ISS of interconnected impulsive systems with and without time-delays

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1 ISS of nterconnected mpulsve systems wth and wthout tme-delays Sergey Dashkovsky,1 Mchael Kosmykov,2 Lars Nauok,1 Unversty of Bremen, Centre of Industral Mathematcs, P.O.Box , Bremen, Germany Abstract: We consder networks of mpulsve systems wth and wthout tme-delays and nvestgate under whch condtons the whole network s nput-to-state stable (ISS). We provde condtons on the sze of the tme ntervals between mpulses and on the nterconnecton structure that guarantee ISS of the overall system, where Lyapunov and Lyapunov-Razumkhn functons for the subsystems are used. The condton on the nterconnecton allows to construct a Lyapunov (-Razumkhn) functon and a correspondng gan for the whole system. Keywords: Impulsve systems, nput-to-state stablty, nterconnected systems, tme-delay systems, Lyapunov methods, small-gan theorems 1. INTRODUCTION Impulsve systems combne contnuous and dscontnuous behavors of a dynamcal system. The contnuous dynamcs s typcally descrbed by ordnary dfferental equatons and the dscontnuous behavor are nstantaneous state umps that occur at gven tme nstants, also referred to as mpulses. Impulsve systems are closely connected to hybrd systems and swtched systems and have wde applcatons n network control, engneerng, bologcal or economcal systems, see van der Schaft and Schumacher [2000], Haddad et al. [2006], Shorten et al. [2007]. In ths paper we study the nput-to-state stablty (ISS) property of mpulsve systems wth external nputs. ISS was frst ntroduced for contnuous systems n Sontag [1989]. A useful tool to verfy the ISS property for contnuous systems are Lyapunov functons (see Sontag and Wang [1995]) as well as for other varants of ISS, namely nput-to-state dynamcal stablty (ISDS) (Grüne [2002]), local ISS (LISS) (Sontag and Wang [1996]) and ntegral-iss (ISS) (Sontag [1998]). Investgatons of ISS for hybrd systems can be found n Ca and Teel [2009]. For tme-delay systems the ISS property can be verfed by Lyapunov-Razumkhn functons (Teel [1998]) or Lyapunov-Krasovsk functonals (Pepe and Jang [2006]). For mpulsve systems the ISS and ISS propertes were studed n Hespanha et al. [2008] for the delay-free case and n Chen and Zheng [2009] for tme-delay systems. By the help of exponental ISS Lyapunov(-Razumkhn) functons and an (reverse) average dwell-tme (ADT) condton the ISS property for mpulsve (tme-delay) systems was proved. 1 Sergey Dashkovsky and Lars Nauok are supported by the German Research Foundaton (DFG) as part of the Collaboratve Research Center 637 Autonomous Cooperatng Logstc Processes: A Paradgm Shft and ts Lmtatons. 2 Mchael Kosmykov s supported by the Volkswagen Foundaton as part of the Proect Nr. I/82684 Dynamc Large-Scale Logstcs Networks. We are nterested n the ISS property for nterconnectons of mpulsve systems wth and wthout tme-delays. The frst results on the ISS property for the delay-free case were gven for two coupled contnuous systems n Jang et al. [1994] and for an arbtrarly large number (n N) of coupled systems n Dashkovsky et al. [2007], usng a smallgan condton. Lyapunov versons of the ISS small-gan theorems were proved n Jang et al. [1996] (two systems) and Dashkovsky et al. [2009] (n systems), for the ISDS property n Dashkovsky and Nauok [2010], for LISS n Dashkovsky and Rüffer [2010] and for ISS n Ito [2006] (two systems) and Ito et al. [2009] (n systems), where Lyapunov functons for the overall system are constructed. ISS for nterconnected hybrd systems was nvestgated n Nesc and Teel [2008] (two systems) and Dashkovsky and Kosmykov [2009] (n systems). A general approach of the verfcaton of the ISS property for nterconnected systems can be found n Karafylls and Jang [2009]. In ths paper we prove that under a small-gan condton and the (reverse) ADT condton for nterconnected mpulsve systems accordng to Hespanha et al. [2008] the overall system s agan ISS and construct the exponental ISS Lyapunov(-Razumkhn) functon and the correspondng gan of the whole system. The paper s organzed as follows: In Secton 2 we note some basc defntons. The frst man result, the ISS smallgan theorem for nterconnected delay-free mpulsve systems can be found n Secton 3. Secton 4 contans the second man result, the ISS small-gan theorem for nterconnected mpulsve systems wth tme-delays. Fnally Secton 5 concludes ths paper wth a short summary. 2. PRELIMINARIES 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

2 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 N ) T defned on an nterval I R, we defne x I := max max t I { x (t) }. V denotes the gradent of a functon V. The upper rght-hand sde dervatve of a locally Lpschtz contnuous functon V : R N R + along x(t) R N s defned by D + V (x(t)) = lm sup h 0 + V (x(t + h)) V (x(t)). h Let θ R + be the maxmum nvolved delay. The functon x t : [ θ, 0] R N s gven by x t (τ) := x(t + τ), τ [ θ, 0] and we defne x t := max t θ s t x(s). For a, b R, a < b, let C ( [a, b] ; R N) denote the Banach space of pecewse rght-contnuous functons defned on [a, b] equpped wth the norm [a,b] and take values n R N. Defnton 2.1. Classes of comparson functons are: K := {γ : R + R + γ s contnuous, γ(0) = 0 and strctly ncreasng}, K := {γ K γ s unbounded}, L := {γ : R + R + γ s contnuous and 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. 3. DELAY-FREE INTERCONNECTED IMPULSIVE SYSTEMS We consder an nterconnecton of n mpulsve subsystems wth nputs ẋ (t) = f (x 1 (t),..., x n (t), u (t)), t t k, x (t) = g (x 1 (t),..., x n (t), u (t)), t = t k, (1) x R N, u R M, where {t 1, t 2, t 3,...} s a strctly ncreasng sequence of mpulsve tmes n (t 0, ) for some ntal tme t 0 of the th subsystem. We assume further t 0 = 0. The set of mpulsve tmes s assumed to be ether fnte or nfnte and unbounded. Gven a sequence {t k} and a par of tmes s, t satsfyng t 0 s < t, N (t, s) denotes the number of mpulsve tmes t k n the semopen nterval (s, t] of the th subsystem. The state x (t) R N of the th subsystem s absolutely contnuous between mpulses; u (t) R M s a locally bounded, Lebesgue-measurable nput. We assume that functons f : R N R M R N and g : R N R M R N are locally Lpschtz contnuous. All sgnals (states x and nputs u ) are assumed to be rght-contnuous and to have left lmts at all tmes and we denote x (t) := lm s t x (s), u (t) := lm s t u (s). If we defne N := N N n, M := M M n, x := (x T 1,..., x T n ) T, u := (u T 1,..., u T n ) T and f := (f1 T,..., fn T ) T and the mpulsve { tme sequence of the whole system {t k } by {t k } := t t = t k } k, N. Furthermore we defne I k := { t k = t k}, I k := { t k t k} Ik and N(t, s) denotes the number of mpulsve tmes n the sem-open nterval (s, t] of the whole system. Then x Ik := (0,..., g 1,...,..., g p,..., 0) T, where I k, = 1,..., p and x Ik := (0,..., x 1,...,..., x l,..., 0) T, where I k, = 1,..., l. Wth these defntons the nterconnected system (1) can be descrbed as a system of the form ẋ(t) = f(x(t), u(t)), t t k, x(t) = x Ik + x Ik =: g(x (t), u (2) (t)), t = t k, k N. The ISS property of mpulsve systems s defned as follows, see Hespanha et al. [2008]. Defnton 3.1. Assume that a sequence {t k } s gven. We call system (2) nput-to-state stable (ISS) f there exst functons β KL, γ K, such that for every ntal condton x(0) and every nput u the correspondng soluton to (2) exsts globally and satsfes x(t) max{β( x(0), t), γ( u [0,t] )}, t 0. (3) The supremum norm of an nput u on the nterval [t 0, t] s redefned by { } u [t0,t] := max ess sup s [t 0,t] u(s), sup t k [t 0,t] u(t k ) The mpulsve system (2) s unformly ISS over a gven class S of admssble sequences of mpulsve tmes f (3) holds for every sequence n S, wth functons β and γ that are ndependent of the choce of the sequence. For a system wth several nputs we use the followng Defnton 3.2. Assume that a sequence {t k} s gven. The th subsystem of (2) s ISS f there exst β KL, γ, γ K {0} such that for every ntal condton x (0) and every nput u the correspondng soluton to (1) exsts globally and satsfes for all t 0 x (t) max{β ( x (0), t), max γ ( x [0,t] ), γ ( u [0,t] )}, Functons γ are called gans. The mpulsve system (1) s unformly ISS over a gven class S of admssble sequences of mpulsve tmes f (4) holds for every sequence n S, wth functons β and γ, γ that are ndependent of the choce of the sequence. For stablty analyss of mpulsve systems we use exponental Lyapunov functons, see Hespanha et al. [2008]. Defnton 3.3. We say that a functon V : R N R s an exponental ISS-Lyapunov functon for (2) wth rate coeffcents c, d R f V s locally Lpschtz, postve defnte, radally unbounded, and satsfes V (x) γ( u ) V (x) f(x, u) cv (x) (5) for almost all x, all u and V (x) γ( u ) V (g(x, u)) e d V (x) (6) for all x, u, where γ s some functon from K. Condton (5) states, that f c s postve then the functon V decreases. On the other hand, f c < 0 then the functon V may ncrease. Condton (6) states, that f d s postve then the ump (mpulse) decreases the magntude of V. On the other hand, f d < 0 then the ump (mpulse) may ncrease the magntude of V. Remark 3.4. Note that n Hespanha et al. [2008] the condtons (5) and (6) are n dsspatve form. By Proposton 2.6. (4)

3 n Ca and Teel [2009] the condtons n dsspatve form are equvalent to the condtons n mplcaton form, used n Defnton 3.3, but coeffcents c, d may be dfferent. Smlarly we defne Lyapunov functons for subsystems. Defnton 3.5. Functon V : R N R s an exponental ISS-Lyapunov functon for the th subsystem of (1) wth rate coeffcents c, d R f there exst functons V, = 1,..., n, whch are contnuous, proper and postve defnte and locally Lpschtz contnuous on R N \{0} and V satsfes V (x ) max{max, γ (V (x )), γ ( u )} V (x ) f (x, u ) c V (x ) for almost all x, all u and V (x ) max{max, γ (V (x )), γ ( u )} V (g (x, u )) e d V (x ) for all x, u, where γ, γ are some functons from K. The rate coeffcents c, d are not requred to be nonnegatve and therefore V may not decrease even f u = 0. Note that n general γ, γ, γ n the defntons of ISS and ISS-Lyapunov functons are dfferent. In Hespanha et al. [2008] the followng theorem was proved whch establshes stablty of a sngle mpulsve system even f one of rate coeffcents s not postve. Theorem 3.6. Let V be an exponental ISS-Lyapunov functon for (2) wth rate coeffcents c, d R wth d 0. For arbtrary constants µ, λ > 0, let S[µ, λ] denote the class of mpulsve tme sequences {t k } satsfyng (7) (8) dn(t, s) (c λ)(t s) µ, t s 0. (9) Then the system (2) s unformly ISS over S[µ, λ]. When d = 0 the theorem can be appled, because (9) holds for every d < 0. Ths case and the case c = 0 were nvestgated n more detal n Hespanha et al. [2008], Secton 6. Condton (9) guarantees stablty of the mpulsve system even f the contnuous or dscontnuous behavor s unstable. For example, f the contnuous behavor s unstable, whch means c < 0, then ths condton assumes that the dscontnuous behavor has to stablze the system (d > 0) and the umps have to occur often enough. Opposte f the dscontnuous behavor s unstable (d < 0) and the contnuous behavor s stable (c > 0) then the umps have to occur rarely, whch stablzes the system. In general even f all subsystems of (1) are ISS, the whole system (2) may be not ISS. Thus we are lookng for the condtons that guarantee ISS of (2). We collect the nonlnear gans γ of the subsystems n a matrx Γ = (γ ) n n,, = 1,..., n denotng γ 0, = 1,..., n for completeness, see Dashkovsky et al. [2007], Rüffer [2007]. Note that ths matrx descrbes n partcular the nterconnecton topology of the whole network, moreover t contans the nformaton about the mutual nfluence between the subsystems. We also ntroduce the gan operator Γ : R n + R n + defned by ( ) n Γ(s) := max γ n 1(s ),..., max γ n(s ), s R n +. (10) =1 =1 To show one of the man results we need the noton of a so called Ω-path, see Dashkovsky et al. [2009], Rüffer [2009]. A functon σ = (σ 1,..., σ n ) T : R n + R n +, where σ K s called an Ω-path, f t possesses the followng propertes: () s locally Lpschtz contnuous on (0, ); () for every compact set P (0, ) there are fnte constants 0 < K 1 < K 2 such that for all ponts of dfferentablty of we have 0 < K 1 ( ) (r) K 2, r P () Γ(σ(r)) < σ(r), r > 0. (11) The next theorem provdes a condton for the exstence of an Ω-path. Theorem 3.7. Let Γ (K {0}) n n be a gan matrx. If Γ satsfes the small-gan condton Γ(s) s, s R n +\ {0}, (12) then there exsts an Ω-path σ wth respect to Γ. Ths path can be chosen pecewse lnear. The proof can be found n Dashkovsky et al. [2009], Theorem 5.2, see also Rüffer [2009], however only the exstence s proved n these works. Now we can formulate one of the man results that s an ISS small-gan theorem for mpulsve systems wthout tmedelays. Ths theorem allows to construct an ISS Lyapunov functon for the whole nterconnecton. 3.1 Man result Theorem 3.8. Assume that each subsystem of (1) has an exponental ISS-Lyapunov functon V wth correspondng ISS-Lyapunov gans γ and rate coeffcents c, d, d 0. Defne c := mn c and d := mn d. For arbtrary constants µ, λ > 0, let S[µ, λ] denote the class of mpulsve tme sequences {t k } of the whole system. If the followng holds ) S[µ, λ] satsfes condton (9), ) Γ = (γ ) n n satsfes the small-gan condton (12), then the mpulsve system (2) s unformly ISS over S[µ, λ] and the exponental ISS-Lyapunov functon s gven by V (x) := max{ (V (x ))}, (13) where σ = (σ 1,..., σ n ) T s a pecewse lnear Ω-path. The gan s gven by γ(r) := max (γ (r)). The small-gan condton ) s used for example n Dashkovsky et al. [2007, 2009] to verfy the ISS property of nterconnected systems. The condton ) s the average dwell-tme condton for nterconnectons of mpulsve systems. For sngle mpulsve systems, whch means there s only one c and one d ths condton can be found n Hespanha et al. [2008]. Proof. As the small gan condton (12) s satsfed t follows from Theorem 3.7 that there exsts an Ω-path σ wth respect to Γ. We can choose ths path to be pecewse lnear. Let us defne V (x) = max (V (x )) (14)

4 and show that ths functon s an exponental ISS- Lyapunov functon for the system (2). It can be easly checked that ths functon s locally Lpschtz, postve defnte and radally unbounded. 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 { σ 1 (V (x )) } }. (15) 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 γ(r) := max (γ (r)), r > 0 and assume V (x) γ( u ). It follows from (11) that V (x ) =σ (V (x)) max{max(γ (σ (V (x)))), σ (γ( u ))} max{max γ (V (x)), γ ( u )}. Let [s l, s l+1 ], s l < s l+1, l = 0, 1,... be an nterval where σ s lnear,.e., σ (t) = a l s, s [s l, s l+1 ]. Then for all ntervals wth σ (s) = a l s and from (7) we obtan for almost all x and V (x ) [s l, s l+1 ] V (x) =( ) (V (x )) V (x ) f (x, u ) 1 c V (x ) = c V (x). a l By the defnton of c := mn c the functon V satsfes (5). As d := mn d t holds V (g(x, u)) = 1 (V (g (x 1,..., x n, u ))) a l 1 e d V (x ) e d V (x), V (x ) [s l, s l+1 ], a l.e., V satsfes condton (6). All condtons of Defnton 3.3 are satsfed and thus V s the exponental ISS-Lyapunov functon of system (2). By assumpton ) there exst µ, λ > 0 such that dn(t, s) c(t s) µ λ(t s), t s 0 we can apply Theorem 3.6 and the overall system s unformly ISS over S[µ, λ]. Usng c = mn c, d = mn d n the average dwelltme condton some knd of conservatveness may occur, whch means we cannot verfy the ISS property for an nterconnected mpulsve system by the applcaton of Theorem 3.8, although the system possesses the ISS property. Remark 3.9. If both dynamcs, the contnuous and the dscontnuous, are stable, whch means that c, d > 0, then the condton (9) n Theorem 3.8 s not necessary, accordng to Theorem 2 n Hespanha et al. [2008]. The followng secton provdes a smlar small-gan result, but for mpulsve systems wth tme-delays. 4. INTERCONNECTED IMPULSIVE SYSTEMS WITH TIME-DELAYS Consder n nterconnected mpulsve systems wth tmedelays of the form ẋ (t) = f (x t 1,..., x t n, u (t)), t t k, x (t) = g ((x t 1),..., (x t n), u (t)), t = t k, (16) k N, = 1,..., n, where the same assumptons on the system as n the delay-free case are consdered wth the followng dfferences: We denote x t (τ) := x (t + τ), τ [ θ, 0], where θ s the maxmum nvolved delay and (x t ) (τ) := x (t + τ) := lm s t x (s + τ), τ [ θ, 0]. We assume that the functonals f : C([ θ, 0], R N1 )... C([ θ, 0], R Nn ) R M R N and g : C([ θ, 0], R N1 )... C([ θ, 0], R Nn ) R M R N are locally Lpschtz contnuous. If we defne t k, N, M, x, u, f and g as n the delay-free case, then (16) becomes the system of the form ẋ(t) = f(x t, u(t)), t t k, k N, x(t) = g(x t, u (t)), t = t k, k N. (17) We assume that the regularty condtons n Ballnger and Lu [1999] for the exstence and unqueness of solutons of systems (16) and (17) are satsfed. For any ξ C([ θ, 0], R N ) the soluton of the th subsystem of (16) s denoted by x (t, t 0, ξ ) or x (t) for short, satsfyng the ntal condton x t0 (τ) = ξ (τ), τ [ θ, 0]. In a smlar way we denote the soluton of the system (17) by x(t, t 0, ξ) or x(t) for short for any ξ C([ θ, 0], R N ) that exsts n a maxmal nterval [ θ, b), 0 < b +, satsfyng the ntal condton x t0 (τ) = ξ(τ), τ [ θ, 0]. The defnton of ISS s then smlar to the delay-free case: Defnton 4.1. Suppose that a sequence {t k } s gven. We call system (17) ISS f there exst functons β KL, γ K, such that for every ntal condton x 0 = ξ C([ θ, 0]) and every nput u the correspondng soluton to (17) exsts globally and satsfes x(t) max{β( ξ [ θ,0], t), γ( u [0,t] )}, t 0. (18) The mpulsve system (17) s unformly ISS over a gven class S of admssble sequences of mpulsve tmes f (18) holds for every sequence n S, wth functons β and γ that are ndependent of the choce of the sequence. For the stablty analyss of systems of the form (17) Razumkhn-type theorems were proved n Chen and Zheng [2009]. In our second man result we use the defnton of an exponental ISS-Lyapunov-Razumkhn functon n mplcaton form and the (reverse) average dwell-tme condton from Hespanha et al. [2008]. Defnton 4.2. A locally Lpschtz contnuous functon V : R N R + s called an exponental ISS-Lyapunov- Razumkhn functon for system (17), f there exst ψ 1, ψ 2 K, γ u K, γ d K and scalars c, d R such that ψ 1 ( x ) V (x) ψ 2 ( x ), (19) V (x) max{γ d ( V d (x) ), γ u ( u(t) )} D + V (x) cv (x), (20) V (x) max{γ d ( V d (x) ), γ u ( u(t) )} V (g(x, u)) e d V (x), (21) hold for all x(t) R N and u(t) R M, where V d (x(t)) := V (x(t + τ)), τ [ θ, 0]. Remark 4.3. Note that n Chen and Zheng [2009] the condtons (20) and (21) are n dsspatve form. By Proposton 2.6 n Ca and Teel [2009] the condtons n dsspa-

5 tve form are equvalent to the condtons n mplcaton form, used n Defnton 4.2, where the coeffcents c, d are dfferent n general. The followng proposton s smlar to Theorem 1 n Hespanha et al. [2008]. Here we adopt the approach used n ths paper to tme-delay systems. In Chen and Zheng [2009] a dfferent approach for the characterzaton of the condton on the tme ntervals of the mpulses s used, but the statement s the same. By combnng the results n the two prevous mentoned papers we can state the followng: Proposton 4.4. Let V be an exponental ISS-Lyapunov- Razumkhn functon for (17) wth c, d R. For arbtrary constants µ, λ R +, let S[µ, λ] denote the class of mpulse tme sequences {t k } satsfyng (9). If γ d < Id, then the system (17) s unformly ISS over S[µ, λ]. The proof s skpped, because t s a combnaton of the proofs of Theorem 1 n Hespanha et al. [2008] and Theorems 1 and 2 n Chen and Zheng [2009] wth an exponental ISS-Lyapunov-Razumkhn functon as n Defnton 4.2. Now we consder nterconnected mpulsve tme-delay systems and defne the ISS property and Lyapunov functons of the subsystems as follows: Defnton 4.5. Suppose that a sequence {t k} s gven. The th subsystem of (16) s ISS f there exst β KL, γ, γ u K {0} such that for every ntal condton x 0 = ξ and every nput u the correspondng soluton to the th subsystem of (16) exsts globally and satsfes x (t) max{β ( ξ [ θ,0], t), max, γ ( x [ θ,t] ), γ u ( u [0,t] )} (22) for all t 0. The th subsystem of (16) s unformly ISS over a gven class S of admssble sequences of mpulsve tmes f (22) holds for every sequence n S, wth functons β, γ and γ u that are ndependent of the choce of the sequence. Defnton 4.6. A locally Lpschtz contnuous functon V : R N R + s called an exponental ISS-Lyapunov- Razumkhn functon of the th subsystem of (16), f there exst functons V, = 1,..., n, whch are contnuous, proper and postve defnte and locally Lpschtz contnuous on R N \{0} and there exst γ u K {0}, γ K {0}, = 1,..., n and scalars c, d R, such that V (x ) max{max γ ( V d (x (t)) ), γ u ( u (t) )} D + V (x ) c V (x ), (23) V (x ) max{max γ ( V d (x (t)) ), γ u ( u (t) )} V (g (x 1,..., x n, u )) e d V (x ), (24) hold for all x = (x T 1,..., x T n ) T R N and u R M, where V d(x (t)) := V (x (t+τ)), τ [ θ, 0]. Furthermore we defne the gan-matrx Γ := (γ ) n n and the map Γ : R n + R n + by Γ(s) := (max γ 1 (s ),..., max γ n (s )) T, s R n +. Now we state our second man result: the ISS small-gan theorem for nterconnected mpulsve systems wth tmedelays. We construct the Lyapunov-Razumkhn functon and the gan of the overall system under a small-gan condton. The necessary ADT condton on the sze of the tme ntervals between mpulses s the same as n the delay-free case. 4.1 Man result Theorem 4.7. Assume each subsystem of (16) has an exponental ISS-Lyapunov-Razumkhn functon wth c, d R, d 0 and gans γ. Defne c := mn c and d := mn d. For arbtrary constants µ, λ > 0, let S[µ, λ] denote the class of mpulsve tme sequences {t k } of the whole system. If the followng holds ) S[µ, λ] satsfes condton (9), ) Γ = (γ ) n n satsfes the small-gan condton (12), then the whole system (17) s unformly ISS over S[µ, λ] and the exponental ISS-Lyapunov-Razumkhn functon s gven by V (x) := max { (V (x ))}, where σ = (σ 1,..., σ n ) T s a pecewse lnear Ω-path. The gans are gven by γ d (r) := max k, k (γ k(σ (r))), γ u (r) := max (γ u(r)). The proof goes along the lne of the proof of Theorem 3.8 wth accordng changes to tme-delay systems. Proof. Let 0 x = (x T 1,..., x T n ) T. We defne V (x) := max{ (V (x ))} and show that V s the exponental ISS-Lyapunov- Razumkhn functon for the overall system. V satsfes (19), whch can be easly checked. Note that V s locally Lpschtz contnuous. For any {1,..., n} consder open domans M R N \{0} defned as n (15). 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 the gans γ d (r) := max k, k (γ k(σ (r))), γ u (r) := max (γ u (r)), r > 0 and assume V (x) max{γ d ( V d (x) ), γ u ( u )}. Note that γ d (r) < r, by (11). It follows V (x ) σ (max{max k max k (γ k(σ ( V d (x) ))), (γ u ( u ))}) max{max γ ( V d (x) ), γ u ( u )}. Let [s l, s l+1 ], s l < s l+1, l = 0, 1,... be an nterval where σ s lnear,.e., σ (s) = a l s, s [s l, s l+1 ]. Then for all ntervals wth σ (s) = a l s and from (23) we obtan D + V (x) = D + 1 V (x ) 1 c V (x ) = c V (x), a l a l for almost all x and for all V (x ) [s l, s l+1 ]. By defnton of c := mn c the functon V satsfes (20). By defnton of d := mn d t holds V (g(x, u)) 1 e d V (x ) e d V (x), V [s l, s l+1 ] a l.e., V satsfes condton (21). All condtons of Defnton 4.2 are satsfed and V s the exponental ISS-Lyapunov-Razumkhn functon of the

6 whole system of the form (17). Usng that there exst µ, λ > 0 such that dn(t, s) c(t s) µ λ(t s), t s 0 we can apply Proposton 4.4 and the whole system s unformly ISS over S[µ, λ]. Remark 4.8. The meanng of c, d R s the same as n the delay-free case. If both dynamcs, the contnuous and the dscontnuous, are stable, whch means that c, d > 0, then the condton (9) n Theorem 4.7 s not necessary, accordng to Theorem 2 n Hespanha et al. [2008]. 5. CONCLUSIONS We have establshed condtons that guarantee nput-tostate stablty of a network of several mpulsve systems wth and wthout tme-delays. These condtons are based on the nterconnecton structure of the network (smallgan condton) and on the szes of tme ntervals between the mpulses (average dwell-tme condton). Our approach provdes a method of constructon of Lyapunov and Lyapunov-Razumkhn functons for the overall system. REFERENCES George Ballnger and Xnzh Lu. Exstence and unqueness results for mpulsve delay dfferental equatons. Dynam. Contn. Dscrete Impuls. Systems, 5(1-4): , Dfferental equatons and dynamcal systems (Waterloo, ON, 1997). C. Ca and Andrew R. Teel. Characterzatons of nputto-state stablty for hybrd systems. Systems Control Lett., 58:47 53, Wu-Hua Chen and We Xng Zheng. Bref paper: Inputto-state stablty and ntegral nput-to-state stablty of nonlnear mpulsve systems wth delays. Automatca (Journal of IFAC), 45(6): , Sergey Dashkovsky and Mchael Kosmykov. Stablty of networks of hybrd ISS systems. In Proceedngs of the 48th IEEE Conference on Decson and Control, Shangha, Chna, Dec , 2009, pages , Sergey Dashkovsky and Lars Nauok. ISDS small-gan theorem and constructon of ISDS Lyapunov functons for nterconnected systems. Accepted n Systems and Control Letters, DOI: /.sysconle Sergey Dashkovsky, Börn S. Rüffer, and Faban R. Wrth. An ISS small gan theorem for general networks. Math. Control Sgnals Systems, 19(2):93 122, Sergey Dashkovsky, Börn 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: Sergey N. Dashkovsky and Börn S. Rüffer. Local ISS of large-scale nterconnectons and estmates for stablty regons. Systems & Control Letters, 59(3-4): , Lars Grüne. Input-to-state dynamcal stablty and ts Lyapunov functon characterzaton. IEEE Trans. Automat. Control, 47(9): , Wassm M. Haddad, Vay Sekhar Chellabona, and Sergey G. Nersesov. Impulsve and hybrd dynamcal systems. Prnceton Seres n Appled Mathematcs. Prnceton Unversty Press, Prnceton, NJ, João P. Hespanha, Danel Lberzon, and Andrew R. Teel. Lyapunov condtons for nput-to-state stablty of mpulsve systems. Automatca J. IFAC, 44(11): , Hrosh Ito. State-dependent scalng problems and stablty of nterconnected ISS and ISS systems. IEEE Transactons on Automatc Control, 51(10): , 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 , 2009, pages , Z.-P. Jang, A. R. Teel, and L. Praly. Small-gan theorem for ISS systems and applcatons. Math. Control Sgnals Systems, 7(2):95 120, 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): , Iasson Karafylls and Zhong-Png Jang. A vector smallgan theorem for general nonlnear control systems, D. Nesc and A.R. Teel. A Lyapunov-based small-gan theorem for hybrd ISS systems. In Proceedngs of the 47th IEEE Conference on Decson and Control, 2008, Cancun, Mexco, Dec. 9-11, 2008, pages , P. Pepe and Z.-P. Jang. A Lyapunov-Krasovsk methodology for ISS and ISS of tme-delay systems. Systems Control Lett., 55(12): , B. S. Rüffer. Monotone nequaltes, dynamcal systems, and paths n the postve orthant of Eucldean n-space. Postvty, To appear, DOI: /s Börn Rüffer. Monotone dynamcal systems, graphs, and stablty of large-scale nterconnected systems. PhD thess, Fachberech 3 (Mathematk & Informatk) der Unverstät Bremen, Robert Shorten, Faban Wrth, Olver Mason, Ka Wulff, and Chrstopher Kng. Stablty crtera for swtched and hybrd systems. SIAM Rev., 49(4): , Eduardo D. Sontag. Smooth stablzaton mples coprme factorzaton. IEEE Trans. Automat. Control, 34(4): , Eduardo D. Sontag. Comments on ntegral varants of ISS. Systems Control Lett., 34(1-2):93 100, Eduardo D. Sontag and Yuan Wang. On characterzatons of the nput-to-state stablty property. Systems Control Lett., 24(5): , Eduardo D. Sontag and Yuan Wang. New characterzatons of nput-to-state stablty. IEEE Trans. Automat. Control, 41(9): , Andrew R. Teel. Connectons between Razumkhn-type theorems and the ISS nonlnear small gan theorem. IEEE Trans. Automat. Control, 43(7): , Aran van der Schaft and Hans Schumacher. An ntroducton to hybrd dynamcal systems, volume 251 of Lecture Notes n Control and Informaton Scences. Sprnger- Verlag London Ltd., London, 2000.