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, 93106-9560 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 F4960-00-1-0106. 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
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. 190-195] 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.
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
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
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 () 4 + + 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
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, 1986. [] N.N. Bogoliubo and Y.A. Miropolkii, Aympoic Mehod in he Theory of Nonlinear Ocillaor, Gordon & Breach, New York, 1961. [3] J. Hale, Ordinary dierenial equaion, Rober, E. Krieger Pub. Co., 1980. [4] H.K. Khalil, Nonlinear yem. Prenice- Hall: New Jerey, 1996. [5] V. Lakhmikanham and S. Leela, Dierenial and inegral inequaliie: Theory and applicaion, Par 1. Academic Pre: New York, 1969. [6] B. Lehman and S. P. Weibel, Parial averaging of funcional dierenial equaion, Proc. 38 h Conf. Deci. Conr., Phoenix, Arizona, 1999, pp. 4684-4689. [7] D. Neic and A.R. Teel, Inpu-o-ae abiliy for ime-varying nonlinear yem via averaging, ubmied for publicaion, 1999. [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, 1998. [9] J.A.Sander and F.Verhul, Averaging mehod in nonlinear dynamical yem. Springer-Verlag: New York, 1985. [10] A.R. Teel, J. Peueman and D. Aeyel. Semi-global pracical aympoic abiliy and averaging, Sy. Conr. Le., 37 (1999), pp. 39-334. [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