The generalized deniion allow u for inance o deal wih parial averaging of yem wih diurbance (for ome claical reul on parial averaging ee [3, pp

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1 Averaging wih repec o arbirary cloed e: cloene of oluion for yem wih diurbance A.R.Teel 1, Dep. of Elec. and Comp. Eng., Univeriy of California, Sana Barbara, CA, D.Neic Dep. of Elec. and Elecronic Eng., The Univeriy of Melbourne, Parkville, 305, Vicoria, Auralia L.Moreau 3 SYSTeMS, Univeriei Gen, Technologiepark-Zwijnaarde 9, 905 Gen, Belgium. Abrac We coniderwo dieren deniion of \average" for yem wih diurbance: he \rong" and \weak" average ha were inroduced in [7]. Our deniion are more general han hoe in [7] a we ue he diance o an arbirary cloed e A inead of he Euclidean norm for ae in he deniion of average. Thi generalizaion allow u o deal wih more general cae of averaging for yem wih diurbance, uch a parial averaging. Under appropriae condiion, he oluion of a ime-varying yem wih diurbance are hown o converge uniformly on compac ime inerval o he oluion of he yem' average a he rae of change of ime increae o inniy. 1 Inroducion Averaging i an imporan approximae mehod for analyi of ime-varying yem. In i claical form (ee, for example, [1,, 4, 9, 1] and reference herein) i applie o ordinary dierenial equaion of he form _x = f (= x ) (1) where > 0 and where f ha an average f av aifying a condiion like 1 f av (x) = lim T!1 T Z +T f( x 0)d 8 0 : 1 Reearch uppored in par by he AFOSR under gran F Reearch uppored by he Auralian Reearch Council under he mall ARC gran cheme. 3 Suppored by BOF gran 011D0696 of he Ghen Univeriy. Thi paper preen reearch reul of he Belgian Programme on Iner-univeriy Pole of Aracion, iniiaed by he Belgian Sae, Prime Minier' Oce for Science, Technology and Culure. The cienic reponibiliy re wih i auhor. Claical averaging reul ae ha, under appropriae moohne aumpion on f, he oluion of (1) converge uniformly on compac ime inerval o he oluion of _x = f av (x) () a end o zero. 1 Moreover, if he yem () ha an exponenially able equilibrium poin p ha i an equilibrium poin of (1) for mall > 0 hen p i an exponenially able equilibrium poin of (1) for mall >0. Excep for [7], we are no aware of any reul on averaging ha conider yem wih exogenou diurbance. However, yem wih diurbance occur frequenly in conrol heory. Recenly, in [7], wo dieren deniion of average for yem wih diurbance (\rong" and \weak" average) were inroduced and reul were preened on deducing inpu-o-ae abiliy (ISS) for a yem from inpu-o-ae abiliy for he yem' rong or weak average. Thee reul generalize, in a ene, he exponenial abiliy reul menioned above, a well a more recen abiliy reul baed on averaging, like in[10]. Given an arbirary cloed e A, which i no necearily compac, in hi noe we udy yem wih diurbance ha poe an A-weak or A- rong average (ee Deniion 1 and, repecively), ha i average dened wih repec o he e A. The deniion ha we ue in hi paper are more general han hoe ued in [7], ince in [7] we only conidered he cae of A- weak and A-rong average when A = f 0 g. 1 Someime averaging i applied o yem of he form dx d = f( x ) inead of he form (1). In hi cae, he convergence of oluion i eablihed on inerval of lengh proporional o 1=. Thi i een o be an equivalen reul by caling ime a = =. p. 1

2 The generalized deniion allow u for inance o deal wih parial averaging of yem wih diurbance (for ome claical reul on parial averaging ee [3, pp ] for ome recen reul on parial averaging of diurbance-free ordinary dierenial equaion ee [8] for parial averaging of diurbance-free funcional dierenial equaion, ee [6] and reference herein). More preciely, we udy yem of he form _x = f (= x w() ) x() =x (3) where w() i he exogenou diurbance and i a parameer vecor wih mall norm. We are inereed in condiion ha guaranee ha he oluion of (3) are cloe, on compac ime inerval, o he oluion of (3)' A-average which i aken when = 0. The average yem ha he form _y = f A av(y w()) y() =y (4) where f A av i aumed o be locally Lipchiz. We will no impoe any abiliy aumpion on he A-average yem. Moreover, for f we will only aume ha i i coninuou in (x w ) uniformly in. In paricular, we do no aume uniquene of oluion for (3). We aume ha w belong o a e of funcion ha i equi-bounded (ee Deniion 4) for he cae of A-rong average, or equi-bounded and equi-uniformly coninuou (ee Deniion 5) for he cae of A-weak average. We will how, among oher hing, ha when a rajecory for he A-averaged yem i dened on a given compac ime inerval, he rajecorie of he acual yem converge o ha average rajecory uniformly on he compac ime inerval. The paper i organized a follow: In Secion we preen ome preliminary deniion including he deniion of A-weak and A-rong average ha generalize he deniion of rong and weak average from [7]. Our main reul are aed formally in Theorem 1 and of Secion 3. In he la Secion we provide he proof of main reul. Preliminarie For our purpoe, a funcion : R0 R0! R0 i of cla-kl if i i nondecreaing from zero in i r argumen and converging o zero in i econd argumen. Given a meaurable funcion w(), we dene i inniy norm kwk 1 := e up 0 jw()j. If we have kwk 1 < 1, henwe wrie w L 1. If w() i aboluely coninuou, i derivaive i dened almo R everywhere and we can wrie w() ; w() = _w()d. Given an arbirary e AR n,we dene he diance of a poin x R n o he e A a: jxj A := inf jx ; j A where jxj i he Euclidean norm of x. We ay ha a funcion f( w) i A-locally Lipchiz if given any riple of ricly poiive real number (R 1 R R 3 ) here exi L>0, uch ha jf(x w) ; f(y w)j L jx ; yj for all jxj A R 1, jyj A R, jwj R 3. The following wo deniion of A-rong and A-weak average for a ime-varying yem wih exogenou diurbance generalize he deniion ha were inroduced in [7]: Deniion 1 (A-weak average) Le A R n be an arbirary cloed e. An A-locally Lipchiz funcion fwa A : R n R m! R n i aid o be he A- weak average of f( x w ) if here exi KL and T > 0 uch ha 8T T and 8 0 we have f A wa(x w) ; 1 T Z +T f( x w 0)d maxfjxj A jwj 1g T : (5) The A-weak average of yem (3) i hen dened a _y = f A wa(y w): (6) If A = f 0 g, hena-weak average i referred o imply a weak average. Deniion (A-rong average) Le AR n be an arbirary cloed e. An A-locally Lipchiz funcion fa A : R n R m! R n i aid o be hearong average of f( x w ) if here exi KL and T > 0 uch ha 8w L 1 8T T and 8 0 he following hold: 1 T Z +T h f A a(x w()) ; f( x w() 0) maxfjxj A kwk 1 1g T i d : (7) The A-rong average of yem (3) i hen de- ned a _y = f A a(y w): (8) If A = f 0 g, hen A-rong average i referred o imply a rong average. Noe ha w in he inegral i a conan vecor. p.

3 Remark 1 Deniion 1 and generalize he deniion inroduced in [7], ince in [7] he e A wa alway choen o be he origin in R n. Thi generalizaion allow u, for inance, o ae reul on parial averaging of yem wih diurbance. Indeed, conider: _~x = ~ f (= ~x w) : (9) If wewanoaverage f ~ only in he r argumen and leave he dependence on in he econd argumen, hen he average yem i ime-varying and we have parially averaged f. ~ Our reul apply o hi iuaion in he following way. Inroduce a new ae variable p = and rewrie (9) a _~x = ~ f (= p ~x w) _p = 1 : (10) Inroducing x := (~x T p) T and f := ( f ~ T 1) T,we can rewrie (10) a _x = f (= x w). Le he cloed non-compac e be dened a A := fx : ~x = 0g. Then, parial weak or rong averaging reul for (9) can be recovered by uing A-weak or A-rong averaging reul repecively. Remark I ha been hown in [7] ha funcion f ha have a rong average are, in eence, funcion of he form f( x w 0) = e f( x) + g(x w) where e f( x) ha a well-dened (weak) average. We alo need deniion of forward compleene, equi-boundedne and equi-uniform coninuiy. Deniion 3 Le F be a e of locally eenially bounded funcion, A an arbirary cloed e and le f av beaconinuou funcion. The yem _x = f av (x w) x(0) = x (11) i aid o be FA-forward complee if for each r> 0 and T > 0 here exi R r uch ha, for all jxj A r and w() F, he oluion x() of (11) exi and jx()j A R for all [0 T]. If A = f 0 g, and he yem i FA-forward complee, hen we imply ay ha he yem i F-forward complee. Deniion 4 Le F be a e of locally eenially bounded funcion. The e F i equi- (eenially) bounded ifhere exi a ricly poiive real number uch ha, for all w() F, jjwjj 1 Deniion 5 Le F be a e of locally eenially bounded funcion. The e F i equi-uniformly coninuou if for each > 0 here exi > 0 uch ha, for all w() F and all 0, [0 ] =) jw( + ) ; w()j : Remark 3 A ucien condiion for F o be equi-uniformly coninuou i ha all w() Fare aboluely coninuou (on [0 1)) and here exi a ricly poiive real number 1 uch ha, for all w() F, jj _wjj 1 1, i.e., w() i Lipchiz. 3 Main Reul Our main reul give condiion under which he oluion of (3) are cloe o he oluion of (3)' A-weak or A-rong average, when hee average exi. Theorem 1 (Cloene o A-weak average) Suppoe an arbirary cloed e A i given and alo: 1. he funcion f( x w ) i: (a) meaurable in for each (x w ), (b) coninuou in x uniformly in for each pair of (w ), (c) for any R > 0, coninuou in (w ), uniformly in and x fx : jxj A Rg, (d) for every R > 0 here exi B > 0 uch ha jf( x 0 0)j B for all and x fx : jxj A Rg. he e F i equi-(eenially) bounded and equi-uniformly coninuou 3. he A-weak average of he yem (3) exi and i FA-forward complee. Then, for each riple (T r) of ricly poiive real number here exi a riple ( ) of ricly poiive real number uch ha, for each (0 ), jj <, 0, jyj A r, w F and each x uch ha jx ; yj, each oluion x ( x w) of (3) and he oluion y( ; y w) of he weak average aify jx ( x w) ; y( ; y w)j (1) for all [ + T ]. p. 3

4 Wihou he aumpion ha F i equi-uniformly coninuou, he concluion of Theorem 1 i no correc, in general. Thi i demonraed by he yem _x = ;0:5x 3 +co(=) x 3 w, which wa dicued in deail in [7]. There i wa hown ha he weak average of hi yem i _y = ;0:5y 3 bu he yem under he inpu w() = co(=) exhibi nie ecape ime. The aumpion ha F i equi-uniformly coninuou can be removed when he rong average exi and i F-forward complee: Theorem (Cloene o A-rong average) Suppoe an arbirary cloed e A i given and alo: 1. he funcion f( x w ) i: (a) meaurable in for each (x w ), (b) coninuou in x uniformly in for each pair of (w ), (c) for any R > 0 coninuou in (w ), uniformly in and x fx : jxj A Rg, (d) for every R > 0 here exi B > 0 uch ha jf( x 0 0)j B for all and x fx : jxj A Rg. he e F i equi-(eenially) bounded 3. he A-rong average of he yem (3) exi and i FA-forward complee. Then, for each riple (T r) of ricly poiive real number here exi a riple ( ) of ricly poiive real number uch ha, for each (0 ), jj <, 0, jyj A r, w F and each x uch ha jx ; yj, each oluion x ( x w) of (3) and he oluion y( ; y w) of he rong average aify jx ( x w) ; y( ; y w)j (13) for all [ + T ]. The above reul can alo be applied o average yem ha are no F-forward complee (ee, for inance, [11] for he appropriae modicaion needed for he cae when A = f 0 g). 4 Proof of Theorem 1 Since he proof of Theorem follow exacly he ame ep a he proof of Theorem 1 wih he appropriae change, we preen below only he proof of Theorem 1 (for more deail on he proof of Theorem for he cae of A = f 0g ee [11]). Sep 1: Deniion of and The riple (T r)igiven. Wihou lo of generaliy, aume <1. Le R r come from FAforward compleene of he weak average (Definiion 1) and le come from equi-(eenial) boundedne of F (Deniion 4). From he definiion of A-weak average (in paricular becaue f A wa i A-locally Lipchiz - ee Deniion 1) i follow ha here exi L>0uch ha, for all (x y) aifying jxj A R +1, jyj A R, and for all w aifying jwj we have f A wa (x w) ; f A wa(y w) Ljx ; yj : (14) Then dene and le be uch ha f x w ; f := exp ; 1 LT x w 0 (15) 0:15 L e LT ; 1 (16) for all jj, jwj, jxj A R +1, 0. Condiion 1(c) of Theorem guaranee ha uch alway exi. In preparaion for dening, le KLand T > 0 come from Deniion 1 and le e T T aify max fr +1 g e T 1T exp(lt ) : (17) According o 1(c) and 1(d) condiion of Theorem and Deniion 1, he quaniy n B := up max jf( x w )j jfwa(y A w)j 8 >< >: 0 9 jxj A R +1 jyj A R jwj jj i nie. Dene g (= ew) :=ew T 1 >= > (18) f (= ew ew 3 0) ; f A wa ( ew ew 3 ) (19) he ew i being componen, of appropriae dimenion, of a vecor ew. Dene F e o be he e of funcion ew() = 4 ew 1() ew () ew 3 () 3 5 (0) o p. 4

5 uch ha ew 3 () F, and ew 1 (), ew () are aboluely coninuou wih jj ew 1 jj 1 1, jj _ ew 1 jj 1 B, jjew jj 1 R+1, and jj _ ew jj 1 B. Le >0 be uch ha, for all ew e F and all i 0, if [ i i + ] hen jexp[l( i ; )]g (= ew()) ; g (= ew( i ))j (1) i le han 1T exp(lt. Thi exi ince g ) i coninuou in ew uniformly in, F e i equiuniformly coninuou and for = i he quaniy being bounded in (1) i zero. Then dene := min ( T e ) : () 1(B T e exp(lt )) Sep : Comparion of oluion Le (0 ), jj <, jyj A r, 0, w F and conider any x uch ha jx ; yj. Dene e () :=x ( x w) ; y( ; y w) (3) and noe ha je ()j < 1 < 1. If je ()j < 1 for all [ + T ] hen dene = + T. Oherwie, dene =inff [ + T ]:je ()j =1g : (4) Noe ha > and e () andx ( x w) are dened and aboluely coninuou on [ ]. Le ew() F e be uch ha, for all [ ], 4 ew 1() ew () ew 3 () 3 5 = 4 e () x ( x w) w() 3 5 : (5) Such a ew() e F exi ince, for all [ ], je ()j 1. Indeed, ince jy( ; y w)j A R for all [ + T ], i follow ha jx ( x w)j A R + 1 for all [ ]. In urn, i follow from (18) ha, for almo all [ ], j _e ()j B and j _x ( x w)j B. For almo all [ ] we have (dropping he argumen of ignal for noaional convenience) _e = f (= x w ) ; f A wa(y w) = [f A wa(x w) ; f A wa(y w)] + f (= x w 0) ; f A wa(x w) +[f (= x w ) ; f (= x w 0)] : (6) For he calar-valued funcion V () := 1 et ()e (), which i alo aboluely coninuou on [ ], we have V ( ) 1 = exp(;lt ) 8 and, for almo all [ ], _V LV ; f x w 0 ; fwa(x A w) +e T L + 8fexp(LT );1g = LV + g (= ew()) + L 8 fexp(lt ) ; 1g (7) where we have ued he deniion of, (16), (19) and (5). By andard comparion heorem i follow ha for all [ ], V () exp(lt )V ()+ 8 + Z exp[l( ; )]g (= ew()) d (8) 4 + Z exp[l( ; )]g (= ew()) d : Fix [ ] and dene k o be he large nonnegaive ineger uch ha k ; T e. For i =0 ::: k, dene i = + ie T and noe ha, from he deniion of k and (), we have ; k e T 1(B exp(lt )) i+1 ; i = e T : (9) We pli he inerval of inegraion in (8) uing he ime i o obain Z V () k;1z i+1 X i=0 k exp[l( ; )]g (= ew()) d i exp[l( ; )]g (= ew()) d : (30) I follow from he deniion of e F, (18), (19) and (9) ha he r inegral on he righ-hand ide of (30) i bounded a Z k e L(;) g ew() d e TBe LT 1 : (31) To bound he econd inegral on he righ-hand ide of (30), we pli he inegrand ino wo piece: one ha will be ued o exploi he wo p. 5

6 ime-cale behavior of g a a funcion of and he oher ha will be ued o exploi he coninuiy properie of g wih repec o. In he calculaion ha follow, we will ue he bound (3), which i a reul of he deniion of e F, (5), (17), (19) and Holder' inequaliy. Then we ue he bound on (1) and inequaliy (9) and he fac ha k ~ T ; T o obain (33). Combining (30), (31) and (33), i follow ha V () for all [ ]. Since V () = 1 et ()e (), i follow ha je ()j < 1 for all [ ]. From he deniion of,ifollow ha = + T o ha je ()j for all [ + T ]. Thi eablihe he reul. ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; Z i+1 i k;1z i+1 X i=0 i = g (= ew( i )) d k;1 X i=0 = exp[l( ; )]g exp[l( ; i )] k exp(lt ) ewt 1 ( i ) Z i+ ~ T i f (= ew ( i ) ew 3 ( i ) 0) ; f A wa ( ew ( i ) ew 3 ( i )) d ~ T max fr +1 g ~ T ~ T ew() Z i+1 i e T d g ew( i) 1T exp(lt ) + e T 1T exp(lt ) Z i+1 n d + exp[l( i ; )]g i 1T exp(lt ) : (3) o ; g ew( i) d ew() 6 : (33) ;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;;; Reference [1] B.D.O. Anderon, R.R. Bimead, C.R. Johnon Jr., P.V. Kokoovic, R. L. Kou, I.M.Y. Mareel, L. Praly and B.D. Riedle, Sabiliy of adapive yem: paiviy and averaging analyi. MIT Pre: Cambridge, Maachue, [] N.N. Bogoliubo and Y.A. Miropolkii, Aympoic Mehod in he Theory of Nonlinear Ocillaor, Gordon & Breach, New York, [3] J. Hale, Ordinary dierenial equaion, Rober, E. Krieger Pub. Co., [4] H.K. Khalil, Nonlinear yem. Prenice- Hall: New Jerey, [5] V. Lakhmikanham and S. Leela, Dierenial and inegral inequaliie: Theory and applicaion, Par 1. Academic Pre: New York, [6] B. Lehman and S. P. Weibel, Parial averaging of funcional dierenial equaion, Proc. 38 h Conf. Deci. Conr., Phoenix, Arizona, 1999, pp [7] D. Neic and A.R. Teel, Inpu-o-ae abiliy for ime-varying nonlinear yem via averaging, ubmied for publicaion, [8] J. Peueman and D. Aeyel, A noe on exponenial abiliy of nonlinear ime-varying differenial equaion and parial averaging, Proc. 3rd Porugee Conference on Auomaic Conrol: Conrolo '98, Sep. 9-11, Coimbra, Porugal, pp. 1-6, [9] J.A.Sander and F.Verhul, Averaging mehod in nonlinear dynamical yem. Springer-Verlag: New York, [10] A.R. Teel, J. Peueman and D. Aeyel. Semi-global pracical aympoic abiliy and averaging, Sy. Conr. Le., 37 (1999), pp [11] A.R. Teel and D. Neic. Semi-global pracical aympoic abiliy and averaging, o appear in Sy. Conr. Le., (000). [1] V.M.Voloov, Averaging in yem of ordinary dierenial equaion, Ruian Mah. Survey, 17, 196. p. 6