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2 Nonlinear Analyi: Real World Applicaion 14 (213) Conen li available a SciVere ScienceDirec Nonlinear Analyi: Real World Applicaion journal homepage: Reduced order dead-bea oberver for he chemoa Iaon Karafylli a,, Zhong-Ping Jiang b a Deparmen of Environmenal Eng., Technical Univeriy of Cree, 731, Chania, Greece b Deparmen of Elecrical and Compuer Eng., Polyechnic Iniue of New York Univeriy, Six Meroech Cener, Brooklyn, NY 1121, USA a r i c l e i n f o a b r a c Aricle hiory: Received 19 Sepember 211 Acceped 1 July 212 Keyword: Oberver deign Chemoa Hybrid yem Thi paper udie he rong obervabiliy propery and he reduced-order deadbea oberver deign problem for a coninuou bioreacor. New relaionhip beween coexience and rong obervabiliy, and checkable ufficien condiion for rong obervabiliy, are eablihed for a chemoa wih wo compeing microbial pecie. Furhermore, he dynamic oupu feedback abilizaion problem i olved for he cae of one pecie. 212 Elevier Ld. All righ reerved. 1. Inroducion The deign of oberver i a very imporan problem in mahemaical conrol heory. In hi work, we focu on he oberver deign problem for he chemoa wih n compeing pecie and one limiing ubrae (ee [1]): ẋ i () = (µ i (()) D() b i ) x i (), i = 1,..., n n ṡ() = D() ( in () ()) g i (())x i () i=1 x() = (x 1 (),..., x n ()) Ω, () A (1.1) wih meauremen y = and inpu u = (D, in ) U = [, + ) 2 R 2. A uual, x i () denoe he concenraion of he ih microbial pecie and Ω = in(r n + ), () A (, + ) denoe he concenraion of he limiing nurien, in() denoe he inle concenraion of he limiing nurien and D() denoe he diluion rae. The pecific growh rae µ i () of he i- h microbial pecie i a coninuouly differeniable, poiive definie, bounded funcion µ i : [, + ) [, + ) (i = 1,..., n) wih µ i () > for all >. The conan b i (i = 1,..., n) are he moraliy rae of he microbial pecie, while he coninuouly differeniable, poiive definie, bounded funcion g i : [, + ) [, + )(i = 1,..., n) wih g i () > for all >, are he produc of he pecific growh rae of he pecie wih he correponding poibly varying yield conan (ee [2,3] for he chemoa wih variable yield). For he open e A (, + ), we will diinguih he following cae: A = (, + ) (he general cae), for he cae in () in, A = (, in ). The lieraure concerning chemoa model of he form (1.1) i va, ince he chemoa appear o be one of he cornerone of Mahemaical Populaion Biology. The dynamic of (1.1) were udied in [4,5,1 3] (ee alo reference herein), where he heory of monoone dynamical yem (ee [6,7,1]) play an imporan role. Feedback abilizaion Correponding auhor. Tel.: addree: ikarafyl@enveng.uc.gr (I. Karafylli), zjiang@poly.edu (Z.-P. Jiang) /$ ee fron maer 212 Elevier Ld. All righ reerved. doi:1.116/j.nonrwa

3 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) problem for he chemoa have been udied in [8 24]. Oberver problem for he chemoa have been udied in [8,25 32] and exiing oberver deign procedure (like hoe in [33]) can be ued for he oberver deign for he chemoa. I hould be noed ha he ucceful oberver deign for he chemoa can lead o he oluion of dynamic oupu feedback conrol problem (ee [8,22]). In hi work, we apply recen reul for he obervabiliy of yem which are linear in he unmeaured ae componen (preened in [34]), in order o deign hybrid dead-bea reduced-order oberver for yem (1.1). More pecifically, we how ha: yem (1.1) wih n = 1 i rongly obervable in ime r > for arbirary r >, he dynamic oupu feedback abilizaion problem for (1.1) wih n = 1 can be olved wih he combinaion of a aic ae feedback abilizer and he propoed dead-bea hybrid reduced order oberver (Propoiion 3.1), coexience implie abence of rong obervabiliy for n = 2 (Propoiion 3.3), i i impoible o deign a mooh oberver for yem (1.1) wih n = 2 which guaranee convergence of he eimae for all inpu (D, in ) L loc (R +; U), under he aumpion of coexience (Propoiion 3.4), yem (1.1) wih n = 2, D (bach culure) and Michaeli Menen kineic for he pecific growh rae i rongly obervable in ime r > for arbirary r > if and only if he pecific growh rae and he moraliy rae of he wo microbial pecie are no idenical (Theorem 3.5); hi reul i imporan becaue bach culure of microbial pecie are ued only for finie ime and he propoed dead-bea hybrid reduced order oberver can provide exac eimae in very hor ime, a e of ufficien condiion (which doe no allow coexience) can guaranee rong obervabiliy of yem (1.1) wih n = 2 in ime r > (Theorem 3.7 and 3.8). The abence of rong obervabiliy and he impoibiliy of deigning an oberver for he cae n = 2 under he aumpion of coexience, juifie he ue of feedback law which necearily depend on he concenraion of he microbial pecie for he abilizaion of he coexience equilibrium poin (ee [9,21]). Finally, i hould be menioned ha he obained reul can be exended in he ame piri o he cae n 3 (ee Lemma 2.5 in he preen work and [22]). Noaion. Throughou hi paper we adop he following noaion: Le I R + := [, + ) be an inerval. By L (I; U)(L loc (I; U)) we denoe he pace of meaurable and (locally) eenially bounded funcion u( ) defined on I and aking value in U R m. For definiion of he funcion clae K, KL ee [31]. By C ( ; Ω), we denoe he cla of coninuou funcion on R n, which ake value in Ω. The erm mooh funcion mean a funcion wih derivaive of all order. A funcion f : R, where R n i a non-empy e wih, i called poiive definie if f (x) > for all x \ {} and f () =. For a vecor x R n, we denoe by x i ranpoe. The deerminan of a quare marix A R n n i denoed by de(a). A R n m denoe he ranpoe of he marix A R m n. By A = diag(l 1, l 2,..., l n ) we mean ha he marix A = {a ij ; i = 1,..., n, j = 1,..., n} i diagonal wih a ii = l i, for i = 1,..., n. By in(u) U we denoe he inerior of a e U R m. By R n + we denoe he e of all x = (x 1,..., x n ) R n wih x i (i = 1,..., n). Under he aumpion decribed above, for every pair of inpu (D, in ) L loc (R +; U) and for every (x, ) Ω A here exi a unique oluion (x(), ()) Ω A of (1.1) defined for all wih iniial condiion (x(), ()) = (x, ) correponding o inpu (D, in ) L loc (R +; U). 2. Review of recen reul Conider an auonomou yem decribed by ordinary differenial equaion of he form: ẋ() = f (x(), u()) x() Ω R n, u() U R m (2.1) where Ω R n i an open e, U R m i a non-empy cloed e and he mapping f : Ω U R n i locally Lipchiz. The oupu of yem (2.1) i given by y() = h(x()) where he mapping h : Ω R k i coninuou. We aume ha for every x Ω and u L (R loc +; U), he oluion x(, x ; u) of (2.1) wih iniial condiion x() = x(, x ; u) = x and correponding o inpu u L (R loc +; U) exi for all, i.e., we aume forward compleene. For yem (2.1) we adop he following noion of obervabiliy. (2.2)

4 342 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) Definiion 2.1. Conider yem (2.1) wih oupu (2.2). We ay ha he inpu u L ([, r]; U) rongly diinguihe he ae x Ω in ime r >, if he following condiion hold max h(x(, x ; u)) h(x(, ξ; u)) >, for all ξ D wih x ξ. (2.3) [,r] We nex define he noion of rongly obervable yem in ime r >. The reader hould noe ha rong obervabiliy i a more demanding noion han imple obervabiliy (ee [35]). Definiion 2.2. Conider yem (2.1). We ay ha (2.1) i rongly obervable in ime r > if every inpu u L ([, r]; U) rongly diinguihe every ae x D in ime r >. Now, we how how he main reul of [34], ha i, Propoiion 2.3 and Corollary 3.3 of [34], can be applied o he coninuou bioreacor model (1.1). More preciely, auming for he ime being ha yem (1.1) i rongly obervable in ime r >, we are in a poiion o define he operaor: P : C ([, r]; A) L ([, r]; U) Ω. For each C ([, r]; A), (D, in ) L ([, r]; U), P(, D, in ) i defined by r P(, D, in ) = Φ(r)Q 1 p(τ)q(τ)dτ where Φ() := diag exp (µ 1((w)) D(w) b 1 ) dw q (τ)dτ, q(τ) = τ,..., exp (µ n((w)) D(w) b n ) dw, Q = r q(τ) Φ ()C()d, C(τ) = (g 1 ((τ)),..., g n ((τ))) R n, p(τ) = (τ) () τ D(w) ( in (w) (w)) dw for all τ [, r]. Propoiion 2.3 in [34] guaranee ha, under he aumpion of rong obervabiliy in ime r > for yem (1.1), hen for every (x, ) Ω A and (D, in ) L loc (R +; U) he following equaliy hold: x() = P(δ r, δ r D, δ r in ), for all r (2.5) where (δ r ) (w) = ( r + w), (δ r D) (w) = D( r + w), (δ r in ) (w) = in ( r + w) for w [, r]. Therefore, if yem (1.1) i rongly obervable in ime r >, hen we are in a poiion o provide a reduced order dead-bea oberver for yem (1.1). Given, z Ω, we calculae z() by he following algorihm. Calculaion of z() for [ + ir, + (i + 1)r], where i i a non-negaive ineger: (1) Calculae z() for [ + ir, + (i + 1)r), he oluion of ż() = M((), D())z(), where M(, D) = diag (µ 1 () D b 1,..., µ n () D b n ). (2) Se z( + (i + 1)r) = P(δ +ir, δ +ird, δ +ir in ), where P : C ([, r]; A) L ([, r]; U) Ω i he operaor defined by (2.4). For i = we ake z( ) = z (iniial condiion). Schemaically, we wrie: (2.4) ż() = M((), D())z(), [, +1 ) z(+1 ) = P(δ τi, δ τi D, δ τi in ) +1 = + r. (2.6) Thu, we obain from Corollary 3.3 in [34] he following reul. Propoiion 2.3. Conider yem (1.1) and aume ha i i rongly obervable in ime r >. Conider he unique oluion (x(), (), z()) Ω A Ω of (1.1), (2.6) wih arbirary iniial condiion (x,, z ) Ω A Ω correponding o arbirary inpu (D, in ) L loc (R +; U). Then he oluion (x(), (), z()) Ω A Ω of (1.1), (2.6) aifie: z() = x(), for all r. (2.7) The explici formulae for he oberver (2.6) for n = 2 are given nex. ż 1 () = (µ 1 (()) D() b 1 ) z 1 () ż 2 () = (µ 2 (()) D() b 2 ) z 2 (), for [, +1 ) and τi+1 z j (+1 ) = N j I1 I 2 I 2 1 1,2 exp µj (()) D() b j d, j = 1, 2

5 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) where and N 1 := N 2 := I j := τi+1 τi+1 τ i τi+1 φ j () := ( ) () + ( ) () + φ 2 j ()d, j = 1, 2 and I 1,2 = g j ((τ)) exp τ I2 D(w) ( in (w) (w)) dw φ 1 () I 1,2φ 2 () d I1 D(w) ( in (w) (w)) dw φ 2 () I 1,2φ 1 () d τi+1 φ 1 ()φ 2 ()d µj ((w)) D(w) b j dw dτ, j = 1, 2. I hould be noed ha he oberver (2.6) i a hybrid oberver which ue delay and guaranee exac knowledge of he concenraion of he microbial pecie afer r ime uni. Finally, we end hi ecion by preening he following lemma, which i o be ued in nex ecion. Lemma 2.4. Suppoe ha yem (1.1) wih n = 2 i no rongly obervable in ime r >. Then here exi (D, in ) L ([, r]; U) and (x, ) in(r 2 + ) A uch ha ṡ() = D() ( in () ()) x 2, + g 1( ) g 2 ( ) x 1, g 2 (()) exp (µ 2 ((w)) D(w) b 2 ) dw, and for almo all [, r] κ(())ṡ() = µ 2 (()) µ 1 (()) + b 1 b 2, for almo all [, r] (2.9) where () A denoe he componen of he oluion (x(), ()) in(r 2 + ) A of (1.1) wih iniial condiion (x, ) in(r 2 + ) A correponding o inpu (D, in) L (R loc +; U), x = (x 1,, x 2, ) in(r 2 + ) and κ() := d g1 d ln (). (2.1) g 2 () Proof. Suppoe ha yem (1.1) i no rongly obervable in ime r >. By virue of Corollary 2.4 in [34] here exi (D, in ) L ([, r]; U), (x, ) in(r 2 + ) A and ξ = (ξ 1, ξ 2 ) R 2, ξ uch ha ξ 1 g 1 (()) exp (µ 1 ((w)) b 1 ) dw + ξ 2 g 2 (()) exp (µ 2 ((w)) b 2 ) dw =, for all [, r] (2.11) where () A denoe he componen of he oluion (x(), ()) in(r 2 + ) A of (1.1) wih iniial condiion (x, ) in(r 2 + ) A correponding o inpu (D, in) L (R loc +; U). Therefore, (2.11) implie ha: g 1 ( ) g 2 ( ) de =, g 1 (()) exp (µ 1 ((w)) b 1 ) dw g 2 (()) exp (µ 2 ((w)) b 2 ) dw or equivalenly, for all [, r] g 1 (()) exp (µ 1 ((w)) b 1 ) dw = g 1( ) g 2 ( ) g 2(()) exp (2.8) (2.12) (µ 2 ((w)) b 2 ) dw, for all [, r]. (2.13) Equaliy (2.13) in conjuncion wih (1.1) and he fac ha x i () = x i, exp (µ i((w)) D(w) b i ) dw, i = 1, 2, implie equaliy (2.8). Equaliy (2.9) i obained by differeniaion of (2.13). The reul of Lemma 2.4 can be exended o a number of microbial pecie (n) greaer han 2. However, in uch a cae, we need addiional regulariy properie and he reul i a differenial equaion for () of order n 1. The following lemma how he exenion of Lemma 2.4 o he cae n = 3.

6 344 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) Lemma 2.5. Conider yem (1.1) wih n = 3 and uppoe ha he funcion µ i : [, + ) [, + ) (i = 1, 2, 3) and g i : [, + ) [, + ) (i = 1, 2, 3) are mooh. Moreover, uppoe ha he applied inpu (D, in ) L ([, r]; U) are aboluely coninuou funcion. If yem (1.1) wih n = 3 i no rongly obervable in ime r > hen here exi aboluely coninuou inpu (D, in ) L ([, r]; U) and (x, ) in(r 3 + ) A uch ha eiher or κ 2,3 (())ṡ() + µ 2,3 (()) d κ1,3 (())ṡ() + µ 1,3 (()) + κ 2,3 (())ṡ() + µ 2,3 (()) d κ 1,3 (())ṡ() + µ 1,3 (()) κ 2,1 (())ṡ() + µ 2,1 (()) + κ 1,3 (())ṡ() + µ 1,3 (()) d d κ2,3 (())ṡ() + µ 2,3 (()) =, for almo all [, r] (2.14) κ 1,2 (())ṡ() + µ 1,2 (()) =, for all [, r] (2.15) where () A denoe he componen of he oluion (x(), ()) in(r 3 + ) A of (1.1) wih iniial condiion (x, ) in(r 3 + ) A correponding o inpu (D, in) L (R loc +; U) and κ i,j () := d gi d ln (), µ i,j () := µ i () µ j () b i + b j (2.16) g j () for all i, j = 1, 2, 3. Proof. Suppoe ha yem (1.1) i no rongly obervable in ime r >. By virue of Corollary 2.4 in [34] here exi (D, in ) L ([, r]; U), (x, ) in(r 3 + ) A and ξ = (ξ 1, ξ 2, ξ 3 ) R 3, ξ uch ha ξ 1 g 1 (()) exp (µ 1 ((w)) b 1 ) dw + ξ 2 g 2 (()) exp (µ 2 ((w)) b 2 ) dw + ξ 3 g 3 (()) exp (µ 3 ((w)) b 3 ) dw =, for all [, r] (2.17) where () A denoe he componen of he oluion (x(), ()) in(r 3 + ) A of (1.1) wih iniial condiion (x, ) in(r 3 + ) A correponding o inpu (D, in) L loc (R +; U). We diinguih he following cae: (1) Cae 1: ξ 3 =. In hi cae, working exacly a in he proof of Lemma 2.4 we can how ha (2.17) implie ha (2.15) hold. (2) Cae 2: ξ 2 =. In hi cae, working exacly a in he proof of Lemma 2.4 we can how ha (2.17) implie he differenial equaion κ 1,3 (())ṡ() + µ 1,3 (()) = for all [, r]. Conequenly, (2.14) hold. (3) Cae 3: ξ 3 and ξ 2. In hi cae, by differeniaing (2.17) and uing definiion (2.16) we obain: ξ 1 κ1,3 (())ṡ() + µ 1 (()) g 1 (()) exp (µ 1 ((w)) b 1 ) dw + ξ 2 κ2,3 (())ṡ() + µ 2 (()) g 2 (()) exp (µ 2 ((w)) b 2 ) dw =, for all [, r]. (2.18) By differeniaing (2.18) and uing definiion (2.16), we obain (2.14). 3. Srong obervabiliy of he chemoa The reul of he previou ecion clearly indicae ha i i imporan o udy under wha condiion he chemoa (1.1) i a rongly obervable yem Cae n = 1 For hi cae we have he yem: ẋ() = (µ(()) D() b) x() ṡ() = D() ( in () ()) g(())x(). (3.1)

7 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) Uing Corollary 2.4 in [34], we conclude ha yem (3.1) i rongly obervable in ime r > for arbirary r >. The oberver (2.6) can be ued for yem (3.1), which for n = 1 ake he form: where ż() = (µ(()) D() b) z(), [, +1 ) τi+1 τi+1 1 τi+1 z(+1 ) = exp (µ((w)) D(w) b) dw q 2 ()d p()q()d +1 = + r p() := () ( ) D(w) ( in (w) (w)) dw and p q() := g((p)) exp (µ((w)) D(w) b) dw dp. The hybrid reduced-order oberver (3.2) guaranee ha for every iniial condiion (x,, z ) Ω A Ω and for every (D, in ) L loc (R +; U) he oluion of (3.1), (3.2) aifie z() = x() for all r. I i imporan o noice ha he oberver (3.2) can be combined wih ae feedback law for he abilizaion of he chemoa model (3.1) by mean of dynamic oupu feedback. More pecifically, we obain he following reul. Propoiion 3.1. Le (, x ) (, in ) (, + ) be an equilibrium poin for (3.1) wih in () in >, D() D >, i.e., µ( ) = D + b and D ( in ) = g( )x. Suppoe ha here exi a locally Lipchiz feedback law k : (, in ) (, + ) (, + ) wih D = k(, x ) uch ha (, x ) (, in ) (, + ) i globally aympoically able for he cloed-loop yem (3.1) wih he feedback law D = k(, x) and (, x) (, in ) (, + ). Moreover, uppoe ha yem (3.1) wih he dynamic feedback law D = k(, z), ż = (µ() D b) z, z (, + ) i forward complee. Then for every r >, he equilibrium poin (, x, z) = (, x, x ) i globally aympoically able for he cloed-loop yem (3.1) wih (3.2) and D = k(, z). Proof. Le r > be arbirary. Fir, he following change of coordinae i performed: (3.2) x = x exp(x 1 ), = in exp(x 2 ) G + exp(x 2 ) where G := in. Under he above change of coordinae, yem (3.1) wih in () in > ake he form: (3.3) ẋ 1 = µ(x 2 ) + D D ẋ 2 = (G exp( x 2 ) + 1) D D g(x 2 ) exp(x 1 ) x = (x 1, x 2 ) R 2 (3.4) where µ(x 2 ) := µ in exp(x 2 ) G+exp(x 2 ) µ( ), g(x 2 ) := G+exp(x 2) G+1 g in exp(x 2 ) G+exp(x 2 ) g(. Since he feedback law D = k(, x) guaranee ha ) (, x ) (, in ) (, + ) i globally aympoically able for he cloed-loop yem (3.1), i follow ha R 2 i globally aympoically able for he cloed-loop yem (3.4) wih D = k in exp(x 2 ) G+exp(x 2 ), x exp(x 1 ). Therefore, here exi σ KL uch ha for every x R 2, he oluion of he cloed-loop yem (3.4) wih D = k in exp(x 2 ) G+exp(x 2 ), x exp(x 1 ) iniiaed from (x 1 (), x 2 ()) = x aifie: x() σ ( x, ). Syem (3.1) wih he dynamic feedback law D = k(, z), ż = (µ() D b) z i ranformed o yem (3.4) wih D = k in exp(x 2 ) G+exp(x 2 ), x exp(x 3 ) and (3.5) ẋ 3 = µ(x 2 ) + D D; x 3 R (3.6) where z = x exp(x 3 ). Moreover, yem (3.4), (3.6) wih D = k follow ha yem (3.4), (3.6) wih D = k (3.7) G+exp(x 2 ), x exp(x 3 ) i forward complee. Uing he main reul in [36], i i Robuly Forward Complee (ee [37]) and conequenly, in exp(x 2 ) in exp(x 2 ) G+exp(x 2 ), x exp(x 3 )

8 346 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) by virue of Lemma 2.3in [37], here exi a K uch ha every x R 3, [, r] he oluion of he cloed-loop yem (3.4), (3.6) wih D = k in exp(x 2 ) G+exp(x 2 ), x exp(x 3 ) iniiaed from (x 1 (), x 2 (), x 3 ()) = x aifie: (x 1 (), x 2 (), x 3 ()) a ( x ). (3.8) Since z() = x() for all r, i follow from (3.3), (3.5), (3.7) and (3.8) ha he cloed-loop yem (3.1) wih (3.2) and D = k(, z) aifie he following eimae: x() ln + (in x ln )() + z() ( in ()) ln x x() 3σ a ln + z() x ln + (in x ln )(), r, for all r (3.9) ( in ()) x() ln + (in x ln )() + z() ( in ()) ln x x() 3a ln + z() x ln + (in x ln )(), for all [, r]. (3.1) ( in ()) Inequaliie (3.8) and (3.1) allow u o conclude ha he equilibrium poin (, x, z) = (, x, x ) i globally aympoically able for he cloed-loop yem (3.1) wih (3.2) and D = k(, z). The proof i complee. Example 3.2. In [15] i i hown ha he feedback law D = µ() (D +b)x x + L max (, ), where L > i a conan, guaranee ha (, x ) (, in ) (, + ) i globally aympoically able for he cloed-loop yem (3.1) wih g() = Kµ(), in () in, where K > i a conan. Here, we will how ha yem (3.1) wih g() = Kµ(), in () in, D = D µ() (D +b)x z + L max (, ), ż = (µ() D b) z i forward complee. Therefore, Propoiion 3.1 guaranee ha for every r >, he equilibrium poin (, x, z) = (, x, x ) i globally aympoically able for he cloed-loop yem D (3.1) wih (3.2) and D = µ() (D +b)x z + L max (, ), g() = Kµ(), in () in. We conider yem (3.1) wih g() = Kµ(), in () in, D = µ() (D +b)x z +L max (, ), ż = (µ() D b) z. Clearly, for every (z, x, ) (, + ) (, + ) (, in ) here exi max > uch ha he oluion (z(), x(), ()) D (, + ) (, + ) (, in ) of yem (3.1) wih g() = Kµ(), in () in, D = µ() (D +b)x z + L max (, ), ż = (µ() D b) z iniiaed from (z(), x(), ()) = (z, x, ) exi for all [, max ). Uing he fac ha µ() µ max for all, i follow ha: z() = x()x 1 z, for all [, max ) (3.11) x() x exp ((µ max b) ), for all [, max ). (3.12) Simple manipulaion how ha ṡ = L max (, ) ( in ) + Kµ()x in c( in ) ( ω + ), where c = x z, ω = c + (1 c). I follow ha ṡ in < for all > max 1, ω 1 and ṡ > for all < min 1, ω 1. A hi poin i hould be noiced ha in > max 1, ω 1. Therefore, he following inequaliie hold: min,, ω 1 () max,, ω 1, for all [, max ). (3.13) Uing (3.11), (3.12), (3.13), i follow ha he following differenial inequaliy hold for all [, max ) ẋ µ max ( in ) min,, ω 1 K z exp ((µ max b) ) L b x D D which direcly implie ha he following eimae hold: µ max K z x() exp (µ max b) ( in ) min,, ω 1 (exp ((µ max b) ) 1) L b x, for all [, max ). (3.14) Inequaliie (3.12) (3.14) in conjuncion wih (3.11) and a andard conradicion argumen how ha we mu have D max = +. Hence, yem (3.1) wih g() = Kµ(), in () in, D = µ() (D +b)x z + L max (, ), ż = (µ() D b) z i forward complee.

9 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) Coexience implie abence of rong obervabiliy for n = 2 Conider yem (1.1) for n = 2 wih conan inpu D() D and in () in. A coexience equilibrium poin for he chemoa model (1.1) i an equilibrium poin (x 1, x 2, ) in(r 2 + ) (, in) of (1.1) aifying (ee [1]) µ 1 ( ) b 1 = D = µ 2 ( ) b 2 (3.15) D( in ) = g 1 ( )x 1 + g 2( )x 2. Propoiion 3.3. If yem (1.1) wih n = 2 admi a coexience equilibrium poin, hen for every r >, yem (1.1) wih n = 2 i no rongly obervable in ime r >. Proof. By virue of Definiion 2.1 and 2.2, i uffice o how ha here exi inpu (D, in ) L ([, r]; U), ae (x, ) Ω A, (ξ, ) Ω A uch ha () = (), for all [, r] where () A denoe he componen of he oluion (x(), ()) Ω A of (1.1), wih iniial condiion (x, ) Ω A correponding o inpu (D, in ) L loc (R +; U) and () A denoe he componen of he oluion ( x(), ()) Ω A of (1.1), wih iniial condiion (ξ, ) Ω A correponding o inpu (D, in ) L loc (R +; U). Conider he conan inpu D() D, in () in and noice ha for every ξ = (ξ 1, ξ 2 ) Ω wih (3.16) (3.17) ξ 2 = g 1( ) x g 2 ( 1 ) ξ 1 + x 2 (3.18) he componen () A of he oluion (x(), ()) Ω A of (1.1), wih iniial condiion (x, ) Ω A correponding o inpu (D, in ) L loc (R +; U) and he componen () A of he oluion ( x(), ()) Ω A of (1.1), wih iniial condiion (ξ, ) Ω A correponding o inpu (D, in ) L loc (R +; U) aify () (). The proof i complee. The abence of rong obervabiliy under he aumpion of coexience implie ha a mooh (convenional) oberver canno be deigned for hi cae. Thi i guaraneed by he following (negaive) reul. Propoiion 3.4. Aume ha a coexience equilibrium poin for he chemoa model (1.1) wih n = 2 exi. Then here are no locally Lipchiz funcion G i : A A Ω U R (i = 1, 2, 3) uch ha he following yem ż 1 () = (µ 1 (()) D() b 1 ) z 1 () + (() ξ()) G 1 ((), ξ(), z 1 (), z 2 (), D(), in ()) ż 2 () = (µ 2 (()) D() b 2 ) z 2 () + (() ξ()) G 2 ((), ξ(), z 1 (), z 2 (), D(), in ()) ξ() = D() ( in () ()) g 1 (())z 1 () g 2 (())z 2 () + (() ξ()) G 3 ((), ξ(), z 1 (), z 2 (), D(), in ()) z() = (z 1 (), z 2 ()) Ω, i an oberver for yem (1.1) wih n = 2. ξ() A Proof. Suppoe he conrary, i.e., uppoe ha here exi locally Lipchiz funcion G i : A A Ω U R (i = 1, 2, 3) uch ha yem (3.2) i an oberver for yem (1.1) wih n = 2. Uing he argumen in he proof of Propoiion 3.3, we conider yem (1.1) wih n = 2, D() D, in () in iniiaed a (x 1, x 2, ), where x = (x 1, x 2 ) Ω aifie D( in ) = g 1 ( )x 1 + g 2 ( )x 2 and yem (3.2) iniiaed a (z 1, z 2, ), where z = (z 1, z 2 ) Ω aifie D( in ) = g 1 ( )z 1 + g 2 ( )z 2 and z x. In hi cae i hold ha ξ() = () and z() x() z() x(). On he oher hand, he aumpion ha (3.2) i an oberver for yem (1.1) wih n = 2 implie ha lim + z() x() =, for every (D, in ) L (R loc +; U). Thu, we have a conradicion. The negaive reul of Propoiion 3.3 and 3.4 do no mean ha no obervaion of yem (1.1) wih n = 2 i poible, under he aumpion of coexience. If pecial inpu (D, in ) L loc (R +; U) are ued hen we may be able o obain an oberver for yem (1.1). The following ubecion how ha hi i he cae Bach culure for n = 2 The model of a bach culure of n pecie in compeiion i given by (1.1) wih D. For n = 2 we obain he inpu-free model: (3.19) (3.2) ẋ 1 = (µ 1 () b 1 ) x 1, ẋ 2 = (µ 2 () b 2 ) x 2 ṡ = g 1 ()x 1 g 2 ()x 2. (3.21)

10 348 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) We will aume conan yield coefficien, i.e., g i () = µ i () for i = 1, 2 and Michaeli Menen kineic for he pecific growh rae, i.e., µ 1 () = a 1 k 1 + ; µ 2() = a 2 k 2 + where a 1, a 2, k 1, k 2 > are poiive conan. Therefore, we obain he model: a1 ẋ 1 = k 1 + a2 b 1 x 1, ẋ 2 = k 2 + b 2 x 2 ṡ = a 1 k 1 + x 1 (x 1, x 2, ) in(r 3 + ). a 2 k 2 + x 2 We are in a poiion o prove he following reul. Theorem 3.5. For every r > he following implicaion hold: Syem (3.23) i no rongly obervable in ime r > a 1 = a 2, b 1 = b 2 and k 1 = k 2. Remark 3.6. Theorem 3.5 guaranee ha if he pecific growh rae and he moraliy rae of he wo pecie are no idenical, hen for every r >, yem (3.23) i rongly obervable in ime r >. Therefore he hybrid oberver (2.6) will be a reduced order dead-bea oberver. I i clear ha he convere implicaion of he one provided by Example 3.2 auomaically hold. Noice ha if a 1 = a 2 = a, b 1 = b 2 = b and k 1 = k 2 = k, hen yem (3.23) can be decompoed ino wo ubyem: he obervable ubyem d a d (x 1 + x 2 ) = k + b (x 1 + x 2 ) ṡ = a k + (x 1 + x 2 ) and he unobervable ubyem d a d (x 1 x 2 ) = k + b (x 1 x 2 ). Therefore he characerizaion provided by Theorem 3.5 i harp. Proof. Suppoe ha yem (3.23) i no rongly obervable in ime r >. Applying Lemma 2.4 wih D and uing (3.22) and he fac ha g i () = µ i () for i = 1, 2, we guaranee he exience of (x, ) in(r 3 + ) uch ha: (3.22) (3.23) (k 2 k 1 ) ṡ = [a 1 a 2 + b 2 b 1 ] 2 + [a 1 k 2 a 2 k 1 + (b 2 b 1 )(k 1 + k 2 )] + k 1 k 2 (b 2 b 1 ), for all [, r] (3.24) ṡ() = x 2, + µ 1( ) µ 2 ( ) x 1, µ 2 (()) exp (µ 2 ((w)) b 2 ) dw <, for all [, r] (3.25) where () > denoe he componen of he oluion (x(), ()) in(r 3 + ) of (3.23) wih iniial condiion (x, ) in(r 3 + ). We nex diinguih he following cae: Cae 1: k 1 = k 2 = k. From (3.24) and (3.25) we conclude ha he polynomial p() = [a 1 a 2 + b 2 b 1 ] 2 + k [a 1 a 2 + 2(b 2 b 1 )] + k 2 (b 2 b 1 ) mu be he zero polynomial. I follow ha b 1 = b 2 and a 1 = a 2. Cae 2: k 1 k 2. We will how ha hi cae canno happen becaue i lead o a conradicion. In hi cae, we ge from (3.24) and (3.25): and ṡ = f () := a 1 a 2 + b 2 b 1 k 2 k a 1k 2 a 2 k 1 + (b 2 b 1 )(k 1 + k 2 ) k 2 k 1 + k 1 k 2 b 2 b 1 k 2 k 1, for all [, r] (3.26) ṡ µ 2 () exp (µ 2 ((w)) b 2 ) dw = A, for all [, r] (3.27)

11 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) where A := x 2, + µ 1( ) µ 2 ( ) x 1,. By virue of (3.27) i follow ha for all [, r]: ṡ2 µ 2 () µ () 2 ṡ (µ 2() b 2 ) = (3.28) and by virue of (3.26) i follow ha for all [, r]: = f ()f (). (3.29) Combining (3.26), (3.28) and (3.29), we obain for all [, r]: f () f () µ 2 () µ () (µ 2 2() b 2 ) f () = or equivalenly, where p()f () = (3.3) p() = 2 (a 1 a 2 + b 2 b 1 ) 3 + [(2a 1 2a 2 + 3b 2 b 1 ) k 2 b 1 (k 1 + k 2 )] 2 + k 2 b 2 (k 2 k 1 ) (b 2 b 1 )k 1 k 2 2. (3.31) The fac ha ṡ < implie ha he polynomial p()f () mu be he zero polynomial. Therefore, b 1 = b 2 = and a 1 = a 2 = a >. However, noice ha in hi cae (3.26) give ṡ = a, which conradic (3.25) and he fac ṡ <. The proof i complee Condiion for rong obervabiliy in ime r > for n = 2 We nex provide condiion for rong obervabiliy of yem (1.1) wih n = 2 for wo differen cae. Theorem 3.7. Conider yem (1.1) wih n = 2, in () in and A = (, in ). Aume ha here exi a conan c > uch ha one of he following hold: (A1) κ() and µ 2() µ 1 ()+b 1 b 2 κ() c, for all (, in ) or (A2) κ() and µ 2() µ 1 ()+b 1 b 2 κ() c, for all (, in ) where κ i defined by (2.1). Then for every r c 1 in, (1.1) i rongly obervable in ime r >. Theorem 3.8. Conider yem (1.1) wih n = 2 and A = (, + ). Aume ha here exi conan a, c > uch ha one of he following hold: (A3) κ() and µ 2() µ 1 ()+b 1 b 2 a 2 c, for all (, + ) κ() or (A4) κ() and µ 2() µ 1 ()+b 1 b 2 a 2 + c, for all (, + ) κ() where κ i defined by (2.1). Then for every r π, yem (1.1) i rongly obervable in ime r >. 2 ac Remark 3.9. For he uual cae where g i () = µ i () for i = 1, 2 and he pecific growh rae aify he Michaeli Menen kineic (3.22), condiion (A1) (A4) are expreed by: (A1 ) k 1 k 2 and a 1 a 2 +b 2 b 1 2 k 2 k 1 + a 1k 2 a 2 k 1 +(b 2 b 1 )(k 1 +k 2 ) b k 2 k 1 + k 1 k 2 b 1 2 k 2 k 1 c, for all (, in ) or (A2 ) k 1 k 2 and a 1 a 2 +b 2 b 1 2 k 2 k 1 + a 1k 2 a 2 k 1 +(b 2 b 1 )(k 1 +k 2 ) b k 2 k 1 + k 1 k 2 b 1 2 k 2 k 1 c, for all (, in ) and (A3 ) k 1 k 2 and a 1 a 2 +b 2 b 1 2 k 2 k 1 + a 1k 2 a 2 k 1 +(b 2 b 1 )(k 1 +k 2 ) b k 2 k 1 + k 1 k 2 b 1 2 k 2 k 1 a 2 c, for all (, + ) or (A4 ) k 1 k 2 and a 1 a 2 +b 2 b 1 2 k 2 k 1 + a 1k 2 a 2 k 1 +(b 2 b 1 )(k 1 +k 2 ) b k 2 k 1 + k 1 k 2 b 1 2 k 2 k 1 a 2 + c, for all (, + ). For Theorem 3.7, if in addiion we have D() [, D max ], where D max, hen condiion (A2) can obain he following, le demanding form: (A2 ) κ() for all (, in ) and µ 2() µ 1 ()+b 1 b 2 κ() c for all (, in ) wih µ 2()+b 1 µ 1 () b 2 κ()( in ) < D max.

12 35 I. Karafylli, Z.-P. Jiang / Nonlinear Analyi: Real World Applicaion 14 (213) Proof of Theorem 3.7 and 3.8. The proof are made by conradicion. Suppoe ha yem (1.1) i no rongly obervable in ime r >. Employing Lemma 2.4, we can guaranee he exience of (D, in ) L ([, r]; U) and (x, ) in(r 2 + ) A uch ha: ṡ() = µ 2(()) µ 1 (()) + b 1 b 2, for almo all [, r]. (3.32) κ(()) I i direc o verify ha if A = (, in ), r c 1 in and hypohei (A1) or hypohei (A2) hold hen he oluion of (3.32) canno aify () A = (, in ) for all [, r], a conradicion. Indeed, if (A1) hold, hen (3.32) give ṡ() c, for almo all [, r], which direcly implie (r) cr. The previou inequaliy for A = (, in ) and r c 1 in how ha (r), i.e., (r) A, a conradicion. Similarly, if (A2) hold, hen (3.32) give ṡ() c, for almo all [, r], which direcly implie (r) + cr. The previou inequaliy for A = (, in ) and r c 1 in how ha (r) in, i.e., (r) A, a conradicion. If hypohei (A3) hold and r π, hen by employing he comparion lemma in [31], we can guaranee ha 2 ac he oluion of (3.32) canno aify () A for all [, r]. Indeed, (3.32) in conjuncion wih hypohei (A3) give ṡ() a 2 () c for almo all [, r]. The fac ha he mapping () i aboluely coninuou in conjuncion wih he inequaliy ṡ() a 2 () c for almo all [, r], implie ha lim up h + (+h) () a 2 () c for all arcan [, r]. The previou differenial inequaliy in conjuncion wih he comparion lemma in [31], implie ha arcan ac, for all [, r]. The previou inequaliy how ha () A for π () a c On he oher hand, if hypohei (A4) hold and r a c π 2 ac. 2 ac hen by uing he comparion lemma in [31], we can guaranee ha he oluion of (3.32) preen a finie ecape ime in he inerval [, r]. Indeed, (3.32) in conjuncion wih hypohei (A3) give ṡ() a 2 () + c for almo all [, r]. The mapping y() = 1 A = (, + ) i an aboluely () coninuou mapping on [, r] which aifie ẏ() = 2 ṡ() a c 2 () = a cy 2 () for almo all [, r]. From hi poin he analyi i exacly he ame a in he cae of hypohei (A3). 4. Concluding remark In hi work, we have applied recen reul for he obervabiliy of yem which are linear wih repec o he unmeaured ae componen in [34] in order o deign hybrid dead-bea reduced-order oberver for he chemoa wih microbial pecie in compeiion, i.e., yem (1.1). We have aumed ha he meaured oupu i he concenraion of he nurien and we are inereed in he eimaion of he ize of he populaion of he compeing microbial pecie. I ha been howed ha he chemoa wih one pecie i rongly obervable in ime r > for arbirary r >. The deign of a reduced-order hybrid dead-bea oberver for he cae wih one pecie allowed u o how ha he dynamic oupu feedback abilizaion problem for (1.1) wih n = 1 can be olved wih he combinaion of a aic ae feedback abilizer and he propoed dead-bea hybrid reduced order oberver. Furhermore, new relaionhip beween coexience and rong obervabiliy are eablihed for a chemoa wih wo microbial pecie (Propoiion 3.3). The propoed dead-bea reduced-order oberver can be ued for yem (1.1) wih n = 2, D (bach culure) and Michaeli Menen kineic for he pecific growh rae: he bach culure i rongly obervable in ime r > for arbirary r > if and only if he pecific growh rae and he moraliy rae of he wo microbial pecie are no idenical. The reul (Theorem 3.5) i imporan becaue bach culure of microbial pecie are ued only for finie ime and he propoed dead-bea hybrid reduced order oberver can provide exac eimae in very hor ime. Finally, a e of ufficien condiion (which do no allow coexience) ha can guaranee rong obervabiliy of yem (1.1) wih n = 2 in ime r > i provided. The obained reul can be exended in he ame piri o he cae of yem (1.1) wih n 3 (ee Lemma 2.5 in he preen work and [22]). Moreover, he obained reul can be ued for he udy of he oberver deign problem of yem (1.1) under periodic inpu (D, in ) L loc (R +; U) and he oluion of he oberver-baed oupu feedback conrol problem. Thi will be he ubjec of fuure reearch. h Acknowledgmen Thi work ha been uppored in par by he NSF gran DMS and DMS Reference [1] H. Smih, P. Walman, The Theory of he Chemoa. Dynamic of Microbial Compeiion, in: Cambridge Sudie in Mahemaical Biology, vol. 13, Cambridge Univeriy Pre, Cambridge, [2] L. Zhu, X. Huang, Muliple limi cycle in a coninuou culure veel wih variable yield, Nonlinear Analyi 64 (26) [3] L. Zhu, X. Huang, H. Su, Bifurcaion for a funcional yield chemoa when one compeior produce a oxin, Journal of Mahemaical Analyi and Applicaion 329 (27) [4] P. De Leenheer, D. Angeli, E.D. Sonag, Crowding effec promoe coexience in he chemoa, Journal of Mahemaical Analyi and Applicaion 319 (26) 48 6.

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