( ) 1 Comparison Functions. α is strictly increasing since ( r) ( r ) α = for any positive real number c. = 0. It is said to belong to

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1 Compaiso Fuctios I this lesso, we study stability popeties of the oautoomous system = f t, x The difficulty is that ay solutio of this system statig at x( t ) depeds o both t ad t = x Thee ae thee special compaiso fuctios kow as class K, class K, ad class KL fuctios to help with edefiig the defiitios of stability ad asymptotic stability so that they hold uifomly i the iitial time t Defiitio 4 A cotiuous fuctio α : [, a) [, ) is said to belog to class K if it is stictly iceasig ad α ( ) = It is said to belog to class K if a= ad α( ) as Defiitio 43 A cotiuous fuctio β :[, a ) [, ) [, ) is said to belog to class KL if, fo each fixed s, the mappig β (, s) belogs to class K with espect to ad, fo each fixed, the mappig β (, s) is deceasig with espect to s ad β(, s) as s Figue shows sketches of class K ad class KL fuctios α( ) Class K 858 Noliea Systems i Mechaical Egieeig Lesso 6, a a β ( s) Class KL Figue : Class K ad class KL fuctios Example []: Detemie whethe the followig fuctios belog to class K, class K, o class KL a) α( ) = ( ) Solutio ( ) ta α is stictly iceasig sice ( ) ( ) belogs to class, c b) ( ), K but ot class K sice α( ) α = fo ay positive eal umbe c α ' = / + > It lim = π / < s Solutio α ( ) is stictly iceasig sice α ( ) lim α( ) ; = thus, it belogs to class K c ' c = > Moeove, Copyight 7 by Withit Chatlataagulchai

2 β, = / +, fo ay positive eal umbe k c) ( s) ( ks ) Solutio (, s) β is stictly iceasig i sice ad stictly deceasig i s sice Moeove, ( s) β = > + ( ks ) β k = < s + ( ks ) β, as s Theefoe, it belogs to class KL The ext lemma states some useful popeties of the compaiso fuctios, which will be eeded late o Lemma 4 Let α ad α be class K fuctios o [, a ), α 3 ad α 4 be class K fuctios, ad β be a class KL fuctio Deote the ivese of α i by ) ) α The, i α is defied o, α( a)) α is defied o [ ) 3 α α belogs to class K α α belogs to class 3) 4) 3 4 5) σ α β α ad belogs to class K, ad belogs to class K ( ) K, s =, s belogs to class KL Poof See Execise 434 of [] Class K ad class KL fuctios ete ito Lyapuov aalysis though the ext two lemmas 858 Noliea Systems i Mechaical Egieeig Lesso 6 Lemma 43 Let V : D R be a cotiuous positive defiite fuctio defied o a domai D R that cotais the oigi Let B D fo some > The, thee exist class K fuctios α ad α, defied o [, ], such that α ( x ) V( α ( x ) fo all x B If D= R, the fuctios α ad α will be defied o, ad the foegoig iequality will hold fo all x R Moeove, if [ ) V x is adially ubouded, the α ad α ca be chose to belog to class K Poof See Appedix C4 of [] Fo a quadatic positive defiite fuctio V( x T Px, follows fom the iequalities λ mi T λ max P x x Px P x = Lemma 43 Lemma 44 Coside the scala autoomous diffeetial equatio, y = α y y t = y whee α is a locally Lipschitz class K fuctio defied o [, a ) Fo all y < a, this equatio has a uique solutio y( t ) defied fo all t t Moeove, = σ(, ) y t y t t whee σ is a class KL fuctio defied o [, a ) [, ) Poof See Appedix C5 of [] Copyight 7 by Withit Chatlataagulchai

3 Specific examples whee a closed-fom solutio ca be foud ae used to illustate the Lemma 44 y = ky, k >, the the solutio is = ( ) σ = ( ) y t y exp k t t, s exp ks Noautoomous Systems Coside the oautoomous system whee f :[, ) D Lipschitz i x o [ ) (, ) = f t x () R is piecewise cotiuous i t ad locally, D, ad D oigi x= The oigi is a equilibium poit fo () at t= if f t, =, t R is a domai that cotais the The followig defiitios edefie stability popeties of the oigi equilibium poit of the oautoomous system x= of x= ε > thee is δ δ( ε, t ) Defiitio 44 The equilibium poit ) stable if, fo each, δ is = > such that x t < x t < ε, t t () ) uifomly stable if, fo each, of t, such that () is satisfied ε > thee is δ δ( ε), = > idepedet 3) ustable if it is ot stable 4) asymptotically stable if it is stable ad thee is a positive costat c= c( t ) such that x( t) as t, fo all x( t ) < c 5) uifomly asymptotically stable if it is uifomly stable ad thee is a positive costat c, idepedet of t, such that fo all c x( t) 858 Noliea Systems i Mechaical Egieeig Lesso 6 x t <, as t, uifomly i t ; that is, fo each η >, thee is T T( η) = > such that η ( η) x t <, t t + T, x t < c 6) globally uifomly asymptotically stable if it is uifomly stable, δ( ε ) ca be chose to satisfy lim ε δ( ε) =, ad, fo each pai of positive umbes η ad c, thee is T = T( η, c) > such that η ( η ) x t <, t t + T, c, x t < c The followig Lemma states a alteative of the Defiitio 44 usig class K ad class KL fuctios Lemma 45 The equilibium poit is x= of x= ) uifomly stable if ad oly if thee exist a class K fuctio α ad a positive costat c, idepedet of t, such that α ( ) ( ) x t x t, t t, x t < c ) uifomly asymptotically stable if ad oly if thee exist a class KL fuctio β ad a positive costat c, idepedet of t, such that β ( ) ( ) x t x t, t t, t t, x t < c (3) 3) globally uifomly asymptotically stable if ad oly if iequality (3) is x t satisfied fo ay iitial state Poof See Appedix C6 of [] 3 Copyight 7 by Withit Chatlataagulchai

4 β A impotat case is stated i the followig defiitio whe, s = ke λs Defiitio 45 The equilibium poit is expoetially x= of x= stable if thee exist positive costats c, k, ad λ such that ( t t ) λ x t k x t e, x t c < (4) ad globally expoetially stable if (4) is satisfied fo ay iitial state x t ( ) The ext two theoems cocetate o uifom stability ad uifom asymptotic stability These ae the cases we ecoute i most oautoomous applicatios of Lyapuov s method Theoem 48 Let D ad x= be a equilibium poit fo x= x= Let V :[, ) D R R be a domai cotaiig cotiuously diffeetiable fuctio such that (, ) W x V t x W x V V + f ( t, x ) be a t ad x D, W x ae cotiuous positive defiite fuctios o D The, x= is uifomly stable Poof See page 5 of [] whee W ( x ) ad (5) 858 Noliea Systems i Mechaical Egieeig Lesso 6 Theoem 49 Suppose the assumptios of Theoem 48 ae satisfied with iequality (5) stegtheed to V V + f t x W 3 x x (, ) t ad x D, W3 x is a cotiuous positive defiite fuctio o D The, x= is uifomly asymptotically stable Moeove, B = x D ad whee if ad c ae chose such that { } c< mi W x, the evey tajectoy statig i { x B W( c} satisfies x = β x t x t, t t, t t fo some class KL fuctio β Fially, if D R W x is adially ubouded, the x= is globally uifomly asymptotically stable Poof See page 53 of [] = ad A fuctio V( t, x ) is positive semidefiite if positive defiite if V( t, W( W ( x ), adially ubouded if W V( t, W( A fuctio (, ) V( t, is positive defiite (semidefiite) V t, x It is fo some positive defiite fuctio x is so, ad decescet if V t x is egative defiite (semidefiite) if The followig theoem states coditios fo expoetial stability Theoem 4 Let D ad x = be a equilibium poit fo x= x= Let V :[, ) D R R be a domai cotaiig cotiuously diffeetiable fuctio such that be a 4 Copyight 7 by Withit Chatlataagulchai

5 a k x V t, x k x, V V + f ( t, x ) k 3 x t ad x D, whee k, k, k 3, ad a ae positive costats The, x= is expoetially stable If the assumptios hold globally, the x= is globally expoetially stable Poof See page 54 of [] a a 858 Noliea Systems i Mechaical Egieeig Lesso 6 Example 3 []: Detemie the stability popety of the oigi of the system = x g t x = x x whee g( t ) is cotiuously diffeetiable ad satisfies g( t) k ad g ( t) g( t), t,, Example []: Detemie the stability popety of the oigi of the system 3 = + g t x, whee g( t ) is cotiuous ad g( t) fo all t Solutio usig the Lyapuov fuctio cadidate 4 4 = + V x = x /, we obtai V t, x g t x x, x R, t The assumptios of Theoem 49 ae satisfied globally with 4 W x W x V x W x = x Hece, the oigi is globally = = ad uifomly asymptotically stable 3 5 Copyight 7 by Withit Chatlataagulchai

6 Solutio Takig V( t, = x + + g t x as a Lyapuov fuctio cadidate, it ca be see that Hece, (, ) x + x V t, x x + + k x, x R V t x is positive defiite, decescet, ad adially ubouded The deivative of V is give by Usig the iequality we obtai V t, x = x + xx + g t g t x + g t g t + g t g t x x T V t, x x + x x x = x Qx x = x whee Q is positive defiite; theefoe, V( t, all coditios of Theoem 49 ae satisfied globally T Recallig that a positive defiite quadatic fuctio λ mi λ P x x x Px P x x is egative defiite Thus, T T T max theefoe all coditios of Theoem 4 ae satisfied with a= T x Px satisfies 858 Noliea Systems i Mechaical Egieeig Lesso 6 3 Liea Time-Vayig Systems ad Lieaizatio The stability behavio of the oigi as a equilibium poit of the liea system = t A t x ca be completely chaacteized i tems of the state tasitio matix of the system =Φ x t t, t x t The ext theoem chaacteizes uifom asymptotic stability i Φ t, t The theoem also shows that, fo liea systems, tems of uifom asymptotic stability of the oigi is equivalet to expoetial stability Theoem 4 The equilibium poit A t x is (globally) x= of x( t) = uifomly asymptotically stable if ad oly if the state tasitio matix satisfies the iequality ( t t ) λ Φ t t ke t t fo some positive costats k ad λ Poof See page 57 of [],, Fo liea time-vayig systems, uifom asymptotic stability caot be chaacteized by the locatio of the eigevalues of the matix A Theoem 4 Let x= be the expoetially stable equilibium poit of x t = A t x Q t be a Suppose A( t ) is cotiuous ad bouded Let cotiuous, bouded, positive defiite, symmetic matix The, thee is a cotiuously diffeetiable, bouded, positive defiite, symmetic matix P t = P t A t + A T t P t + Q t Hece, P( t ) that satisfies 6 Copyight 7 by Withit Chatlataagulchai

7 T (, ) V t x = x P t x is a Lyapuov fuctio fo the system that satisfies the coditios of Theoem 4 Poof See page 58 of [] The ext theoem states Lyapuov s idiect method fo showig expoetial stability of the oigi i the oautoomous case Theoem 43 Let x= be a equilibium poit of the oliea system (, ) = f t x whee f :[, ) D R D= x R x <, ad the Jacobia matix [ f / x] { } Lipschitz o D, uifomly i t Let is cotiuously diffeetiable, f = ( t A t, x x= is bouded ad The, the oigi is a expoetially stable equilibium poit fo the oliea system if it is a expoetially stable equilibium poit fo the liea system 4 Covese Theoems 858 Noliea Systems i Mechaical Egieeig Lesso 6 Fo example, a covese theoem fo uifom asymptotic stability would cofim that if the oigi is uifomly asymptotically stable, the thee is a Lyapuov fuctio that satisfies the coditios of Theoem 49 The mee kowledge that a Lyapuov fuctio exists is bette tha othig At least, we kow that ou seach is ot hopeless Theoem 44 Let x= be a equilibium poit fo the oliea system (, ) = f t x whee f :[, ) D R D= x R x <, ad the Jacobia matix [ f / x] { } is cotiuously diffeetiable, is bouded o D, uifomly i t Let k, λ, ad be positive costats with < / k Let D { x R x } satisfy = < Assume that the tajectoies of the system ( t t ) λ x t k x t e x t D t t,, The, thee is a fuctio V :[, ) D R that satisfies the iequalities Poof See page 6 of [] = A t x c x V t, x c x, V V + f ( t, x ) c 3 x, V c4 x fo some positive costats,, 3, c c c ad c 4 Moeove, if = ad the oigi is globally expoetially stable, the V( t, x ) is defied ad satisfies 7 Copyight 7 by Withit Chatlataagulchai

8 the afoemetioed iequalities o R Futhemoe, if the system is autoomous, V ca be chose idepedet of t Poof See page 63 of [] Theoem 45 Let x= be a equilibium poit fo the oliea system (, ) = f t x whee f :[, ) D R D= x R x <, ad the Jacobia matix [ f / x] { } Lipschitz o D, uifomly i t Let is cotiuously diffeetiable, f = ( t A t, x x= is bouded ad The, x= is a expoetially stable equilibium poit fo the oliea system if ad oly if it is a expoetially stable equilibium poit fo the liea system Poof See page 66 of [] = A t x Coollay 43 Let x= be a equilibium poit of the oliea system x= f x f x is cotiuously diffeetiable i some, whee x= Let A [ f x] eighbohood of = / The, x= is a expoetially stable equilibium poit fo the oliea system if ad oly if A is Huwitz Theoem 46 Let x= be a equilibium poit fo the oliea system 858 Noliea Systems i Mechaical Egieeig Lesso 6 = f t, x whee f :[, ) D R D= x R x <, ad the Jacobia matix [ f / x] { } costat such that ( ) is cotiuously diffeetiable, is bouded o D, uifomly i t Let β be a class KL fuctio ad be a positive β, < Let D = { x R x < } Assume that the tajectoy of the system satisfies β x t x t, t t, x t D, t t The, thee is a cotiuously diffeetiable fuctio V :[, ) D R that satisfies the iequalities α α α α ad 4 ( x ) V( t, α ( x ) V V + f t x x V α4( x ) (, ) α3 α ae class K fuctios defied o [ ] whee,, 3, system is autoomous, V ca be chose idepedet of t Poof See Appedix C7 of [], If the Theoem 47 Let x= be a asymptotically stable equilibium poit fo the oliea system = f x whee f : D R is locally Lipschitz ad D R is a domai that cotais the oigi Let R D be the egio of attactio of x= A 8 Copyight 7 by Withit Chatlataagulchai

9 The, thee is a smooth, positive defiite fuctio V( x ) ad a cotiuous, positive defiite fuctio W( x ), both defied fo all x R A, such that ad fo ay, R R, V( A V( as x R V f x W x x R { }, A c> V x c is a compact subset of R Whe = is adially ubouded Poof See Appedix C8 of [] 5 Boudedess ad Ultimate Boudedess Lyapuov aalysis ca be used to show boudedess of the solutio of the state equatio, eve whe thee is o equilibium poit at the oigi ae ) uifomly bouded if thee exists a positive costat c, idepedet of t, ad fo evey a (, c), thee is β = β( a) >, idepedet of Defiitio 46 The solutios of x= t, such that x( t) a x( t) β t t A, (6) ) globally uifomly bouded if (6) holds fo abitaily lage a 3) uifomly ultimately bouded with ultimate boud b if thee exist positive costats b ad c, idepedet of t, ad fo evey a (, c), thee is T T( a b) =,, idepedet of t, such that A 858 Noliea Systems i Mechaical Egieeig Lesso 6 4) globally uifomly ultimately bouded if (7) holds fo abitaily lage a Theoem 48 Let D R be a domai that cotais the oigi ad, D R be a cotiuously diffeetiable fuctio such that V :[ ) α ( x ) V( t, α ( x ) V V + f ( t, x ) W 3( x ), x µ > t ad x D, whee α ad α ae class K fuctios ad W3( x ) is a cotiuous positive defiite fuctio Take > such that B D ad suppose that µ α α ( ) < The, thee exists a class KL fuctio β ad fo evey iitial state x ( t ), satisfyig x t α α, x( t ) ad µ ) such that the solutio of x= Moeove, if D β thee is T (depedet o satisfies x t x t, t t, t t t + T (8) α α ( µ ) x t, t t + T (9) = R ad, fo ay iitial state Poof See Appedix C9 of [] α belogs to class, K the (8) ad (9) hold x t with o estictio o how lage µ is x t a x t b, t t + T (7) 9 Copyight 7 by Withit Chatlataagulchai

10 Lesso 6 Homewok Poblems 439, 445, Noliea Systems i Mechaical Egieeig Lesso 6 Homewok poblems ae fom the equied textbook (Noliea Systems, by Hassa K Khalil, Petice Hall, ) Refeeces [] Noliea Systems, by Hassa K Khalil, Petice Hall, Copyight 7 by Withit Chatlataagulchai

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