ Reserch Aricle Wiringer-ype Inequliy nd he Siliy Anlysis of Delyed Lur e Syse Zixin Liu, 1, Jin Yu, 3 Doyun Xu, 1 nd Dingo Peng 3 1 College of Copuer Science nd Inforion, GuiZhou Universiy, Guiyng, Guizhou 555, Chin School of Mheics nd Sisics, Guizhou Universiy of Finnce nd Econoics, Guiyng, Guizhou 554, Chin 3 College of Science, Guizhou Universiy, Guiyng, Guizhou 555, Chin Correspondence should e ddressed o Zixin Liu; xinxin95@163.co Received 11 My 13; Revised 6 July 13; Acceped 7 July 13 Acdeic Edior: M. De l Sen Copyrigh 13 Zixin Liu e l. his is n open ccess ricle disriued under he Creive Coons Ariuion License, which peris unresriced use, disriuion, nd reproducion in ny ediu, provided he originl work is properly cied. his pper proposes new dely-depended siliy crierion for clss of delyed Lur e syses wih secor nd slope resriced nonliner perurion. he proposed ehod eploys n iproved Wiringer-ype inequliy for consrucing new Lypunov funcionl wih riple inegrl ies. By using he convex expression of he nonliner perurion funcion, he originl nonliner Lur e syse is rnsfored ino liner uncerin syse. Bsed on he Lypunov sle heory, soe novel dely-depended siliy crieri for he reserched syse re eslished in ers of liner rix inequliy echnique. hreenuericl exples represenedoillusrehevlidiyofheinresuls. 1. Inroducion Lur e conrol syse is n iporn nonliner conrol syse. Since he noion of solue siliy ws firs ie inroduced y Lur e in [1],heproleofhesoluesiliy of Lur e conrol syse hs een widely sudied for severl decdes (see [ 6]). However, ecuse of he exisence of ie delys, sochsic disurnces, preer uncerinies, nd so on, he convergence of Lur e syse y ofen e desroyed. his kes he design or perfornce for he corresponding closed-loop syses ecoe difficul. herefore, he siliy nlysis of delyed Lur e syse ecoes very iporn. Up o now, vrious siliy condiions hve een oined, nd ny excellen ppers nd onogrphs hve een ville (see [7 1]). Recenly, gre del of effor hs een done o he siliy nlysis of delyed Lur e syse wih secor nd slope resriced nonlineriies. o enlrge he fesiiliy region of he siliy crieri, y inroducing vriles in crosser, Prk reserched new ounding echnique in [13]. Concerning he descripor ehod for delyed syse, n exensive work ws developed y Fridn nd Shked in [14]. By eploying liner rix inequliy nd rix decoposing echnique, Co nd Zhong [6] reserched he solue siliy prole of Lur e conrol syses wih uliple ie delys nd nonlineriies nd eslished soe iproved dely-dependen crieri. In order o furher reduce siliy crierion s conservis, secor ounds nd slope ounds re eployed o Lypunov-Krsovskii funcionl hrough convex represenion of he nonlineriies so h soe new iproved crieri were eslished y Lee nd Prk in [1]nd Yin e l. in [15], respecively. On he oher hnd, hese previous works only focused on he relionship eween x (s)qx(s)ds nd ( x(s)ds) Q( x(s)ds) or eween +θ x (s)qx(s)ds dθ nd ( x(s)ds +θ dθ) Q( x(s)ds dθ). Onenurlquesion is wheher here exiss relionship ong +θ +θ z(s) Rz(s)ds dθ, z(s)ds dθ, +θ z (s)rz(s)ds, z( τ), nd z(s)ds. hisideoiveshissudy.by using he Jensen inegrl inequliy, we firs eslish soe iproved vecor Wiringer-ype inequliies. On he sis of hese new eslished inequliies, new Lypunov

2 Discree Dynics in Nure nd Sociey funcionl including riple inegrl ies is proposed, nd soe less conservive dely-dependen siliy crieri re derived. Finlly, hree nuericl exples re presened o illusre he vlidiy of he in resuls. Noion. henoionsreusedinourpperexcepwhere oherwise specified. R, R n re rel nd n-diension rel nuer ses, respecively; dig( ) denoes he lock digonl rix. I is ideniy rix; represens he eleens elow he in digonl of syeric lock rix; rel rix P > (<) denoes P is posiive-definie (negive-definie) rix.. Preliinries Consider he following delyed Lur e syse: y () =Ay() +By(()) +Cf(σ ()), σ () =H y (),, y (s) =φ(s), s [ u,], where y() R n denoes he se vecor; σ() R is he oupu vecor; H=(h 1,h,...,h ) n R n ; A, B, nd C re consn known rices of pproprie diensions. he delyτ() is ssued o sisfy <τ l τ() τ u, (1) τ () τ<1. () f(σ()) R denoes he nonliner funcion in feedck ph, which hs he following for: f (σ ()) =[f 1 (σ 1 ()),f (σ ()),...,f (σ ())] σ () =[σ 1 (),σ (),...,σ ()] [h 1 y (),h y (),...,h y ()], which sisfies secor condiion wih f i ( ),(i = 1,,...,) elonging o secor [l i,l+ i ],wherel i,l+ i re known consn sclrs; h is, l i f i (σ i ()) σ i () l + i (3),,,...,. (4) Noice h he nonliner funcion f i ( ) cn e wrien s convex coinion of he secor ounds s follows: f i (σ i ()) =(λ i (σ i ())l i +(1 λ i (σ i ())) l + i )σ i (),,,...,, where λ i (σ i ) = (f i (σ i ()) l i σ i())/(l + i l i )σ i() sisfying λ i (σ i ) 1. Nely, f i (σ()) = Λ i (σ i ())σ i (), where Λ i (σ i ()) is n eleen of convex hull Co{l i,l+ i }.Se L=dig{l 1,l,...,l },wherel i = x{ l i, l+ i }. Oviously, 1 (λ i (σ i ())l i +(1 λ i (σ i ()))l + i )/l i 1.DefineΔ i = (λ i (σ i ())l i +(1 λ i (σ i ()))l + i )/l i, i = 1,,...,, Δ = dig(δ 1,Δ,...,Δ ) hen nonliner funcion f(σ()) cn e (5) expressed s f(σ()) = LΔH y(),whereδ sisfies Δ Δ I. And he syse (1)cnerewrienshefollowingdelyed uncerin syse: y () =(A+CLΔH )y() +By(()), y (s) =φ(s), s [ u,]. Rerk 1. Differen fro previous work [6, 9, 1, 1, 15], in his pper, y using he convex expression f i (σ()) = Λ i (σ i ())σ i (), we rnsfor he originl nonliner syse (1) ino liner uncerin syse (6). As resul, he siliy prole of nonliner Lur e syse (1) cn e rnsfored ino he rous siliy prole of liner uncerin syse (6). Le L = dig(l 1,l,...,l ) nd L+ = dig(l + 1,l+,...,l+ ). For furher discussion, he following les re needed. Le (see [16]). For ny posiive definie syeric consn rix Q nd sclr τ>, such h he following inegrions re well defined, hen y (s) Qy (s) ds dθ +θ 1 τ ( Q( +θ (6). +θ (7) Le 3 (see [17]). Given syeric rix P 1 nd ny rel rices P,P 3 of pproprie diensions, P 1 +P ΔP 3 +P 3 Δ P <, (8) for ll Δ Θ sisfying Δ Δ I if nd only if here exiss S S Δ such h P 1 +P 3 [ SP 3 P ] <, (9) [ P S] where S Δ =: {dig(s 1 I,...,s k I, S 1,...,S l ):S i >,k,l N}. On he sis of Jensen inegrl inequliy, we firs give ou n iproved Wiringer-ype vecor inequliy s follows. Le 4. Le z() R n hve coninuous derived funcion z() on inervl [, ]. Assue h z() = ;henfornyn nrix R>,hefollowinginequliyholds: z (s) Rz (s) ds ( ) z (s) Rz (s) ds. (1)

3 Discree Dynics in Nure nd Sociey 3 Proof. Since z() =,onecngehz(s) = s z (s) Rz (s) ds = = = s ( z () d) s (s ) R( s z () d) ds z () Rz () d ds (s ) z () Rz () ds d z () Rz () ( ( ) ( ) his coplees he proof. z () Rz () d. z()d, ( ) )d (11) On he sis of Le 4, wefurhergiveousoe iproved Wiringer-ype inequliies s follows. Le 5. Le z() R n hve coninuous derived funcion z() on inervl [, ].henfornyn n-rix R>,sclrτ>, he following inequliy holds: (1) z (s) Rz (s) ds () τ 3 ( z (s) ds) R( τ z () Rz ()+ 4 τ ( +θ z(s) Rz (s) ds dθ z (s) ds) τ 4 ( z (s) ds dθ) R( +θ τ ( z (s) ds) R( + 4 τ 3 ( z (s) ds) Rz () ; (1) z (s) ds dθ) +θ z (s) ds) z (s) ds dθ) R( +θ z (s) ds). (13) Proof. Se x(s) = z(s) z(), s [,].Noiceh x()=z() z()=;frole 4,wehve x (s) Rx (s) ds = (z (s) z()) R (z (s) z()) ds τ = τ ( d (z (s) z()) ) ds R( d (z (s) z()) )ds ds z(s) Rz (s) ds. Addiionlly, fro Jensen inequliy, one cn ge hus, we hve (z (s) z()) R (z (s) z()) ds 1 τ ( (z (s) z()) ds) τ 3 ( R( (z (s) z()) ds). z(s) Rz (s) ds (z (s) z()) ds) R( (z (s) z()) ds) = τ 3 ( z (s) ds) R( z (s) ds) τ z () Rz ( τ)+ 4 τ ( (14) (15) z (s) ds) Rz (). (16) Siilrly, le x(s) = z(s) z( + θ); fro inequliy(16), Les,nd4,onecnoin +θ z(s) Rz (s) ds dθ +θ 1 τ ( R( 1 τ ( R( ( θ) x(s) Rx (s) ds dθ +θ +θ +θ +θ (z (s) z(+θ)) ds dθ) θ (z (s) z(+θ)) ds dθ) θ z(s) τ z(s) τ ds dθ z (s) ds) ds dθ z (s) ds)

4 4 Discree Dynics in Nure nd Sociey = τ 4 ( z (s) ds dθ) R( +θ τ ( z (s) ds) R( + 4 τ 3 ( his coplees he proof. z (s) ds dθ) +θ z (s) ds) z (s) ds dθ) R( +θ z (s) ds). (17) Rerk 6. Copred wih rdiionl Wiringer-ype inegrl inequliy, Jensen inegrl inequliy [18], nd doule inegrl Jensen inequliy [16], Le 5 gives ou rnsiionl for ong +θ z(s) Rz(s)ds dθ, z(s)ds dθ, +θ z (s)rz(s)ds, z( τ),nd z(s)ds. hese inequliy relionships cn e used s hndy ool o del wih he siliy prole. 3. Min Resuls In his secion, we ep o eslish soe new prciclly copule siliy crieri for syse (1). By consrucing new Lypunov funcionl including riple inegrl ies, we oin he following siliy resul. heore 7. For given sclrs τ l >, τ u >, τ < 1, L = dig(l 1,l,...,l ), L+ = dig(l + 1,l+,...,l+ ),syse (1) is glolly sypoiclly sle if here exis posiive definie digonl rices D 1 = dig{d 1,d,...,d }, D = dig{λ 1,λ,...,λ }, D 3 = dig{α 1,α,...,α },syeric posiive definie rices Q 1, Q, Q 3, Q 4, nd rirry rices M 1,M of pproprie diensions such h he following condiion holds: [ Ξ+Ψ SΨ Φ ]<, (18) S where S S Δ,Ξ=(Ξ ij ),Φ,Ψ R 7 7, i,j=1,,...,7, Ξ 11 = (τ l +τ u )HD 3 (L + L )H +M 1 A+A M 1 +Q 4, Ξ 1 =M 1 B, Ξ 13 = M 1 +A M +Q 1 L HH D 1 +L + HH D, Ξ = (1) Q 4, Ξ 3 =B M, Ξ 33 = M M + τ l Q + τ u Q 3, Ξ 44 = τl Q, Ξ 46 = 4 τ 3 Q, l Ξ 55 = τl Q 3, Ξ 57 = 4 τu 3 Q 3, Ξ 66 = τl 4 Q τl (HD 3 (L + L )H ), Ξ 77 = τu 4 Q 3 τu (HD 3 (L + L )H ), Φ=dig {M 1 CL,,H,,,,}, Ψ=( H,, M CL + (D 1 D ) HL,,,, ). O (19) Proof. Choose new clss of Lypunov funcionl cndide s follows: V(y()) =V 1 (y ())+V (y ())+V 3 (y ())+V 4 (y ()), () where V 1 (y ()) =y () Q 1 y () + V (y ()) = + { + + l θ { h i y() { λ i (f i (s) l i s) ds h i y() + d i (l + i s f i (s))ds}; α i σ i (s) [f i (σ i (s)) l i σ i (s)] +μ ds dμ dθ} l θ { ds dμ dθ} u θ { ds dμ dθ} u θ V 3 (y ()) = α i σ i (s) [l + i σ i (s) f i (σ i (s))] +μ ds dμ dθ} ; α i σ i (s) [f i (σ i (s)) l i σ i (s)] +μ α i σ i (s) [l + i σ i (s) f i (σ i (s))] +μ l θ +μ u θ +μ + y (s) Q y (s) ds dμ dθ y (s) Q 3 y (s) ds dμ dθ; V 4 (y ()) = y (s) Q 4 y (s) ds. () (1)

5 Discree Dynics in Nure nd Sociey 5 he ie derivive of V(y()) long he rjecory of syse (1)isgivens V(y()) = where V 1 (y ()) Consider V (y ()) = V 1 (y ())+ V (y ())+ V 3 (y ())+ =y () Q 1 y () +[f (σ ()) H y () L HH ]D 1 y () +[y () L + HH f (σ ()) H ]D y (), =y () (Q 1 L HH D 1 +L + HH D ) y () +y () HΔLH (D 1 D ) y (). { l θ { α i σ i () [f i (σ i ()) l i σ i ()]dμdθ l θ l θ { l θ u θ { u θ u θ u θ α i σ i ( + μ) [f i (σ i ( + μ)) l i σ i ( + μ)] dμ dθ} α i σ i () [l + i σ i () f i (σ i ())] dμ dθ α i σ i ( + μ) [l + i σ i ( + μ) V 4 (y ()), () (3) f i (σ i ( + μ))] dμ dθ} α i σ i () [f i (σ i ()) l i σ i ()]dμdθ α i σ i ( + μ) [f i (σ i ( + μ)) l i σ i ( + μ)] dμ dθ} α i σ i () [l + i σ i () f i (σ i ())] dμ dθ α i σ i ( + μ) [l + i σ i ( + μ) f i (σ i ( + μ))] dμ dθ} = + {α i σ i () [f i (σ i ()) l i σ i ()] τ l + + u +θ = l +θ α i σ i (s) [f i (σ i (s)) l i σ i (s)]dsdθ} {α i σ i () [l + i σ i () f i (σ i ())] τ l l +θ α i y i (s) [l + i y i (s) f i (σ i (s))] ds dθ} {α i σ i () [f i (σ i ()) l i σ i ()] τ u u +θ α i σ i (s) [f i (σ i (s)) l i σ i (s)]dsdθ} {α i σ i () [l + i σ i () f i (σ i ())] τ u α i σ i (s) [l + i σ i (s) f i (σ i (s))] ds dθ} { α iτ l σ i () [l + i l i ]σ i () + n Noice h α i τ u σ i () [l + i l i ]σ i () l +θ u +θ α i τ l σ i () [l + i α i σ i (s) [l + i l i ]σ i (s) ds dθ α i σ i (s) [l + i l i ]σ i (s) ds dθ}. l i ]σ i () =τ l y () HD 3 [L + L ]H y (), α i τ u σ i () [l + i Fro Le,wehve { l +θ = τ l l i ]σ i () =τ u y () HD 3 [L + L ]H y (). l +θ ( α i σ i (s) [l + i l +θ [L + L ]H ( l i ]σ i (s) ds dθ} y (s) HD 3 [L + L ]H y (s) ds dθ HD 3 l +θ. (4) (5) (6)

6 6 Discree Dynics in Nure nd Sociey Siilrly, Nely, V (y ()) { u +θ u +θ τu ( α i σ i (s) [l + i [L + L ]H ( l i ]σ i (s) ds dθ} HD 3 u +θ. y () [(τ l +τ u )HD 3 (L + L )H ]y() τ l ( ( l +θ l +θ τu ( ( V 3 (y ()) = l θ + u +θ u +θ u θ [HD 3 (L + L )H ] [HD 3 (L + L )H ] ; [ y () Q y () y ( + μ) Q y(+μ)]dμdθ [ y () Q 3 y () y ( + μ) Q 3 y(+μ)]dμdθ = τ l y () Q y () + τ u y () Q 3 y () l +θ u +θ y (s) Q y (s) ds dθ y (s) Q 3 y (s) ds dθ (7) + 4 τ 3 l V 4 (y ()) ( τu 4 ( l +θ u +θ Q ( y (s) ds) l Q 3 ( u +θ τu ( y (s) ds) Q 3 ( y (s) ds) u u + 4 τu 3 ( u +θ =y () Q 4 y () (1 ()) y (()) Q 4 y (()) Q 3 ( y (s) ds), u y () Q 4 y () (1) y (()) Q 4 y (()). (8) Addiionlly, for rirry rices M 1, M of pproprie diensions, we hve y() M 1 [ y () +(A+CLΔH )y() +By(())] =, y () M [ y () +(A+CLΔH )y() +By(())] =. (9) Hence, V(y()) ξ ()(Ξ + Φ ΔΨ + Ψ Δ Φ )ξ(), where ξ() = [y (), y ( τ()), y (), y (s)ds, y (s)ds, l u l +θ y (s)ds dθ, u +θ y (s)ds dθ],nd Δ = Δ = dig(δ,,δ,,,,).sinceδ Δ I,hus Δ Δ I.Fro Le 3, if he condiions in heore 7 re sisfied, hen, Ξ+Φ ΔΨ + Ψ Δ Φ < ;ylypunovsleheory, hedelyedlur esyse(1) issypoicllysle,which coplees he proof. Rerk 8. Differen fro previous work, in he proof of Le 5, we eslish he relionship ong +θ z(s) Rz(s)ds dθ, z(s)ds dθ, nd z(s)ds. +θ On he sis of his new relionship, in heore 7, ies y (s)ds, y (s)ds, l u l +θ y (s)ds dθ, nd u +θ y (s)ds dθ reinroducedsheseofξ(); his y reduce crierion s conservis. τ l y () Q y () + τ u y () Q 3 y () τ 4 l τ l ( l +θ Q ( l +θ ( y (s) ds) Q ( y (s) ds) l l For furher use of he inforion of nonliner funcion f(σ()), le us define W() = (w ij ()) n = Λ()H,where Λ() = dig(λ 1 (σ 1 ()), Λ (σ ()),...,Λ (σ ())). Since Λ i (σ i ()) = λ i (σ i ())l i +(1 λ i (σ i ()))l + i [l i,l+ i ], i N, here us exis W = (w ij ) n, W = (w ij ) n such h w ij w ij () w ij, [,+ ),i,j N.Se H =(W+ W)/ = ( h ij ) n, H =(W W)/ = (ĥij) n. Fro he resul

7 Discree Dynics in Nure nd Sociey 7 oined in [19], W() = (w ij ()) n = Λ()H cn e rewrien s Λ()H = H+EΣF,where E=[ ĥ 11 e 1,..., ĥ 1 e 1, ĥ 1 e,..., ĥ e ] R, F=[ ĥ 11 e 1,..., ĥ 1 e, ĥ 1 e 1,..., ĥ e ] R, (3) e i (,,...,)denoes he ih colun vecors of he ideniy rix, nd Σ=dig(ε 11,...,ε 1,...,ε 1,...,ε,), where ε ij (i,j=1,,...,)sisfies ε ij 1.hisensh syse (1)cnerewriens y () =[(A+C H) + CEΣF] y () +By(()), y (s) =φ(s), s [ u,], (31) where Σ sisfies Σ Σ=ΣΣ I.Froheore 7, wecn ge he following resul. heore 9. For given sclrs τ l >,τ u >,τ < 1, L = dig(l 1,l,...,l ), L+ = dig(l + 1,l+,...,l+ ),syse (1) is glolly sypoiclly sle if here exis posiive definie digonl rices D 1 = dig{d 1,d,...,d }, D = dig{λ 1,λ,...,λ }, D 3 = dig{α 1,α,...,α },syeric posiive definie rices Q 1, Q, Q 3, Q 4, nd rirry rices M 1,M of pproprie diensions such h he following condiion holds: [ Ξ +Ψ SΨ Φ ]<, (3) S where S S Δ,Ξ =(Ξ ij ), Φ,Ψ R 7 7, i,j=1,,...,7, Ξ 11 = (τ l +τ u )HD 3 (L + L )H +M 1 (A + C H) +(A+C H) M 1 +Q 4, Ξ 13 = M 1 +(A+C H) M +Q 1 Ξ 1 =M 1B, +( H H L HH )D 1 +(L + HH H H )D, Ξ = (1) Q 4, Ξ 3 =B M, Ξ 33 = M M + τ l Q + τ u Q 3, Ξ 44 = τl Q, Ξ 46 = 4 τ 3 Q, l Ξ 55 = τl Q 3, Ξ 57 = 4 τu 3 Q 3, Ξ 77 = τu 4 Q 3 τu (HD 3 (L + L )H ), Φ = dig {M 1 CE,,F,,,,}, Ψ =( F,,M CE + (D 1 D ) HE,,,, ). O (33) Rerk 1. Due o he exisence of ies Ψ SΨ nd Ψ SΨ, he resuls eslished in heores 7 nd 9 re no LMI crieri. In order o overcoe his flw, y using he le derived in [], we furher eslish he following ore prcicle sle rules. Corollry 11. For given sclrs τ l >,τ u >,τ < 1, L = dig(l 1,l,...,l ), L+ = dig(l + 1,l+,...,l+ ),syse (1) is glolly sypoiclly sle if here exis posiive definie digonl rices D 1 = dig{d 1,d,...,d }, D = dig{λ 1,λ,...,λ }, D 3 = dig{α 1,α,...,α },syeric posiive definie rices Q 1, Q, Q 3, Q 4,posiivesclrδ>, nd rirry rices M 1,M of pproprie diensions such h he following condiion holds: δξ Φ δψ [ I ] <. (34) [ I ] Corollry 1. For given sclrs τ l >,τ u >,τ < 1, L = dig(l 1,l,...,l ), L+ = dig(l + 1,l+,...,l+ ),syse (1) is glolly sypoiclly sle if here exis posiive definie digonl rices D 1 = dig{d 1,d,...,d }, D = dig{λ 1,λ,...,λ }, D 3 = dig{α 1,α,...,α },syeric posiive definie rices Q 1, Q, Q 3, Q 4,posiivesclerδ>, nd rirry rices M 1,M of pproprie diensions such h he following condiion holds: 4. Nuericl Exple δξ Φ δψ [ I ] <. (35) [ I ] In order o show he effeciveness of he echnique proposed in his pper, we revisi he exple in [1], nd copre our crieri wih exising dely-dependen crieri. Exple 1. In order o copre wih preview resuls esily, consider he delyed syse (1) wih preers given y A=[.5 ], B=[ ], C=[. ], H=[ ]. ie-vrying dely τ() = τ is consn. (36) Ξ 66 = τl 4 Q τl (HD 3 (L + L )H ), his syse hs een invesiged in [19, 4]. And he xiu vlues of τ x for he siliy of syse (1) re

8 8 Discree Dynics in Nure nd Sociey τ x =.353, τ x =.33, τ x =.978, nd τ x =.55, respecively. In order o deduce he conservis of hose crieri eslished in [19, 4], in e l. derived new iproved resul y decoposing rix B s B = B 11 +B 1 in [1], where B 11 =[ ],B 1 =[ ] ndgveouhexiudelyounds τ x = However, one cn see h, when ie-vrying dely τ() is no consn, hen he resuls eslished in [19, 1 4] reinvlid.ifc = [..3 ],H = [.6.8 ] f(σ()) = [f 1 (σ 1 ()), f (σ ())], f i (σ i ()) =.5( σ i () + 1 σ i () 1 ), i = 1,. Oviously, nonliner funcion f(σ()) sisfies σ()f(σ()), f() =, nd f i (σ i ())/σ i ().5. If τ l =4, τ u =8, τ() =.9. Leδ =.1; one cn ge h he resuls oined in Corollries 11 nd 1 refesiiliy.his en h he resuls oined in his pper re ore generl ndlessconservivehnhosein[19, 1 4]. Exple. Consider he syse descried in (1) wih preers given y A=[ 1. ], B=[ ], 1.6 C=[ ], H=[ ]. l 1 =l =, l+ 1 =1, l+ =3. (37) his syse hs een invesiged in [5, 6], nd he xiu vlue of τ u forhesiliyofsyse(1) re τ x u =.585, ndτ x u =.678, respecively.in[7], y using he secor ounds nd slope ounds o he Lypunov- Krsovskii funcionl hrough convex represenion of he nonlineriies, Choi e l. iproved he upper ound of τ() o Le δ =.1, τ l =.1; y using he resuls oined in Corollries 11 nd 1, one cn ge he xiu vlues of τ x u re 1.13 nd 1.314, respecively. his ens h he resulsoinedinhispperrelessconservivehnhose in [5 7]. Exple 3. Consider he syse descried in (1) wih preers given y A=[ ], B=[ ], C=[. ], H=[ ]; (38) f(σ()) = ( sin())σ(). ie-vrying dely τ() = τ is consn. Oviously, he ounds of he secor nonlineriy re l 1 =., l + 1 =.5. For his syse, he xiu vlues of τ x forhesiliyofsyse(1) re τ x =.4859 nd τ x =.549 in [5, 6], respecively. By using vrious convex opiizion echniques, he resuls oined in [5, 6] were furher iproved y Lee nd Prk in [1], nd τ x = FroCorollry 11, we find he xiu llowle ie dely ound cn e.581, which ens h he resul eslished in Corollry 11 is less conservive hn he ones oined in [5, 6]. 5. Conclusions Coined wih Lypunov sle heory nd doule inegrl inequliy, his pper reserches clss of delyed Lur e syses wih inervl ie-vrying delys. Differen fro previous work on his opic, his pper firs eslishes soe new vecor Wiringer-ype inequliies. hen, y using he propery of convex funcion, he originl nonliner Lur e syse is rnsfored ino liner uncerin syse. A ls, y consrucing new Lypunov funcionl including riple inegrl ies, soe new less conservive dely-dependen siliy crieri re eslished. Nuericl exples show h he new crieri derived in his pper re less conservive hn soe previous resuls oined in he references cied herein. Acknowledgens his work ws suppored y Chin Posdocorl Science Foundion Grn (1 M51718) nd Sof Science Reserch Projec in Guizhou Province ([11]LKC4). References [1] A. I. Lur e, Soe Nonlier Prole in he heory of Auoic Conrol, H.M. Sionry Office, London, UK, [] Y. Wng, X. Zhng, nd Y. He, Iproved dely-dependen rous siliy crieri for clss of uncerin ixed neurl nd Lur e dynicl syses wih inervl ie-vrying delys nd secor-ounded nonlineriy, Nonliner Anlysis: Rel World Applicions,vol.13,no.5,pp ,1. [3] S. Lun nd Z. i, Dissipiviy for Lur e disriued preer conrol syses, Applied Mheics Leers, vol. 5, no. 4, pp , 1. [4] Y. He nd M. Wu, Asolue siliy for uliple dely generl Lur e conrol syses wih uliple nonlineriies, Journl of Copuionl nd Applied Mheics,vol.159,no.,pp.41 48, 3. [5] Q.-L. Hn, Asolue siliy of ie-dely syses wih secorounded nonlineriy, Auoic, vol. 41, no. 1, pp , 5. [6] J. Co nd S. Zhong, New dely-dependen condiion for solue siliy of Lurie conrol syses wih uliple ie-delys nd nonlineriies, Applied Mheics nd Copuion, vol. 194, no. 1, pp. 5 58, 7. [7] A. I. Zečević, E. Cheng, nd D. D. Šiljk, Conrol design for lrge-scle Lur e syses wih rirry inforion srucure consrins, Applied Mheics nd Copuion,vol.17,no. 3,pp ,1. [8] V. A. Ykuovich, G. A. Leonov, nd A. Kh. Gelig, Siliy of Sionry Ses in Conrol Syses wih Disconinuous Nonlineriies, vol.14ofseries on Siliy, Virion nd Conrol of Syses, World Scienific, River Edge, NJ, USA, 4. [9] F. Ho, Asolue siliy of uncerin discree Lur e syses nd xiu dissile perured ounds, Journl of he Frnklin Insiue,vol.347,no.8,pp ,1.

9 Discree Dynics in Nure nd Sociey 9 [1] Z. i nd S. Lun, Asoluely exponenil siliy of Lur e disriued preer conrol syses, Applied Mheics Leers,vol.5,no.3,pp.3 36,1. [11] J. Hung, Z. Hn, X. Ci, nd L. Liu, Adpive full-order nd reduced-order oservers for he Lur e differenil inclusion syse, Counicions in Nonliner Science nd Nuericl Siulion,vol.16,no.7,pp ,11. [1] S. M. Lee nd J. H. Prk, Dely-dependen crieri for solue siliy of uncerin ie-delyed Lur e dynicl syses, Journl of he Frnklin Insiue,vol.347,no.1,pp ,1. [13] J. G. Prk, A dely-dependen siliy crierion for syses wih un-cerin liner se-delyed syses, IEEE rnscions on Auoic Conrol,vol.35,pp ,1999. [14] E. Fridn nd U. Shked, H -conrol of liner se-dely descripor syses: n LMI pproch, Liner Alger nd is Applicions, vol , pp. 71 3,. [15] C. Yin, S.-. Zhong, nd W.-f. Chen, On dely-dependen rous siliy of clss of uncerin ixed neurl nd Lur e dynicl syses wih inervl ie-vrying delys, Journl of he Frnklin Insiue,vol.347,no.9,pp ,1. [16] J. Sun, G. P. Liu, nd J. Chen, Dely-dependen siliy nd silizion of neurl ie-dely syses, Inernionl Journl of Rous nd Nonliner Conrol,vol.19,no.1,pp , 9. [17] L. El Ghoui nd H. Lere, Rous soluions o les-squres proles wih uncerin d, SIAM Journl on Mrix Anlysis nd Applicions, vol. 18, no. 4, pp , [18] K. Gu, An inegrl inequliy in he siliy prole of ie-dely syses, in Proceedings of he 39h IEEE Confernce on Decision nd Conrol (CDC ), pp , Sydney, Ausrli, Deceer. [19] B. J. Xu nd X. X. Lio, Asolue siliy crieri of delydependen for Lurie conrol syses, Ac Auoic Sinic, vol. 8, no., pp ,. [] S. Zhou nd J. L, Rous silizion of delyed singulr syses wih liner frcionl preric uncerinies, Circuis, Syses, nd Signl Processing,vol.,no.6,pp , 3. [1]J.in,S.Zhong,ndL.Xiong, Dely-dependensolue siliy of Lurie conrol syses wih uliple ie-delys, Applied Mheics nd Copuion,vol.188,no.1,pp , 7. [] X. H. Nin, Dely dependen condiions for solue siliy of Lur e ype conrol syses, Ac Auoic Sinic, vol.5, no. 4, pp , [3] D.-Y. Chen nd W.-H. Liu, Dely-dependen rous siliy for Lurie conrol syses wih uliple ie-delys, Conrol heory nd Applicions,vol.,no.3,pp.499 5,5. [4] B. Yng nd M. Chen, Dely-dependen crierion for solue siliy of Lurie ype conrol syses wih ie dely, Conrol heory & Applicions,vol.18,no.6,pp ,1. [5] Y. He, M. Wu, J.-H. She, nd G.-P. Liu, Rous siliy for dely Lur e conrol syses wih uliple nonlineriies, Journl of Copuionl nd Applied Mheics,vol.176,no.,pp , 5. [6] S. Xu nd G. Feng, Iproved rous solue siliy crieri for uncerin ie-dely syses, IE Conrol heory nd Applicions,vol.1,no.6,pp ,7. [7]S.J.Choi,S.M.Lee,S.C.Won,ndJ.H.Prk, Iproved dely-dependen siliy crieri for uncerin Lur e syses wih secor nd slope resriced nonlineriies nd ie-vrying delys, Applied Mheics nd Copuion, vol.8,no., pp. 5 53, 9.

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