In the last decades, due to the increasing progress in genome

Transcription

1 Se Esimion for Geneic Regulory Neworks wih Time-Vrying Delys nd Recion-Diffusion Terms Yunyun Hn, Xin Zhng, Member, IEEE, Ligng Wu, Senior Member, IEEE nd Yno Wng rxiv:59.888v [mh.oc] Sep 5 Absrc This pper is concerned wih he se esimion problem for geneic regulory neworks wih ime-vrying delys nd recion-diffusion erms under Dirichle boundry condiions. I is ssumed h he nonliner regulion funcion is of he Hill form. The purpose of his pper is o design se observer o esime he concenrions of mrna nd proein hrough vilble mesuremen oupus. By inroducing new inegrl erms in novel Lypunov Krsovskii funcionl nd employing Wiringer-bsed inegrl inequliy, Wiringer s inequliy, Green s ideniy, conve combinion pproch, nd reciproclly conve combinion pproch, n sympoic sbiliy crierion of he error sysem is esblished in erms of liner mri inequliies (LMIs. The obined sbiliy crierion depends on he upper bounds of he delys nd heir derivives. I should be highligh h if he se of LMIs re fesible, he desired observer eiss nd cn be deermined. Finlly, wo numericl emples re presened o illusre he effeciveness of he proposed designed scheme. Inde Terms Geneic regulory neworks, Recion-diffusion erms, Se esimion, Wiringer-bsed inegrl inequliy. I. INTRODUCTION In he ls decdes, due o he incresing progress in genome sequencing nd gene recogniion, geneic regulory neworks (GRNs hve become significn re in biologicl nd biomedicl sciences. However, here sill eiss lrge gp beween he genome sequencing nd he undersnding of gene funcions which hve become chllenge problems in sysem biology. A gre moun of eperimenl resuls show h mhemicl modeling of GRNs cn be powerful ool for reserching he gene regulion process nd discovering comple srucure of biologicl orgnism [] []. Generlly, here re wo bsic models for GRNs: Boolen model (discree-ime model [], [5] nd differenil equion model (coninuous-ime model [6] [8]. Differenil equion model describes he chnge res of he concenrions of mrnas nd proeins. Furhermore, differenil equion model hs been mos frequenly uilized since i cn more precisely describe Y. Hn, X. Zhng (Corresponding uhor nd Y. Wng re wih he School of Mhemicl Science, Heilongjing Universiy, Hrbin 58, Chin, Emil: inzhng@ieee.org. L. Wu is wih he Spce Conrol nd Ineril Technology Reserch Cener, Hrbin Insiue of Technology, Hrbin, 5, Chin, Emil: ligngwu@hi.edu.cn. This work ws suppored in pr by he Nionl Nurl Science Foundion of Chin (76, 676 nd 6, he Nionl Nurl Science Foundion of Heilongjing Province (F6,A6, he Fund of Heilongjing Educion Commiee (56, he Fundmenl Reserch Funds for he Cenrl Universiies (HIT.BRETIV., nd he Heilongjing Universiy Innovion Fund for Grdues (YJSCX5-HLJU. he whole nework nd mde i possible o undersnd he dynmic behvior of whole nework in deil. In biologicl sysems, priculrly in GRNs, sbiliy is he mos significn nd essenil dynmicl behviors [9] []. I is reled o no only he srucure nd funcion of n orgnism bu lso he srengh nd chrcerisics of he eernl disurbnces. In ddiion, s i is well known, ime delys cused by he slow processes of rnscripion nd rnslion in rel GRNs. I hs been well shown from presen reserch resuls h ime dely my led o insbiliy, bifurcion or oscillion for sysems [] [6]. However, mhemicl modeling of GRNs wihou inroducing delys will led o wrong predicions of he concenrions of mrnas nd proeins. Therefore, he problem of sbiliy nlysis for biologicl sysems wih ime delys hs sirred incresing reserch ineress nd gre del of ecellen resuls hs been repored in lierure in recen decdes (see, e.g., [7], [8], [7] []. In some mhemicl modeling, i is implicily ssumed h he geneic regulory sysems re spilly homogeneous, nmely, he concenrions of mrna nd proein re homogenous in spce ll imes. However, here re some siuions in which hese ssumpions re no resonble. For insnce, i migh be necessry o consider he diffusion of regulory proeins from one comprmen o noher [], [] [5]. In his siuion, he generl funcionl differenil equion model cn no precisely describe geneic regulory process more or less. Hence, i is imperive o inroduce reciondiffusion erms in mhemicl modeling of GRNs. To he bes of uhors knowledge, he delyed GRNs wih reciondiffusion erms re only sudied in [5] [8]. M e l. [7] inroduced recion-diffusion erms o GRNs for he firs ime nd esblished dely-dependen sympoic sbiliy crieri. Bsed on he Lypunov funcionl mehod, Zhou, Xu nd Shen [6] invesiged finie-ime robus sochsic sbiliy crieri for uncerin GRNs wih ime-vrying delys nd reciondiffusion erms. Hn nd Zhng [5], [8] grdully improved M e l. s resuls by inroducing novel Lypunov Krsovskii funcionl nd employing Jensen s inequliy, Wiringer s inequliy, Green s ideniy, conve combinion pproch nd reciproclly conve combinion (RCC pproch. In comple biologicl neworks such s neurl neworks nd GRNs, i is ofen he cse h only pril informion bou he ses of he nodes is vilble in he nework oupus. In order o undersnd biologicl neworks beer, i is indispensble o esime he ses of he nodes hrough

2 vilble mesuremens. Hence, he problem of se esimion for biologicl neworks hs been one of he invesiged dynmicl behviors in recen yers [5], [9] []. However, o he bes of he uhors knowledge, here is sill no ny published resul on he se esimion problem for GRNs wih ime-vrying delys nd recion-diffusion erms, which rouses our reserch ineress. Moived by bove discussion, we im o invesige he se esimion problem for GRNs wih ime-vrying delys nd recion-diffusion erms. By inroducing new inegrl erms ino novel Lypunov Krsovskii funcionl nd employing Wiringer-bsed inegrl inequliy, Wiringer s inequliy, Green s ideniy, conve combinion pproch nd RCC pproch, n sympoic sbiliy crierion of he error sysem is esblished in erms of LMIs. Thereby, se observer is designed, nd he observer gin mrices re described in erms of he soluion o se of LMIs. The res of he pper is orgnized s follows: he problem is formuled nd some preliminries re given in Secion ; in Secion, n sympoic sbiliy crierion for he error sysem is esblished, nd n pproch o design se observer is proposed; wo numericl emples re provided in Secion ; nd finlly, we conclude his pper in Secion 5. Noion We now se some sndrd noions, which will be used in he res of he pper. I is he ideniy mri wih pproprie dimension,a T represens he rnspose of he mri A. For rel symmeric mrices X nd Y, X > Y(X Y mens h X Y is posiive definie (posiive semi-definie. is compc se in he vecor spce R n wih smooh boundry. Le C k (X,Y be he Bnch spce of funcions which mp X ino Y nd hve coninuous k-order derivives. For posiive ineger n, le n be he se {,,...,n}. II. MODEL DESCRIPTION AND PRELIMINARIES This pper considers he following GRNs wih ime-vrying delys nd recion-diffusion erms [5]: m i (, = ( l m i (, D ik i m i (, k + n j= w ijg j ( p j ( σ(,+q i, p i (, = ( l D p i (, ik c i p i (, k +b i m i ((,, ( where i n, = col(,,..., l R l, = { k L k,k l }, L k is given consn; D ik > nd Dik > denoe he diffusion re mrices; m i(, nd p i (, re he concenrions of mrna nd proein of he ih node, respecively; i nd c i re degrdion res of he mrna nd proein, respecively; b i represens he rnslion re; W := [w ij ] R n n is he coupling mri of he geneic neworks, which is defined s follows: γ ij, if j is n civor of gene i, w ij =, if here is no link from gene j o i, γ ij, if j is repressor of gene i, here γ ij is he dimensionless rnscripionl re of rnscripion fcor j o gene i; g j represens he feedbck regulion funcion of proein on rnscripion, which is he monoonic funcion in Hill form, i.e., g j (s = sh +s, where H is he Hill H coefficien; q i = Σ j Ii γ ij, I i is he se of ll he nodes which re repressors of gene i; σ( nd τ( re ime-vrying delys sisfying τ( τ, τ( µ, ( σ( σ, σ( µ, where τ, σ, µ nd µ re non-negive rel numbers. The iniil condiions ssocied wih GRN ( re given s follows: m i (s, = φ i (s,,, s [ d,],i n, p i (s, = φ i (s,,, s [ d,],i n, where d = m{σ,τ}, nd φ i (s,, φ i (s, C ([ d,],r. In his pper, he following ype of boundry condiions (Dirichle boundry condiions is considered: nd m i (, =,, [ d,+, p i (, =,, [ d,+. Now, we ssume h m ( := col(m (,m (,...,m n ( p ( := col(p (,p (,...,p n( re he unique equilibrium soluion of GRN (, h is, = ( l m i D ( k i m i( k + n j= w ijg j (p j (+q i, = ( l D p i ( k c i p i (+b im i ( k fori n. Obviously, he rnsformions, m i = m i m i nd p i = p i p i, i n, rnsform GRN ( ino he following mri form: m(, = ( l m(, D k k A m(,+wf( p( σ(,, p(, = ( ( l D p(, k k C p(,+b m((,, where A = dig(,,..., n,c = dig(c,c,...,c n, B = dig(b,b,...,b n, D k = dig(d k,d k,...,d nk, D k = dig(d k,d k,...,d nk, m(, = col( m (,, m (,,..., m n (,, p(, = col( p (,, p (,,..., p n (,, f( p( σ(, = col(f ( p ( σ(,,,f n ( p n ( σ(,, f i ( p i ( σ(, = g i ( p i ( σ(,+p i g i(p i, i n.

3 Becuse of he compleiy of GRN (, i is normlly of he cse h only pril informion bou he ses of he nodes is vilble in he nework oupus. In order o obin he rue se of (, i becomes necessry o esime he ses of he nodes hrough nework mesuremens. The vilble mesuremens re given s follows: { zm (, = M m(,, ( z p (, = N p(,, where z m (, nd z p (, re he cul mesuremen oupus, nd M nd N re known consn mrices wih pproprie dimensions. To esime he ses of GRN ( hrough vilble mesuremen oupus in (, we consruc he following se observer: ˆm(, ˆp(, = ( l ˆm(, D k +Wf(ˆp( σ(, +K [z m (, M ˆm(,], = ( l D ˆp(, k Aˆm(, Cˆp(, +Bˆm((, +K [z p (, Nˆp(,], (5 where ˆm(, nd ˆp(, re he esimions of m(, nd p(,, respecively, nd K nd K re he observer gin mrices o be designed ler. The iniil condiions for he se observer (5 re ssumed o be (ˆm i (,, ˆp i (, = (φ i (s,,φ i (s,. Our im is o find suible observer gins K nd K, so h ˆm(, nd ˆp(,, respecively, pproch o m(, nd p(, s +. Le he error se vecors be m(, = m(, ˆm(, nd p(, = p(, ˆp(,. Then i follows from (, ( nd (5 h m(, where p(, = ( l m(, D k (A+K Mm(,+W f(p( σ(,, = ( l D p(, k (C +K Np(,+Bm((,, (6 f(p( σ(, = f( p( σ(, f(ˆp( σ(,. From he relionship mong f i, f i nd g i, one cn esily obin h nmely, f i ( =, f i (y y ξ i, y R,y,i n, We inroduce he following lemms which ply key roles in obining he min resuls of his pper. Lemm (Jensen s Inequliy: [], [5] For ny consn mri M T = M > of pproprie dimension, ny sclrs nd b wih < b, nd vecor funcion w : [,b] R n such h he inegrls concerned re well defined, hen he following inequliy holds: ( T ( b b w(sds M w(sds (b b w T (smw(sds, ( b b T ( θ w(sdsdθ b b M θ w(sdsdθ (b b b θ wt (smw(sdsdθ. Lemm (Wiringer-bsed Inegrl Inequliy: [6] For given symmeric posiive definie mri Q R n n, nd differenible funcion ω : [,b] R n, he following inequliy holds: b ẇ T (uqẇ(udu [ ] T [ ] Q, b where Q = dig(q,q, = w(b w( nd = w(b+w( b b w(udu. Lemm (Wiringer s Inequliy: [7] Assume h he funcion f C ([,b],r n sisfies f( = f(b =. Then b f (vdv (b π b [f (v] dv. Lemm : [5] Le N > nd N > be pir of digonl mrices. Then he ses of (6 sisfy m T (s, ( l N m(, D k = mt (,N l [ ( D m(, k p T (s, l N (Dk p(, = pt (,N l [ ( Dk p(, ], ]. Lemm 5 (RCC Lemm: [8] Lef,f,...,f N : D R hve posiive finie vlues, where D is open subse of R m. Then he RCC of f i over D sisfies f( =, f T (z( f(z Kz, z R n, (7 where K = dig(ξ,ξ,...,ξ n >. In his pper, we ssume h error sysem (6 sisfies Dirichle boundry condiions: m i (, =,, [ d,+, p i (, =,, [ d,+. subjec o min {αi:α i>, i i αi=} α i f i ( = i f i(+m gi,j( i j g i,j( g ij : R m R, g j,i ( = g i,j (, [ fi ( g i,j ( g i,j ( f j ( ].

4 III. OBSERVER DESIGN In his secion, we will design se observer (5 for GRN (, h is, find pir of observer gin mrices K nd K such h he rivil soluion of sysem (6 is sympoiclly sble under Dirichle boundry condiions. For his end, we define e = n n, e i = col( n (i n,i n, n (n in T,i, ( ϕ(,s, = col m(s,, m(s, ds, τ ( ψ(,s, = col p(s,, p(s, ds, σ ς(, =col(m(,,m(,,m((,,p(, p( σ,,p( σ(,, f(p(,, f(p( σ(,, m(, τ( σ(, p(,, ( τ( σ( σ ( m(s,ds, σ( p(s,ds, σ σ( m(s, ds, p(s, ds. Theorem : For given sclrs τ, σ, µ nd µ sisfying (, he rivil soluion of error sysem (6 under Dirichle boundry condiions is sympoiclly sble if here eis mrices Q T i = Q i > (i 5, R T j = R j > (j, M T j = M j > (j, digonl mrices P j >, Λ j > (j, nd mrices G, G, W nd W of pproprie sizes, such h he following LMIs hold for τ {, τ} nd σ {, σ}: [ ] Rj G ˆR j := j, j, (8 R j G T j Φ(τ,σ = Φ +Φ +Φ (τ,σ+φ +Φ (τ,σ+φ 5 (τ,σ <, (9 where Φ = e 7 Λ e T 7 +e Λ Ke T 7 +e 7 KΛ e T e 8 Λ e T 8 +e 6 KΛ e T 8 +e 8Λ Ke T 6 e 9(P A+W Me T e (P A+W M T e T 9 +e 9 P We T 8 +e 8 W T P e T 9 e 9P e T 9 e (P C +W Ne T e (P C +W N T e T +e P Be T +e B T P e T e P e T, Φ =.5π e P D L e T e (P A+W Me T +e P We T 8 +e 8 W T P e T.5π e P D L et e (P C +W Ne T +e P Be T +e B T P e T, Φ (τ,σ = e Q e T ( µ e Q e T +e Q e T ( µ e 6 Q e T 6 + Q T +τ( Q T + Q T Q T ( Q T + Q T + Q T 6 + 6Q T +τ( 5 Q T Q T 5+Θ Q Θ T +σ(θ Q Θ T +Θ Q Θ T Θ Q Θ T σ(θ Q Θ T +Θ Q Θ T +Θ Q Θ T 6 +Θ 6 Q Θ T +σ(θ 5Q Θ T 6 +Θ 6Q Θ T 5, Φ = e 7 Q 5 e T 7 ( µ e 8 Q 5 e T 8, Φ (τ,σ=φ Φ (τ Φ (σ [ 7 8 ]ˆR [ 7 8 ] T [Θ 7 Θ 8 ]ˆR [Θ 7 Θ 8 ] T, Φ = τ e 9 R e T 9 + σ e R e T + τ e R e T + σ e R e T, Φ (τ = τ( τ e R e T + ττe R e T, Φ (τ = σ( σ σe R e T + σσe R e T, Φ 5 (τ,σ = Φ 5 Φ 5 Φ 5 (τ τ 8 M T 8 (σ σ σ Θ 8 M Θ T 8, Φ 5 = τ e 9M e T 9 + σ e M e T, Φ 5 = (e e M (e e T +(e e M (e e T, Φ 5 = (e e M (e e T +(e 6 e M (e 6 e T, = [e τe ], = [e e e ], = [e τe ], = [ τe τ e ], 5 = [e e τ(e e ], 6 = [e e e ], 7 = [e e e +e e ], 8 = [e e e +e e ], Θ = [e σe ], Θ = [e e e ], Θ = [e 5 σe ], Θ = [ σe σ e ], Θ 5 = [e e σ(e e ], Θ 6 = [e e e 5 ], Θ 7 = [e 6 e 5 e 6 +e 5 e ], Θ 8 = [e e 6 e +e 6 e ], R = dig(r,r, R = dig(r,r, M = τ dig(m,m, M = σ dig(m,m, D L = dig ( l D k L k, l D k L k,..., l D nk L k,

5 5 D L = dig ( l D k L k, l D k L k,..., l D nk L k nd L k, D ik, Dik, A, B, C, W nd K re he sme wih previous ones. Moreover, he observer gin mrices re given by K = P W nd K = P W. Proof: Consruc Lypunov-Krsovskii funcionl for error sysem (6 s follows: where V(,m,p = 5 V i (,m,p, i= V (,m,p = mt (,P m(, + pt (,P p(, + l m T (, m(, P D k + l P Dk p(,, p T (, V (,m,p = ( mt (s,q m(s, + ϕt (,s,q ϕ(,s, + σ( pt (s,q p(s, + σ ψt (,s,q ψ(,s,, V (,m,p = σ( f T (p(s,q 5 f(p(s,, m T (s, m(s, +θ s R s p T (s, +θ s V (,m,p =τ +σ p(s, R σ s +τ +θ mt (s,r m(s, +σ σ +θ pt (s,r p(s,, V 5 (,m,p = σ m T (s, θ +λ s p T (s, θ +λ s m(s, M s dsdλdθ M p(s, s dsdλdθ. Tking he ime derivives of V i (,m,p (i 5 long he rjecory of error sysem (6 yields V (,m,p = [ l ( mt (,P m(, D k (A+K Mm(,+W f(p( σ(, ] + [ l pt (,P (Dk p(, ] (C +K Np(,+Bm((, + l m T (, P D k + l p T (, ( m(, P D k ( p(,,, ( V (,m,p = mt (,Q m(, ( τ( mt ((,Q m((, + pt (,Q p(, ( σ( pt ( σ(,q p( σ(, + ϕt (,,Q ϕ(,, ϕt (,,Q ϕ(,, + ϕt ϕ(,s, (,s,q + ψt (,,Q ψ(,, ψt (, σ,q ψ(, σ,, + σ ψt ψ(,s, (,s,q ςt (,Φ (τ(,σ(ς(,, V (,m,p = ( σ( f T (p( σ(,q 5 f(p( σ(, + f T (p(,q 5 f(p(, ςt (,Φ ς(,, V (,m,p = τ +σ σ m T (, m(, R m T (s, s p T (, p(, R σ m(s, R s p T (s, p(s, s R s +τ mt (,R m(, mt (s,r m(s, ( ( +σ pt (,R p(, σ σ pt (s,r p(s,, ( V 5(,m,p = τ m T (, m(, M m T (s, m(s, +θ s M s + σ p T (, p(, M +θ p T (s, p(s, σ s M s. ( From Green formul, Dirichle boundry condiions nd Lemm, we hve l ( mt m(, (,P D k = l ( m T m(, (,P D k l m T (, m(, P D k = l ( l m T m(, (,P D k n ds l m T (, m(, P D k = l m T (, m(, P D k π mt (,P D L m(,, (5

6 6 where = Similrly, ( l m T m(, (,P D k ( m T m(, (,P D,...,m T m(, (,P D l l. l pt (,P (Dk p(, π pt (,P DL p(,. (6 The combinion of (, (5 nd (6 gives V (,m,p = mt (,P [ π D Lm(, (A+K Mm(,+W f(p( σ(, ] + [ pt (,P π D L p(, (C +K Np(,+Bm((,] + l m T (, P D k ( m(, + l p T (, P Dk ( p(, = ςt (,Φ ς(, + l m T (, P D k + l ( m(, p T (, P Dk ( p(,. (7 Noe h he second erm on he righ of ( cn be wrien s: = m T (s, m(s, s R s ( m T (s, m(s, s R s m T (s, m(s, ( s R s Applying Lemm, one cn obin h nd ( m T (s, s. m(s, R s τ τ( ςt (, 7 R T 7 ς(, m T (s, m(s, ( s R s τ τ( ςt (, 8 R 8 ς(,. (8 (9 On he oher hnd, by Lemm, i yields = τ τ( τ τ( mt (s,r m(s, ( m T (s,r m(s, ( mt (s,r m(s, ( m T ( (s,dsr m(s, ( mt (s,dsr ( mt (s, ςt (,Φ (τ(ς(,. In he sme wy, σ σ pt (s,r p(s, ςt (,Φ (σ(ς(,. Combining (, (-( yields ( ( V (,m,p ςt (,Φ (τ(,σ(ς(,. ( The second erm on he righ of ( cn be divided ino hree prs: = ( ( m T (s, m(s, +θ s M s m T (s, m(s, +θ s M s ( m T (s, m(s, +θ s M s m T (s, ( s (τ ( m(s, M s. (5 By using Lemm, we cn esime he following inequliies m T (s, m(s, ( +θ s M s ( ( m T (s, m(s, +θ s M s τ( ( +θ m(s, ( +θ s ( ( (τ( +θ ( ( m(s, +θ s m T (s, s M m T (s, s M = ςt (,Φ 5 ς(,. (6 As in (9, he ls erm on he righ of (5 cn be bounded by pplying he sme procedure, (τ ( (τ( τ ( m T (s, s m(s, M s This, ogeher wih (8, (8 nd Lemm 5, implies h In similr mnner, ςt (, 8 M T 8 ς(,. (7 m T (s, s m(s, R s ςt (,[ 7 8 ]ˆR [ 7 8 ] T ς(,. Similrly, ( nd σ( σ( σ P T (s, P(s, +θ s M s σ( +θ ςt (,Φ 5 ς(,, p T (s, p(s, s M s (8 σ σ p T (s, p(s, s R s ςt (,[Θ T 7 Θ 8 ]ˆR [Θ T 7 Θ 8 ] T ς(,. ( σ( p T (s, p(s, s M s (σ σ( (σ σ( σ ςt (,Θ 8 M Θ T 8ς(,. (9

7 7 Combining ( nd (5-(9, we cn obin V 5(,m,p ς T (,Φ 5 (τ(,σ(ς(,. ( nd I is lso esy o see h [ m T (, l P m(, D k (A+K Mm(,+W ] f(p( σ(, = m(, ( [ p T (, l P (Dk p(, (C +K Np(,+Bm((, p(, ] =. ( ( According o Lemm, Green formul nd Dirichle boundry condiions, we hve m T (, ( l P m(, D k = mt (,P l = l Similrly, m T (, P D k [ ( D m(, k ( m(,. p T (, l P (Dk p(, = l pt (,P = l [ ( Dk p(, p T (, P Dk ( p(,. ] ] ( ( Finlly, for he digonl mrices Λ > nd Λ >, i cn be obined from (7 h f T (p(,λ f(p(, p T (,KΛ f(p(,, (5 f T (p( σ(,λ f(p( σ(, p T ( σ(,kλ f(p( σ(,. (6 From (, (, (7, ( nd (-(6, one cn obin V(,m,p = 5 i= V i(,m,p ςt (,Φ(τ(,σ(ς(,. Since Φ(τ(, σ( depends ffinely on τ( nd σ(, respecively, i follows from (9 h V(,m,p < for ll τ( nd σ( sisfying (. Therefore, he rivil soluion of error sysem (6 is sympoiclly sble. This complees he proof. We end he secion by he following remrks on Theorem. Remrk : Compred wih [5] [8], he dvnges of his pper re s follows: We inroduce new inegrl iems like τ m T (s,r m(s, nd θ +λ +θ m T (s, s m(s, M dsdλdθ s ino Lypunov-Krsovskii funcionl nd employ Wiringer-bsed inegrl inequliy (insed of Jensen s inequliy o esime he derivive of he second one, which will ge more ccure resul. The so-clled conve combinion pproch nd RCC pproch re employed simulneously, which will improve he precision of esimion o he concenrions of mrna nd proein. The coefficiens of some iems in ς(,, like τ( τ( nd, ply very imporn role in simplificion of he LMI condiion (9. We use ϕ(,s, nd ψ(,s, insed of m(s, nd p(s, in V (,m,p, respecively. This will highly minin consisen wih V (,m,p. Remrk : The pproch proposed in his pper cn esily be pplied o esblish dely-dependen nd dely-redependen sympoic sbiliy crierion for GRN (. Due o Remrk bove, he crierion is cerinly less conservive hn ones in [5], [7], [8]. IV. ILLUSTRATIVE EXAMPLES In his secion, wo numericl emples re provided o demonsre he effeciveness nd pplicbiliy of he proposed se observer. Emple : Consider GRN ( wih mesuremens (, he deerminisic prmeers re given s: A = dig(.,.,., B = dig(.,.,.7, C = dig(.,.7,., L = L = L =, W =.5.5,.5 D = D = D = dig(.,.,., D = D = D [ = dig(.,.,., ].5.6 M =,..8. [ ].7.5. N =.... Here he regulion funcion is ken s f( = +. One cn ge (7 holds when K =.65I. When τ = σ = nd µ = µ =, by using he MATLAB YALMIP Toolbo, one cn see h he LMIs given in Theorem re fesible wih he following fesible soluion mrices. To sve spce, we only lis some of he fesible soluion mrices s follows: P = dig(57.656,.,5.577, P = dig(5.799,9.68,.957, Q = , Q 5 = ,

8 8 R = R = M = W = W = Moreover, we cn ge he corresponding observer gin mrices s follows:.68.5 K = P W =.5.65, K = P W =, Emple : When l = n =, GRN ( is simplified ino ( m(, = m(, D A m(, +Wf( p( σ(,, ( (7 p(, = D p(, C p(, +B m((,. We choose he vlues of prmeers in (7 re s follows, A =., B =, C =., L =, W =.5, D =., D =., M =, N =.7. When µ = µ =,K =.65 nd τ = σ =, for Dirichle boundry condiions, by using he MATLAB YALMIP Toolbo o solve he LMIs given in Theorem, we obin he following fesible soluion mrices. To sve spce, we only lis some of he fesible mrices s follows: P =.8, Q =., R =.87, M =.556, W =, W =.78. Moreover, we cn ge he corresponding observer gin mrices s follows: K = P W =, K = P W =.557. Furher, when σ( = τ( =, he se responses of GRN (7, observer (5 nd he corresponding error sysem re given in Figures 6.,,,. Figure. Figure. Figure. Figure. mrna concenrion.... The rel rjecory of mrna ( m(, Esime of mrna concenrion.... The esimed rjecory of mrna ( ˆm(, Error of mrna concenrion The esimion error of mrna ( m(, ˆm(, proein concenrion..... The rel rjecory of proein ( p(,

9 9 Figure 5. Figure 6. Esime of proein concenrion..... The esimed rjecory of proein (ˆp(, Error of proein concenrion.... The esimion error of proein ( p(, ˆp(, V. CONCLUSIONS In his pper, he se esimion problem for clss of GRNs wih ime-vrying delys nd recion-diffusion erms re sudied. An se observer is designed o esime he gene ses hrough vilble sensor mesuremens, nd gurnee h he error sysem is sympoiclly sble. By inroducing new inegrl erms in novel Lypunov Krsovskii funcionl nd employing he so-clled Wiringer-bsed inegrl inequliy, Wiringer s inequliy, Green s second ideniy nd, conve combinion pproch, RCC pproch, sufficien condiion gurneeing he eisence of se observers is esblished in erms of LMIs. The concree epression of he desired se observer hs been presened in Theorem. Finlly, wo numericl emples re given o illusre he effeciveness of he heoreicl resuls. REFERENCES [] H. De Jong, Modeling nd simulion of geneic regulory sysems: lierure review, J. Compu. Biology, vol. 9, no., pp. 67,. 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