NEW RESULTS IN TRAJECTORY-BASED SMALL-GAIN WITH APPLICATION TO THE STABILIZATION OF A CHEMOSTAT Iao Karafyll * ad Zhog-Pg Jag ** * Dept. of Evrometal Eg., Techcal Uverty of Crete, 73, Chaa, Greece, emal: arafyl@eveg.tuc.gr ** Dept. of Electrcal ad Computer Eg., Polytechc Ittute of New Yor Uverty, Sx Metrotech Ceter, Brooly, NY, U.S.A., emal: zjag@cotrol.poly.edu Abtract New trajectory-baed mall-ga reult are obtaed for olear feedbac ytem uder relaxed aumpto. Specfcally, durg a traet perod, the oluto of the feedbac ytem may ot atfy ome ey equalte that prevou mall-ga reult uually utlze to prove tablty properte. The reult allow the applcato of the mall-ga perpectve to varou ytem whch atfy le demadg tablty oto tha the Iput-to-Output Stablty property. The robut global feedbac tablzato problem of a ucerta tme-delayed chemotat model olved by mea of the trajectorybaed mall-ga reult. Keyword: Iput-to-Output Stablty, Feedbac Sytem, Small-Ga Theorem, Chemotat Model.. Itroducto Small-ga reult are mportat tool for robute aaly ad robut cotroller deg Mathematcal Cotrol Theory. A olear, geeralzed mall-ga theorem wa developed [4], baed o the oto of Iput-to-State Stablty (ISS orgally troduced by Sotag [38]. Recetly, olear mall-ga reult were developed for mootoe ytem, a mportat cla of olear ytem mathematcal bology (ee [,7]. Further exteo of the mall-ga perpectve to the cae of o-uform tme tablty, dcrete-tme ytem ad Lyapuov characterzato are purued by everal author depedetly; ee, for tace, [,4,5,9,3,5,6,7,8,9,,3,4,4]. A geeral vector mall-ga reult, whch ca be appled to a wde cla of cotrol ytem, wa developed [8]. Oe of the mot mportat obtacle applyg olear mall-ga reult the repreetato of the orgal compote ytem a the feedbac tercoecto of ubytem whch atfy the Iput-to-Output Stablty (IOS property. More pecfcally, ometme the ubytem do ot atfy the IOS property: there a traet perod after whch the oluto eter a certa rego of the tate pace. Wth th rego of the tate pace the ubytem atfy the mall-ga requremet. I other word, the eetal equalte, whch mall-ga reult utlze order to prove tablty properte, do ot hold for all tme: th feature exclude all avalable mall-ga reult from poble applcato. Partcularly, th feature mportat ytem of Mathematcal Bology ad Populato Dyamc. Ideed, the dea of developg tablty reult whch utlze certa Lyapuov-le codto after a tal traet perod wa ued [6,7] wth prmary motvato from addreg robut feedbac tablzato problem for certa chemotat model. I th wor we preet mall-ga reult whch ca allow a traet perod durg whch the oluto do ot atfy the IOS equalte (Theorem.5 ad Theorem.6. The obtaed reult are drect exteo of the recet vector mall-ga reult [8] ad f the tal traet vahe the the reult cocde wth Theorem 3. [8]. The gfcace of the obtaed reult twofold: t allow the applcato of the mall-ga perpectve to varou ytem whch atfy le demadg tablty oto tha ISS, t allow the tudy of ytem Mathematcal Bology ad Populato Dyamc. Th wor ha bee upported part by the NSF grat DMS-5446 ad DMS-96659.
To emphaze the latter pot, we how how the obtaed mall-ga reult ca be ued for the feedbac tablzato of ucerta chemotat model. Chemotat model are ofte adequately repreeted by a mple dyamc model volvg two tate varable, the mcrobal boma cocetrato ad the lmtg utret cocetrato S (ee [37]. The commo delay-free model for mcrobal growth o a lmtg ubtrate a chemotat of the form: ( = S ( = D( ( S ( μ( D( ( Kμ( ( ( (,, (, S, D( (. where S the feed ubtrate cocetrato, D the dluto rate (whch ued a the cotrol pu, μ (S the pecfc growth rate ad K > a boma yeld factor. The lterature o cotrol tude of chemotat model of the form (. exteve. I [6], feedbac cotrol of the chemotat by mapulatg the dluto rate wa tuded for the promoto of coextece. Other teretg cotrol tude of the chemotat ca be foud [3,8,,4,9,3,3]. The tablty ad robute of perodc oluto of the chemotat wa tuded [3,33]. The problem of the tablzato of a o-trval teady tate (, S of the chemotat model (. wa codered [9], where t wa how that the mple feedbac law D = μ( S / a globally tablzg feedbac. See alo the recet wor [4] for the tudy of the robute properte of the cloed-loop ytem (. wth D = μ( S / for tme-varyg let ubtrate cocetrato S. The recet wor [6] tuded the ampled-data tablzato of the o-trval teady tate (, S of the chemotat model (., whle ucerta chemotat model were codered [7]. I th wor we coder the robut global feedbac tablzato problem for the chemotat model wth delay: ( = r S ( = D( ( S ( p( T ( S D( b ( (,, (, S ( K( μ( (, D( (. where T r ( S = { t θ : θ [ r,]} the r htory of S, b the cell mortalty rate, r the maxmum delay, K ( S > a pobly varable yeld coeffcet ad p : C ([ r,]; (, S (, a cotuou fuctoal that atfe ( T ( S max μ( τ m μ( τ p (.3 r The fucto μ :[, S ] [, μ max ], K :[, S ] (, wth μ ( =, μ ( S > for all S >, are aumed to be locally Lpchtz fucto. The chemotat model (. uder (.3 very geeral, ce we may have: p( T r ( S μ( p( T r ( S ( t r =, whch gve the tadard chemotat model wth o delay, = μ, whch gve the tme-delayed chemotat model tuded [37], λ h( τ r μ( τ dτ, where λ [,], h C ([, r];[, p( Tr ( S = w μ( t r ( λ r = t t r wth h ( d =, w, r [, r] ( =,..., wth w =. Moreover, t hould be oted that the cae of varable yeld coeffcet ha bee tuded recetly (ee [46,47] ad ha bee propoed for the jutfcato of expermetal reult. The reader hould otce that chemotat model wth tme delay were codered [44,45]. We aume the extece of a o-trval equlbrum pot for (.,.e., the extece of ( S, (, S (, uch that = D ( S S μ ( S = D b, = (.4 K( S ( D b
where D > the equlbrum value for the dluto rate. The tablzato problem for the equlbrum pot ( S, (, S (, crucal: [37] t how that the equlbrum pot utable eve f μ : (, (, μ max S ] mootoe (e.g., the Mood pecfc growth rate. Moreover, a remared [37] the chemotat model (. uder (.3 allow the expreo of the effect of the tme dfferece betwee coumpto of utret ad growth of the cell (ee the dcuo o page 38-4 [37]. We olve the feedbac tablzato problem for the chemotat by provdg a delay-free feedbac whch acheve global tablzato (ee Theorem 4. below. The proof of the theorem rele o the trajectory-baed mall-ga reult of the paper. No owledge of the maxmum delay r aumed. The tructure of the preet wor a follow: Secto cota the tatemet of the trajectory-baed mall-ga reult (Theorem.5 ad Theorem.6. Secto 3 provde llutratve example whch how the applcablty of the obtaed reult to ytem whch atfy le demadg tablty oto tha ISS. Secto 4 devoted to the developmet of the oluto of the feedbac tablzato problem for the ucerta chemotat (.. The cocluo are provded Secto 5. The proof of the trajectory-baed mall-ga reult are gve Appedx A. Fally, for reader coveece, the defto of the ytem-theoretc oto ued th wor are gve Appedx B. Notato Throughout th paper, we adopt the followg otato: We deote by K the cla of potve, cotuou fucto defed o R : = { x R: x }. We ay that a fucto ρ : R R potve defte f ρ ( = ad ρ ( > for all >. By K we deote the et of potve defte, creag ad cotuou fucto. We ay that a potve defte, creag ad cotuou fucto ρ : R R of cla K f lm ρ ( =. By KL we deote the et of all cotuou fucto R R σ = σ (, : R wth the properte: ( for each t the mappg σ (, of cla K ; ( for each, the mappg σ (, o-creag wth lm σ (, =. t By, we deote the orm of the ormed lear pace. By we deote the Eucldea orm of R. Let U wth U [, r]: = u U ; u r we deote the terecto of U wth the cloed ball of. By { } B U radu r, cetered at U. If x deote the trapoe of x. = ( R = {(,..., x R : x,..., x } U R the t(u deote the teror of the et U R. R : x. { e} = deote the tadard ba of R. Z deote the et of o-egatve teger. Let y R. We ay that x y f ad oly f ( x R. We ay that a fucto ρ : R of cla y N, f ρ cotuou wth ρ ( = ad uch that ρ( x ρ( for all For t t let [ V ] [ t : = up (,..., up (, t] V τ V τ τ [ t, t] τ [ t, t] e up x( t [ a, b] [ t, t] V ( τ = ( V ( τ,..., V ( τ x y R, wth x y. R τ R be a bouded map. We defe deote the eetal upremum of x (. For a meaurable ad eetally bouded fucto. Gve a fucto we defe ( x : = x( t θ ; θ [ r,], for t [ a, b, to be the r htory of x. T r m We ay that Γ : R R o-decreag f Γ ( x Γ( for all ( we defe Γ ( x = Γ Γ... Γ( x, whe m =. tme We defe = (,,..., R. If u, v R ad u v the u v. x :[ a, b] R x :[ a r, b R, where r >, a < b, x y R, wth x y. For a teger, Let U be a ubet of a ormed lear pace U, wth U. By M (U we deote the et of all locally bouded fucto u : R U. By u we deote the detcally zero put,.e., the put that atfe u ( = U for all t. If U R the M U deote the pace of meaurable, locally bouded fucto u : R U. Let A, B Y, where, Y are ormed lear pace. We deote by C ( A; B the cla of cotuou mappg f : A B. For x C ([ r,]; R we defe x : = max x( θ. r θ [ r,] 3
. New Trajectory-Baed Small-Ga Theorem I th Secto we tate the ma reult of the preet wor. The proof of the ma reult (Theorem.5 ad Theorem.6 are provded at Appedx A. For the tatemet of the ma reult oe eed to ow the abtract ytem theoretc framewor troduced [,,3] ad ued [8]. For the coveece of the reader, all defto of the bac oto are provded Appedx B. The followg techcal defto were ued [8] ad are eeded here. Defto.: Let z x = ( x,..., R, x y y = ( y,..., R. We defe z = MA{ y}, where z = ( z,..., R atfe z = max{ x, y } for =,...,. Smlarly for z = MA u,..., u } a vector { m Defto.: We ay that x y R, the followg equalty hold: z = ( z,..., z R wth z = max{ u,..., u m }, =,...,. R Γ : R MA-preervg f ( MA{ y} = MA{ Γ( x, Γ( } R u m u,..., R we have Γ : R o-decreag ad for every Γ (. The above defed MA-preervg map ejoy the followg mportat property (ee [8]. Propoto.3: Γ : R R wth Γ( x = ( Γ ( x,..., Γ ( x MA-preervg f ad oly f there ext odecreag fucto γ, j : R R,, j =,..., wth Γ ( x = max γ, j ( x j for all x R, =,...,. j=,..., The followg cla of MA-preervg mappg play a mportat role what follow. Defto.4: Let Γ : R R wth Γ( x = ( Γ ( x,..., Γ ( x be a MA-preervg mappg for whch there ext fucto γ, j N,, j =,..., wth Γ ( x = max γ, j ( x j for all x R, =,...,. We ay that Γ : R R j=,..., atfe the cyclc mall-ga codto f the followg equalte hold: ad f > the for each r =,..., t hold that: for all j {,..., }, j f j. γ (, >, =,..., (., < ( γ γ... ( <, > (.3,, γ 3 r, Propoto.7 [8] how that the MA-preervg cotuou mappg Γ : R R atfe the cyclc mallga codto f ad oly f R Globally Aymptotcally Stable for the dcrete-tme x( = Γ( x(, where x ( R, Z. The followg fact are coequece of the related reult [5,8,36,43] ad defto (., (., (.4 ad wll be ued repeatedly the proof of the ma reult of the preet ecto. Fact I: If ( R Γ : R atfe the cyclc mall-ga codto, the lm Γ ( x = ( ( { Γ( x, Γ ( x,..., Γ ( x } Γ ( x Q( x = MA for all ad x R. ( for all x R ad Fact II: If R ( Γ : R a MA-preervg mappg, the the mappg ( { Γ( x, Γ ( x,..., Γ ( x } Q( x = MA a MA-preervg mappg. Fact III: If x R R Γ : R atfe the cyclc mall-ga codto, the Γ ( Q( x Q( x ad Q( x x for all ( (, where Q( x MA { Γ( x, Γ ( x,..., Γ ( x } =. 4
Fact IV: If r R p N ad R R R : a o-decreag mappg, the the followg equalty hold for all, : p( MA { R( R( r } max( p( R(, p( R( r, =. R Fact V: If Γ : R atfe the cyclc mall-ga codto ad ( ( x Q(, where Q( x = MA { Γ( x, Γ ( x,..., Γ ( x }., atfy x MA { y Γ( x } x y R,, the We coder a abtract cotrol ytem Σ : = (,Y, M U, M D, φ, π, H wth the BIC property for whch a robut equlbrum pot from the put u M U (ee Appedx B for the oto of a abtract cotrol ytem, the BIC property ad the oto of a robut equlbrum po. We uppoe that there ext a et-valued map R t wth for all t, map t ( =,..., ad a MA-preervg cotuou map hypothee hold: (H There ext fucto σ KL, ζ N, t R t ( t, t, x t V : { t} U R, wth V ( t,, = for all Γ : R wth Γ ( = uch that the followg L : { t} R wth L ( t, = for all t, uch that for every t, x R M U M D wth φ for all t [ t, t max the mappg ( = ( V ( t, φ ( t, t, x, u(,..., V ( t, ( t, t, x, u( ad t L = L( t, φ( t, t, x t V ( φ locally bouded o t, ad the followg etmate hold: [ t max ( L( t, t t, Γ [ V ] (, [ u ] are ( d V ( MA σ [ t, ] t ζ, for all t [ t, U [ t, t] max t (.4 where t max the maxmal extece tme of the trato map of Σ. (H For every ( t, x R M U M D there ext ξ π ( t, x uch that φ ( t, t, x for all t [ ξ, t max. Moreover, there ext fucto ν, c, c K, a, η, η, p u, g u N, p N, uch that the followg equalte hold for every t, x R M U M ( D : u ( x, p( [ V ], p [ u ] L( max ν ( t t, c( t, a [ ξ, t], for all t [ ξ, t U [ t, t] max (.5 ( x, η [ u ] φ ( t, t, x max ν ( t t, c ( t, a, for all t [ t, ] U [ t, t] ξ (.6 ( x c ξ t (.7 a ( t ( c( t x, η [ u ] H ( t, φ ( t, t, x, u( max a, for all t [ t, ] Y U [ t, t] ξ (.8 L ( u ξ, φ( ξ, t, x max a( c( t x, g [ u ] U [ t, ξ ] (.9 (H3 There ext fucto b N, g N, μ, κ K uch that the followg equalte hold: ( L( t, x g( V ( t, u κ( μ ( x b t where V t, u = ( V ( t, u,..., V ( t x u,, (. 5, for all ( t, u { t} U (. t
(H4 There ext q N uch that the followg equalty hold: ( V( t, H( t, u q u Y, for all ( t, u { t} U (. t Dcuo of Hypothee (H, (H: Hypothee (H, (H hold automatcally whe hypothee (H-3 of Theorem 3. [8] hold (hypothee (H-3 [8] correpod to the pecal cae S ( = ad ξ = t. Coequetly, Hypothee (H, (H are le retrctve hypothee. Ideed, equalte (.4, (.5 are ot aumed to hold for all tme t [ t, t max but oly after the oluto map φ ( t, t, x ha etered the et S (. Moreover, the et-valued map S ( ot aumed to be potvely varat. We are ow ready to tate the ma reult. Theorem.5 (Trajectory-Baed Small-Ga Reult for IOS: Coder ytem Σ : = (,Y, M U, M D, φ, π, H uder the above hypothee. Aume that the MA-preervg cotuou map Γ : R R wth Γ ( = atfe the cyclc mall-ga codto. The ytem Σ atfe the IOS property from the put u M U wth ga γ ( : = max η, q( G(, where G( = ( G (,..., G ( defed by: { ( } u u u G( Q( max{ σ ( p (,, σ ( g (,, σ ( p( Q( σ ( g (,,, σ ( p( Q( ζ (,, ζ ( } ( ( wth Q( x = MA { Γ( x, Γ ( x,..., Γ ( x } for all R. Moreover, f = (. atfe the UIOS property from the put M U x u wth ga ( : max{ η(, q( G( } γ =. c K bouded, the ytem Σ Remar: It of teret to ote that Theorem.5 a ew trajectory-baed mall-ga reult for IOS becaue equalte (.4, (.5 are ot aumed to hold for all tme. Itead, we aume that for each trajectory there ext a tme ξ π t, x after whch equalte (.4, (.5 hold. O the other had, order to be able to coclude ( d IOS for the ytem, we have to aume addtoal equalte whch hold for the traet perod t [ t, ξ ],.e., equalte (.6, (.7, (.8, (.9 are requred to hold. We coder ext a abtract cotrol ytem Σ : = (,Y, M U, M D, φ, π, H wth U = {} ad the BIC property for whch a robut equlbrum pot from the put u M U. Suppoe that there ext a et-valued map R t wth for all t, map t ( =,..., ad a MA-preervg cotuou map hypothe hold: (H5 There ext fucto σ KL, t ( t, x, R M D wth ( t, t, x, u, 6 t R V : { t} R, wth V ( t, = for all Γ : R wth Γ ( = uch that the followg L : { t} R wth L ( t, = for all t, uch that for every φ for all t t, t the mappg = ( V ( t, φ ( t, t, x, u,,..., V ( t, ( t, t, x, u, ad t L = L( t, φ( t, t, x, u, t V ( φ locally bouded o t, ad the followg etmate hold: [ t max V { σ ( L( t, t t, Γ( [ V ] } ( MA [ t, t] where t max the maxmal extece tme of the trato map of Σ. (H6 For every ( t, x, R M D there ext ( t, x, u, t [ ξ, t max. Moreover, there ext fucto, c K ( t, x, R M D the followg equalte hold: are ( d, for all t t, t (.3 ξ π uch that φ t, t, x, u, for all ( t ν, a N, p N, uch that for every
L { ν ( t t, c( t, a( x p( [ V ] }, for all t ξ, t (.4 ( max, [ ξ, t] { ν ( t t, c( t a( }, x, u, max, φ ( t, t x, for all t [ t, ξ ] (.5 ( x c ξ t (.6 a ( t ( x c( L ( ξ a t (.7 Dcuo of Hypothe (H6: Hypothe (H6 almot the ame wth Hypothe (H appled to the cae U = {}. Noethele, otce the dfferece that the etmate for L (ξ equalty (.7 le tght tha the etmate eeded equalty (.9 of Hypothe (H. Ideed, whe x, etmate (.7 doe ot yeld L ( ξ =, cotrary to the etmate (.9, whch gve L ( ξ =. Fally, the aalogue of equalty (.8 for U = {} ot eeded hypothe (H6. = U D R Theorem.6 (Trajectory-Baed Small-Ga for Robut Global Aymptotc Output Stablty (RGAOS: Coder ytem Σ : = (,Y, M, M, φ, π, H wth U = {} uder hypothee (H3-6. Aume that the MApreervg cotuou map Γ : R wth Γ ( = atfe the cyclc mall-ga codto. The ytem Σ RGAOS. Moreover, f Σ : = (,Y, M U, M D, φ, π, H T perodc for certa T > the ytem Σ Uformly RGAOS (URGAOS. It clear that Hypothee (H5, (H6 le demadg tha Hypothee (H, (H. O the other had the cocluo of Theorem.6 weaer tha the cocluo of Theorem.5: Theorem.6 guaratee RGAOS whle Theorem.5 guaratee IOS. The proof of Theorem.5 ad Theorem.6 are provded at Appedx A ad are mlar prt to the proof of Theorem 3. [8]. 3. Example ad Dcuo The frt example dcate that the trajectory-baed mall-ga reult of the prevou ecto ca be ued to tudy the feedbac tercoecto of ytem whch do ot ecearly atfy the IOS property. Example 3.: Coder the ytem x = f ( d, y = g( d, x R, y R, d D R l (3. where l D R a o-empty compact et, f : D R R R, g : D R R R are locally Lpchtz mappg wth f ( d,, =, g ( d,, = for all d D. Suppoe that there ext potve defte, cotuouly dfferetable ad radally ubouded fucto V : R, W : R atfyg the followg equalte for all ( R R : R R V ( x W ( max V ( x f ( d, d D V ( x ( V ( x( W ( W ( max W ( g( d, V ( x d D W ( (3. (3.3 7
It clear that the ubytem y = g( d, doe ot atfy ecearly the ISS property from the put x R. Coequetly, the clacal mall-ga theorem [4] caot be appled becaue the y ubytem (3. ot ISS but tegral ISS wth x R a put. Recet mall-ga approache have bee ued for ytem (3., where t how that R R Globally Aymptotcally Stable (ee [,] for the dturbace-free cae. Here we wll how, by mag ue of Theorem.6 that R R Uformly Robutly Globally Aymptotcally Stable (URGAS. Ideed, the equalte (3., (3.3 mply that ytem (3. RFC. Notce that equalty (3. mple for all ( R R : max V ( x f ( d, d D The above dfferetal equalty mple that the oluto x( = x R, t, t : ( x(, R R of (3. wth tal codto = y R correpodg to arbtrary d M D atfe the followg equalty for all V ( x( V ( x t (3.4 Moreover, equalty (3.3 mple for all ( R R : max W ( g( d, V ( x d D The above dfferetal equalty cojucto wth (3.4 mple that the oluto ( x(, R R of (3. wth tal codto x( = x R, = y R correpodg to arbtrary d M D atfe the followg equalty for all t [, t max : t W ( W ( y tv ( x (3.5 Iequalte (3.4, (3.5 mply that ytem (3. Robutly Forward Complete (ee Appedx B. Iequalte (3., (3.3 mply for every ε > the extece of a potve defte fucto ρ C ( R ; R uch that: ( ε W ( V ( x ( W ( max V ( x f ( d, ρ( V ( x d D (3.6 ( ε V ( x W ( V ( x ad V ( x < max W ( g( d, ρ( W ( d D (3.7 Lemma 3.5 [5] cojucto wth mplcato (3.6, (3.7 mple that the extece of followg equalte hold: ε W ( τ V ( x( max σ ( V ( x,, up τ W ( τ V ( x( max σ ( V ( x( ξ, t ξ σ KL uch that the, for all t (3.8 ε W ( τ, up, for all ξ t (3.9 ξ τ W ( τ W ( max σ ( W ( ξ, t ξ V ( x( τ, up ( ε, for all ξ t wth up V ( x( τ < ξ τ V ( x( τ ξ τ (3. Iequalty (3.8 how that for every ( x(, S for ξ x( x R, = y R, d M D there ext ξ uch that = t wth S : = {( R R : V ( x ε }. Th follow from Propoto 7 [39] whch 8
mple the extece of a a K σ (, a exp( a ( for all, t. Let a 3 K be a fucto ( x a x for all x R. The reader ca verfy that ( x(, S for t ξ wth atfyg ( V 3, uch that ( a ξ : = l a ( a ( x 3 ( ε (3. Moreover, equalty (.6 hold for approprate a K ad c (. Defe V ( V ( x =, V ( x V ( = W (. The reader hould otce that for ε (, equalte (3.4, (3.5, (3.9, (3. ad defto y (3. guaratee that equalte (.3, (.4, (.5 ad (.7 hold for approprate σ KL, wth c (, p, ν K, a K ( ε γ, ( : =, γ, : ( 4 ( 3 ε ( = ε, γ, ( = γ, (, L ( : = V ( x W ( ad : R R H ( t, : = (. Fally, otce that the MA-preervg mappg Γ wth Γ x = max γ ( x ( =, atfe the cyclc mall-ga codto for ε (, automatcally atfed for approprate fucto b N, g, q N, (, j j=,. The reader ca verfy that hypothee (H3, (H4 are μ, κ K. j By vrtue of Theorem.6, we coclude that the autoomou ytem (3. URGAS. The followg example deal wth the robut global ampled-data tablzato of a olear plaar ytem. Example 3.: Coder the followg plaar ytem x = ( y x y y = f ( x g( x y u ( R, u R (3. where f, g : R R are locally Lpchtz fucto wth f ( =. We wll how that there ext cotat M > uffcetly large ad r > uffcetly mall o that ytem (3. cloed loop wth the feedbac law u = My appled wth zero order hold,.e., the cloed-loop ytem x ( = ( y ( x( y ( = M τ f ( x( g( x(, t [ τ, τ (3.3 τ = τ exp ( w( τ r, w( R atfe the UISS property wth zero ga whe w codered a put. Frt, otce that there ext a fucto x = ( y x y wth tal codto x ( = x correpodg to put y L loc ( R ; atfe the followg etmate for all t : x( max σ ( x,, up ( ( t γ y τ (3.4 τ σ KL uch that for all ( x, R L ( R ; the oluto of wth γ ( : =. Ideed, equalty (3.4 ca be verfed by ug the Lyapuov fucto V ( x = x whch 4 atfe the followg mplcato: y f V ( x = x the V V ( x 4y 4 The above mplcato cojucto wth Lemma 3.5 [5] guaratee that (3.4 hold for approprate σ KL. Next we how the followg clam. loc 9
Clam : For every ε, a > there ext σ KL, M > uffcetly large ad r > uffcetly mall uch that for ( loc loc every y, w R L ( R ; B[, a] L ( R ; R the oluto of y ( = M τ f ( x( g( x(, t [ τ, τ τ = τ exp ( w( τ r, w( R (3.5 wth tal codto y ( = y correpodg to put ( w L loc ( R ; B[, a] Lloc ( R ; R atfe the followg equalty: max σ ( y,, ε up x( τ τ (3.6 Proof of Clam : Let ε, a > be arbtrary. Sce f, g : R R are locally Lpchtz fucto wth f ( =, there ext cotat P, Q > uch that Let M > ad r > be choe o that: f ( x P x ad g( x Q, for all x B[, a] (3.7 9P M Q ad 3( M Q r exp( Qr (3.8 ε Coder a oluto y ( of (3.5 correpodg to arbtrary ( w L ( R ; B[, a] L ( R ; R wth tal codto y ( = y R. By vrtue of Propoto.5 [], there ext a maxmal extece tme for the oluto deoted by t max. Moreover, let π : = { τ, τ,...} the et of amplg tme (whch may be fte f t max < mp( : = max τ π : τ t. Let x : = up x( ad τ = mp(. Iequalte (3.7, (3.8 ad the fact that ad { } t τ r cojucto wth the Growall-Bellma equalty mple: ( M Q r exp( Qr P r y ( τ exp( Qr x (3.9 ( M Q r exp( Qr ( M Q r exp( Qr Defe V ( = y ( o [, t max. Let I [, t max be the zero Lebegue meaure et where y ( ot dfferetable or where y ( M τ f ( x( g( x(. Ug (3.7, (3.8 ad (3.9 we obta for all t, t \ I : loc ε V ( x, for all t [, t max \ I V loc (3. Drect tegrato of the dfferetal equalty (3. ad the fact that V ( = y ( mple that: { exp( y, ε x } y max, for all t, t (3. ( Clearly, equalty (3. mple that a log a the oluto of (3.5 ext, y ( bouded. A tadard cotradcto argumet cojucto wth the Boudede-Imple-Cotuato property for (3.5 (ee Propoto.5 [], mple that t max =. Iequalty (3.6 a drect coequece of equalty (3.. The proof complete. We elect M > uffcetly large ad r > uffcetly mall uch that equalty (3.6 hold wth ε < / ad a = /. The oluto of the cloed-loop ytem (4.a, (4., where defed by (4.5, ext for all t. The extece of the oluto guarateed by the followg clam.
Clam : For every M >, r > ad ( y, x, w R R L loc ( R ; R, the oluto of (3.3, wth tal codto ( x (, = ( x, y correpodg to put w L loc ( R ; R ext for all t. Moreover, for M > uffcetly large ad r > uffcetly mall, there ext g K ad ξ π uch that (( x, y x ( t y ( t g, for all t [, ξ ] (3. x( a, for all t ξ (3.3 where a = /. ( ξ r g x (3.4 Proof of Clam : Let M >, r > ad y, x, w R R L ( R ; R be arbtrary. Coder a oluto ( ( x (, of (3.3 correpodg to arbtrary w L loc ( R ; R wth tal codto ( x (, = ( x, y. By vrtue of Propoto.5 [], there ext a maxmal extece tme for the oluto deoted by t max. Moreover, let π = { τ, τ,...} the et of amplg tme (whch may be fte f t < ad : { π t} mp( : = max τ : τ. By vrtue of (3.4 we have for all t, t Defe loc { σ ( x, a} x max, (3.5 ( = max{ f ( x : x max{ σ ( x, a } ad Q max{ g( x : x max{ σ ( x,, a } P, : : max = (3.6 Ug (3.5, (3.6 cojucto wth Growall-Bellma lemma we obta the followg equalty for all t, t : y ( τ exp(( M Q( t τ P( t τ exp( Q( t τ (3.7 where τ = mp(. Ug (3.7 ad by ducto we ca how the followg equalty for all τ π : y τ y exp(( M Q τ Pτ exp( Qr exp(( M Q τ (3.8 ( where we have ued the fact that τ r. Etmate (3.7 cojucto wth (3.8 gve for all t, t : τ [ y Pt exp( Qr ] exp(( M Q y ( t (3.9 A tadard cotradcto argumet cojucto wth the Boudede-Imple-Cotuato property for (3.3 (ee Propoto.5 [], mple that t max =. The extece of ξ π uch that (3.3 hold a drect coequece of (3.4 ad defto γ ( : =, 4 a = /. By vrtue of (4.3 ad Propoto 7 [39] there ext β K uch ( ξ r β x (3.3 Fally, let M > uffcetly large ad r > uffcetly mall o that (3.6 hold for a = / ad ε < /. For x R wth σ ( x, a we obta from (3.6 ad (3.4 for all t : ( ε σ ( x, σ ( y, ( x( (3.3 Ug (3.5, (3.6, (3.9, (3.3 ad (3.3 we guaratee the extece of β K ( ( x, y uch that x t y ( t β, for all t [, ξ ] (3.3 (
The extece of complete. g K atfyg (3. ad (3.4 a drect coequece of (3.3 ad (3.3. The proof The fact that the robut global tablzato problem for (3.3 wth ampled-data feedbac appled wth zero order hold olvable wth M > uffcetly large ad r > uffcetly mall a coequece from all the above ad Theorem.5. Ideed, we apply Theorem.5 wth =, V = x, V = y, L = x y, H = (, u : = {( R R : x a }, γ, ( : =, γ,( : = ε, γ,, γ,, ζ, g, η, η, u p, c ( = c ( = ν ( = μ( = κ( r, g, p (, w := w, for approprate a, b K, σ KL ad q N. All hypothee (H-4 are atfed by ug the above defto ad prevou reult. Therefore, we coclude that the cloed-loop ytem (3.3 wth M > beg uffcetly large ad r > beg uffcetly mall, atfe the UISS property from the put w wth zero ga. The reader hould otce that alteratve ampled-data feedbac deg for ytem (3. appled wth zero order hold ad potve amplg rate ca be obtaed by ug the reult [34,35], whch, however, acheve emglobal ad practcal tablzato. It hould be emphazed that the feedbac deg obtaed by ug the trajectory-baed mallga reult of the preet wor guaratee global ad aymptotc tablzato. Moreover, robute to perturbato of the amplg chedule guarateed (that the reao for troducg the put w the cloed-loop ytem (3.3. 4. A Delayed Chemotat Model I th ecto we tudy the robut global feedbac tablzato problem for ytem (. uder (.3. More pecfcally, order to emphaze the fact that the mappg p : C ([ r,]; (, S (, uow, we wll coder the tablzato problem of the equlbrum pot ( S, (, S (, atfyg (.4 for the ucerta chemotat model ( = m μ( τ d( max ( ( m ( ( μ S τ μ S τ D( b ( S ( = D( ( S K( μ( ( (4. ( (,, (, S where d ( [,] the ucertaty. We wll aume that:, D(, d( [,] (H There ext S < S [ uch that μ ( S > b for all S S, S ]. Hypothe (H automatcally atfed for the cae of a mootoe pecfc growth rate. Hypothe (H ca be atfed for o-mootoe pecfc growth rate (e.g., Haldae or geeralzed Haldae growth expreo. By ug the trajectory-baed mall-ga Theorem.6 we ca prove the followg theorem. Theorem 4.: Let a > be a cotat that atfe S m μ ( S b > ad (4. S S S S The the locally Lpchtz delay-free feedbac law: K( μ( ( ad D( = S m ( S m(, S (, S (4.3 acheve robut global tablzato of the equlbrum pot ({ θ = S, θ [ r,] }, C ([ r,]; (, S (,, for the ucerta chemotat model (4. uder hypothe (H. It hould be oted that the chage of coordate:
S exp( = exp( x, S = (4.4a G exp( S where G : = ad the put traformato S map the et (, S (, oto = S, θ [ r,], C ([ r,]; (, S ({ } (, D = D exp(u (4.4b R ad the equlbrum pot θ of ytem (4. to the equlbrum pot R C ([ r,]; of the traformed cotrol ytem: where x ( = m τ d( max y ( = D ( G exp( ( R, u( R, d( [,] t τ m τ D [ exp( u( ( G exp( g( exp( x( ] S exp( μ( : = μ G exp( S exp( S exp( g( : = K μ D SG G exp( G exp( exp( u( b (4.5 (4.6 I the ew coordate the feedbac law (4.3 tae the form: a u ( = l g( exp( x( m( G exp(, G max( exp(, (4.7 G The feedbac law (4.7 (or (4.3 a delay-free feedbac, whch acheve global tablzato of R C ([ r,]; for ytem (4.5 o matter how large the delay. Furthermore, o owledge of the maxmum delay r eeded for the mplemetato of (4.7. The proof of Theorem 4. therefore equvalet to the proof of robut global aymptotc tablty of the equlbrum pot R C ([ r,]; for ytem (4.5. Before we gve the proof of Theorem 4., t mportat to udertad the tuto that lead to the cotructo of the feedbac law (4.7 ad the dea behd the proof of Theorem 4.. To expla the procedure we follow the followg argumet:. For the tablzato of the equlbrum pot R C ([ r,];, we frt tart wth the tablzato of y ( = D ( G exp( exp( u( ( G exp( g( exp( x( wth x a put. Ay ubytem [ ] feedbac law whch atfe ( = l( g( exp( x( ( G u ( > l( g( exp( x( ( G exp( for ( < a put. u for y ( ad y acheve ISS tablzato of the ubytem wth x. I order to prove URGAS for the compote ytem by mea of mall-ga argumet oe ha to how the ISS property of the x ubytem x ( = m τ d( y y max ( τ m ( τ D exp( u( b wth y a put. Notce that the feedbac electo from prevou tep gve x ( = m τ d( y y max ( τ m ( τ D g( exp( x( ( G b for y ( 3
ad x ( < m τ d( y y max ( τ m ( τ D g( exp( x( ( G exp( b for y ( <. The etmato of the dervatve x ( how that the ISS equalty for the x ubytem doe ot hold ule we have m τ > b for all t uffcetly large. By vrtue of hypothe (H there ext y <, uch that the ISS equalty for the the x ubytem hold f t uffcetly large. 3. The feedbac law u ( l( g( exp( x( ( G exp( equalty m τ y m τ y hold for all > for y ( < elected uch that the hold for all tal codto after a traet perod. Sce the ISS equalte wll hold oly after th traet perod the trajectory-baed mall-ga reult Theorem.6 mut be ued for the proof of URGAS of the cloed-loop ytem. Schematcally, we have: Fgure : The tuto that lead to the cotructo of the feedbac law (4.7 ad the dea behd the proof of Theorem 4. Proof of Theorem 4.: Coder the oluto ( x (, R of (4.5 wth (4.7 wth arbtrary tal codto x ( = x R, T r ( y = y C ([ r,]; ad correpodg to arbtrary put d M D. The followg equato hold for ytem (4.5 wth (4.7: G exp( y ( = ad ( exp(, G y ( = D g( ( G exp( exp( x( f ( exp(, f > (4.8 Equato (4.8 mply that the fucto V ( = y ( o-creag ad coequetly, we obta: 4
y, for all t [, t max (4.9 r Ug the fact that μ : (, S (, μ max ] ad defto (4.6 of μ, we get that y μ max for all y R. Th mple the followg dfferetal equalty: x ( μ b whch by drect tegrato yeld the etmate: max x( x ( μ bt, for all t, t (4. max Defe κ ( : = ( G D max g(. Iequalte (4.9, (4. mply that the followg dfferetal equalty hold: y ad x b κ G whch by drect tegrato yeld the etmate: x( x ( y exp( x ( μ b ( max t r exp( ( μ max b ( y exp( x ad b t κ G r, for all t [, t max (4. μ max b Iequalte (4.9, (4., (4. ad a tadard cotradcto argumet how that ytem (4.5 wth (4.7 forward complete,.e., t max =. Therefore, equalte (4.9, (4., (4. hold for all t ad ce ytem (4.5 wth (4.7 autoomou, t follow that ytem (4.5 wth (4.7 Robutly Forward Complete (RFC, ee Appedx B. By coderg (4.8 ad the fucto fucto ρ C ( R ; R uch that: y ( W ( = f, we obta the extece of a potve defte f > ( W ( W ( ρ t, for all t (4. Lemma 3.5 [5] mple the extece of σ KL uch that for every x ( = x R, T r ( y = y C ([ r,]; ad d M D t hold that: W ( σ W (, t, for all t (4.3 ( Iequalty (4.3 cojucto wth Propoto 7 [39], how the extece of x ( = x R, T r ( y = y C ([ r,]; ad d M D there ext r where : l S S c = > S G ad S < S ξ wth a( y a K uch that for every ξ atfyg: r c t r, for all t ξ (4.4 the cotat volved hypothe (H. Defe S : = R C ([ r,];[ c,. Iequalty (4.4 how that ( x(, Tr ( S for all t ξ ad that equalty (.6 hold for approprate a K ad c (. r Notce that for ( x(, Tr ( S, the fuctoal V ( σθ ( = max exp z( t θ θ [ r,], V = x( (4.5 where σ > ad ( exp( z( y ( = c t (4.6 5
are well-defed. Moreover, by coderg the dfferetal equato: z ( = ad z ( = c D ( exp( z( G exp( c c( G g exp( z( ( exp( c( exp( z( z( ( c( exp( z( ( G exp( c( exp( z( exp( x( z( ( exp( c( exp( z(, f z( > we coclude from Lemma 3.5 [5] that for every γ K there ext σ KL uch that: V max σ, ( V ( t, t t, up γ ( V ( τ (, t τ, f, for all t t (4.7 Fally, ug hypothe (H ad defto (4.5, we guaratee that there ext a potve defte fucto ρ C ( R ; R uch that the followg mplcato hold for every ε > : ( G exp( c( exp( V ( D max g c( exp( z If ( l z V ( ε m μ z exp( σ r V ( G max μ( c( exp( z b z exp( σ r V ( ( ε l G exp( c( exp( V ( D m g c exp( z z V ( x( x ( ρ x ( ( ( ( a ( c( exp( z b D exp c( exp( V ( ( ( ( ( Therefore, Lemma 3.5 [5] mple that there ext σ KL uch that: x( x( ad the where V max σ ( V ( t, t t, up γ ( V ( τ (, t τ γ, ( : = ( ε g ( : = g ( : = m z exp( σ r ( l( max{ g (, g ( } ( G exp( c( exp( D max g c( exp( z μ, for all t t (4.8 ( ( ( ad ( c( exp( z b exp c( exp( max z exp( σ r μ G ( c( exp( z ( G exp( c( exp( D m g c( exp( z z z b ( (4.9 Iequalte (4.9, (4., (4., (4.7 ad (4.8 guaratee that equalte (., (., (.3, (.4, (.5 ad (.7 hold for approprate σ KL, ν K, a K wth c (, p, γ, ( : = γ,, γ, ( = γ, (, L : = V V ad r : R R H ( t, : = x y. Fally, otce that the MA-preervg mappg Γ wth Γ x = max γ ( x ( =, atfe the cyclc mall-ga codto. (, j j=, j By vrtue of Theorem.6 we coclude that the autoomou ytem (4.5 wth (4.7 URGAS. 6
5. Cocluo Oe of the mot mportat obtacle applyg olear mall-ga reult the fact that the eetal equalte, whch mall-ga reult utlze order to prove tablty properte, do ot hold for all tme: th feature exclude all avalable mall-ga reult from poble applcato. I th wor ovel mall-ga reult are preeted, whch ca allow a traet perod durg whch the oluto do ot atfy the uual equalte requred by prevou mall-ga reult (Theorem.5 ad Theorem.6. The obtaed reult allow the applcato of the mall-ga methodology to varou clae of ytem whch atfy le demadg tablty oto tha the Iputto-Output Stablty property. The robut global feedbac tablzato problem of a ucerta tme-delay chemotat model olved by mea of the trajectory-baed mall-ga reult. Future reearch wll focu o the applcato of the trajectory-baed mallga reult to Lota-Volterra ytem Mathematcal Bology (ee []. Appedx A-Proof of Theorem.5 ad Theorem.6 Proof of Theorem.5: The proof mlar to the proof of Theorem 3. [8] ad cot of four tep: Step : We how that for every ξ : ( t, x R M U M D the followg equalty hold for all t [, t max V ( MA Q (A. U [ ξ, t] ( σ ( L( ξ,, Q ζ [ u ] where ξ π t, x the tme uch that φ t, t, x for all t ξ, t (recall Hypothe (H. ( d ( t Th tep proved exactly the ame way a the proof of Theorem 3. [8] (ug Fact V. Step : We how that for every ( t, x R M U M D, t hold that max = t. The proof of Step exactly the ame wth the proof of Theorem 3. [8]. The oly dfferece the addtoal ue of equalty (.6, whch guaratee that the trato map bouded durg the traet perod t [ t, ξ ]. Step 3: We how that Σ RFC from the put u M U. Aga the proof of Step 3 exactly the ame wth the proof of Theorem 3. [8]. The oly dfferece the addtoal ue of equalte (.6 ad (.9. Step 4: We prove the followg clam. Clam: For every ε >, Z, R, T there ext τ ( ε, R, T uch that for every ( t, x R M U M D wth t [, T ] ad x R the followg equalty hold: ( Q( ( L( ξ,, G [ u ] ( V ( MA Q( ε, Γ σ, for all t ξ τ U [ t, t] (A. Moreover, f c K bouded the for every ε >, Z, R there ext τ ( ε, uch that for every ( t, x R M U M D wth x R equalty (A. hold. Proof of Step 4: The proof of the clam made by ducto o Z. Iequalty (A. for = a drect coequece of equalte (.4, (.9 ad defto (.. 7
We otce that equalty (.5 cojucto wth equalty (A. ad Fact IV mply for all t ξ : ( ( ( ( [ ] u x, p Q σ L( ξ,, p Q ζ u, p [ u ] L( max ν ( t t, c( t, a (A.3 U [ t, t] U [ t, t] Next uppoe that for every ε >, R, T there ext τ ( ε, R, T uch that for every t, x R M U M D wth t [, T ] ad x R (A. hold for ome Z. Let arbtrary ε >, ( [, T R, T, ( t, x R M U M D wth t ] ad x R be gve. Notce that the wea emgroup property mple that π t, x [ ξ τ, ξ τ ]. Let t π ( t, x [ ξ τ, ξ τ r]. The (.4 mple: ( L( t, t t, Γ [ V ] ( r ( [ u ] V ( MA σ [ t, t], ζ, for all t t U [ t, t] (A.4 Ug equalte (A., (A.3, (A.4, (.9 ad worg the ame way a the proof of Theorem 3. [8] we ca derve the followg equalty: ( V ( MA ( ( ( σ L( t,, (, Γ (, t ξ τ r Q ε Q σ L ξ, U [ t, t] for all t ξ τ r (A.5 (, G [ u ] Defto (. cojucto wth (.7, (.9, equalty (A.3 ad the fact that x R mple that ( L( t, t ξ τ r MA σ ( f ( ε, T,, t ξ τ r, G [ u ] t ξ τ r, t [, ] ad T σ, U [ t, t] for all t ξ τ r (A.6 where ad f ( ε, T, : = max max a( C( T τ ( ε, R, T r ν (, C( T, a ( R, p( Q( σ ( a( RC( T, (A.7 C( T : = max c( (A.8 T The reader hould otce that f c K bouded ad τ depedet of T the f ca be choe to be depedet of T a well. The ret of proof of the clam follow from a combato of equalte (A.5 ad (A.6 ad approprate electo of τ (et τ ( ε, R, T = τ ( ε, R, T r τ ( ε, R, T, where τ ( ε, R, T atfe σ f ( ε, T,, τ. ( ε To fh the proof, let arbtrary ε >, R, T, ( t, x R M U M D ad deote Y( = H( t, φ( t, t, x, u( for t t. Ug Fact IV, (. ad (A. we obta for all t ξ : Y( Y max q Q ( ( σ( L( ξ,, q Qζ [ u ] U [ ξ, t] The above equalty cojucto wth (.9 mple that Y( Y max q Q ( ( ( ( u σ a c( t x,, q Q [ ] σ g u,, q Q ζ [ u ] for all U [ t, t] U [ t, t], t ξ (A.9 8
Ug (.8 ad (A.9, we coclude that the followg etmate hold for all t t : Y( Y q Q max η ( ( σ ( a( c( t x,, a( c( t x, [ ] [ ] [ ] u u, q Qσ g u,, qq ζ u U [ t, t] U [ t, t] U [ t, t] (A. Iequalty (A. how that properte P ad P of Lemma.6 [3] hold for ytem Σ wth V = H( t, u Y ad γ ( : = max{ η(, q( G( }. Moreover, f c K bouded the (A. mple that properte P ad P of Lemma.7 [3] hold for ytem Σ wth V = H( t, u Y ad γ ( : = max{ η(, q( G( }. Iequalty (A. cojucto wth Fact III, (.9, (A.8 ad defto (. guaratee that for every ε >, Z, R, T there ext τ ( ε, R, T uch that for every ( t, x R M U M D wth t [, T ] ad x R the followg equalty hold: ( V ( MA Q( ε, Γ σ, for all t ξ τ U [ t, t] (A. ( Q( ( a( RC ( T,, G [ u ] ( l (ε uch that ( ( ( ( Notce that Fact I guaratee the extece of, T, Z Q( ε Γ Q σ a RC( T, for all l. If c K bouded the depedet of T. Therefore by vrtue of (A. ad (.7, property P3 of Lemma.6 [3] hold for ytem Σ wth V = H( t, u Y ad γ ( : = max{ η(, q( G( }. Moreover, f c K bouded the (A. ad (.7 mply that property P3 of Lemma.7 [3] hold for ytem Σ wth V = H( t, u Y ad γ ( : = max{ η(, q( G( }. The proof of Theorem.4 thu completed wth the help of Lemma.6 (or Lemma.7 [3]. Proof of Theorem.6: By vrtue of Lemma 3.3 [] we have to how that Σ Robutly Forward Complete (RFC ad atfe the Robut Output Attractvty Property,.e. for every ε >, T ad R, there ext a τ : = τ ε, T, R, uch that: ( x R, t [, T ] H ( t, φ( t, t, x, u,, ε, t [ t τ, Y, d M D The reader hould otce that Lemma 3.3 [] aume the clacal emgroup property; however the emgroup property ot ued the proof of Lemma 3.3 []. Coequetly, Lemma 3.3 [] hold a well for ytem atfyg the wea emgroup property. Moreover, Lemma 3.5 [] guaratee that ytem Σ URGAOS cae that Σ : = (,Y, M U, M D, φ, π, H T perodc for certa T >. Aga the proof cot of four tep: Step : We how that for every ξ : ( t, x, R M D the followg equalty hold for all t [, t max ( σ ( L(, V ( Q ξ (A. where ξ π t, x, u, the tme uch that φ t, t, x, u, for all t ξ, t (recall Hypothe (H. ( d ( t Step : We how that for every ( t, x, R M D, t hold that max = t. 9
Step 3: We how that Σ RFC. Step 4: We prove the followg clam. Clam: For every ε >, Z, R, T there ext τ ( ε, R, T uch that for every ( t, x, R M D wth t [, T ] ad x R the followg equalty hold: ( { Q( ε, Γ ( Q( σ ( L( ξ, } V ( MA, for all t ξ τ (A.3 The proof of the above tep are exactly the ame wth the proof of Theorem.5 ad are omtted. The dfferece betwee equalte (.9 ad (.7 doe ot play ay role. To fh the proof, let arbtrary ε >, R, T, ( t, x, R M D ad deote Y( = H( t, φ( t, t, x, u,, for t t. Iequalty (A.3 cojucto wth (.7 ad (A.8 guaratee that for every ε >, Z, R, T there ext τ ( ε, R, T uch that for every ( t, x, R M D wth t [, T ] ad x R the followg equalty hold: ( V ( MA { Q( ε, Γ ( Q( σ ( a( C( T, }, for all t ξ τ (A.4 ( l (ε uch that ( ( ( ( Notce that Fact I guaratee the extece of, T, Z Q( ε Γ Q σ a R C( T, for all l. Therefore, (A.4 mple that for every ε >, R, T, there ext τ ( ε, R, T uch that for every ( t, x, R M D wth [, T ] t ad x R, t hold that V ( Q( ε, for all t ξ τ (A.5 It follow from equalte (. ad (A.5 that for every ε >, R, T, there ext τ ( ε, R, T uch that for every ( t, x, R M D wth t [, T ] ad x R, t hold that Y ( q( Q( ε Y, for all t ξ τ (A.6 Therefore by vrtue of (A.6 ad (.6, the Robut Output Attractvty Property hold for ytem Σ. The proof complete. Appedx B-Bac Noto To mae our wor elf-cotaed, we troduce ome oto whch are eetal to the ytem theoretc framewor preeted [,,3]. The abtract ytem theoretc framewor ued [,,3] utlzed the preet wor. The oto of a Cotrol Sytem-Defto. [3]: A cotrol ytem Σ : = (,Y, M, M, φ, π, H wth output cot of ( a et U (cotrol e whch a ubet of a ormed lear pace U wth (allowable cotrol put whch cota at leat the detcally zero put u, U D U ad a et M U M(U ( a et D (dturbace e ad a et M D M( D, whch called the et of allowable dturbace, ( a par of ormed lear pace, Y called the tate pace ad the output pace, repectvely, (v a cotuou map H : R U Y that map bouded et of R U to bouded et of Y, called the output map, (v a et-valued map R M U M D ( t, x π ( t, x [ t,, wth t π ( t, x for all ( t, x R M U M D, called the et of amplg tme (v ad the map φ : A φ where A φ R R M U M D, called the trato map, whch ha the followg properte:
Extece: For each ( t, x R M U M D, there ext t > t uch that t, t] ( t, x Aφ [. Idetty Property: For each ( t, x R M U M D, t hold that φ ( t, t, x = x. 3 Caualty: For each ( t, t, x Aφ wth t > t ad for each ( u, d M U M D wth ( u ( τ, d ( τ = ( u( τ, d( τ for all τ [ t, t], t hold that ( t, t x u d,,, Aφ wth φ t, t, x = φ( t, t, x, u,. ( d 4 Wea Semgroup Property: There ext a cotat r >, uch that for each t t wth t, t, x Aφ (a ( τ, t, x d Aφ for all [ t, t] τ, (b φ t, τ, φ( τ, t, x, u, = φ( t, t, x for all τ t, t] π ( t, x, ( d ( t r, t, x Aφ π ( t, x d ( τ, t, u, [ d ( t, x [ t, t r] A π ( τ, φ( τ,, x, u, π ( t, x ( : (c f, the t hold that π. ( for all τ wth φ we have t = [ τ,. The BIC ad RFC properte-defto.4 [3]: Coder a cotrol ytem Σ : = (,Y, M U, M D, φ, π, H wth output. We ay that ytem Σ ( ha the Boudede-Imple-Cotuato (BIC property f for each ( t, x R M U M D, there ext a maxmal extece tme,.e., there ext t = t ( t, x ( t, ], uch that A = t, t {( t, x }. I max : max φ [ max ( t, x, u, d R M U M D [ t, t max (, t, x addto, f t < the for every M > there ext t wth φ t > M. max ( robutly forward complete (RFC from the put u M U f t ha the BIC property ad for every r, T, t hold that { φ ( t, t, x ; u M( B [, r] M, [, T ], x r, t [, T ], d M } < up U U D The oto of a robut equlbrum pot-defto.5 [3]: Coder a cotrol ytem Σ : = (,Y, M U, M D, φ, π, H ad uppoe that H ( t,, = for all t. We ay that a robut equlbrum pot from the put u M U for Σ f ( for every ( t, t, R R M D wth t t t hold that φ ( t, t,, u, =. ( for every ε >, T, h R there ext δ : = δ ( ε, T, h > uch that for all ( t, u [, T ] M U, τ t, t ] wth x up u( < δ t hold that ( τ, t, u, A for all d M D ad [ h t U { φ ( τ, t, u, ; d M, τ [ t, t h], t [, ] } < ε up D T Next we preet the Iput-to-Output Stablty property for the cla of ytem decrbed prevouly (ee alo [4,4] for fte-dmeoal, tme-varat dyamc ytem. The oto of IOS, UIOS, ISS ad UISS-Defto.4 [3]: Coder a cotrol ytem Σ : = (,Y, M U, M D, φ, π, H wth output ad the BIC property ad for whch a robut equlbrum pot from the put u M U. Suppoe that Σ RFC from the put u M U. If there ext fucto σ KL, β K, γ N uch that the followg etmate hold for all u M U, ( t, x, R M D ad t t : H ( t, φ( t, t ( β ( t x, t t up γ ( u(, x, u( σ Y τ t τ the we ay that Σ atfe the Iput-to-Output Stablty (IOS property from the put u M U wth ga γ N. Moreover, f β K may be choe a β ( the we ay that Σ atfe the Uform Iput-to-Output Stablty (UIOS property from the put u M U wth ga γ N. φ U
For the pecal cae of the detty output mappg,.e., H ( t, u : = x, the (Uform Iput-to-Output Stablty property from the put u M U called (Uform Iput-to-State Stablty ((U ISS property from the put u M U. Whe U = {} (the o-put cae ad Σ atfe the (UIOS property, the we ay that Σ atfe the (Uform Robut Global Aymptotc Output Stablty (RGAOS property. Whe U = {} (the o-put cae ad Σ atfe the (Uform ISS property, the we ay that Σ atfe the (Uform Robut Global Aymptotc Stablty (RGAS property. Referece [] Agel, D., P. De Leeheer ad E. D. Sotag, A Small-Ga Theorem for Almot Global Covergece of Mootoe Sytem, Sytem ad Cotrol Letter, 5(5, 4, 47-44. [] Agel, D. ad A. Atolf, A Tght Small-Ga Theorem for ot ecearly ISS Sytem, Sytem ad Cotrol Letter, 56, 7, 87-9. [3] Atoell, R. ad A. Atolf, Nolear Cotroller Deg for Robut Stablzato of Cotuou Bologcal Reactor, Proceedg of the IEEE Coferece o Cotrol Applcato, Achorage, AL, September. [4] Che, Z. ad J. Huag, A mplfed mall ga theorem for tme-varyg olear ytem, IEEE Traacto o Automatc Cotrol, 5(, 5, 94-98. [5] Dahovy, S., B. S. Ruffer ad F. R. Wrth, A ISS Small-Ga Theorem for Geeral Networ, Mathematc of Cotrol, Sgal ad Sytem, 9, 7, 93-. [6] De Leeheer, P. ad H. L. Smth, Feedbac Cotrol for Chemotat Model, Joural of Mathematcal Bology, 46, 3, 48-7. [7] Eco, G. A. ad E. D. Sotag, Global Attractvty, I/O Mootoe Small-Ga Theorem, ad Bologcal Delay Sytem, Dcrete ad Cotuou Dyamcal Sytem, 4(3, 6, 549-578. [8] Gouze, J. L. ad G. Robledo, Robut Cotrol for a Ucerta Chemotat Model, Iteratoal Joural of Robut ad Nolear Cotrol, 6(3, 6, 33-55. [9] Grue, L., Iput-to-State Dyamcal Stablty ad t Lyapuov Fucto Characterzato, IEEE Traacto o Automatc Cotrol, 47(9,, 499-54. [] Harmard, J., A. Rapaport ad F. Mazec, Output tracg of cotuou boreactor through recrculato ad by-pa, Automatca, 4, 6, 5-3. [] Hofbauer, J. ad K. Sgmud, Evolutoary Game ad Populato Dyamc, Cambrdge Uverty Pre,. [] Ito, H., Stablty Crtera for Itercoected ISS ad ISS Sytem Ug Scalg of Supply Rate, Proceedg of the Amerca Cotrol Coferece,, 4, 55-6. [3] Ito, H. ad Z.-P. Jag, Neceary ad Suffcet Small-Ga Codto for Itegral Iput-to-State Stable Sytem: A Lyapuov Perpectve, IEEE Traacto o Automatc Cotrol, 54(, 9, 389-44. [4] Jag, Z.P., A. Teel ad L. Praly, Small-Ga Theorem for ISS Sytem ad Applcato, Mathematc of Cotrol, Sgal ad Sytem, 7, 994, 95-. [5] Jag, Z.P., I.M.Y. Mareel ad Y. Wag, A Lyapuov Formulato of the Nolear Small-Ga Theorem for Itercoected Sytem, Automatca, 3, 996, -4. [6] Jag, Z.P. ad I.M.Y. Mareel, A Small-Ga Cotrol Method for Nolear Cacaded Sytem wth Dyamc Ucertate, IEEE Traacto o Automatc Cotrol, 4, 997, 9-38. [7] Jag, Z.P., Y. L ad Y. Wag, Nolear Small-Ga Theorem for Dcrete-Tme Feedbac Sytem ad Applcato, Automatca, 4(, 4, 9-34. [8] Jag, Z.P. ad Y. Wag, A Geeralzato of the Nolear Small-Ga Theorem for Large-Scale Complex Sytem, Proceedg of the 7 th World Cogre o Itellget Cotrol ad Automato, Chogqg, Cha, 8, 88-93. [9] Karafyll, I. ad J. Ta, No-Uform Tme ISS ad the Small-Ga Theorem, IEEE Traacto o Automatc Cotrol, 49(, 4, 96-4. [] Karafyll, I., The No-Uform Tme Small-Ga Theorem for a Wde Cla of Cotrol Sytem wth Output, Europea Joural of Cotrol, (4, 4, 37-3. [] Karafyll, I., A Sytem-Theoretc Framewor for a Wde Cla of Sytem I: Applcato to Numercal Aaly, Joural of Mathematcal Aaly ad Applcato, 38(, 7, 876-899. [] Karafyll, I., A Sytem-Theoretc Framewor for a Wde Cla of Sytem II: Iput-to-Output Stablty, Joural of Mathematcal Aaly ad Applcato, 38(, 7, 466-484. [3] Karafyll, I. ad Z.-P. Jag, A Small-Ga Theorem for a Wde Cla of Feedbac Sytem wth Cotrol Applcato, SIAM Joural Cotrol ad Optmzato, 46(4, 7, 483-57. [4] Karafyll, I., C. Kravar, L. Syrou ad G. Lyberato, A Vector Lyapuov Fucto Characterzato of Iputto-State Stablty wth Applcato to Robut Global Stablzato of the Chemotat, Europea Joural of Cotrol, 4(, 8, 47-6.
[5] Karafyll, I. ad C. Kravar, Global Stablty Reult for Sytem uder Sampled-Data Cotrol, Iteratoal Joural of Robut ad Nolear Cotrol, 9, 9, 5-8. [6] Karafyll, I., ad C. Kravar, Robut Global Stablzablty by Mea of Sampled-Data Cotrol wth Potve Samplg Rate, Iteratoal Joural of Cotrol, 8(4, 9, 755-77. [7] Karafyll, I., C. Kravar ad N. Kalogera, Relaxed Lyapuov Crtera for Robut Global Stablzato of Nolear Sytem, Iteratoal Joural of Cotrol, 8(, 9, 77-94. [8] Karafyll, I. ad Z.-P. Jag, A Vector Small-Ga Theorem for Geeral Nolear Cotrol Sytem, ubmtted to IEEE Traacto o Automatc Cotrol. A hort vero wa publhed the Proceedg of the 48 th IEEE Coferece o Deco ad Cotrol 9, Shagha, Cha, 9, pp. 7996-8. A prelmary vero avalable at http://arxv.org/ab/94.755. [9] Malleret, L. ad O. Berard, A Smple Robut Cotroller to Stablze a Aaerobc Dgeto Proce Proceedg of the 8 th Coferece o Computer Applcato Botechology,, 3-8. [3] Maloff, M. ad F. Mazec, Cotructo of Strct Lyapuov Fucto, Sprger Verlag, Lodo, 9. [3] Mazec, F., M. Maloff, J. Harmad, Stablzato ad Robute Aaly for a Chemotat Model wth Two Spece ad Mood Growth Rate va a Lyapuov Approach, Proceedg of the 46th IEEE Coferece o Deco ad Cotrol, New Orlea, 7. [3] Mazec, F., M. Maloff ad P. De Leeheer, O the Stablty of Perodc Soluto the Perturbed Chemotat, Mathematcal Bocece ad Egeerg, 4(, 7, 39-338. [33] Mazec, F., M. Maloff ad J. Harmad, Further Reult o Stablzato of Perodc Trajectore for a Chemotat wth Two Spece, IEEE Traacto o Automatc Cotrol, 53(, 8, 66-74. [34] Nec, D. ad A.R. Teel, Sampled-Data Cotrol of Nolear Sytem: A Overvew of Recet Reult, Perpectve o Robut Cotrol, R.S.O. Mohema (Ed., Sprger-Verlag: New Yor,, -39. [35] Nec, D. ad A. R. Teel, Stablzato of Sampled-Data Nolear Sytem va Bacteppg o ther Euler Approxmate Model, Automatca, 4, 6, 8-88. [36] Ruffer, B. S., Mootoe Dyamcal Sytem, Graph ad Stablty of Large-Scale Itercoected Sytem, PhD The, Uverty of Breme, Germay, 7. [37] Smth, H. ad P. Waltma, The Theory of the Chemotat. Dyamc of Mcrobal Competto, Cambrdge Stude Mathematcal Bology, 3, Cambrdge Uverty Pre: Cambrdge, 995. [38] Sotag, E.D., Smooth Stablzato Imple Coprme Factorzato, IEEE Traacto o Automatc Cotrol, 34, 989, 435-44. [39] Sotag, E.D., Commet o Itegral Varat of ISS, Sytem ad Cotrol Letter, 34, 998, 93-. [4] Sotag, E.D. ad Y. Wag, Noto of Iput to Output Stablty, Sytem ad Cotrol Letter, 38, 999, 35-48. [4] Sotag, E.D. ad B. Igall, A Small-Ga Theorem wth Applcato to Iput/Output Sytem, Icremetal Stablty, Detectablty, ad Itercoecto, Joural of the Fral Ittute, 339,, -9. [4] Teel, A., A Nolear Small Ga Theorem For the Aaly of Cotrol Sytem Wth Saturato, IEEE Traacto o Automatc Cotrol, 4, 996, 56-7. [43] Teel, A.R., Iput-to-State Stablty ad the Nolear Small Ga Theorem, Preprt, 5. [44] Wag, L. ad G. Wolowcz, A delayed chemotat model wth geeral omootoe repoe fucto ad dfferetal removal rate, Joural of Mathematcal Aaly ad Applcato, 3, 6, 45-468. [45] Wolowcz, G. ad H. a, Global Aymptotc Behavor of a chemotat model wth dcrete delay, SIAM Joural o Appled Mathematc, 57, 997, 9-43. [46] Zhu, L. ad. Huag, Multple lmt cycle a cotuou culture veel wth varable yeld, Nolear Aaly, 64, 6, 887-894. [47] Zhu, L.,. Huag ad H. Su, Bfurcato for a fuctoal yeld chemotat whe oe compettor produce a tox, Joural of Mathematcal Aaly ad Applcato, 39, 7, 89-93. 3