Stability of Switched Linear Systems on Cones: A Generating Function Approach

Transcription

1 49th IEEE Conference on Decson and Control December 15-17, 2010 Hlton Atlanta Hotel, Atlanta, GA, USA Stablty of Swtched Lnear Systems on Cones: A Generatng Functon Approach Jngla Shen and Jangha Hu Abstract Ths paper extends the recent study of the generatng functon approach to stablty analyss of swtched lnear systems from the Eucldean space to a closed convex cone. Examples of the latter class of swtched systems nclude swtched postve systems that model varous bologc and economc systems wth postve states. Strong and weak stablty notons are consdered n ths paper. In partcular, t s shown that asymptotc and exponental stablty are equvalent for both notons. Strong and weak generatng functons on cones are ntroduced and ther propertes are establshed. Necessary and suffcent condtons for strong/weak exponental stablty of swtched lnear systems on cones are obtaned n terms of the rad of convergence of strong/weak generatng functons. I. INTRODUCTION There has been a surgng nterest n swtched and hybrd systems and ther applcatons across a number of felds, such as engneerng, robotcs, and systems bology. A fundamental ssue n the analyss and desgn of swtched dynamcal systems are ther stablty [11], [14], [15]. Numerous technques have been proposed for the stablty analyss, e.g., the Lealgebrac approach [10] and the Lyapunov framework [3], [7]. In the vast lterature on swtched systems, swtched lnear systems have receved partcular attenton due to ther relatvely smple structure and yet rch dynamcal behavors. Recently ntroduced n [8], the generatng functons have been proven to be an effcent and unfed tool for studyng the exponental stablty of dscrete-tme swtched lnear systems. Roughly speakng, generatng functons are sutably defned power seres wth coeffcents determned from the systems traectores under certan swtchng polces. Ther rad of convergence characterze the exponental growth rates of the system traectores. Therefore, the exponental stablty of a swtched lnear system can be completely descrbed n terms of the rad of convergence of ts generatng functons. Furthermore, generatng functons are closely related to the value functons of properly defned optmal control problems and admt effcent numercal computaton. Ths allows one to develop effectve algorthms to determne the exponental growth rates, and n turn the exponental stablty, under dfferent swtchng polces, e.g., arbtrary swtchng, optmal swtchng, or random swtchng. The Jngla Shen s wth the Department of Mathematcs and Statstcs, Unversty of Maryland, Baltmore County, Baltmore, MD 21250, USA shen@umbc.edu. Ths research was supported by the NSF grant ECCS Jangha Hu s wth the School of Electrcal and Computer Engneerng, Purdue Unversty, West Lafayette, IN 47906, USA angha@purdue.edu. Ths research was partally supported by the NSF grant CNS generatng functon approach can also be extended to handle state-dependent swtchngs [13]. Most lterature on swtched systems concentrates on those on the Eucldean space. However, a varety of appled systems have ther states confned wthn certan regons. A promnent example s postve systems [5] that model a wde range of ndustral, bologcal, economc, and socal systems. Stablty of swtched postve systems and ther extenson,.e., swtched systems over cones, has also receved ncreasng attenton due to applcatons n such areas as communcaton and mult-agent systems; certan stablty tools have been studed, e.g. the common Lyapunov functon approach [2], [4], [6], [12]. In ths paper, we carry out the stablty analyss for swtched lnear systems on closed convex cones by usng the generatng functon approach. Specfcally, we consder the strong and weak stablty notons on a cone, and show that for both notons, the asymptotc and exponental stablty are equvalent. Analytc propertes of strong and weak generatng functons on a cone, as well as ther stablty mplcatons and numercal approxmatons, are establshed. The paper s organzed as follows. In Secton II, swtched lnear systems on cones are ntroduced and ther stablty notons are defned. In Secton III, the equvalence of the asymptotc and exponental stablty s proven. Sectons IV and V treat the strong and the weak generatng functons of swtched lnear systems on closed convex cones, respectvely. II. STABILITY OF SWITCHED LINEAR SYSTEMS ON CONES The dynamcs of a dscrete-tme autonomous swtched lnear system SLS) s gven by xt + 1) = A t) xt), t = 0, 1,.... 1) Here xt) R n s the state; {A 1,...,A m } s a set of subsystem dynamcs matrces; and t) M := {1,...,m} for all t, or smply, s the swtchng sequence. Gven the ntal state x0) = z, the traectory of the SLS under the swtchng sequence s denoted by xt; z, ). Let C be a closed convex cone. Throughout ths paper, we assume that the SLS 1) s postvely nvarant wth respect to C, namely, each subsystem defned by A satsfes A z C whenever z C. Ths assumpton ensures that a traectory xt; z, ) startng from z C wll reman n C at all subsequent tmes regardless of the swtchng sequence. Because of the postve nvarance assumpton, the restrcton of the SLS 1) on the cone C s a vald dynamcal system, whch we refer to as the SLS 1) on C. A partcular example of SLSs on cones s swtched postve systems, where C = /10/$ IEEE 420

2 R n + s the nonnegatve orthant of R n, and A, M, are all postve matrces. A. Stablty Notons of SLSs on Cones The stablty of the SLS 1) on the cone C can be defned as follows. Defnton 1: The SLS 1) on the cone C s called exponentally stable under arbtrary swtchng wth the parameters κ and r) f there exst κ 0 and r [0, 1) such that startng from any ntal state z C and under any swtchng sequence, the traectory xt; z, ) satsfes xt; z, ) κr t z for all t Z + ; exponentally stable under optmal swtchng wth the parameters κ and r) f there exst κ 1 and r [0, 1) such that startng from any ntal state z C, there exsts a swtchng sequence for whch xt; z, ) satsfes xt; z, ) κr t z, for all t Z +. Smlar to lnear systems, we can defne the notons of stablty n the sense of Lyapunov) and asymptotc stablty for the SLS on C under both arbtrary and optmal swtchngs. Due to the homogenety of the SLS, the local and global stablty notons are equvalent. For smplcty, we also refer to the stablty under arbtrary swtchng as strong stablty and stablty under optmal swtchng as weak stablty. In Defnton 1, by replacng the cone C wth R n, we obtan the correspondng notons of strong and weak exponental stablty for the orgnal SLS 1) on R n ). It s easly seen that stablty of the SLS on R n n any partcular sense such as strongly or weakly exponentally stable) mples that on C but not vce versa. As a result, the stablty study for SLSs on cones poses new challenges beyond that for SLSs on R n. Before endng ths secton, we brefly revew some basc notons of cones [1]. A cone C s called ponted f the condton that x x k = 0 wth x C, = 1,..., k, mples that x = 0 for all. A convex cone C s ponted f and only f C C) = {0}, or equvalently, f C does not contan a nontrval subspace. For example, R n + s ponted but the half space {x R n x 1 0} s not. A convex cone C can always be decomposed as C = K + V, where K s a ponted cone and V = C C) s a subspace called the lnearty space of C) orthogonal to K: K V. A cone C s sold f t has nonempty nteror. For example, R n + s sold and hence proper.e. closed, convex, sold, and ponted.) III. ASYMPTOTIC AND EXPONENTIAL STABILITY OF SLSS ON CONES It s well known that the notons of asymptotc stablty and exponental stablty are equvalent for lnear systems. In ths secton, we wll extend ths result to SLSs on cones, frst for strong stablty and then for weak stablty. The prevous result n [13] on the strong stablty of SLSs on R n then becomes a specal case of our proof. A. Equvalence of Strong Asymptotc and Exponental Stablty for SLSs on Cones To show the equvalence of asymptotc and exponental stablty for the SLS 1) on the cone C under arbtrary swtchng, we consder a more general settng: conewse lnear nclusons CLIs) on the cone C. SLSs on C are specal nstances of CLIs on C. Let Ξ := {X } l =1 be a fnte famly of nonempty closed cones whose unon s C, namely, l =1 X = C. Each X s nether necessarly polyhedral nor convex, and two cones n Ξ may overlap. Assocated wth each cone X s a lnear dynamcs x A x f x X, for some matrx A R n n postvely nvarant on C. The conewse lnear ncluson on C s the dynamcal system defned by: xt + 1) fxt)), t Z +. 2) Here, f : C C s the set-valued map defned by fx) := {A x for all such that x X }. Thus, at any tme t, each X whch the current state xt) = x belongs to offers a possble locaton A x to whch the state may evolve at the next step. Obvously, by settng l = m and X = C for all = 1,...,m, the CLI 2) on C reduces to the SLS 1) on C. Startng from an ntal state z C, denote by xt, z) a soluton traectory of the CLI 2). Due to the set-valued nature of the dynamcs, there are n general nfntely) many choces of xt, z). The local) stablty notons of the CLI 2) at the equlbrum pont x e = 0 are defned as follows. Defnton 2: At x e = 0, the CLI 2) on C s called strongly stable f, for each ε > 0, there s a δ ε > 0 such that xt, z) < ε, t Z +, for any traectory xt, z) startng from z C wth z δ ε ; strongly asymptotcally stable f t s strongly stable and there s a δ > 0 such that xt, z) 0 as t for any traectory xt, z) startng from z C wth z < δ; strongly exponentally stable f there exst δ > 0, κ 1, and r [0, 1) such that xt, z) κr t z, t Z +, for any xt, z) startng from z C wth z < δ. Due to homogenety of the dynamcs 2), the local and global stablty notons of the CLI 2) are equvalent. In other words, n the above defntons we can equvalently set δ =. In the Appendx, we shall prove the followng result. Theorem 1: The CLI 2) on C s strongly asymptotcally stable f and only f t s strongly exponentally stable. Snce the SLS 1) on C s a specal nstance of CLIs on C, Theorem 1 mples the followng. Corollary 1: The asymptotc and exponental stablty of the SLS 1) on C under arbtrary swtchng are equvalent. B. Equvalence of Weak Asymptotc and Exponental Stablty for SLSs on Cones It has been shown through a counter example n [13, Example 5] that weak asymptotc and weak exponental stablty are not equvalent for general CLIs on C or even on R n ). In ths subsecton, however, we establsh the equvalence of these two weak stablty notons for SLSs on cones. The underlyng reason for the dfference n these two cases, as evdenced by the followng proof, s that solutons to SLSs on C under a fxed swtchng sequence depend contnuously on ntal states, whle ths s not the case for CLIs on C. Theorem 2: The SLS 1) on C s weakly asymptotcally stable f and only f t s weakly exponentally stable. 421

3 Proof: It suffces to show that weak asymptotc stablty mples weak exponental stablty as the other drecton s trval. Assume that the SLS 1) on C s asymptotcally stable under optmal swtchng, namely, for any ntal state z C, the state traectory xt; z, ) 0 as t for at least one swtchng sequence. For each z C S n 1, where S n 1 := {z R n z = 1}, there exst a swtchng sequence z and a tme T z Z + such that xt z ; z, z ) 1 4. Snce under the fxed swtchng sequence z, the soluton xt; z, z ) at tme T z depends contnuously on the ntal state z, we can fnd a neghborhood U z of z n C S n 1 such that xt z ; y, z ) 1 2 for all y U z. The unon of all such neghborhoods, {U z z C S n 1 }, s an open coverng of the compact set C S n 1 ; hence there must exst a fnte sub-coverng: C S n 1 l =1 U z for some l < and z1,..., z l C Sn 1. The above obtaned fnte coverng enables us to construct a state-feedback swtchng polcy that leads to an exponentally convergng state traectory. To see ths, defne T := max T z. For any ntal state z C S n 1, the above argument mples that z U z for some 1 l. By our constructon, xt 1 ) := xt z ; z, z ) satsfes xt 1 ) 1 2. Assume wthout loss of generalty that xt 1 ) 0. Then xt 1 )/ xt 1 ) U z for some 1 l, and as a result, xt 2 ) := xt z ; xt 1 ), z ) satsfes xt 2 ) 1 2 xt 1). Repeatng ths process nductvely, we obtan a swtchng sequence z concatenated by z, z,... and a sequence of tmes 0 = T 0 T 1 T 2 wth at most T between successve ones such that the resultng traectory xt; z, z ) satsfes xt k+1 ; z, z ) 1 2 xt k; z, z ) for all k. Let κ := T =0 max M A ). Then t s easly seen that xt; z, z ) κ0.5) t/t 1 z for all t Z +. Snce nether κ nor T depends on z, the SLS 1) on C s exponentally stable under optmal swtchng. Remark 1: We call the SLS 1) on C weakly convergent f for any z C, a swtchng sequence z exsts such that xt; z, z ) 0 as t. Ths condton seems weaker than weak asymptotc stablty as weak Lyapunov stablty s not requred. However, the proof of Theorem 2 essentally shows that weak convergence s equvalent to weak exponental thus asymptotc) stablty. Ths observaton wll be exploted n Theorems 5 and 6 n Secton V. IV. STRONG GENERATING FUNCTIONS OF SLSS ON CONES A. Strong Generatng Functons of SLSs on R n In [8], the noton of strong generatng functons s proposed to study the exponental stablty under arbtrary swtchng of SLSs. The strong generatng functon of the SLS 1) on R n s the map G : R + R n R + { } defned as follows: for each z R n and λ 0, G λ z) := Gλ, z) := sup λ t xt; z, ) q, 3) where the supremum s taken over all the possble swtchng sequences, q s a postve nteger, and s an arbtrary norm on R n. The radus of convergence of the strong generatng functon on R n s defned as λ R n := sup{λ 0 G λz) <, z R n }. 4) The followng result s proved n [8]. Theorem 3: The SLS 1) on R n s exponentally stable under arbtrary swtchng f and only f λ R n > 1. Thus the radus of convergence of the strong generatng functon on R n fully characterzes the strong exponental stablty of the SLS on R n. B. Strong Generatng Functons of SLSs on Cones For the closed convex cone C, defne W to be the smallest subspace of R n nvarant wth respect to {A } M contanng C, or equvalently, the set of all vectors generated from elements of C through repeated operatons of multplcaton by matrces n {A 1,..., A m } and lnear combnatons: { } W := span C, M A C,, M A A C,.... In partcular, f C s sold, then W = R n. If C s polyhedral,.e. t s fntely and postvely) generated such that C = { l k=1 α k v k α 0} for some vectors v k R n, k = 1,, l, then W = span{{v k } l k=1, MA {v k } l k=1,, MA A {v k } l k=1,...}. Note that C W R n form a cascade of sets nvarant wth respect to {A } M. Hence, the SLS 1) restrcted to each set s well defned and the defnton of the generatng functon n 3) can be extended to C and W as well. In partcular, the strong generatng functon of the SLS 1) on the cone C s defned as, for λ 0, G λ z) := sup λ t xt; z, ) q, z C. 5) Here, the same notaton G λ ) s used as n 3) as 5) s exactly the restrcton of 3) on C. For ths reason, we smply refer to 5) as the strong generatng functon on C. Smlarly, we can defne G λ ) on W as the restrcton of 3) on W. Defne the rad of convergence of the strong generatng functons on C and W respectvely as λ C := sup{λ 0 G λz) <, z C}, λ W := sup{λ 0 G λ z) <, z W}. For each λ 0, defne the three subsets G λ C) := {z C G λ z) < } C, G λ W) := {z W G λ z) < } W, G λ R n ) := {z R n G λ z) < } R n, whch satsfy G λ C) G λ W) G λ R n ), and G λ C) = G λ R n ) C, G λ W) = G λ R n ) W. 6) These sets wll be useful n the next subsecton. 422

4 C. Propertes of Strong Generatng Functons Obtaned through restrcton, the strong generatng functons on C and W nhert many of the propertes of ther counterpart on R n establshed n [8], as lsted below. Proposton 1: For any q N and any vector norm, the strong generatng functons G λ z) of the SLS 1) on C and W have the followng propertes. 1. Bellman Equaton): For all λ 0 and z C or W), G λ z) = z q + λ max M G λa z). 2. Sub-addtvty): For each λ 0, we have Gλ z 1 + z 2 ) ) 1/q Gλ z 1 ) ) 1/q + Gλ z 2 ) ) 1/q, for all z 1, z 2 C or W). 3. Convexty): For each λ 0, the functon G λ z) ) 1/q s convex on C or W). 4. Invarant Cone): Let λ 0 be arbtrary. The set G λ C) s a closed convex cone n C nvarant wth respect to {A } M. Partcularly, f C s polyhedral, so s G λ C). 5. Invarant Subspace): Let λ 0 be arbtrary. The set G λ W) s a subspace of W nvarant wth respect to {A } M. 6. For 0 λ < max M A q ) 1, where the matrx norm s nduced from the vector norm, G λ z) s fnte everywhere on C. Proof: 1. Ths follows drectly from the dynamc programmng prncple. 2. For any z 1, z 2 C, z 1 + z 2 C as C s a convex cone. Then, by defnton, G λ z 1 + z 2 ) = sup λ t xt; z 1, ) + xt; z 2, ) q sup sup λ t ) q xt; z 1, ) + xt; z 2, ) [ ) 1/q λ t xt; z 1, ) q ) 1/q ] q + λ t xt; z 2, ) q [ Gλ z 1 ) ) 1/q + Gλ z 2 ) ) ] q 1/q, where the second nequalty s due to the Mnkowsk nequalty. The case for W s entrely smlar. 3. Ths s due to the subaddtvty and the postve homogenety of G λ z) ) 1/q. 4. The conc property and the convexty of G λ C) follow from the postve homogenety and the convexty of Gλ z) ) 1/q, respectvely. The nvarance wth respect to {A } M s a consequence of the Bellman equaton. To show that G λ C) s a closed convex cone, we note that G λ C) = G λ R n ) C, and G λ R n ) s easly shown to be a subspace [8]. Thus, G λ C) as the ntersecton of the convex cone C and a subspace s convex and closed, and s polyhedral whenever C s polyhedral. 5. G λ W) = G λ R n ) W s evdently a subspace. 6. The proof s smlar to that n [8], hence omtted. Besdes the above nherted propertes, the strong generatng functons on C and W also have some other shared propertes. Obvously, the former s the restrcton of the latter on the cone C. Less obvously, we have the followng. Proposton 2: For any λ 0, the strong generatng functons G λ z) of the SLS 1) on C and W satsfy: G λ z) <, z C G λ z) <, z W. Proof: It suffces to show drecton. Suppose G λ ) s fnte on C,.e., G λ C) = C. Snce G λ R n ) s a subspace of R n nvarant wth respect to {A } M and contans G λ C), hence C, t must also contan W, as W s the smallest nvarant subspace contanng C. In other words, W G λ R n ). Hence G λ z) s fnte for all z W. As a result, the rad of convergence of the strong generatng functons on C, W, and R n have the followng relaton. Corollary 2: λ C = λ W λ Rn. In partcular, f C s sold, then λ C = λ W = λ R n. Proposton 3: The strong generatng functons G λ z) of the SLS 1) on C and W have the followng propertes. 1. If λ [0, λ C ) hence G λz) < for all z C), then there exsts a constant c [1, ) such that z G λ z) ) 1/q c z, z W. 2. Relatve Lpschtz Property) Let λ [0, λ C ). Then Gλ z) ) 1/q s relatvely Lpschtz on W thus on C),.e., there exsts L > 0 such that for any x, y W, G λ x) ) 1/q Gλ y) ) 1/q L x y. Proof: 1. The frst nequalty s obvous as G λ z) z q follows drectly from the defnton. To show the second nequalty, by homogenety, t suffces to show that Gλ z) ) 1/p c, z W S n 1, for some constant c 1. Let {u } l =1 be a bass of W. Snce W Sn 1 s bounded, we can fnd γ > 0 such that for each z W S n 1, there exsts a unque real tuple {α 1,, α l } satsfyng z = l =1 α u and l =1 α < γ. Therefore, by vrtue of the subaddtvty and postve homogenety of G λ z) ) 1/q, we conclude that G λ z) ) 1/q c := γ l =1 Gλ u ) ) 1/q for all z W S n 1. It s easy to verfy that c It follows from the subaddtvty of G λ z) ) 1/q on W that for any x, y W, Gλ x) ) 1/q Gλ y) ) 1/q Gλ x y) ) 1/q. Swtchng x and y, we have Gλ y) ) 1/q Gλ x) ) 1/q Gλ y x) ) 1/q. Combnng the above two nequaltes, we obtan G λ x) ) 1/q Gλ y) ) 1/q Gλ x y) ) 1/q c x y, where the last step s due to x y) W and the frst property. 423

5 Remark 2: By the results of ths subsecton, G λ C) s a closed convex sub-cone of C: suppose C admts the decomposton C = K + V where K s a ponted cone and V s a subspace, then G λ C) = K λ + V λ wth K λ K a ponted cone and V λ V a subspace. As λ ncreases, G λ wll ncrease, hence the nvarant subsets G λ C), G λ W), and G λ R n ) wll shrnk. In partcular, f C s not ponted.e., V {0} s nontrval), then as λ ncreases, G λ C) wll change from non-ponted to ponted, or equvalently, V λ wll shrnk to {0}, at exactly λ V := nf{λ 0 G λz) =, z V}. D. Strong Exponental Stablty Characterzaton The rad of convergence of the strong generatng functons characterze the strong exponental stablty of the SLS on C, as stated by the followng theorem. Theorem 4: The followng are equvalent: 1. the SLS 1) on the cone C or on the subspace W) s exponentally stable under arbtrary swtchng; 2. λ C = λ W > 1; 3. G 1 z) s fnte for all z C or W). Proof: In vew of Corollary 1 and Proposton 3, the proof s essentally the same as that of [8, Theorem 1]. E. Numercal Computaton of Strong Generatng Functons The algorthm developed n [8] for computng strong generatng functons on R n can be extended to compute those on closed convex cones. We brefly dscuss ths n ths secton. For any λ [0, λ C ), defne g λ := sup z C, z =1 G λ z). Moreover, defne the followng functons that approxmate the strong generatng functon G λ z) on C: G k λ z) := max k λt xt; z, ) q, z C. It s easy to see that G k λ z) satsfes Gk λ z) = z q + λmax M G k 1 λ A z), z C, wth G 0 λ z) = z q. Ths yelds a recursve procedure to compute these functons. Applyng Propostons 1 and 3 as well as the smlar argument as n [8, Proposton 6], we have the followng. Proposton 4: The functons G k λ z) satsfy 1) G 0 λ z) G1 λ z) G λz), λ 0, z C. 2) G k λ z) G λz) g λ 1 1/g λ ) k+1 z q, k Z +, z C, for any λ [0, λ C ). These results show that the sequence of functons {G k λ } converges unformly and exponentally fast to G λ and as such can provde numercal approxmatons of the latter. To effcently mplement ths numercal procedure, an overapproxmaton can be developed usng the convex, conc structure of C; we refer the nterested reader to Algorthm 1 n [8] for further detals. V. WEAK GENERATING FUNCTIONS OF SLSS ON CONES Smlar to the strong generatng functons, weak generatng functons can be defned to address weak asymptotc/exponental stablty of SLSs on cones,.e. the stablty of the SLSs on cones under optmal swtchng. Specfcally, for the SLS 1) on the closed convex cone C, defne ts weak generatng functon H : R + C R + { } as H λ z) := Hλ, z) := nf λ t xt; z, ) q, 7) where λ 0, z C, and the nfmum s over all swtchng sequences of the SLS on C. In addton, q N, and s an arbtrary norm n R n. The radus of convergence for the weak generatng functon of the SLS on C s defned as λ C := sup{λ 0 H λ z) <, z C}. Proposton 5: For any q N and any vector norm, the weak generatng functon H λ z) of the SLS 1) on C has the followng propertes. 1. Bellman Equaton): For any λ 0 and z C, H λ z) = z q + λ mn M H λa z). 2. Invarant Cone): For any λ 0, the set H λ C) := {z C H λ z) = } s a cone n C not contanng 0. Further, H λ C) s nvarant wth respect to {A } M,.e. A H λ C) H λ C), M. 3. For 0 λ < mn M A q ) 1, where the matrx norm s nduced from the vector norm, H λ z) s fnte everywhere on C. Proof: The proofs of these propertes are smlar to those n [8] for the correspondng propertes of the weak generatng functon on R n ; hence they are omtted. The subsequent theorem shows two mportant results: ) As λ ncreases, λ C s the exact value at whch H λ z) starts to have the nfnte value; ) f for some λ 0, the weak generatng functon H λ ) s fnte everywhere on C, then t must be bounded by a homogeneous functon c z q unformly on C. Theorem 5: For each λ 0, the followng are equvalent: a) H λ z) c z q, z C, for some constant c > 0 generally dependent on λ); b) H λ z) < for all z C; c) λ [0, λ C ). Proof: It s obvous that a) b) and c) b). We shall show b) a) and b) c) as follows, whch leads to the equvalence of the three statements. To prove b) a), consder λ 0 such that b) holds. Then for any z C, there exsts a swtchng sequence z such that λt xt; z, z ) q <. Defne à := λ 1/q A, M. For the ntal state z C and the swtchng sequence z, let xt; z, z ) denote the traectory from z under z wth the dynamcs matrces A n the correspondng xt; z, z ) replaced by Ã. Then, xt; z, z ) q = λ t xt; z, z ) q for all t Z +. Snce xt; z, z) q < for each z C, xt; z, z ) 0 as t. In other words, the SLS defned by subsystem dynamcs matrces {Ã} M s weakly convergent on C. We thus deduce from Remark 1 that t s weakly exponentally stable on C. Thus there exst κ > 0 and ρ 0, 1) such that xt; z, z ) q κρ t z q, t Z + for each z C. Hence H λ z) λ t xt; z, z ) q = κ 1 ρ z q, z C. xt; z, z ) q 424

6 Lettng c := κ/1 ρ) yelds a). Further, c) a) holds. To show b) c), t suffces to show that H λ C z) = for some z C. To ths end, defne h λ := sup z C, z =1 H λ z), λ [0, λ C ). It follows from the mplcaton c) a) and a smlar argument as n [8, Proposton 9] that for any λ 0, λ C ), λ/1 1/h λ) λ C, or equvalently, 0 1/h λ 1 λ/λ C. Therefore, h λ tends to nfnty as λ λ C. Ths n turn mples that H λ C z) = for some z C, snce otherwse, by usng b) a), we would have h λ C c for some c > 0, a contradcton to the fact that h λ as λ λ C. The next result shows that the radus of convergence λ C characterzes the weak exponental stablty of the SLSs on C. Theorem 6: The SLS 1) on C s exponentally stable under optmal swtchng f and only f λ C > 1. Proof: The proof for necessty s straghtforward. To show suffcency, suppose λ C > 1. Thus for λ = 1, H 1 z) s fnte for any z C. Ths mples that the SLS on C s weakly convergent and thus weakly exponentally stable, n vew of Remark 1 and Theorem 2. Usng Theorem 6, t can be shown as n [8] that Corollary 3: For any r > λ C ) 1/q, there exsts κ r > 0 such that for any z C, there exsts a swtchng sequence z such that xt; z, z ) κ r r t z, t Z +. Smlar numercal approxmatons for the weak generatng functon on C can be obtaned; see [8] for more detals. VI. APPENDIX: PROOF OF THEOREM 1 The followng techncal lemma s easy to show and ts proof s omtted. Lemma 1: Let {S } l =1 be a fnte famly of closed sets n R n and let S := l =1 S. For any x S, there exsts a neghborhood U of x such that U S) Ix )S, where the ndex set Ix ) := { x S }. Assume that the CLI 2) s strongly stable on C. Then for any gven r > 0, there exsts δ > 0 such that xt, x 0 ) < r, t Z +, for any traectory xt, x 0 ) startng from x 0 C wth x 0 δ. Proposton 6: If the CLI 2) s asymptotcally stable on C, then for the δ > 0 obtaned above and a gven c 0, 1), there s T δ, c Z + dependng on δ and c only) such that [x 0 C wth x 0 δ] xt, x 0 ) c δ, t T δ, c for any xt, x 0 ) startng from x 0. Proof: For the gven δ > 0 and a gven c 0, 1), suppose the proposton fals. Then there exst an ntal state sequence {x 0 k } B δ C, the correspondng traectores {xt, x 0 k )}, and a strctly ncreasng tme sequence {t k } Z + wth lm t k = such that xt k, x 0 k ) > c δ. k Furthermore, t follows from the stablty of x e = 0 that a postve scalar µ exsts wth µ < δ), along wth the postve scalar r > δ, such that ) xt, x 0 k ) r, t Z +, for all k; ) x 0 B µ C) xt, x 0 ) c δ, t Z +. By ) and the sem-group property, we have xt, x 0 k ) µ for all t {0, 1,, t k }. Snce µ x 0 k δ for all k and C s closed, there exsts a subsequence of {x 0 k } convergng to x 0 C wth µ x 0 δ. Wthout loss of generalty, let {x 0 k } be that subsequence convergng to x0. In vew of )- ) and the constructon of {t k }, we see that the sequence {x1, x 0 k )} k t 1 C satsfes µ x1, x 0 k ) r for all k t 1. Thus t has a subsequence convergng to x 1 C wth µ x 1 r. By Lemma 1, a neghborhood N of x 0 can be found such that N C) Ix 0 )X. Note that x1, x 0 k ) = A x 0 k for some and x0 k N for all large k. Furthermore, snce the ndex set Ix 0 ) s fnte, we deduce that there exst a subsequence {x1, x 0 k )} of {x1, x0 k )} k t 1 and an ndex 1 Ix 0 ) such that x1, x0 k ) = A 1 x 0 k for all k wth x1, x 0 k ) x1 and x 0 k x0. Ths shows that x 1 = A 1 x 0. Recallng 1 Ix 0 ), we have x 1 fx 0 ). Repeatng ths argument and usng nducton, we obtan {x t } t Z+ C such that ) µ x t r for all t Z + ; ) for each t Z +, x t+1 fx t ). Ths shows that the traectory xt, x 0 ) = {xt } t Z + s such that xt, x 0 ) µ, t Z +. Ths contradcts the assumpton of asymptotc stablty of the CLI on C. 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