Lyapunov-Razumikhin and Lyapunov-Krasovskii theorems for interconnected ISS time-delay systems

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1 Proceedngs of the 19th Internatonal Symposum on Mathematcal Theory of Networks and Systems MTNS July 2010 Budapest Hungary Lyapunov-Razumkhn and Lyapunov-Krasovsk theorems for nterconnected ISS tme-delay systems Sergey Dashkovsky and Lars Naujok Abstract We consder an arbtrary number of nterconnected nonlnear systems wth tme-delays and nvestgate them n vew of nput-to-state stablty (ISS). The useful tools for sngle tme-delay systems the ISS Lyapunov-Razumkhn functons and ISS Lyapunov-Krasovsk functonals are redefned and appled to nterconnected systems. By the help of a smallgan condton we prove that the whole system wth tme-delays has the ISS property f each subsystem has an ISS Lyapunov- Razumkhn functon or ISS Lyapunov-Krasovsk functonal. Furthermore we construct the ISS Lyapunov-Razumkhn (- Krasovsk) functon(al) and the correspondng gans of the whole system. I. INTRODUCTION In ths paper we study the nput-to-state stablty (ISS) property ntroduced n [18] of systems wth tme-delays. ISS and ts varants for example nput-to-state dynamcal stablty (ISDS) [6] local ISS (LISS) [21] and ntegral-iss (ISS) [19] became mportant durng the recent years for the stablty analyss of dynamcal systems and were appled n network control engneerng bologcal or economcal systems for example. A useful tool to verfy the ISS property for contnuous systems are Lyapunov functons (see [20]) as well as for other varants of ISS. For tme-delay systems the ISS property can be verfed by ISS Lyapunov-Razumkhn functons ([22]) or ISS Lyapunov-Krasovsk functonals ([15]). We are nterested n the ISS property for nterconnectons of systems wth tme-delays. The frst results on the ISS property for the delay-free case were gven for two coupled contnuous systems n [10] and for an arbtrarly large number (n N) of coupled systems n [2] usng a small-gan condton. Lyapunov versons of the ISS small-gan theorems were proved n [11] (two systems) and [3] (n systems) for the ISDS property n [4] for LISS n [5] and for ISS n [8] (two systems) and [9] (n systems) where Lyapunov functons for the overall system are constructed. A general approach of the verfcaton of the ISS property for nterconnected systems can be found n [12]. In ths paper we utlze on the one hand ISS Lyapunov- Razumkhn functons and on the other hand ISS Lyapunov- Krasovsk functonals to prove that a network of ISS systems wth tme-delays has the ISS property under a small-gan condton provded that each subsystem has an ISS Lyapunov-Razumkhn functon and an ISS Lyapunov- Krasovsk functonal respectvely. To prove ths we construct the ISS Lyapunov-Razumkhn functon and ISS S. Dashkovsky and L. Naujok are wth the Centre of Industral Mathematcs Unversty of Bremen P.O.Box Bremen Germany dsnlarsnaujok@math.un-bremen.de Lyapunov-Razumkhn functonal respectvely and the correspondng gans of the whole system. The paper s organzed as follows: In Secton 2 we note some basc defntons. The man results the ISS smallgan theorems for nterconnected tme-delay systems can be found n Secton 3 where Subsecton 3.1 contans the ISS Lyapunov-Razumkhn type theorem and Subsecton 3.2 the ISS Lyapunov-Krasovsk type theorem. In Secton 4 an example s gven to llustrate the results. Fnally Secton 5 concludes ths paper wth a short summary. II. NOTATIONS AND DEFINITIONS By x T we denote the transposton of a vector x R N N N furthermore R + := [0 ) and R N + denotes the postve orthant { x R N : x 0 } where we use the partal order for x y R N gven by x y x y = 1... N and x y : x < y x > y x > y = 1... N. We denote the Eucldean norm by. For x = (x 1... x k ) T defned on an nterval I we let x I = max 1 k { x I }. Let θ R +. The functon x t : [ θ 0] R N s gven by x t (τ) := x(t + τ) τ [ θ 0]. For a b R a < b let C ( [a b] ; R N) denote the Banach space of contnuous functons defned on [a b] equpped wth the norm [ab] and take values n R N. For functons x t we defne x t (τ) := max τ θ s τ x(s). Defnton 2.1: We defne followng classes of functons: K := {γ : R + R + γ s contnuous γ(0) = 0 K := {γ K γ s unbounded} and strctly ncreasng} L := {γ : R + R + γ s contnuous and strctly decreasng wth lm γ(t) = 0} t KL := {β : R + R + R + β s contnuous β( t) K β(r ) L t r 0} Note that for γ K the nverse functon γ 1 always exsts and γ 1 K. We recall the defnton of ISS for sngle tme-delay systems and note the man results of prevous works. Sngle nonlnear tme-delay systems are of the form ẋ(t) = f(x t u(t)) t 0 x(τ) = ξ 0 (τ) τ [ θ 0] where t R + x(t) R N and u(t) R M s an essentally bounded measurable nput. θ s the maxmum nvolved delay (1) ISBN

2 S. Dashkovsky and L. Naujok Lyapunov-Razumkhn and Lyapunov-Krasovsk Theorems for Interconnected ISS Tme-Delay Systems and f : C ( [ θ 0] ; R ) N R M R N s a locally Lpschtz contnuous functonal on any bounded set to guarantee that the system (1) admts a unque soluton x(t) on a maxmal nterval [ θ b) 0 < b + where x(t) s locally absolutely contnuous (see [7] [13] [15]). We denote the soluton by x(t 0 ξ) or x(t) for short satsfyng the ntal condton x 0 = ξ for any ξ C([ θ 0] R N ). Defnton 2.2: The system (1) s called nput-to-state stable (ISS) f there exst β KL and γ K such that for all t 0 t holds ( ) ) x(t) β ξ [ θ0] t + γ ( u [0 ). Next we defne ISS Lyapunov-Razumkhn functons ntroduced n [22]. Defnton 2.3: A locally Lpschtz functon V : R N R + s called an ISS Lyapunov-Razumkhn functon for system (1) f there exst ψ 1 ψ 2 K χ d χ u and α K such that the followng condtons hold: ψ 1 ( x ) V (x) ψ 2 ( x ) (2) V (x) χ d ( V d (x) )) + χ u ( u(t) ) D + V (x) α(v (x)) for all x(t) R N and all essentally bounded measurable nputs u(t) R M where V d (x(t)) := V (x(t + τ)) τ [ θ 0] and D + V (x) denotes the upper rght-hand dervatve along the soluton x(t) whch s defned as D + V (x(t + h)) V (x(t)) V (x(t)) = lm sup. h 0 h + Wth ths defnton we state the followng: Theorem 2.4: If there exsts an ISS Lyapunov- Razumkhn functon V for system (1) and χ d (s) < s s R + then the system (1) s ISS from u to x wth gan γ = ψ1 1 χ d. The proof can be found n [22]. Another approach to check f a system of the form (1) has the ISS property was ntroduced n [15]. There ISS Lyapunov-Krasovsk functonals are used. Gven a locally Lpschtz contnuous functonal V : C ( [ θ 0] ; R N) R + the upper rght-hand dervate D + V of the functonal V s defned for all φ C([ θ 0]; R N ) (see [1] Defnton pp. 258) as follows D + 1 V (φ u) := lm sup h 0 h (V (φ h) V (φ)) + where φ h C ( [ θ 0] ; R N) s gven by { φ φ(s + h) s [ θ h] h(s) = φ(0) + f(φ u)(h + s) s [ h 0]. Wth the symbol a we ndcate any norm n C ( [ θ 0] ; R N) such that for some postve reals b c the followng nequaltes hold b φ(0) φ a c φ [ θ0] φ C ( [ θ 0] ; R N ). (3) Defnton 2.5: A locally Lpschtz contnuous functonal V : C ( [ θ 0] ; R N ) R + s called an ISS Lyapunov- Krasovsk functonal for system (1) f there exst functons ψ 1 ψ 2 K and functons χ α K such that ψ 1 ( φ(0) ) V (φ) ψ 2 ( φ a ) (4) V (φ) χ ( u ) D + V (φ u) α (V (φ)) (5) φ C ( [ θ 0] ; R N) u R M. The next theorem s a counterpart to Theorem 2.4 wth accordng changes to Lyapunov-Krasovsk functonals. Theorem 2.6: If there exsts an ISS Lyapunov-Krasovsk functonal V : C ( [ θ 0] ; R N) R + for system (1) then system (1) s ISS. Proof: Ths follows by Theorem 3.1 n [15] by defnton of ρ := ψ 1 2 χ and D + V (φ u) α 3 ( φ a ) α(v (φ)) where α := α 3 ψ 1 2 and the functonal s chosen locally Lpschtz contnuous accordng to results n [14] [16]. In the next secton we consder nterconnected tme-delay systems and nvestgate under whch condtons the network has the ISS property. III. MAIN RESULTS In ths secton we state our two man results the ISS Lyapunov-Razumkhn and the ISS Lyapunov-Krasovsk small-gan theorem for general networks wth tme-delays. We consder n N nterconnected systems of the form ẋ (t) = f ( x t 1... x t n u(t) ) = 1... n (6) where x t (τ) := x (t + τ) τ [ θ 0] x R N. θ denotes the maxmal nvolved delay and x t can be nterpreted as the nternal nputs of a subsystem. The functonals f : C ( ) ( ) [ θ 0] ; R N1... C [ θ 0] ; R N n R M R N are locally Lpschtz contnuous on any bounded set. We denote the soluton of a subsystem by x (t 0 ξ ) or x (t) for short satsfyng the ntal condton x 0 = ξ for any ξ C([ θ 0] R N ). Defnton 3.1: The -th subsystem of (6) s called ISS f there exst β KL and γj d γu K {0} j = 1... n j such that x (t) β ( ξ [ θ0] t) + γj( x d j [ θt] ) + γ u ( u [0 ) ). If we defne N := N x := (x T 1... x T n ) T and f := (f T 1... f T n ) T then (6) becomes the system of the form (1) whch we call the whole system. We nvestgate under whch condtons the whole system has the ISS property and utlze Lyapunov-Razumkhn functons as well as Lyapunov- Krasovsk functonals. 1180

3 Proceedngs of the 19th Internatonal Symposum on Mathematcal Theory of Networks and Systems MTNS July 2010 Budapest Hungary A. Lyapunov-Razumkhn theorem for nterconnected systems In ths subsecton we state the frst man result of ths paper the ISS Lyapunov-Razumkhn small-gan theorem for nterconnected networks wth tme-delays. Defnton 3.2: A locally Lpschtz contnuous functon V : R N R + s called an ISS Lyapunov-Razumkhn functon for the -th subsystem of (6) f there exst functons V j j = 1... n whch are contnuous proper postve defnte and locally Lpschtz contnuous on R Nj \{0} functons χ u K {0} χd j K {0} α K j = 1... n such that the followng condton holds: V (x ) χ d j( Vj d (x j ) ) + χ u ( u ) j D + V (x ) α (V (x )) x R N and all essentally bounded measurable nputs u R M. The gan-matrx s defned by Γ := (χ d j ) n n j = 1... n and the map Γ : R n + R n + by Γ(s) := χ d 1j(s j )... χ d nj(s j ) s R n +. (8) j j Note that we get for v w R n +: v w Γ(v) Γ(w). We say that for a dagonal operator D : R n + R n + d = (Id +µ) µ K d j = 0 j the matrx Γ satsfes the small-gan-condton f for all s R n + s 0 we have T (7) D Γ(s) := D(Γ(s)) s. (9) More nformaton about the condton (9) can be found n [2] [17] [3]. For the proof of the results n ths secton we wll need the followng: Defnton 3.3: A contnuous path σ K n s called an Ω-path wth respect to D Γ where Γ s a gan-matrx and D a dagonal operator f () for each the functon s locally Lpschtz contnuous on (0 ); () for every compact set K (0 ) there are constants 0 < c < C such that for all ponts of dfferentablty of and = 1... n we have () t holds 0 < c ( ) (r) C r K; D(Γ(σ(r))) < σ(r) r > 0. (10) If the gan-matrx Γ satsfes the small-gan condton (9) then there exsts an Ω-path σ wth respect to D Γ. Ths path can be chosen pecewse lnear. Ths s Theorem 5.2 n [3]. We can now formulate our frst man result: Theorem 3.4: (ISS Lyapunov-Razumkhn theorem for general networks wth tme-delays) Consder the nterconnected system (6) where each subsystem has an ISS Lyapunov-Razumkhn functon V. If the correspondng gan-matrx Γ gven by (8) satsfes the smallgan condton (9) where D s a dagonal operator then the functon V (x) = max{ (V (x ))} s the ISS Lyapunov-Razumkhn functon for the whole system of the form (1) whch s ISS from u to x where σ = (σ 1... σ n ) T s an Ω-path as n Defnton 3.3. The gans are gven by χ d (r) := max j χ u (r) := max ρ 1 (χ u (r)) j ((χ d j) 1 ((Id + µ 2 ) 1 )(χ d j(σ j (r)))) for r 0 where ρ(r) := mn k ρ k (r) ρ k (r) := µ 2 ( χ d kj (σ k(r))). Remark 3.5: The defnton of ISS and ISS Lyapunov- Razumkhn functons gven here n terms of sums s equvalent to the defnton f one uses a maxmum nstead of a sum. Then the gans are gven by χ d (r) := max j (χ d j(σ j (r))) χ u (r) := max (χ u (r)). All subsystems of (6) have an ISS Lyapunov- Proof: Razumkhn functon V = 1... n.e. V satsfes (7). From the small-gan condton (9) for Γ gven by (8) and D s a dagonal operator t follows by Theorem 5.2 n [3] that there exsts an Ω-path σ = (σ 1... σ n ) T as n Defnton 3.3. Note that K = 1... n. Let 0 x = (x T 1... x T n ) T. We defne V (x) := max{ (V (x ))} as the ISS Lyapunov-Razumkhn functon canddate for the overall system. Note that V s locally Lpschtz contnuous. V satsfes (2) whch can be easly checked. For any {1... n} consder open domans M R N \{0} defned by M := { ( x T 1... x T n ) T R N \ {0} : (V (x )) > max j { σ 1 j (V j (x j )) } }. Now for any ˆx = (ˆx T 1... ˆx T n ) T R N \{0} there s at least one {1... n} such that ˆx M and t follows that there s a neghborhood U of ˆx such that V (x) = (V (x )) holds for all x U. We defne χ d (r) := max j j ((χ d j ) 1 ((Id + µ 2 ) 1 )(χ d j (σ j(r)))) χ u (r) := max ρ 1 (χ u (r)) r > 0 where ρ(r) := mn k ρ k (r) ρ k (r) := µ 2 ( χ d kj (σ k(r))) and assume V (x) χ d ( V d (x) ) + χ u ( u ). 1181

4 S. Dashkovsky and L. Naujok Lyapunov-Razumkhn and Lyapunov-Krasovsk Theorems for Interconnected ISS Tme-Delay Systems Note that χ d (r) < r. It follows from (10) V (x ) = σ (V (x)) > (Id +µ) χ d j(σ j (V (x))) From (7) we obtan χ d j( V d j (x j ) ) + χ u ( u ). D + V (x) = D + (V (x )) = ( ) (V (x ))D + V (x ) ( ) (V (x ))α (V (x )) = α (V (x)) where α (r) := ( ) (σ (r))α (σ (r)) r > 0. By defnton of α := mn α the functon V satsfes (3). All condtons of Defnton 2.3 are satsfed and V s the ISS Lyapunov-Razumkhn functon of the whole system of the form (1). By Theorem 2.4 the whole system s ISS from u to x. B. Lyapunov-Krasovsk theorem for nterconnected systems In ths subsecton we provde a counterpart to Theorem 3.4 where we use ISS Lyapunov-Krasovsk functonals. Defnton 3.6: A locally Lpschtz contnuous functonal V : C([ θ 0]; R N ) R + s called an ISS Lyapunov- Krasovsk functonal of the -th subsystem of (6) f there exst functonals V j j = 1... n whch are contnuous proper postve defnte and locally Lpschtz contnuous on C([ θ 0]; R Nj )\{0} functons χ j χ K {0} α K j = 1... n j such that V (φ ) χ j (V j ( φ j )) + χ ( u ) (11) D + V (φ u) α (V (φ )) φ C ( ) [ θ 0] R N u R M χ 0 = 1... n. The gan-matrx s defned by Γ := (χ j ) n and the map Γ : R n + R n + by T Γ(s) := χ 1j (s j )... χ nj (s j ) s R n +. (12) The next theorem s the second man result of ths paper. Theorem 3.7: (ISS Lyapunov-Krasovsk theorem for general networks wth tme-delays) Consder the nterconnected system (6). Assume that each subsystem has an ISS Lyapunov-Krasovsk functonal V whch satsfes the condtons n Defnton 3.6 = 1... n. If the correspondng gan-matrx Γ gven by (12) satsfes the small-gan condton (9) where D s a dagonal operator then the functonal V (φ) := max{ (V (φ ))} s the ISS Lyapunov-Krasovsk functonal for the whole system of the form (1) whch s ISS from u to x where σ = (σ 1... σ n ) T s an Ω-path as n Defnton 3.3 and φ = (φ... φ n ) T C([ θ 0]; R N ). The gan s gven by χ(r) := max ρ 1 (χ (r)) wth ρ := mn k=1...n ρ k ρ k (r) := µ n k j χ kj(σ j (r)). Proof: All subsystems of (6) have an ISS Lyapunov- Krasovsk functonal V = 1... n.e. V satsfes (11). From the small-gan condton (9) for Γ there exsts an Ω- path σ = (σ 1... σ n ) T. Let 0 x t = ((x t 1) T... (x t n) T ) T C([ θ 0]; R N ). We defne V (x t ) := max { (V (x t ))} as the ISS Lyapunov-Krasovsk functonal canddate. Note that V s locally Lpschtz. V satsfes (4) whch can be easly checked. For any {1... n} consder open domans M R N \{0} defned by M := { ( (x t 1) T... (x t n) T ) T R N \ {0} : (V (x t { )) > max σ 1 j (V j (x t j)) } }. Now for any ˆx t = ((ˆx t 1) T... ( ˆφ t n) T ) T R N \{0} there s at least one {1... n} such that ˆx t M and t follows that there s a neghborhood U of ˆx t such that V (x t ) = j (V (x t )) holds for all x t U. From (10) we get σ (r) > (Id +µ) χ j (σ j (r)) r > 0 σ (r) > µ j j j χ j (σ j (r)) χ j (σ j (r)) =: ρ (r) r > 0. If we defne ρ := mn ρ and assume V (x t ) ρ 1 (χ ( u )) t follows ρ(v (x t )) χ ( u ) σ (V (x t )) χ j (σ j (V (x t ))) > χ ( u ) and we get j V (x t ) = σ (V (x t )) > χ ( u ) + = χ ( u ) + From (11) we obtan j j χ j (V j (x t j)). χ j (σ j (V (x t ))) D + V (x t u) = D + V (x t u) ( ) (V (x t ))α (V (x t )) = α (V (x t )) where α (r) := ( ) (σ (r))α (σ (r)) r > 0. By defnton of χ := max ρ 1 χ and α := mn α the functon V satsfes (5). All condtons of Defnton 2.5 are satsfed and V s the Lyapunov-Krasovsk functonal of the whole system of the form (1). By Theorem 2.6 the whole system s ISS from u to x. 1182

5 Proceedngs of the 19th Internatonal Symposum on Mathematcal Theory of Networks and Systems MTNS July 2010 Budapest Hungary Fg. 1. The gven producton network IV. EXAMPLE In ths secton we provde an example to apply the man results of ths paper. We consder a logstc network consstng of three producton locatons whch are connected by transport routes as shown n Fgure 1. In the followng we call a producton locaton only subsystem. Subsystems one and three get some raw materal from an external source denoted by u 1 and u 3 R +. Subsystem three produces the materal wth some producton rate p 3 (x 3 (t)) where x (t) R + = denotes the amount of unprocessed parts wthn subsystem. 50% of the producton wll be send to subsystem one and two n each case. There the parts enter the subsystems wth the tme-delay T 31 and T 32 whch denotes the transportaton tme from subsystem three to one and two respectvely. Subsystem one processes the parts wth the rate p 1 (x 1 (t)) and sends the processed parts to subsystem two where they arrve wth the tme-delay T 12 and wll be processed wth the rate p 2 (x 2 (t)). 50% of the processed parts of subsystem two wll be send to subsystem three (tme-delay T 23 ) and 50% wll leave the system. Ths can be nterpreted as customer supply. The producton rates are gven by p (x ) := x 2 and we have ẋ 1 (t) = u 1 (t) p 3(x 3 (t T 31 )) p 1 (x 1 (t)) ẋ 2 (t) = p 1 (x 1 (t T 12 )) p 3(x 3 (t T 32 )) p 2 (x 2 (t)) ẋ 3 (t) = u 3 (t) p 2(x 2 (t T 23 )) p 3 (x 3 (t)). It s easy to check that for ξ (τ) 0 τ [ θ 0] where θ := max T j t holds x (t) 0 t R +. At frst we use Lyapunov-Razumkhn functons to nvestgate the network n vew of stablty. We choose V (x ) := x 2 = as ISS Lyapunov-Razumkhn functon canddates of the subsystems whch are contnuous postve defnte and proper and locally Lpschtz contnuous. At frst we nvestgate subsystem one and choose the gans χ u 1( u 1 ) := u1 χ d 1 13( (x ε1 3 (t + τ)) 2 ) := (x3(t+τ))2 2 2(1 ε1 2 ) where 1 > ε 1 > 0 and τ [ T 31 0]. By the assumpton V 1 (x 1 (t)) χ d 13( (x 3 (t + τ)) 2 ) + χ 1 ( u 1 (t) ) and the defnton of the gans t follows D + V 1 (x 1 (t)) = 2(u 1 (t) (x 3(t T 31 )) 2 x 2 1(t)) α 1 (V 1 (x 1 (t))) where α 1 (r) := ε 1 r r 0. Therefor V 1 satsfes the condton (7) and we conclude that V 1 s the ISS Lyapunov- Razumkhn functon for subsystem one. By defnton of the gans χ d 21( (x 1 (t + τ)) 2 ) := (x1(t+τ))2 τ [ T 1 ε2 12 0] 2 χ d 23( (x 3 (t + τ)) 2 ) := (x3(t+τ))2 τ [ T 2(1 ε2 32 0] 2 ) χ u 3( u 3 (t) ) := u3(t) 1 ε3 2 χ d 32(( x 2 (t + τ) ) 2 ) := (x2(t+τ))2 τ [ T 2(1 ε3 23 0] 2 ) 1 > ε 2 > 0 1 > ε 3 > 0 we can prove that V 2 and V 3 are the ISS Lyapunov-Razumkhn functons of the subsystems two and three. Now we check f the small-gan condton s satsfed where Γ := 0 0 χ d 13 χ d 21 0 χ d 23 0 χ d We choose µ(r) := εr r > 0where ε > 0 s arbtrarly small and the dagonal operator s then gven wth ts dagonal elements d (r) = (1+ ε)r. The Ω-path canddate σ(r) = (σ 1 (r) σ 2 (r) σ 3 (r)) T s chosen as σ 1 (r) = σ 3 (r) := r and σ 2 (r) := 7 4r. Note that the condtons () and () of Defnton 3.3 are satsfed. Let us check the condton (): D Γ(σ(s)) = 1 (1 + ε)( 2(1 ε σ 3) 3(s)) (1+ ε) 1 1 ε 1 σ 1 (s) + (1 + ε) 2(1 ε σ 3) 3(s) 1 (1 + ε) 2(1 ε σ 2) 2(s) and by the choce of the Ω-path canddate above we have D Γ(σ(s)) < σ(s) s > 0 for suffcent small ε ε = such that σ s the Ω-path whch s equvalent to the satsfacton of the small-gan condton. By the applcaton of Theorem 3.4 the whole network s ISS where the ISS Lyapunov-Razumkhn functon s gven by V (x) = max{x x2 2 x 2 3}. We now utlze Lyapunov-Krasovsk functonals to nvestgate the network n vew of stablty. We choose V (x t ) = x 2 (t) = as the ISS Lyapunov-Krasovsk functonal canddates. By χ 1 ( u 1 (t) ) := u1(t) 1 ε1 2 χ 13 (V 3 (x t 3)) := ( x3 [t T 31 t]) 2 2(1 ε1 2 ) where 1 > ε 1 > 0 and the assumpton V 1 (x t 1) χ 13 (V 3 (x t 3)) + χ 1 ( u 1 (t) ) we get for the frst subsystem D + V 1 (x t 1) = 2(u 1 (t) (x 3(t T 31 )) 2 x 2 1(t)) α 1 (V 1 (x t 1)) 1183

6 S. Dashkovsky and L. Naujok Lyapunov-Razumkhn and Lyapunov-Krasovsk Theorems for Interconnected ISS Tme-Delay Systems where α 1 (r) := ε 1 r K r 0. By χ 21 (V 1 (x t 1)) := ( x1 [t T 12 t]) 2 1 ε2 2 χ 23 (V 3 (x t 3)) := ( x3 [t T 32 t]) 2 2(1 ε2 2 ) χ 3 ( u 3 (t) ) := u3(t) 1 ε3 2 χ 32 (V 2 (x t 2)) := ( x2 [t T 23 t]) 2 2(1 ε3 2 ) 1 > ε 2 > 0 1 > ε 3 > 0 and smlar calculatons for the other subsystems as for the frst subsystem we conclude that V (x t ) = x2 (t) = are the ISS Lyapunov-Krasovsk functonals for the subsystems. The small-gan condton s satsfed (see above) and by applcaton of Theorem 3.7 for the ISS property the whole network s ISS where the Lyapunov-Krasovsk functonal of the whole system s gven by V (x t ) = max{x x2 2 x 2 3}. V. CONCLUSIONS We have proved two theorems: an ISS Lyapunov- Razumkhn and an ISS Lyapunov-Krasovsk small-gan theorem. They state that a network of tme-delay systems has the ISS property provded that the small-gan condton s satsfed and that each subsystem has an ISS Lyapunov- Razumkhn functon and ISS Lyapunov-Krasovsk functonal respectvely. Furthermore we showed how to construct the ISS Lyapunov-Razumkhn functon the ISS Lyapunov- Krasovsk functonal and the correspondng gans of the whole system. Ths was llustrated by a short example from the logstcs. [8] Hrosh Ito. State-dependent scalng problems and stablty of nterconnected ISS and ISS systems. IEEE Transactons on Automatc Control 51(10): [9] Hrosh Ito Sergey Dashkovsky and Faban Wrth. On a small gan theorem for networks of ISS systems. In Proceedngs of the 48th IEEE Conference on Decson and Control Shangha Chna Dec pages [10] Z.-P. Jang A. R. Teel and L. Praly. Small-gan theorem for ISS systems and applcatons. Math. Control Sgnals Systems 7(2): [11] Zhong-Png Jang Iven M. Y. Mareels and Yuan Wang. A Lyapunov formulaton of the nonlnear small-gan theorem for nterconnected ISS systems. Automatca J. IFAC 32(8): [12] Iasson Karafylls and Zhong-Png Jang. A vector smallgan theorem for general nonlnear control systems [13] V. Kolmanovsk and A. Myshks. Introducton to the theory and applcatons of functonal-dfferental equatons volume 463 of Mathematcs and ts Applcatons. Kluwer Academc Publshers Dordrecht [14] P. Pepe. On Lapunov-Krasovsk functonals under Carathéodory condtons. Automatca J. IFAC 43(4): [15] P. Pepe and Z.-P. Jang. A Lyapunov-Krasovsk methodology for ISS and ISS of tme-delay systems. Systems Control Lett. 55(12): [16] Perdomenco Pepe. The problem of the absolute contnuty for Lyapunov-Krasovsk functonals. IEEE Trans. Automat. Control 52(5): [17] Björn Rüffer. Monotone dynamcal systems graphs and stablty of large-scale nterconnected systems. PhD thess Fachberech 3 (Mathematk & Informatk) der Unverstät Bremen [18] Eduardo D. Sontag. Smooth stablzaton mples coprme factorzaton. IEEE Trans. Automat. Control 34(4): [19] Eduardo D. Sontag. Comments on ntegral varants of ISS. Systems Control Lett. 34(1-2): [20] Eduardo D. Sontag and Yuan Wang. On characterzatons of the nputto-state stablty property. Systems Control Lett. 24(5): [21] Eduardo D. Sontag and Yuan Wang. New characterzatons of nputto-state stablty. IEEE Trans. Automat. Control 41(9): [22] Andrew R. Teel. Connectons between Razumkhn-type theorems and the ISS nonlnear small gan theorem. IEEE Trans. Automat. Control 43(7): VI. ACKNOWLEDGMENTS Ths research was supported by the German Research Foundaton (DFG) as part of the Collaboratve Research Center 637 Autonomous Cooperatng Logstc Processes. REFERENCES [1] T. A. Burton. Stablty and perodc solutons of ordnary and functonal-dfferental equatons volume 178 of Mathematcs n Scence and Engneerng. Academc Press Inc. Orlando FL [2] Sergey Dashkovsky Björn S. Rüffer and Faban R. Wrth. An ISS small gan theorem for general networks. Math. Control Sgnals Systems 19(2): [3] Sergey Dashkovsky Björn S. Rüffer and Faban R. Wrth. Small gan theorems for large scale systems and constructon of ISS Lyapunov functons Accepted n SIAM Journal on Control and Optmzaton avalable electroncally: [4] Sergey N. Dashkovsky and Lars Naujok. Input-to-state dynamcal stablty of nterconnected systems. In Proceedngs of the 48th IEEE Conference on Decson and Control Shangha Chna Dec pages [5] Sergey N. Dashkovsky and Björn S. Rüffer. Local ISS of large-scale nterconnectons and estmates for stablty regons. Systems & Control Letters 59(3-4): [6] Lars Grüne. Input-to-state dynamcal stablty and ts Lyapunov functon characterzaton. IEEE Trans. Automat. Control 47(9): [7] Jack K. Hale and Sjoerd M. Verduyn Lunel. Introducton to functonaldfferental equatons volume 99 of Appled Mathematcal Scences. Sprnger-Verlag New York