Stability Criteria via Common Non-strict Lyapunov Matrix for Discrete-time Linear Switched Systems

Stability Criteria via Common Non-strict L yapunov Mat rix for Discrete-time Linear Switched Systems Xiongpin g Dai a , Y u Huang b , Mingqin g Xiao c a Department of Mathemat ics, Nanjing U niver sity , Nanjing 210093, P eo ple’ s Republic of China b Department of Mathemat ics, Zhongshan (Sun Y at-Sen) University , Guangzhou 510275, P eople’ s R epubli c of China c Department of Mathemat ics, Southern Illinois Universit y , Carbondale , IL 62901-4408, USA Abstract Let S = { S 1 , S 2 } ⊂ R d × d have a comm on, but not necessarily strict, L y apunov matrix (i.e. there exists a symmetric positive-definite matrix P such th at P − S T k PS k ≥ 0 for k = 1 , 2). Based on a splitting th eorem o f the state space R d (Dai, Huang and X iao, ar Xiv:1107.0132 v1[math.PR]), we establish sev e ral stability criteria for the discrete-time linear switched dynamics x n = S σ n · · · S σ 1 ( x 0 ) , x 0 ∈ R d and n ≥ 1 governed b y the s witching signal σ : N → { 1 , 2 } . More s pecifically , let ρ ( A ) stand for th e spectral radius of a matrix A ∈ R d × d , then the outlin e o f r esults obtain ed in th is paper are: (1) For the case d = 2, S is absolu tely stab le (i.e ., k S σ n · · · S σ 1 k → 0 driven b y all switching signals σ ) if and o nly if ρ ( S 1 ) , ρ ( S 2 ) and ρ ( S 1 S 2 ) all are less tha n 1; (2) For th e case d = 3, S is ab solutely stable if an d on ly if ρ ( A ) < 1 ∀ A ∈ { S 1 , S 2 } ℓ for ℓ = 1 , 2 , 3 , 4 , 5 , 6, and 8 . This further implies that for a ny S = { S 1 , S 2 } ⊂ R d × d with the generalized spectral rad ius ρ ( S ) = 1 whe re d = 2 or 3, if S has a common, but not strict in g eneral, L y apunov matrix, then S po ssesses the spe ctral finiteness prop erty . K eywor ds: Linear switched / inclusion dynamics, non-strict L y apunov matrix, asymptotic stability, finiteness prop erty 2010 MSC: 93D20, 37N35 1. Introduction 1.1. Motivations Let R d × d be the standar d topolog ical space of all d -by- d real matrices wh ere 2 ≤ d < + ∞ , and for any A ∈ R d × d , by ρ ( A ) we denote the spectral radius of A . In addition, we identify A with ✩ Project was support ed partly by Nati onal Natural Science Foundatio n of China (Grant Nos. 11 071112 and 11071263), the NSF of Guangdong Provinc e and in part by NSF 0605181 and 1021203 of the United States. Email addr esses: xpdai@nju.edu. cn (Xiongping Dai), stshyu@mail. sysu.edu. cn (Y u Huang), mxiao@ma th.siu.ed u (Mingqing Xiao) Prep rint submitted to xxx Nove m ber 20, 2018 1 INTR ODUCTI ON 2 its induced o perator A ( · ) : x 7→ A x for x ∈ R d . Let S = { S 1 , . . . , S K } ⊂ R d × d be a finite set with 2 ≤ K < + ∞ . W e consider the s tability and stabilization of the linear inclusion / contro l dyn amics x n ∈ { S 1 , . . . , S K } ( x n − 1 ) , x 0 ∈ R d and n ≥ 1 . (1.1) As in [ 12 , 10 ], we denote by Σ + K the set of a ll ad missible co ntrol signals σ : N → { 1 , . . . , K } , equippe d with the stan dard pr oduct topo logy . Here and in the sequel N = { 1 , 2 , . . . } and for any σ ∈ Σ + K we will simply write σ ( n ) = σ n for all n ≥ 1. For any input ( x 0 , σ ), wher e x 0 ∈ R d is an initial state an d σ = ( σ n ) + ∞ n = 1 ∈ Σ + K a c ontrol (switching) signal, there is a unique outpu t h x n ( x 0 , σ ) i + ∞ n = 1 , called an orbit of the system ( 1.1 ), which correspo nds to the unique solution of the discrete-time linear switched dyn amics x n = S σ n · · · S σ 1 ( x 0 ) , x 0 ∈ R d and n ≥ 1 (1.2) driven / governed by the switching signal σ . Then a s usual, S is c alled (asympto tically) stable driven by σ if lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k = 0 ∀ x 0 ∈ R d ; or equiv alently , k S σ n · · · S σ 1 k → 0 as n → + ∞ . S is said to be a bsolutely stable if it is stable driven b y all switchin g signals σ ∈ Σ + K ; see, e.g ., [ 16 ]. W e note that the stability of S is indepen dent of the norm k · k u sed here. It is a well-kn own fact that if each mem ber S k of S shares a commo n L y apunov matrix; i.e., there exists a symmetric positiv e -definite matrix Q ∈ R d × d such that Q − S T k QS k > 0 (1 ≤ k ≤ K ) , then S is absolutely stable. Here T stands for the tran spose operator of matr ices or vectors. An essentially weak con dition is that each mem ber S k of S shares a common , “but not nece ssarily strict, ” L yap unov matrix; that is, there exists a symmetric positi ve-definite matrix P such that P − S T k PS k ≥ 0 , 1 ≤ k ≤ K . (1.3a) Here “ A ≥ 0” means x T A x ≥ 0 ∀ x ∈ R d . Associate d to the weak L yapu nov m atrix P as in ( 1.3a ), we define the vector norm on R d as k x k P = √ x T P x ∀ x ∈ R d . (1.3b) (W e a lso write its indu ced operato r / matrix no rm on R d × d as k · k P .) Then, k S k k P ≤ 1 for all 1 ≤ k ≤ K . Condition ( 1.3a ) is both practically importan t and academically challengin g, for example, [ 20 , 1 , 18 , 2 , 25 ] for the contin uous-time case an d [ 1 6 ] for discrete case. Indeed, it is desirable in many practical issues and is closely r elated to periodic solutio ns and limit cycles, see, e.g., [ 5 , 6 ] and [ 22 , Proposition 18]; in additio n, if S k , 1 ≤ k ≤ K , are p aracontra cti ve (i.e., x T S T k S k x ≤ x T x for all x ∈ R d , and “ = ” h olds if and o nly if S k ( x ) = x , see, e.g. , [ 24 ]), then condition ( 1.3a ) holds. In this pap er , we will stud y the stability of S that satisfies conditio n ( 1.3a ). Even under condition ( 1.3a ), the s tability of e very subsystems S k does not implies the absolute stability of S , as shown by Ex ample 6.6 constructed in Section 6 . So, ou r stability cr iteria — The orems A, B, C, and D — established in this paper, are nontrivial. 1 INTR ODUCTI ON 3 1.2. Stability driven by nonchao tic swi tching signals Under condition ( 1.3a ), in [ 3 ] f or the continuou s-time case, Balde an d J ouan p rovided a large class o f switching signals for which a large class of switche d systems ar e stable, by conside ring nonch aotic in puts and the geometry of ω -limit sets of the matrix sequences h S σ n · · · S σ 1 i + ∞ n = 1 . Recall f rom [ 3 , Defin ition 1 ] that a switchin g sign al σ = ( σ n ) + ∞ n = 1 ∈ Σ + K is said to be non- chaotic , if to any sequence h n i i i ≥ 1 ր + ∞ and any m ≥ 1 there correspo nds some in teger δ with 2 ≤ δ ≤ m + 1 such that ∀ ℓ 0 ≥ 1, ∃ ℓ ≥ ℓ 0 so th at σ is con stant restricted to some su binterval of [ n ℓ , n ℓ + m ] of length gr eater th an or equal to δ . A switching signal σ ∈ Σ + K is said to be generic [ 16 ] ( or r egular in [ 3 ]) if eac h a lphabet in { 1 , . . . , K } app ears in finitely many times in the sequence σ = ( σ n ) + ∞ n = 1 . Then our first stability criterion obtained in this paper can be stated as follows: Theorem A. Let S = { S 1 , . . . , S K } ⊂ R d × d satisfy c ondition ( 1.3a ) with ρ ( S k ) < 1 for all 1 ≤ k ≤ K . Then k S σ n · · · S σ 1 k → 0 as n → + ∞ for any non chaotic switching signal σ = ( σ n ) + ∞ n = 1 ∈ Σ + K . W e note that in Theo rem A, if σ is additionally generic ( regular), then this statement is a direct con sequence of [ 3 , The orem 3 ]. Howe ver , withou t the genericity of σ , here we ne ed to explore an essential pro perty of a n onchao tic switchin g signal; see Lemma 2.1 b elow . In the case of d = 2 and K = 2, an ergodic version o f Th eorem A will be stated in Corollary 5.3 in Section 5 . As is shown by Example 6.6 mentione d befor e, un der the assumption of T heorem A , one cannot expect the stability of S dri ven by a n arbitrary switching signal. 1.3. A splitting theorem d riven by r ecurrent signals Next, we consid er an other type of switchin g signal — rec urrent switching sign al, which doe s not need to be no nchaotic and balanced and which seems m ore general fro m th e viewpoint o f ergodic theory . In fact, all recurren t switchin g signals form a set of total measure 1. Correspon ding to a switching signal σ = ( σ n ) + ∞ n = 1 ∈ Σ + K , for the s ystem S we defin e two importan t s ubspaces of the state space R d : E s ( σ ) =  x 0 ∈ R d : k S σ n · · · S σ 1 ( x 0 ) k P → 0 as n → + ∞  and E c ( σ ) = n x 0 ∈ R d : ∃ h n i i + ∞ i = 1 ր + ∞ such tha t lim i → + ∞ S σ n i · · · S σ 1 ( x 0 ) = x 0 o ; called, respectiv e ly , th e st able and central manifolds of S d riv en by σ . Here E s ( σ ) and E c ( σ ) are indeed independ ent of the norm k · k P . A switching signa l σ = ( σ n ) + ∞ n = 1 ∈ Σ + K is called recurr ent under the classical one-sided Mar kov shift tran sformation , θ : σ ( · ) 7→ σ ( · + 1) , of Σ + K , if for any ℓ ≥ 1 there exists some m su ffi ciently large such that ( σ 1 , . . . , σ ℓ ) = ( σ 1 + m , . . . , σ ℓ + m ) . W e have then, for S , the fo llowing important sp litting theorem of the state space R d based on a recurren t swit ching signal: 1 INTR ODUCTI ON 4 Splitting Theorem ([ 13 ]) . Let S = { S 1 , . . . , S K } ⊂ R d × d satisfy con dition ( 1 .3a ). Then , for any r ecurrent sw itching signal σ ∈ Σ + K it holds R d = E s ( σ ) ⊕ E c ( σ ) and S σ 1 ( E s / c ( σ )) = E s / c ( σ ( · + 1 )) . This the orem is a special ca se of a more general result [ 13 , Th eorem B ′′ ]. So in this case, if the central man ifold E c ( σ ) = { 0 } then S is stable d riv en by the re current switching signal σ . T his splitting is in fact unique under the L y apunov norm k · k P . 1.4. Almost sur e stability Under condition ( 1.3a ), let K k·k P ( S k ) = { x ∈ R d : k S k ( x ) k P = k x k P } for 1 ≤ k ≤ K . W e n ote that if k S k k P < 1 then K k·k P ( S k ) = { 0 } . Next, using the above splitting theore m, we can obta in the following almost sure stab ility criterion: Theorem B. Let S = { S 1 , S 2 } ⊂ R d × d satisfy ( 1.3 a ) a nd K k·k P ( S 1 ) ∩ K k·k P ( S 2 ) = { 0 } , wher e d = 2 or 3 . Then, if P is a non- atomic ergodic pr ob ability measur e of the one-sided Ma rkov shift transformation θ : Σ + 2 → Σ + 2 defined by σ ( · ) 7→ σ ( · + 1) , ther e holds k S σ n · · · S σ 1 k P → 0 a s n → + ∞ for P -a.e. σ ∈ Σ + 2 . W e consider a simple example. Let S = { S 1 , S 2 } with S 1 = diag( 1 2 , 1 2 ) and S 2 = diag(1 , 1 ). Then, K k·k 2 ( S 1 ) = { 0 } and K k·k 2 ( S 2 ) = R 2 , where k · k 2 stands fo r the usual E uclidean n orm. So, K k·k P ( S 1 ) ∩ K k·k P ( S 2 ) = { 0 } . Clearly , S is no t absolutely stable. This shows th at unde r th e situatio n of Theorem B, it is necessary to consider the almost sure stability . 1.5. Absolute stability and finiteness pr ope rty For absolute stability , we can obtain the following two criteria T heorems C and D, which show t he stability is decidab le in t he cases of d = 2 , 3 under condition ( 1.3a ). Theorem C. Let S = { S 1 , S 2 } ⊂ R 2 × 2 satisfy con dition ( 1.3a ). Then, S is absolutely stable if an d only if ρ ( A ) < 1 for a ll A ∈ { S 1 , S 2 } ℓ for ℓ = 1 , 2 . Theorem D. Let S = { S 1 , S 2 } ⊂ R 3 × 3 satisfy condition ( 1.3a ). Then, S is absolutely sta ble if and only if ρ ( A ) < 1 for a ll A ∈ { S 1 , S 2 } ℓ for ℓ = 1 , 2 , 3 , 4 , 5 , 6 , and 8 . On the o ther h and, the accurate computation of the gene ralized spectral radius of S , intro- duced by Daubech ies and Lagarias in [ 15 ] as ρ ( S ) = lim n → + ∞ max σ ∈ Σ + K n p ρ ( S σ n · · · S σ 1 )  = sup n ≥ 1 max σ ∈ Σ + K n p ρ ( S σ n · · · S σ 1 )  , is very impor tant fo r many subjects. If one can find a finite-length word ( w 1 , . . . , w n ) ∈ { 1 , . . . , K } n for some n ≥ 1, which realizes ρ ( S ), i.e., ρ ( S ) = n p ρ ( S w n · · · S w 1 ) , then S is said to h av e the spectral finiteness pr operty . A b rief survey for some recent pro gresses regarding the finiteness property can be found in [ 14 , § 1.2]. 2 SWITCHED SYSTEMS DRIVEN BY NONCHA OTIC SWITCHING SIGN ALS 5 Under condition ( 1.3a ), we have ρ ( S ) ≤ 1. If ρ ( S ) < 1 the n S is absolutely stable; see, e .g., [ 16 ]. If ρ ( S ) = 1 then k · k P is just an extremal n orm f or S (see [ 4 , 28 , 9 ] f or more details). In [ 16 ], Gurvits proved that if S h as a poly tope 1 extremal norm on R d , then it has the spectral fin iteness proper ty . Howe ver, the L y apunov norm k · k P defined as in ( 1.3 b ) does not need to be a poly tope norm, for e xample, P = I d the identity m atrix which is associated with the usual Euclidean n orm k · k 2 on R d . As a co nsequen ce o f the statements of Theorems C and D, we can easily obtain the following spectral finiteness result. Corollary . Let S = { S 1 , S 2 } ⊂ R d × d satisfy conditio n ( 1.3 a ) with ρ ( S ) = 1 . Then the fo llowing two statements hold. (1) F or th e case d = 2 , ther e follows 1 = max  ρ ( S 1 ) , ρ ( S 2 ) , √ ρ ( S 1 S 2 )  . (2) In the case d = 3 , there h olds 1 = max  n p ρ ( S w n · · · S w 1 ) | w ∈ { 1 , 2 } n , n = 1 , 2 , 3 , 4 , 5 , 6 , 8  . Pr oof. Let d = 2. Assume max  ρ ( S 1 ) , ρ ( S 2 ) , √ ρ ( S 1 S 2 )  < 1 . Then Th eorem C implies that S is absolutely stable and s o ρ ( S ) < 1, a contradiction. Similarly , we can prove th e statement in the case d = 3. It should be poin ted o ut tha t if ρ ( S ) < 1 , then ρ ( S ) does no t need to b e attained by these maximum values defined as in the abov e corollary . 1.6. Outline The p aper is organized as follows. W e shall p rove Th eorem A in Section 2 . In fact, we will prove a more general result (Theorem 2.3 ) than Theorem A there. Since the above Splitting Theorem is very im portant fo r th e pr oofs of Theorems B, C, and D, we will give some notes on it in Section 3 . The n, Theorem B will b e proved in Section 4 . Section 5 will be de voted to provin g Theorem s C and D. W e will construc t some examp les in Section 6 to illustrate app lications of ou r Theorem s stated her e. Finally , we will end this paper with som e c oncludin g remarks in Section 7 . 2. Switched systems driven by nonchaotic switching si gnals This section is devoted to proving Theorem A stated in Section 1.2 under th e guise o f a mo re general result. For any integer 2 ≤ K < + ∞ , we reca ll that a switchin g signal σ = ( σ n ) + ∞ n = 1 ∈ Σ + K is called nonchaotic , if to any sequ ence h n i i i ≥ 1 ր + ∞ and any m ≥ 1 the re co rrespond s some δ with 2 ≤ σ ≤ m + 1 such th at for all ℓ 0 ≥ 1, there exists an ℓ ≥ ℓ 0 so that σ is constant restricted to some s ubinterval of [ n ℓ , n ℓ + m ] of length greater than or equal to δ . Clearly , a c onstant switchin g signal σ with σ ( n ) ≡ k is no nchaotic. Then fr om d efinition, we can obtain the following lemm a, which discovers the essential proper ty o f a nonch aotic switching signal. Lemma 2.1. Let σ = ( σ n ) + ∞ n = 1 ∈ Σ + K be a non chaotic switching signal. Th en, there exis ts some alphab et k ∈ { 1 , . . . , K } such that for a ny ℓ ≥ 1 and a ny ℓ ′ ≥ 1 , ther e exists an n ℓ ≥ ℓ ′ so that σ n ℓ + 1 = · · · = σ n ℓ + ℓ = k . 1 A norm k · k on R d is calle d a (real) polytope norm , if the unit s phere S k·k =  x ∈ R d : k x k = 1  is a polytope in R d ; see, e.g., [ 16 ]. 3 ω -LIMIT SETS FOR PR ODUCT BOUNDED SYSTEMS 6 Pr oof. First, we can choose a sequence h n i i i ≥ 1 ր + ∞ and some k ∈ { 1 , . . . , K } , which are such that n i + 1 − n i ր + ∞ and σ n i = k f or a ll i ≥ 1. Now from the defin ition of no nchaotic p roperty with m = 1, it follows tha t we can ch oose a sub sequence of h n i i i ≥ 1 , still write, without loss of generality , a s h n i i i ≥ 1 , such that σ n i = σ n i + 1 = k for all i ≥ 1 . Repeating this pr ocedure f or h n i + 1 i i ≥ 1 proves the statement. Lemma 2 .1 shows that the ω -limit set of a nonchao tic switching sign al con tains at least on e constant switching signal, under the sense of the classical Markov shift transformation. The following fact is a s imple consequen ce of the classical Gel’fand spectr al formula, which will be refined in Section 5 for the L yapunov norm k · k P . Lemma 2.2. F or any A ∈ R d × d and a ny matrix norm k · k on R d × d , if ρ ( A ) < 1 then there is an inte ger N ≥ 1 such that k A N k < 1 . For S = { S 1 , . . . , S K } ⊂ R d × d , it is said to be pr oduc t bounded , if there is a uni versal co nstant β ≥ 1 such that k S σ n · · · S σ 1 k ≤ β ∀ σ ∈ Σ + K and n ≥ 1 . This prop erty does not d epend upon the norm k · k used here. If S is pro duct bo unded , then one always can cho ose a vector n orm k · k on R d such that its induced op erator norm k · k on R d × d is such th at k S k k ≤ 1 fo r all 1 ≤ k ≤ K . Then the norm k · k on R d acts as a L yapunov fun ction for S . Howe ver, there does n ot need to exist a com mon, n ot strict in gen eral, “quadratic” L yapu nov function / matrix P as in ( 1.3a ). So, the follo win g theo rem is more gener al than Theorem A stated in Section 1.2 . Theorem 2.3. Let S = { S 1 , . . . , S K } ⊂ R d × d be pr odu ct bounded . If ρ ( S k ) < 1 for all 1 ≤ k ≤ K , then S is stable driven by any nonchao tic switching signals σ ∈ Σ + K . Pr oof. W ithout lo ss of ge nerality , let k · k be a matrix n orm on R d × d such that k S k k ≤ 1 for all 1 ≤ k ≤ K . Let σ = ( σ n ) + ∞ n = 1 ∈ Σ + K be an arbitrar y nonchaotic switching signal. Let k be given by Lemma 2.1 . Since ρ ( S k ) < 1 , by Lemma 2.2 we have some m ≥ 1 such that k S m k k < 1 . Thus, f or an arb itrary ε > 0 there is an ℓ ≥ 1 suc h that k S m ℓ k k < ε . From Lem ma 2. 1 , it follows that as n → + ∞ , k S σ n · · · S σ 1 k ≤ k S σ n m ℓ + m ℓ · · · S σ n m ℓ + 1 k < ε . So, k S σ n · · · S σ 1 k → 0 as n → + ∞ , since ε > 0 is arbitr ary . This completes the proof of Theorem 2.3 . Under co ndition ( 1.3a ), the statemen t of Theor em 2.3 will b e streng thened by Corollary 5.3 in Section 5 . 3. ω -limit sets for pr o duct bounded systems In this section, we will introdu ce ω -limit sets and g i ve so me notes on o ur splitting theo rem stated in Section 1.3 that is very important for our arguments in the next sections. 3 ω -LIMIT SETS FOR PR ODUCT BOUNDED SYSTEMS 7 3.1. ω -limit sets of a trajectory W e now co nsider the linear inclu sion ( 1.1 ) genera ted by S = { S 1 , . . . , S K } ⊂ R d × d where 2 ≤ K < + ∞ , as in Section 1 . The classical one-sided Markov shift transforma tion θ : Σ + K → Σ + K is defined as σ = ( σ n ) + ∞ n = 1 7→ θ ( σ ) = ( σ n + 1 ) + ∞ n = 1 ∀ σ ∈ Σ + K . Definition 3 .1 ([ 23 , 24 , 3 ]) . Let x 0 ∈ R d be an initial state and σ = ( σ n ) + ∞ n = 1 ∈ Σ + K a switching signal. The set of all limit p oints o f the sequ ence h S σ n · · · S σ 1 ( x 0 ) i + ∞ n = 1 in R d is ca lled th e ω -limit set of S at the input ( x 0 , σ ) . W e denote it by ω ( x 0 , σ ) her e. It is easy to see that for any switching signal σ , the correspo nding switched system is asymp- totically stable if and only if ω ( x 0 , σ ) = { 0 } ∀ x 0 ∈ R d . Thus we n eed to consider the structu re of ω ( x 0 , σ ) in order to study the stability of the switched dynamics induced by S . Lemma 3.2. Assume S is pr oduct bounded ; th at is, th er e is a matrix n orm k · k on R d × d such tha t k S k k ≤ 1 for a ll 1 ≤ k ≤ K . Then, for any in itial data x 0 ∈ R d and any switching signal σ , th e following two statements hold. (1) The ω -limit set ω ( x 0 , σ ) is a compa ct subset contain ed in a sph er e { x ∈ R d ; k x k = r } , fo r some r ≥ 0 . (2) The trajectory h x n ( x 0 , σ ) i + ∞ n = 1 in R d tends to 0 a s n → ∞ if an d o nly if ther e e xists a subsequen ce o f it which t ends to 0 . Pr oof. Since th e sequence hk S σ n · · · S σ 1 ( x 0 ) ki + ∞ n = 1 is non increasing in R f or any σ ∈ Σ + K , it is conv ergent as n → + ∞ . Denoted b y r its limit, we have the statemen t (1). The statement (2) follows immediately from the s tatement (1). This proves Lemma 3.2 . In the case (2) o f th is lem ma, we call the orbit h x n ( x 0 , σ ) i + ∞ n = 1 with initial value x 0 is asymp- totically stable. W e note here that Lem ma 3.2 is actually proved in [ 24 , 3 ] f or the contin uous-time case, but [ 3 ] is unde r the con dition that each member of S sha res a co mmon, no t strict in general, quadra tic L y apunov f unction and [ 24 ] und er an ad ditional assumptio n of “pa racontractio n” ex- cept the L yap unov f unction. In Section 3.3 , w e will c onsider th e ω -limit s et o f a matrix trajecto ry h S σ n · · · S σ 1 i + ∞ n = 1 . In addition , in the c ontinuo us-time case, ω ( x 0 , σ ) is a conn ected set. This is an importan t property needed in [ 24 , 3 ]. For a given s witching sign al, to con sider the stability of the corresp onding switched system, we need to classify which kin d of initial values in R d makes the corresp onding o rbits asym p- totically stab le. It is d i ffi cult to have such classification fo r a g eneral switching signal. In th e following, for the recurren t switching signal, we have a classification resu lt. 3.2. Decomposition for general e xtremal norm In th is sub section, we will introd uce a preliminar y sp litting theorem of the state space R d which plays the key in our classification. First, we recall fro m [ 21 , 27 ] that for a to pologica l dynamical s ystem T : Ω → Ω on a separable metriza ble spac e Ω , a point w ∈ Ω is called “r ecurren t”, pr ovided that one can find a 3 ω -LIMIT SETS FOR PR ODUCT BOUNDED SYSTEMS 8 positive integer seq uence n i ր + ∞ such that T n i ( w ) → w as i → + ∞ . And w ∈ Ω is said to be “weakly Birkho ff recurrent” [ 29 ] (also see [ 1 0 ]), p rovided that for any ε > 0, there exists an integer N ε > 1 such that jN ε − 1 X i = 0 I B ( w ,ε ) ( T i ( w )) ≥ j ∀ j ∈ N , where I B ( w ,ε ) : Ω → { 0 , 1 } is the characteristic function of the open b all B ( w , ε ) o f radius ε cen- tered at w in Ω . W e denote by R ( T ) and W ( T ), respectively , th e set of all recur rent points a nd weakly Birkho ff recu rrent p oints of T . It is easy to see that R ( T ) and W ( T ) both are inv a riant under T and W ( T ) ⊂ R ( T ). In the qu alitati ve theory o f ordinar y di ff erential eq uation, th is ty pe o f rec urrent p oint is also called a “Poisson stable” motion, for instance, in [ 21 ]. For th e one-sided Markov shift ( Σ + K , θ ), it is easily checked that every perio dically switched signal is recu rrent. And σ = ( σ n ) + ∞ n = 1 ∈ R ( θ ) means that there exists a subsequence n i ր + ∞ such that θ n i ( σ ) → σ as i → + ∞ . This imp lies that S σ n i + n · · · S σ n i + 1 → S σ n · · · S σ 1 as i → + ∞ for any n ≥ 1. W e should note that f or any two finite-length words w , w ′ , the switching sig nal σ = ( w ′ , w , w , w , . . . ) is not recurrent. For any fu nction A : Ω → R d × d , the cocycle A T : N × Ω → R d × d driven by T is defined as A T ( n , w ) = A ( T n − 1 w ) · · · A ( w ) for any n ≥ 1 and all w ∈ Ω . Now , our basic decomp osition theo rem can be stated as follows: Theorem 3.3 ([ 13 , T heorem B ′ ]) . Let T : Ω → Ω b e a continu ous transformation o f a separable metrizable space Ω . Let A : Ω → R d × d be a continu ous family of matrices with the pr o perty that ther e e xists a norm k · k such that k A T ( n , w ) k ≤ 1 ∀ n ≥ 1 and w ∈ Ω . Then for any r ecurrent point w of T , ther e corres pond s a splitting of R d into subspaces R d = E s ( w ) ⊕ E c ( w ) , such that lim n → + ∞ k A T ( n , w )( x ) k = 0 ∀ x ∈ E s ( w ) and k A T ( n , w )( x ) k = k x k ∀ n ≥ 1 ∀ x ∈ E c ( w ) . Here k · k do es n ot n eed to be a L yapunov nor m k · k P as in ( 1.3b ) and further the central manifold E c ( σ ) is not necessarily u nique an d inv ariant. Althoug h k A T ( n , w ) | E s ( w ) k conv erges to 0, yet k A T ( n , w ) | E s ( w ) k does no t n eed to conver g e exponentially fast, as is shown by [ 13 , Example 4.6]. Howe ver, un der the assump tions o f Theorem 3.3 , if w is a weakly B irkho ff rec urrent point of T , we have the follo w ing alternati ve results: 3 ω -LIMIT SETS FOR PR ODUCT BOUNDED SYSTEMS 9 Theorem 3. 4. Let T : Ω → Ω be a continu ous transformatio n of a sep arable metrizable spa ce Ω . Let A : Ω → R d × d be a continu ous family of matrices with the p r operty tha t there exists a norm k · k such th at k A T ( n , w ) k ≤ 1 for all n ≥ 1 and w ∈ Ω . I f w ∈ Ω is a weakly B irkho ff r ecurrent point of T , Then either k A T ( n , w ) k exponentially fast − − − − − − − − − − − − → 0 as n → + ∞ , or k A T ( n , T i ( w )) k = 1 ∀ i ≥ 0 for n ≥ 1 . Pr oof. If there exist i ≥ 0 and n ≥ 1 such that k A T ( n , T i ( w )) k < 1 then from T i ( w ) ∈ W ( T ) and [ 10 , Theorem 2.4], it follows that k A T ( m , T i ( w )) k exponentially f ast − − − − − − − − − − − − → 0 as m → + ∞ . This comp letes the proof of Theor em 3 .4 . 3.3. Decomposition unde r a weak Lyapunov matrix For a recur rent switching signal σ = ( σ n ) + ∞ n = 1 of S , to consider its stability , it is essential to compu te the stable m anifold E s ( σ ). From th e proof of Theor em 3.3 presented in [ 13 ], we know that E s ( σ ) is the kernel of an idempo tent matrix that is a limit poin t o f S σ n i · · · S σ 1 with θ n i ( σ ) → σ as i → + ∞ . Howe ver, in ap plications, it is not easy to identif y which subsequen ce h n i i i ≥ 1 with this p rop- erty . In this subsection, in stead of the p roduct bo unded ness, we assum e th e m ore stron g con dition ( 1.3a ) with induced norm k · k P on R d . In this case, we can calculate the stable manifold E s ( σ ) fo r any switching sig nal σ (not necessarily recurrent) of S . T o do this en d, we first consider the geometry of the limit sets ω ( x 0 , σ ) of S driven b y σ . For t he s imilar results in continuou s-time switched linear sy stems, see [ 3 ]. For an y switching signal σ = ( σ n ) + ∞ n = 1 ∈ Σ + K , on the other hand, we w ill consider the sequence h S σ n · · · S σ 1 i + ∞ n = 1 of m atrices and let ω ( σ ) d enote the set of all limit points of this sequ ence in R d × d . Definition 3.5 ([ 28 , 3 ]) . The set ω ( σ ) is called the ω -limit set of S driven by σ , for any σ ∈ Σ + K . From co ndition ( 1.3a ), it follows immediately that ω ( σ ) is n on-emp ty an d com pact. But it may not be a semigr oup in the sense of m atrix mu ltiplication wh en σ is not a recurre nt switch ing signal. W e note that if σ ∈ R ( θ ) th en fro m the p roof of [ 13 , Theor em 4. 2], ω ( σ ) con tains a nonemp ty compact semigroup and so there is an idempo tent element in ω ( σ ). Parallel to Lemma 3.2 , we can obtain the follo wing result. Lemma 3.6. Under conditio n ( 1.3a ), ther e follows the following statements. (a) F or a ny switching signal σ ∈ Σ + K of S , it holds that ω ( σ ) ⊂ { M ∈ R d × d : k M k P = r } , for some constant 0 ≤ r ≤ 1 ; if σ is further r ecurr en t, then either r = 0 or 1 . 3 ω -LIMIT SETS FOR PR ODUCT BOUNDED SYSTEMS 10 (b) F or a ny input ( x 0 , σ ) ∈ R d × Σ + K for S , we have ω ( x 0 , σ ) = { M ( x 0 ) | M ∈ ω ( σ ) } = ω ( σ )( x 0 ) . (c) F or any two elements M and N in ω ( σ ) , it holds that M T P M = N T P N . W e note that the co ntinuous- time ca ses o f the statements (b) and ( c) of Lemma 3.6 ha ve been proved in [ 3 , § 3] using the polar decomp osition of matrices. W e h ere present a simple treatmen t for the sake of self-closeness. Pr oof. W e fir st note that fr om ( 1 .3a ) and ( 1.3 b ), it follows imm ediately th at k S k k P ≤ 1 fo r a ll indices 1 ≤ k ≤ K . For the statemen t (b) , we let ( x 0 , σ ) ∈ R d × Σ + K be ar bitrary . I f M ∈ ω ( σ ), it is clear th at M ( x 0 ) ∈ ω ( x 0 , σ ). Con versely , let y ∈ ω ( x 0 , σ ) be arbitrary . By the definition of ω ( x 0 , σ ) there exists an increasing sequence { n i } such that y = lim i →∞ S σ n i · · · S σ 1 ( x 0 ) . The prod uct bound edness cond ition implies that the sequenc e h S σ n i · · · S σ 1 i + ∞ i = 1 has a conv e rgent subsequen ce, wh ose limit element is denoted by M . Thus y = M ( x 0 ). For the statement (c) of Lemma 3.6 , let M , N ∈ ω ( σ ) b e arb itrary . As k S k k P ≤ 1 fo r all 1 ≤ k ≤ K , from Lemma 3.2 we have k M ( x ) k P = k N ( x ) k P ∀ x ∈ R d . That is x T ( M T P M − N T P N ) x = 0 ∀ x ∈ R d . It follows, from the symmetr y o f the matrix M T P M − N T P N , that M T P M = N T P N . This proves th e statement (c) of Lemma 3.6 . Finally , the statemen t (a) of Lem ma 3.6 com es from the stateme nt (c) and Theorem 3.3 . In fact, let M , N ∈ ω ( σ ) be ar bitrary . Then there are vectors x , y ∈ R d such that k x k P = k y k P = 1 , k M k P = k M ( x ) k P , and k N k P = k N ( y ) k P . So, from (c) it follows that k M k P = √ x T M T P M x = √ x T N T P N x ≤ k N k P = p y T N T P N y = p y T M T P M y ≤ k M k P . This togethe r with Theorem 3.3 proves th e statement (a) of Lemma 3.6 . Thus the proof of Lemma 3.6 is completed. Let M ∈ ω ( σ ). T hen √ M T P M is a no nnegative-definite matrix w hich does no t depen d on the cho ice of the ma trix M ∈ ω ( σ ) by the statement (c) of L emma 3.6 an d is uniqu ely de cided by the switching signal σ . So, we write Q σ = √ M T P M ∀ M ∈ ω ( σ ) . (3.1) The continuo us-time case of the following statement (1) of Propo sition 3.7 has alread y been proved b y Balde and Jouan [ 3 , Theorem 1] using the polar decomp osition of ma trices. 3 ω -LIMIT SETS FOR PR ODUCT BOUNDED SYSTEMS 11 Proposition 3 .7. Under cond ition ( 1.3a ), for any switching signa l σ = ( σ n ) + ∞ n = 1 of S , there hold the following two statements. (1) The switc hing signal σ is asymptotica lly stable for S ; that is , lim n →∞ S σ n · · · S σ 1 ( x 0 ) = 0 ∀ x 0 ∈ R d , if and only if Q σ = 0 ; (2) If Q σ , 0 , then lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k P = k Q σ ( x 0 ) k 2 ∀ x 0 ∈ R d . So, the stable manifold of S at σ is such that E s ( σ ) = kernel of Q σ ; that is lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k P = 0 ∀ x 0 ∈ E s ( σ ) . Her e k · k 2 denotes the Euclidean vector norm on R d . Pr oof. The statement (1) holds trivially from th e statemen t (a) of Lemma 3.6 or f rom the state- ment (2) to be p roved soon. W e next will prove the statement (2) . F o r that, let Q σ , 0. For an arbitrary x 0 ∈ R d , by the definition of Q σ as in ( 3.1 ) there exists a subsequence h n i i i ≥ 1 and some M ∈ ω ( σ ) such that lim i → + ∞ k S σ n i · · · S σ 1 ( x 0 ) k P = k M ( x 0 ) k P = q x T 0 Q 2 σ x 0 = q x T 0 Q T σ Q σ x 0 = k Q σ ( x 0 ) k 2 . Therefo re, by ( 1.3a ) we hav e lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k P = k Q σ ( x 0 ) k 2 . This thus complete s the proof of Proposition 3.7 . W e note here th at if Q σ is idem potent, then from Prop osition 3.7 we have E c ( σ ) = Im( Q σ ) and R d = E s ( σ ) ⊕ E c ( σ ). Because in gen eral ther e lacks the recurrence of σ , one cannot defin e a central m anifold E c ( σ ) satisfy ing R d = E s ( σ ) ⊕ E c ( σ ) as done in The orem 3.3 . However , we will establish an other type o f splitting theorem in the case d = 2 fo r S driven by a gener al switching signal, not necessarily recur rent. For that, we first intr oduce several notations for the sake of o ur co n venience. For any g i ven A ∈ R d × d and any vector n orm k · k o n R d , write k A k co = min  k A ( x ) k : x ∈ R d with k x k = 1  , (3.2) called the co-n orm (also minimum n orm in some literature) of A under k · k . Definition 3.8. Under cond ition ( 1.3a ), for any switching signal σ ∈ Σ + K the numbers r E ( σ ) : = k M k P and r I ( σ ) : = k M k P , c o , for M ∈ ω ( σ ) , ar e called the ω -exterior and ω -interior radii of S driven by σ , r espectively . 3 ω -LIMIT SETS FOR PR ODUCT BOUNDED SYSTEMS 12 According to the statement (c) o f Lemma 3.6 , r E ( σ ) and r I ( σ ) both are well defined inde- penden t of the choice o f M . Motiv ated by E c ( σ ) in [ 10 , § 5.2.2 ] and by V i in [ 3 , Lem ma 1 ], fo r any giv e n A ∈ R d × d and any vector norm k · k on R d , let K k·k ( A ) =  x ∈ R d : k A ( x ) k = k A k · k x k  (3.3a) and K k·k co ( A ) =  x ∈ R d : k A ( x ) k = k A k co · k x k  . (3.3b) Clearly , if ker( A ) , { 0 } , then k A k co = 0 and so K k·k co ( A ) = ker( A ) in this case. For a g eneral norm k · k on R d , K k·k co ( A ) and K k·k ( A ) are not n ecessarily line ar subspaces. Howe ver, for a L yapu nov no rm, we can obtain the following. Lemma 3.9. Under the L yap unov no rm k · k P as in ( 1.3b ), ther e K k·k P , co ( A ) and K k·k P ( A ) both ar e linear subspaces of R d for any A ∈ R d × d . Pr oof. Let A ∈ R d × d be arbitrarily given. By de finitions, we hav e x ∈ K k·k P ( A ) ⇔ x T k A k P P x − x T A T P A x = 0 ⇔ x T ( k A k P P − A T P A ) x = 0 ⇔ k G ( x ) k 2 = 0 ⇔ x ∈ ker( G ) . Here G 2 = k A k P P − A T P A ≥ 0 is sym metric. Sin ce ker( G ), th e kern el o f x 7→ G x , is a linear subspace of R d , K k·k P is also a linear subspace of R d . On the other hand, for any x ∈ R d we hav e k A ( x ) k P ≥ k A k P , c o · k x k P . So , x T ( A T P A − k A k P , c o P ) x ≥ 0 ∀ x ∈ R d . Let H 2 = A T P A − k A k P , c o P , which is sym metric and nonnegative-definite. Then it h olds th at K k·k P , co ( A ) = ker( H ), a linear subspace. Thus, the proof of Lemma 3.9 is completed . Now , the improved splitting theo rem can be stated as follo ws: Theorem 3.10. Let S = { S 1 , . . . , S K } ⊂ R 2 × 2 satisfy co ndition ( 1.3a ). Then, for any switching signal σ ∈ Σ + K , not necessarily r ecurr ent, ther e exists a splitting of R 2 into subspaces R 2 = K k·k P , co ( σ ) ⊕ K k·k P ( σ ) such that lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k P = r I k x 0 k P ∀ x 0 ∈ K k·k P , co ( σ ) , lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k P = r E k x 0 k P ∀ x 0 ∈ K k·k P ( σ ) , and r I k x 0 k P < lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k P < r E k x 0 k P ∀ x 0 ∈ R 2 − K k·k P , co ( σ ) ∪ K k·k P ( σ ) . 4 ASYMPTO TICAL ST ABILITY UNDER A WEAK L Y APUNO V MA TRIX 13 Pr oof. Let r I < r E and M ∈ ω ( σ ). Defin e K k·k P , co ( σ ) = K k·k P , co ( M ) and K k·k P ( σ ) = K k·k P ( M ). From the stateme nt (2 ) of Pro position 3.7 , it follows that K k·k P , co ( σ ) an d K k·k P ( σ ) bo th are indepen dent of the cho ice o f M . So, R 2 = K k·k P , co ( σ ) ⊕ K k·k P ( σ ) from Lemma 3.9 . W e note that if r I = r E , then K k·k P , co ( σ ) = K k·k P ( σ ) = R 2 . This completes the proo f o f Theorem 3.10 . In the case where σ is re current, one can easily see that E s ( σ ) = K k·k P , co ( σ ) and E c ( σ ) = K k·k P ( σ ) . 4. Asymptotical stability under a weak L ya punov matrix In this section, we will discu ss the stability o f switch ed linear system with a comm on, but not necessarily strict, quad ratic L yap unov function. In this case, a cr iteria for stability is derived without compu ting the limit matr ix Q σ as in ( 3.1 ). W e still assume S is com posed of finitely many subsystems. Th at is, S = { S 1 , . . . , S K } with 2 ≤ K < + ∞ . 4.1. Stability of generic r ecurr ent switching signals Now for σ = ( σ n ) + ∞ n = 1 ∈ Σ + K , if Card { n | σ n = k } = ∞ for all 1 ≤ k ≤ K then σ is called “generic. ” Recall that a switching signal σ = ( σ n ) + ∞ n = 1 ∈ Σ + K is said to be stable for S if k S σ n · · · S σ 1 k → 0 as n → + ∞ . (Note that the stability is independent of th e chosen norm k · k .) As is kn own, a switching system which is asymp totically stable for all pe riodically switch ing sign als does not need to be asymptotically stable for all switchin g sign als in general [ 8 , 7 , 19 , 17 ]. Ho w e ver we can obtain the following result. Lemma 4 .1. If all r ec urr ent switching signa ls ar e stab le for S , then it is asymptotica lly stable driven by all switching signals in Σ + K . Pr oof. Since the set R ( θ ) of all recurren t switching signals has f ull measu re 1 fo r all e rgodic measures with respect to ( Σ + K , θ ), the result follows from [ 11 , Lemma 2.3]. By L emma 4.1 , to obtain the asymptotic stability of S , it su ffi c es to prove that it is on ly asymptotically stable driven by all r ecurrent switching signals. In addition, we need the following lem ma. Lemma 4.2. Under condition ( 1.3a ), if k S k k P = 1 and K k·k P ( S k ) is S k -in variant, then ρ ( S k ) = 1 . Here K k·k P ( S k ) is defined as in ( 3.3 ). Pr oof. The statemen t comes obviously from Lemma 3.9 . In the following, for simplicity , we just c onsider a switch ed system which is co mposed of two subsystems. Th at is, K = 2. Lemma 4.3. Under conditio n ( 1.3a ) with K = 2 (i.e., S = { S 1 , S 2 } ), if k S 1 k P = k S 2 k P = 1 and K k·k P ( S 1 ) ∩ K k·k P ( S 2 ) = { 0 } , (4.1) and at least one o f th em is in va riant (i.e., S 1 ( K k·k P ( S 1 )) = K k·k P ( S 1 ) or S 2 ( K k·k P ( S 2 )) = K k·k P ( S 2 ) ), then every generic s witching signal is stable for S . 4 ASYMPTO TICAL ST ABILITY UNDER A WEAK L Y APUNO V MA TRIX 14 Pr oof. Assume that K k·k P ( S 1 ) is S 1 -in variant. (Otherwise, if K k·k P ( S 2 ) is S 2 -in variant, the p roof is the same.) L et σ = ( σ n ) + ∞ n = 1 be a gener ic switching signal; th at is, in ( σ n ) + ∞ n = 1 , both 1 and 2 appear infinitely many times. Then th ere e xists a subsequen ce { σ n i } such that σ n i = 1 and σ n i + 1 = 2 ∀ i ≥ 1 . For a g iv e n initial value x 0 ∈ R d , consider the subsequen ce { S σ n i − 1 · · · S σ 1 ( x 0 ) } + ∞ i = 1 . By the assumption ( 1.3a ), it has a con vergent sub sequence in R d . Without loss of generality , we assume that S σ n i − 1 · · · S σ 1 ( x 0 ) → y ∈ R d as i → + ∞ . Thus S σ n i S σ n i − 1 · · · S σ 1 ( x 0 ) → S 1 ( y ) , S σ n i + 1 S σ n i S σ n i − 1 · · · S σ 1 ( x 0 ) → S 2 S 1 ( y ) , as i → + ∞ . By the statement (1) of Lemma 3.2 , we have k S 2 S 1 ( y ) k P = k S 1 ( y ) k P = k y k P . Thus y ∈ K k·k P ( S 1 ) and S 1 ( y ) ∈ K k·k P ( S 2 ). From the S 1 -in variance of K k·k P ( S 1 ) it follows that S 1 ( y ) ∈ K k·k P ( S 1 ) ∩ K k·k P ( S 2 ) . So S 1 ( y ) = 0 and so is y . Fro m the statement (2) of Lemma 3.2 , we ha ve S σ n · · · S σ 1 ( x 0 ) → 0 a s n → + ∞ . That is, σ is a stable switching signal for S . This proves L emma 4.3 . Both E c ( σ ) in [ 10 , § 5.2.2] an d V i in [ 3 , Lemma 1] are inv ariant. Unfo rtunately , here our subspace K k·k P ( S k ) does not need to be S k -in variant in g eneral. See Example 6.2 in Section 6 . I f this is the case, we still have, h owe ver, th e following criter ion. Theorem 4.4 . Und er cond itions ( 1.3a ) a nd ( 4 .1 ) with S = { S 1 , S 2 } ⊂ R d × d , the follo wing two statements hold . (1) If d = 2 , then all generic r ecu rr ent switching signals σ ∈ Σ + 2 , which satisfy σ , ( c 1 , 2 , c 1 , 2 , . . . ) , ar e stable for S ; (2) if d = 3 , then all generic r ecurr en t switc h ing signals σ ∈ Σ + 2 such that σ , ( w , w , w , . . . ) , wher e w ∈ { (1 , 2) , ( 2 , 1) , (1 , 2 , 2) , (2 , 1 , 1) } , ar e stable for S . 4 ASYMPTO TICAL ST ABILITY UNDER A WEAK L Y APUNO V MA TRIX 15 Pr oof. First, if k S 1 k P < 1 or k S 2 k P < 1, then every gen eric switchin g signal is stable for S and hence the statemen ts (1) and (2 ) tri vially hold. So, w e next assume k S 1 k P = k S 2 k P = 1. T his implies that dim K k·k P ( S k ) ≥ 1 for k = 1 , 2. For the st atement ( 1) of Th eorem 4.4 , f rom ( 4.1 ) it follo ws that dim K k·k P ( S k ) = 1 for k = 1 , 2. Let σ = ( σ n ) + ∞ n = 1 be a given gen eric recurrent switching signal such that σ ( · + n ) , ( c 1 , 2 , c 1 , 2 , . . . , c 1 , 2 , . . . ) ∀ n ≥ 1 . (4.2) From Theore m 3.3 , there correspo nds a splitting of R 2 into subspaces R 2 = E s ( σ ) ⊕ E c ( σ ) , such that lim n → + ∞ k S σ n · · · S σ 1 ( x 0 ) k P = 0 ∀ x 0 ∈ E s ( σ ) and k S σ n · · · S σ 1 ( x 0 ) k P = k x 0 k P ∀ n ≥ 1 ∀ x ∈ E c ( σ ) . T o p rove that σ is a stable switching signal for S , we need to prov e that E c ( σ ) = { 0 } . By the genericity of σ and ( 4.2 ), σ must contains the word (1 , 1 , 2) or (2 , 2 , 1). W ithout lo ss of generality , we assume that ( σ 1 , σ 2 , σ 3 ) = (1 , 1 , 2) . Thus we have k S 2 S 1 S 1 ( x 0 ) k P = k S 1 S 1 ( x 0 ) k P = k S 1 ( x 0 ) k P = k x 0 k P ∀ x 0 ∈ E c ( σ ) These imply that { x 0 , S 1 ( x 0 ) } ⊂ K k·k P ( S 1 ) , S 1 S 1 ( x 0 ) ∈ K k·k P ( S 2 ) . Suppose that x 0 , 0. It follows fr om dim K k·k P ( S 1 ) = 1 th at th ere exists a rea l nu mber λ with | λ | = 1 such that S 1 ( x 0 ) = λ x 0 . This means that x 0 is an eigen vector of S 1 with eigenv alu e λ . So S 1 S 1 ( x 0 ) = λ 2 x 0 ∈ K k·k P ( S 1 ) . Therefo re S 1 S 1 ( x 0 ) ∈ K k·k P ( S 1 ) ∩ K k·k P ( S 2 ) = { 0 } . Thu s we have S 1 S 1 ( x 0 ) = 0, wh ich implies x 0 = 0, a co ntradiction . Next, for pr oving the statement ( 2) of Theor em 4.4 that d = 3 , by ( 4.1 ), we have that one o f K k·k P ( S 1 ) , K k·k P ( S 2 ) has dimension 1 and the other has dimension at least 1 and at most 2. If both K k·k P ( S 1 ) an d K k·k P ( S 2 ) h a ve dimension 1, then by the sam e argument as in the state- ment (1), all generic recurren t swit ching signals satisfying ( 4.2 ) are stable for S . Next, we assume that, for e xample, dim K k·k P ( S 1 ) = 1 and dim K k·k P ( S 2 ) = 2 . 4 ASYMPTO TICAL ST ABILITY UNDER A WEAK L Y APUNO V MA TRIX 16 W e claim that for any generic recurrent switching signal σ = ( σ n ) + ∞ n = 1 ∈ Σ + 2 , if σ ( · + n ) < n ( c 1 , 2 , c 1 , 2 , . . . , c 1 , 2 , . . . ) , ( [ 1 , 2 , 2 , [ 1 , 2 , 2 , . . . , [ 1 , 2 , 2 , . . . ) o ∀ n ≥ 1 . (4.3) then σ is stable for S . Ther e is no lo ss o f gen erality in assuming σ 1 = 1; oth erwise replacin g σ by σ ( · + n ) for some n ≥ 1 . T hen, K k·k P ( S 1 ) = E c ( σ ) if E c ( σ ) , { 0 } , where E c ( σ ) is gi ven by Theorem 3.3 . Whenever the w o rd 11 appears in the s equence ( σ n ) + ∞ n = 1 , K k·k P ( S 1 ) is S 1 -in variant. The n, Lemma 4.3 follows th at σ is stable for S . Next, we assume 11 does not appear in ( σ n ) + ∞ n = 1 . If 121 appears in ( σ n ) + ∞ n = 1 then b 12 b 12 b 12 · · · must a ppear too, a con tradiction. So , 12 1 cann ot a ppear in ( σ n ) + ∞ n = 1 . Then 1 22 must ap pear . If 1 221 appea rs in ( σ n ) + ∞ n = 1 then d 122 d 122 d 122 · · · m ust appear too , a contrad iction. Thu s, t he word 1222 must appear in ( σ n ) + ∞ n = 1 . When σ contains the word (2 , 2 , 2 , 1), ass ume that, for example, ( σ n + 1 , σ n + 2 , σ n + 3 , σ n + 4 ) = (2 , 2 , 2 , 1) . Then we have k S 1 S 2 S 2 S 2 ( x 0 ) k P = k S 2 S 2 S 2 ( x 0 ) k P = k S 2 S 2 ( x 0 ) k P = k S 2 ( x 0 ) k P = k x 0 k P ∀ x 0 ∈ E c ( σ ( · + n )) , which show that for all x 0 ∈ E c ( σ ( · + n )), { x 0 , S 2 ( x 0 ) , S 2 S 2 ( x 0 ) } ⊂ K k·k P ( S 2 ) , S 2 S 2 S 2 ( x 0 ) ∈ K k·k P ( S 1 ) . If x 0 and S 2 ( x 0 ) are linear dependen t, th at is, S 2 ( x 0 ) = λ x 0 , for some λ with | λ | = 1, then S 2 S 2 S 2 ( x 0 ) = λ 3 x 0 ∈ K k·k P ( S 2 ). So S 2 S 2 S 2 ( x 0 ) ∈ K k·k P ( S 1 ) ∩ K k·k P ( S 2 ) = { 0 } , which implies that x 0 = 0. On the other hand ,if x 0 and S 2 ( x 0 ) are linear indep endent, then S 2 S 2 ( x 0 ) = λ x 0 + α S 2 ( x 0 ) , for some λ and α , since dim K k·k P ( S 2 ) = 2. Th us S 2 S 2 S 2 ( x 0 ) is a linear com bination o f S 2 ( x 0 ) and S 2 S 2 ( x 0 ). So it is also in K k·k P ( S 2 ). Therefor e S 2 S 2 S 2 ( x 0 ) ∈ K k·k P ( S 1 ) ∩ K k·k P ( S 2 ) = { 0 } , which shows x 0 = 0. Thus E c ( σ ( · + n )) = { 0 } and then E c ( σ ) = { 0 } . Similarly , wh en dim K k·k P ( S 1 ) = 2 and dim K k·k P ( S 2 ) = 1, we can p rove that all gene ric recurren t swit ching signals, but the follo wing four periodic switching signals (1 , 1 , 1 , . . . ) , (2 , 2 , 2 , . . . ) , ( c 2 , 1 , c 2 , 1 , . . . ) , ( [ 2 , 1 , 1 , [ 2 , 1 , 1 , . . . ) , are stable for S . This completes the proof of Theorem 4.4 . 5 ABSOLUTE ST ABILITY OF A P AIR OF MA TRICES 17 W e hav e the following r emarks on Theorem 4.4 . Remark 1 . Similarly , we can con sider a switched linear system co mposed o f two sub systems on R d with d ≥ 4. In this case, under th e assump tions ( 1.3a ) and ( 4.1 ), if eith er K k·k P ( S 1 ) or K k·k P ( S 2 ) has dim ension 1 , th en all generic rec urrent switching signals but fin itely m any perio dic signals are stable for S . Remark 2. Under the assumption s on Theorem 4 .4 , in ord er to obtain the stability fo r all re current switching sign als, we just need to check finitely many periodic sign als to see whether they are stable for S . Remark 3. T heorem 4. 4 su ggests a easy com putable su ffi cient condition of a symptotically s table for switche d lin ear systems which are composed of tw o subsystems. In fact, Remark 2 provides a direct way to ch eck the stability of all r ecurrent signals, which implies the asymptotically stable of the systems by Lemma 4.1 . W e can also discuss the stability of switched linear systems compo sed of finite many sub sys- tems similarly . But it is troubleso me to fo rmulate the cor respondin g assumptions. Here we will giv e an example to i llustrate such condition s in Section 6 . 4.2. Almost sur e stability Let ( Σ + K , B ) be the Borel σ -field of the space Σ + K and then th e one- sided Markov shift map θ : σ ( · ) 7→ σ ( · + 1) is measurab le. A Borel probability m easure P on Σ + K is said to be θ - in va riant , if P = P ◦ θ − 1 , i.e. P ( B ) = P ( θ − 1 ( B )) for a ll B ∈ B . A θ -inv ariant probability mea sure P is called θ -er godic , provid ed that f or B ∈ B , P  ( B \ θ − 1 ( B )) ∪ ( θ − 1 ( B ) \ B )  = 0 implies P ( B ) = 1 or 0. An ergodic measure P is called non- atomic , if ev ery singleton set { σ } h as P -measure 0. Using Theorem 4.4 , we can easily prove Theorem B stated in Section 1.4 . Pr oof of Theo r em B. Let P b e an arb itrary non- atomic θ -ergodic measure on Σ + 2 . Then f rom the Poincar ´ e recu rrence theor em (see, e .g., [ 27 , Theo rem 1.4]) , it f ollows that P -a. e. σ ∈ Σ + 2 are recurren t. In addition, sine P is non -atomic, we o btain th at P -a .e. σ ∈ Σ + 2 are non- periodic and generic. This completes the proof of Theor em B from Theorem 4.4 . W e note that in th e proof of Theo rem B p resented above, th e deduction of the gene ricity o f σ needs the assumption K = 2. 5. Absolute stability of a pair of matrices with a weak L y apunov matrix W e now deal with the case S = { S 1 , S 2 } ⊂ R d × d , where S 1 and S 2 both are stable and sha re a common , but n ot ne cessarily strict, qu adratic L yap unov function . For any A ∈ R d × d , we denote by ρ ( A ) the spectral radius of A . Our first absolute stability result Theorem C is restated as follows: Theorem 5.1. Let S = { S 1 , S 2 } ⊂ R 2 × 2 satisfy condition ( 1.3a ). Then, S is absolutely stable (i.e ., k S σ n . . . S σ 1 k → 0 as n → + ∞ , for all switching signa ls σ ∈ Σ + 2 ) if an d only if there holds that ρ ( S 1 ) < 1 , ρ ( S 2 ) < 1 , and ρ ( S 1 S 2 ) < 1 . 5 ABSOLUTE ST ABILITY OF A P AIR OF MA TRICES 18 Pr oof. W e on ly need to pr ove the su ffi ciency . Let ρ ( S 1 ) < 1 , ρ ( S 2 ) < 1, and ρ ( S 1 S 2 ) < 1. Let σ = ( σ n ) + ∞ n = 1 ∈ Σ + 2 be an arbitrary r ecurren t switching signal. Cl early , if σ is not generic, then it is stable for S . So we assume σ is generic and r ecurren t. Th en, fro m Theorem 3.3 there exists a splitting of R 2 into subspaces: R 2 = E s ( σ ) ⊕ E c ( σ ) . If dim E c ( σ ) = 0, then σ is stable for S ; and if dim E c ( σ ) = 2 then either ρ ( S 1 ) = 1 or ρ ( S 2 ) = 1, a contrad iction. W e now assume dim E c ( σ ) = 1. Then, d im K k·k P ( S 1 ) = 1 and dim K k·k P ( S 2 ) = 1. It mig ht be assumed, without loss of g ener- ality , that σ 1 = 1 and then we hav e K k·k P ( S 1 ) = E c ( σ ). From this, we see σ 2 = 2 , σ 3 = 1 , . . . , σ 2 n = 2 , σ 2 n + 1 = 1 , . . . . This contra dicts ρ ( S 1 S 2 ) = ρ ( S 2 S 1 ) < 1. Therefo re, E c ( σ ) = { 0 } and S is absolutely stable from Lemma 4.1 . So, Theorem C is proved. Next, we need a simple f act for considering higher dimensional cases. Lemma 5.2 ([ 26 , Corollary]) . Let A ∈ R d × d be a stable matrix (i.e., ρ ( A ) < 1 ) such that D − A T DA ≥ 0 for some symmetric, positive-defi nite matrix D. Th en D − ( A d ) T DA d > 0 . This lemma refines Lemm a 3.2 . From it, we can obtain a simp le result which improves th e statement of Theorem A in the case of d = 2 an d K = 2. Corollary 5.3. Let S = { S 1 , S 2 } ⊂ R 2 × 2 satisfy co ndition ( 1.3a ). If ρ ( S 1 ) < 1 and ρ ( S 2 ) < 1 , then for a ny θ -er g odic pr ob ability measur e P on Σ + 2 , S is stable driven by P -a.e. σ ∈ Σ + 2 as long as P satisfies P ( { (12 , 12 , 12 , . . . ) , (2 1 , 21 , 2 1 , . . . ) } ) = 0 . Pr oof. Since P is ergod ic an d P ( { (12 , 12 , 12 , . . . ) , (21 , 21 , 21 , . . . ) } ) = 0, we have P ( { σ ∈ Σ + 2 | σ ( · + n ) = (1 2 , 12 , 12 , . . . ) or (21 , 21 , 21 , . . . ) for some n ≥ 1 } ) = 0 . Now , let σ = ( σ n ) + ∞ n = 1 ∈ Σ + 2 be arbitrary . Th en, σ can con sist of the following 2-len gth words: 11 , 22 , 12 , 21 . If 11 (or 22) appe ars infinitely m any times in ( σ n ) + ∞ n = 1 , then from Lemm a 5.2 it fo llows that S is stable dr iv e n by σ . Next, assume 11 and 22 both only appea r finitely many times in ( σ n ) + ∞ n = 1 and let a = 12 and b = 21 . T hen, one can find some N ≥ 1 such that σ ( · + N ) = ( a , a , a , . . . ) . Note h ere that if a b appear s m times in ( σ n ) + ∞ n = 1 then 22 must appea r m times; if ba appears m times in ( σ n ) + ∞ n = 1 then 11 must appear m times. So, S is stable driven b y P -a.e. σ ∈ Σ + 2 . This completes the proof of Corollary 5.3 . 5 ABSOLUTE ST ABILITY OF A P AIR OF MA TRICES 19 The c ondition P ( { (1 2 , 12 , 12 , . . . ) , (21 , 21 , 21 , . . . ) } ) = 0 mean s that P is n ot d istributed o n the perio dic o rbit of the one-sided Markov shift ( Σ + K , θ ): { (12 , 12 , . . . ) , (21 , 21 , . . . ) } . This c orollary shows that S is “comp letely” alm ost sur e stable u p to only o ne ergod ic measure supported on a period ic orbit g enerated by the word 12. In addition, Theorem C can be directly deduced from Corollary 5.3 and Lemma 4.1 . For the sake o f o ur conv enience, we now restate our second abso lute stab ility re sult Theo- rem D as follows: Theorem 5.4. Let S = { S 1 , S 2 } ⊂ R 3 × 3 satisfy conditio n ( 1 .3a ). Then, S is ab solutely stable if and only if ther e holds the following conditions: ρ ( S 1 ) < 1 , ρ ( S 2 ) < 1 , (C1) ρ ( S 1 S 2 ) < 1 , (C2) ρ ( S w 1 S w 2 S w 3 ) < 1 ∀ ( w 1 , w 2 , w 3 ) ∈ { 1 , 2 } 3 , (C3) ρ ( S w 1 · · · S w 4 ) < 1 ∀ ( w 1 , . . . , w 4 ) ∈ { 1 , 2 } 4 , (C4) ρ ( S w 1 · · · S w 5 ) < 1 ∀ ( w 1 , . . . , w 5 ) ∈ { 1 , 2 } 5 , (C5) ρ ( S w 1 · · · S w 6 ) < 1 ∀ ( w 1 , . . . , w 6 ) ∈ { 1 , 2 } 6 , (C6) ρ ( S w 1 · · · S w 8 ) < 1 ∀ ( w 1 , . . . , w 8 ) ∈ { 1 , 2 } 8 . (C8) W e note here th at it is somewhat surprising that we do not nee d to consider the words of length 7. Pr oof. W e need to con sider only the su ffi ciency . Let condition s (C1) – (C8) all hold. Acco rding to Lemm a 4.1 , we let σ = ( σ n ) + ∞ n = 1 ∈ Σ + 2 be an arb itrary recurren t switching signal. There is no loss of generality in assuming σ 1 = 1. It is easily seen that 0 ≤ d im K k·k P ( S 1 ) ≤ 2 an d 0 ≤ dim K k·k P ( S 2 ) ≤ 2 by conditio n (C1). Then from Theor em 3.3 with k · k = k · k P , there exists a splitting of R 3 into subspaces: R 3 = E s ( σ ) ⊕ E c ( σ ) such that dim E c ( σ ) ≤ dim K k , k·k P for k = 1 , 2 . There is only one of the following three cases occurs. • dim E c ( σ ) = 2; • dim E c ( σ ) = 1; • dim E c ( σ ) = 0. Clearly , if σ is not generic, then it is stable fo r S . So we le t σ be gen eric in wha t follows. W e also note that E c ( σ ) ⊆ K k·k P ( S 1 ). Case (a): Let d im E c ( σ ) = 2. Then dim K k·k P ( S 1 ) = dim K k·k P ( S 2 ) = 2 and furth er we have K k·k P ( S 1 ) = E c ( σ ). If σ 2 = 1 then it follows that K k·k P ( S 1 ) is S 1 -in variant and so ρ ( S 1 ) = 1 by Lemma 4.2 , a co ntradiction . Thus, σ 2 = 2. If σ 3 = 2 it f ollows that K k·k P ( S 2 ) is S 2 -in variant 5 ABSOLUTE ST ABILITY OF A P AIR OF MA TRICES 20 and so ρ ( S 2 ) = 1 by Lemm a 4.2 , also a contradictio n. So, σ 3 = 1. Repeating th is, we can see σ = (1 , 2 , 1 , 2 , 1 , 2 , . . . ), a contradiction to condition (C2). T hus, the case (a) cannot occur . Case (b): Let d im E c ( σ ) = 1. ( This is the most co mplex case needed to discussion.) W e first claim that σ does not contain any one of the following two words: (1 , 1 , 1) , (2 , 2 , 2) . In fact, with out loss o f gen erality , we let ( σ n + 1 , σ n + 2 , σ n + 3 ) = (2 , 2 , 2). Choose a vector x ∈ E c ( σ ) with k x k P = 1. Then, v : = S σ n · · · S σ 1 ( x ) ∈ K k·k P ( S 2 ) with k v k P = 1. Moreover , S 2 ( v ) an d S 2 ( S 2 ( v )) both b elong to K k·k P ( S 2 ) such that with k S 2 ( v ) k P = k S 2 ( S 2 ( v )) k P = 1. Sin ce S 2 ( v ) , ± v (oth erwise ρ ( S 2 ) = 1 ), we see K k·k P ( S 2 ) is S 2 -in variant. So, ρ ( S 2 ) = 1 by Lemma 4.2 , a contradictio n to cond ition (C1). Secondly , we claim that if σ co ntains the w ord of the form (1 , 1 , w 1 , . . . , w m , 1 , 1) then ρ ( S w m . . . , S w 1 S 1 S 1 ) = 1; and if σ contain s the word of the form (2 , 2 , w 1 , . . . , w m , 2 , 2) then ρ ( S w m . . . , S w 1 S 2 S 2 ) = 1 . In fact, without loss of generality , we ass ume that σ = ( 1 , σ 2 , . . . , σ n , 2 , 2 , w 1 , . . . , w m , 2 , 2 , . . . ) . Then, take arbitrar ily a vector x ∈ E c ( σ ) with k x k P = 1 and write v n : = S σ n · · · S σ 1 ( x ). So, v n and S 2 ( v n ) bo th belo ng to K k·k P ( S 2 ) such that k v n k p = k S 2 ( v n ) k P = 1. On the other hand, v ′ : = S w m · · · S w 1 S 2 S 2 ( v n ) and S 2 ( v ′ ) both b elong to K k·k P ( S 2 ) with k v ′ k P = k S 2 ( v ′ ) k P = 1. If v n , ± v ′ then K k·k P ( S 2 ) is S 2 -in variant an d so ρ ( S 2 ) = 1 by Lemma 4.2 , a co ntradiction to condition (C1). Thus, we hav e v n = ± v ′ and then ρ ( S w m . . . , S w 1 S 2 S 2 ) = 1. Thirdly , we sho w the case (b), i.e., dim E c ( σ ) = 1, does no t occur too. In fact, from the above claims, it follows t hat σ = ( σ n ) + ∞ n = 1 only possesses the following forms: 1 →      12 → · · · (case (A)) 2 → ( 1 → · · · (case (B)) 21 → · · · (case (C)) (5.1) Here and in the sequel, “ a → b ” mean s th at b follows a ; i.e., σ n = a and σ n + 1 = b for some n . For example, in the a bove figure, “1 → 2 → 21” mean s σ 1 = 1 , σ 2 = 2 and ( σ 3 , σ 4 ) = (2 , 1 ). In addition, in th e following three figu res, the symbol “ × ” means “T his case does not happ en. ” For 5 ABSOLUTE ST ABILITY OF A P AIR OF MA TRICES 21 the case (A) in the above fig ure ( 5.1 ), we hav e the following: 112 →                                                      1 →                              1 ( × by (C3)) 2 →                          1 → ( 1 ( × by (C5)) 2 ( × by (C2) and Lemma 5.2 ) 21 →                1 ( × b y (C6)) 2 →          1 →      1 ( × b y (C8)) 2 → ( 1 ( × by (C2) and Lemma 5.2 ) 2 ( × by (C5)) 2 ( × by (C3)) 21 →                1 ( × b y (C4)) 2 →          1 →      1 ( × b y (C6)) 2 → ( 1 ( × by (C2) and Lemma 5.2 ) 2 ( × by (C5)) 2 ( × by (C3)) Thus, ( σ 1 , σ 2 , σ 3 ) , (1 , 1 , 2) and then ( σ 1 , σ 2 ) , (1 , 1) . (5.2) For the case (C) in the figure ( 5.1 ), we hav e 1221 →                                                      12 →                1 →          1 ( × b y (C3)) 2 →      1 → ( 1 ( × b y (C5)) 2 ( × b y (C6)) 2 ( × by (C6)) 2 ( × b y (C4)) 2 →                              1 →                          12 →                1 →          1 ( × b y (C3)) 2 →      1 → ( 1 ( × b y (C5)) 2 ( × Le mma 5.2 ) 2 ( × by (C8)) 2 ( × by (C6)) 2 → ( 1 ( × b y Lemma 5.2 ) 2 ( × b y (C5)) 2 ( × b y (C3)) Thus ( σ 1 , σ 2 , σ 3 , σ 4 ) , (1 , 2 , 2 , 1 ) and then ( σ 1 , σ 2 , σ 3 ) , (1 , 2 , 2) . (5.3) 5 ABSOLUTE ST ABILITY OF A P AIR OF MA TRICES 22 Finally , for the case (B) in the figure ( 5.1 ), 121 →                                                                                                                                                                                      12 →                                                      1 →                              1 ( × by (C3)) 2 →                          1 → ( 1 ( × by (C5)) 2 ( × by Lemma 5.2 ) 21 →                1 ( × b y (C6)) 2 →          1 →      1 ( × b y (C8)) 2 → ( 1 ( × by Lemma 5.2 ) 2 ( × by (C5)) 2 ( × by (C3)) 21 →                1 ( × b y (C4)) 2 →          1 →      1 ( × b y (C6)) 2 → ( 1 ( × by Lemma 5.2 ) 2 ( × by (C5)) 2 ( × by (C3)) 2 →                                                                                                                      1 →                                                            12 →                                                      1 →                              1 ( × by (C3)) 2 →                          1 → ( 1 ( × by (C5)) 2 ( × by Lemma 5.2 ) 21 →                1 ( × b y (C6)) 2 →          1 →      1 ( × b y (C8)) 2 → ( 1 ( × by Lemma 5.2 ) 2 ( × by (C5)) 2 ( × by (C3)) 21 →                1 ( × b y (C4)) 2 →          1 →      1 ( × b y (C6)) 2 → ( 1 ( × by Lemma 5.2 ) 2 ( × by (C5)) 2 ( × by (C3)) 2 ( × b y Lemma 5.2 ) 21 →                                                  12 →          1 →      1 ( × b y (C3)) 21 → ( 1 ( × b y (C5)) 2 ( × b y Lemma 5.2 ) 2 ( × by (C4)) 2 →                              1 →                          12 →                1 →          1 ( × b y (C3)) 2 →      1 → ( 1 ( × b y (C5)) 2 ( × b y Lemma 5.2 ) 2 ( × by (C8)) 2 ( × by (C6)) 2 → ( 1 ( × b y Lemma 5.2 ) 2 ( × b y (C5)) 2 ( × b y (C3)) 6 EXAMPLES 23 Thus, ( σ 1 , σ 2 , σ 3 ) , (1 , 2 , 1). Fur ther , from ( 5.3 ) it follows ( σ 1 , σ 2 ) , (1 , 2). This tog ether with ( 5.2 ) implies that ( σ 1 , σ 2 ) < { (1 , 1) , (1 , 2 ) } , a contradictio n. So, dim E c ( σ ) , 1 and hence case (b) does not occur . Therefo re, dim E c ( σ ) = 0. This implies that σ is stable for S . Therefo re S is absolutely stab le from Lemma 4.1 . This completes the proof of Theorem 5.4 . 6. Examples W e in this section shall give several examples to illustrate application s of our results. In what follows, let k · k 2 be the usual Euclidean norm on R d ; that is, P = I d in ( 1.3b ). First, a very simple e x ample is the following. Example 6.1. Let S = { S 1 , S 2 } with S 1 =  1 0 0 α  , S 2 =  β 0 0 1  , where | α | < 1 , | β | < 1 . I t is easy to see that k S 1 k 2 = k S 2 k 2 = 1 , and that K k·k 2 ( S 1 ) = { ( x 1 , 0) T ∈ R 2 | x 1 ∈ R } , K 2 , k·k 2 ( S 2 ) = { (0 , x 2 ) T ∈ R 2 | x 2 ∈ R } . So, we can obtain that K k·k 2 ( S 1 ) T K k·k 2 ( S 2 ) = { 0 } a nd K k·k 2 ( S k ) is S k -in variant. Thu s the switched linear system S is asymptotically stab le for all switching signals in which each k in { 1 , 2 } is stable by Lemma 4.3 . Also, fro m Theo rem 4.4 , it f ollows that all r ecurrent signals but the fixed sig nals (1 , 1 , 1 , . . . ) and (2 , 2 , 2 , . . . ) are stable for S . W e no te her e th at the period ic switching signal (1 , 2 , 1 , 2 , . . . ) is stable for S . A more interesting example is the follo win g Example 6.2. Let S = { S 1 , S 2 } with S 1 = α  1 0 1 1  , S 2 = β  1 3 2 0 1  , where α = s 3 − √ 5 2 , β = 1 2 . Then, k S 1 k 2 = k S 2 k 2 = 1. A direct computation shows that K k·k 2 ( S 1 ) = ( ( x 1 , x 2 ) T ∈ R 2 | x 1 = √ 5 + 1 2 x 2 ) K k·k 2 ( S 2 ) =  ( x 1 , x 2 ) T ∈ R 2 | x 2 = 2 x 1  . Thus K k·k 2 ( S 1 ) T K k·k 2 ( S 2 ) = { 0 } . But they are n ot in variant. Th us S is asymptotically stable for all generic recurrent s witching signals b ut the periodic signal (1 , 2 , 1 , 2 , . . . ) by Theore m 4.4 . Note that the two subsystems themselves are asymptotically stable. 6 EXAMPLES 24 Next, we give an example wh ich is the d iscretization of the switched linear co ntinuou s system borrowed from [ 3 ]. Example 6.3. Let S = { S 1 , S 2 , S 3 } with S 1 =   α 0 0 0 0 − 1 0 1 0   , S 2 =   α 0 0 0 α 0 0 0 1   , S 3 =   α 0 0 0 1 0 0 0 α   , where | α | < 1. I t is easy to see that k S 1 k 2 = k S 2 k 2 = k S 3 k 2 = 1 and K k·k 2 ( S 1 ) =  ( x 1 , x 2 , x 3 ) T ∈ R 3 | x 1 = 0  , K k·k 2 ( S 2 ) =  ( x 1 , x 2 , x 3 ) T ∈ R 3 | x 1 = x 2 = 0  , K k·k 2 ( S 3 ) =  ( x 1 , x 2 , x 3 ) T ∈ R 3 | x 1 = x 3 = 0  . Since K k·k 2 ( S 2 ) T K k·k 2 ( S 3 ) = { 0 } a nd they ar e in variant respect to S 2 and S 3 , resp ecti vely , we have that any gener ic switchin g signal in which e ither the word ( 2 , 3) o r th e (3 , 2) appe ars infinitely many times are stable by Lemma 4.3 . For th e any other generic switching signals σ = ( σ 1 , σ 2 , . . . ), that is, in wh ich b oth the word ( 2 , 3) an d the (3 , 2 ) appear at most finite many times, the matrix Q σ defined in ( 3.1 ) is Q σ =   0 0 0 0 α k 0 0 0 0 α j 0   . for some no nnegative in tegers k 0 and j 0 which depen d o n the times of ap pearance of (2 , 3 ) and (3 , 2) in σ . Thus by Proposition 3.7 , we have lim n →∞ k S σ n · · · S σ 1 x k 2 = 0 , ∀ x ∈ { ( x 1 , 0 , 0) T | x 1 ∈ R } = ker( Q σ ) , lim n →∞ k S σ n · · · S σ 1 x k 2 = k Q σ ( x ) k 2 , ∀ x ∈ { (0 , x 2 , x 3 ) T | x 2 , x 3 ∈ R } = Im ( Q σ ) , for such kind of generic switching signals. The following E xample 6.4 is associated to Theorem C. Example 6.4. Let S = { S 1 , S 2 } with S 1 = 1 2  1 0 3 2 − 1  , S 2 = s 3 − √ 5 2  1 1 0 1  . Then, using p ρ ( A T A ) = k A k 2 we have ρ ( S 1 ) = 1 2 < 1 , k S 1 k 2 = 1 and ρ ( S 2 ) = s 3 − √ 5 2 < 1 , k S 2 k 2 = 1 . In addition, ρ ( S 1 S 2 ) = s 3 − √ 5 2 = ρ ( S 2 ) < 1 . Therefo re, S is absolutely stable by Theorem C. 7 CONCLUDING REMARKS 25 The interesting [ 22 , Prop osition 1 8] implies that if S = { A 1 , . . . , A m } ⊂ R d × d is symmetr ic (i.e. A T ∈ S whenever A ∈ S ), then S h as the spectral finiteness property ; in fact, it holds that ρ ( S ) = p ρ ( A T A ) for som e A ∈ S . This natur ally motivates us to extend an arbitrar y S into a symmetric set S = S ∪ S T . L et us see a simple example. Example 6.5. Let S =  A = q 3 − √ 5 2  1 1 0 1  . Th en, S satisfies ( 1.3a ) with k A k 2 = 1 such that ρ ( S ) = q 3 − √ 5 2 < 1. But f or S = { A , A T } , ρ ( S ) = p ρ ( A T A ) = 1 , ρ ( S ) . This example shows that the extension S do es not work for the original system S needed to be considered here. Finally , the following Exam ple 6.6 is simple. Y et it is very i nteresting to the stability analysis of switched systems. Example 6.6. Let S = { S 1 , S 2 } ⊂ R 2 × 2 with S 1 = s 3 − √ 5 2  1 0 1 1  , S 2 = s 3 − √ 5 2  1 1 0 1  . Then, using p ρ ( A T A ) = k A k 2 we have ρ ( S 1 ) = ρ ( S 2 ) = s 3 − √ 5 2 < 1 , k S 1 k 2 = k S 2 k 2 = 1 , and ρ ( S 1 S 2 ) = 1 . So, S is not ab solutely stable. Y et fro m Corollary 5.3 , S is stable driven by P -a.e. σ ∈ Σ + 2 , for any θ -ergodic prob ability m easure P on Σ + 2 , as long as P is no t the ergodic measure distributed on the periodic orbit { (12 , 12 , 1 2 , . . . ) , (2 1 , 21 , 21 , . . . ) } . 7. Concluding remarks In this paper, we have considered the asymptotic stability of a discrete-time linear switched system, wh ich is indu ced by a set S = { S 1 , . . . , S K } ⊂ R d × d such that eac h S k shares a common, but not necessarily strict, L yap unov m atrix P as in ( 1.3a ). W e have shown that if every sub system S k is stable then S is stable driven by a no nchaotic switching signal. P a rticularly , in the cases K = 2 and d = 2 , 3, we have proven that S has the spectral finiteness prop erty and so the stability is decidable. Recall that S is called p eriodically switched stable , if ρ ( S w n · · · S w 1 ) < 1 for all finite-leng th words ( w 1 , . . . , w n ) ∈ { 1 , . . . , K } n for n ≥ 1; see, e.g., [ 16 , 12 , 10 ]. Finally , we end this paper with a problem for further study . Conjecture. Let S = { S 1 , S 2 } ⊂ R d × d , d ≥ 4 , be an arb itrary pair such tha t co ndition ( 1.3a ). If S is p eriodically switched stab le, then it is ab solutely stable. 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