SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY. J. E. Palomar Tarancón. A.M.S. (MOS) Subject Classification Codes. 44A05, 34A99, 18B99

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1 Elecronic Journal: Souhwes Journal of Pure an Applie Mahemaics Inerne: hp://raler.cameron.eu/swjpam.hml ISSN Issue 2 December 2003 pp Submie: February Publishe: December SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY J. E. Palomar Tarancón Absrac. In his paper he auhor inrouces he concep of smooher. Roughly speaking a smooher is a pair s K consising of a coninuous map s sening each poin p of is omain ino a close neighborhoo V p of p an an operaor K ha ransforms any funcion f ino anoher Kf being smooher han f. This propery allows us o remove he effec of a perurbaion P from he soluions of an auonomous sysem he vecor fiel of which is moifie by P. A.M.S. MOS Subjec Classificaion Coes. 44A05 34A99 18B99 Key Wors an Phrases. Smooher co-algebra inegral ransform perurbaion 0. Inroucion The main aim of his paper consiss of inroucing he concep of smooher ogeher wih an applicaion in ifferenial equaion heory. Roughly speaking a smooher is an operaor ransforming an arbirary funcion f 1 ino a similar one f 2 being smooher han f 1. In general smoohers perform inegral ransforms in funcion spaces. To ge a firs approximaion o he smooher concep consier he following facs. Le y = fx be any inegrable funcion efine in R an σ : R C a map such ha C sans for he collecion of all close subses of R he inerior of each of which is non-voi. For every x R le λ x be any non-negaive real number an le σx = [x λ x x + λ x ]. Wih hese assumpions consier he linear ransform K efine as follows. If λ x 0 is a finie number hen Kfx = 1 λx fx + τ τ 2λ x λ x Deparmen of Mahemaics Insiu Jaume I C/. Europa 2-3E BurrianaCasellón Spain Aress: [email protected] c 2003 Cameron Universiy 36 Typese by AMS-TEX

2 SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY 37 Figure 1. Conversely for λ x infinie 1 k Kfx = fx + τ τ k 2k k Finally if λ x = 0 hen Kfx = k 0 2k k k fx + τ τ = fx Of course assuming ha such a i exiss. Thus he inegral ransform K sens he value of fx a x ino he average of all values of fx in a close neighborhoo [x λ x x + λ x ] of x. Obviously in general he ransform Kfx is smooher han fx. To see his fac consier he following cases. If for every x σx = R hen x R : λ x = an Kfx is a consan funcion ha is he smoohes one ha can be buil. If for every x R σx = {x} hen x : λ x = 0 herefore Kfx = fx an consequenly boh funcions have he same smoohness egree. Thus beween boh exreme cases one can buil several egrees of smoohness. In he former example wha we erm smooher is nohing bu he pair σ K.

3 38 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS Perhaps he mos naural smooher applicaion consiss of removing from a given funcion he noise arising from some perurbaion. For insance consier he curves C1 an C2 in Figure 1. Suppose ha he ifferences beween C1 an C2 are consequence of some perurbaion working over C2. If boh curves are he plos of wo funcions f 1 x an f 2 x respecively in general one can buil a smooher σ K such ha Kf 1 x = f 2 x. Now consier a vecor fiel X an he resul Y of a perurbaion P working over X an assume σ K o saisfy he relaion KY = X. If x an y are he general soluions for he orinary ifferenial equaions x = X x an y = Y y respecively hen we shall say he smooher σ K o be compaible wih he vecor fiel Y provie ha he following relaion hols: Ky = x. Thus one can obain he corresponing perurbaion-free funcion from soluions of y = Y y using he smooh vecor fiel X = KY insea. The main aim of his paper consiss of invesigaing a compaibiliy crierion. 1. Smoohers Le T op san for he caegory of all opological spaces an le N : T op T op be he enofuncor carrying each objec E T in T op ino he opological space NE T = E \ {{ }} T he unerlying se of which E \ {{ }} consiss of all nonempy subses of E. Le T be he opology a subbase S of which is efine as follows. Denoe by C he collecion of all T -close subses of E an for every pair A B C T le K AB = {C E A C B}. Wih hese assumpions efine he subbase S as follows. S = {K AB A B C T } Obviously if A B hen K AB =. Likewise if A = an B = E hen K AB = E \ {{ }}. Le he arrow-map of N be he law sening each coninuous map f : E 1 T 1 E 2 T 2 ino he map Nf carrying each subse A E 1 ino f[a] E 2. I is no ifficul o see Nf o be a coninuous map wih respec o he associae opology T. Definiion Le calgn enoe he concree caegory of N-co-algebras. Thus every objec in calgn is a pair E T σ ET consising of a opological space E T ogeher wih a coninuous map σ ET : E T NE T. Recall ha a coninuous mapping f : E 1 T 1 E 2 T 2 is a morphism in calgn from E 1 T 1 σ E1T 1 ino E2 T 1 σ E2T 2 provie ha he following iagram commues E 1 T 1 σ E1 T 1 NE 1 T 1 f E 2 T 2 σ E2 T 2 Nf NE 2 T 2

4 SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY 39 Now le T vec be he opological vecor space caegory an le T opv ec enoe he caegory he objecs of which are proucs of he form NE T C 0 E V T where V T is a opological vecor space an T he poinwise opology for he se C 0 E V of all coninuous maps from E T ino V T ; where he funcor : T vec T op forges he vecor space srucure an preserves he opological one. In aiion a T opv ec-morphism wih omain NE 1 T 1 C 0 E 1 V T1 an coomain NE 2 T 2 C 0 E 2 V T2 is of he form Nf g where f lies in hom T op E 1 E 2 an g is a coninuous mapping wih omain C 0 E 1 V an coomain C 0 E 2 V. Given any opological space E T le P ET : T vec T opv ec enoe he funcor carrying each T vec-objec V T ino he prouc NE T C 0 E V T an sening every T vec-morphism f : V 1 T 1 V 2 T 2 ino I f ; where f = hom T op E T f sans for he morphism carrying each g C 0 E V 1 ino f g C 0 E V 2 an as usual hom T op E T enoes he covarian hom-funcor. Finally le AlgP ET enoe he caegory of P ET -algebras ha is each objec is a pair of he form V T K VT where K VT : P ET V T = NE T C 0 E V T V T is a coninuous map. In aiion a given coninuous linear mapping f : V 1 T 1 V 2 T 2 is an AlgP ET -morphism whenever he following quarangle commues NE T C 0 E V 1 T1 K V1 T 1 V 1 T 1 I hom T opet f NE T C 0 E V 2 T2 K V2 T 2 f V 2 T 2 Definiion A smooher will be any pair E T σet V T KVT such ha E T σ ET is a co-algebra lying in calgn an V T KVT is an algebra in AlgP ET saisfying he following coniions. a For every p E: p σ ET p. b For every p f E C 0 E V : K VT σet p f C Nf σ ET p

5 40 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS where for any subse A E C A enoes he convex cover of A. c K VT is linear wih respec o is secon argumen ha is o say for every couple of scalars α β an each pair of maps f g he following hols K VT σet pαf + βg = αk VT σet p f + βk VT σet p g Transformaion associae o a smooher. Given a homeomorphism ϕ : E T V T a smooher S = E T σ ET V T KVT inuces a ransformaion S ϕ carrying each poin p E ino ϕ 1 K VT σet p ϕ which will be sai o be inuce by S. Likewise for every one-parameer coninuous map family h : E I E R E one can efine he inuce ransformaion by S hp p = ϕ 1 K VT σet hp ϕ Orering. Smoohers form a caegory Smr he morphism-class of which consiss of every calgn-morphism f : E 1 T 1 E 2 T 2 such ha he following quarangle commues NE1 T 1 C 0 E 1 V K VT V T Nf hom T opf VT NE2 T 2 C 0 E 2 V KVT I V T where hom T op V T : T op T op op sans for he conravarian homfuncor. Regaring Smr as a concree caegory over Se via he forgeful funcor such ha E T σet V T KVT E wih he obvious arrow-map in each fibre one can efine an orering as follows. For any smooher E T σ ET V T KVT le Ω E T σ ET V T KVT enoe he inersecion of all opologies for E conaining he se family K = { σ ET p p E } hen E1 T 1 σ E1T 1 V T KVT E2 T 2 σ E2T 2 V T KVT

6 SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY 41 if an only if he opology Ω E 1 T 1 σ E1T 1 V T KVT is finer han he opology Ω E 2 T 2 σ E2T 2 V T KVT. For a maximal elemen he opology Ω E T σ ET V T KVT mus be iniscree. In his case for every p in E σ ET p = E an consequenly for every p q E K ET σ ET p ϕ = K ET E ϕ = Kσ ET q ϕ herefore S ϕ hp = ϕ 1 K VT σet hp ϕ ransforms hp ino a consan map which is he smoohes funcion one can buil. Conversely a minimal elemen correspons o he iscree opology. In his case by virue of boh coniions a an b he ransformaion is he ieniy so hen hp remains unalere. Beween boh exremes one can buil several egrees of smoohness. 1.1 Smoohers in smooh manifols.. Le M A n be a smooh manifol ϕ : U M R n a char an T he inuce opology for U. Henceforh he pair U T will be assume o be a Hausorff space. In he mos naural way one can buil a smooher S = U T σ UT R n T K R n T over U T he associae se of coninuous maps C 0 U R n conains each smooh one like he iffeomorphism associae o each char. For hose smooh manifols such ha for each p M n each angen space T p is isomorphic o R n ha is o say here is an isomorphism λ p : T p R n one can associae a map ω X : U R n o every smooh vecor fiel X leing p U : ω X p = λ p Xp Accoringly he image of he vecor fiel X uner S is λ p Y = KR n T σut p ω X herefore Y = λ 1 p KRn T σut p ω X From he former equaions i follows immeiaely ha if h : U U is he oneparameer group associae o X hen ω X p = λ p X ϕ h p ϕp p = R n 0 accoringly K Rn T σut p ω X = K R n T σut p ϕ h KR n T σut p ϕ 0

7 42 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS Definiion Le X be a smooh vecor fiel he coorinaes of which are X 1... X n an consier he ifferenial equaion x1 γp = X 1 x 1 γp x 2 γp x2 γp = X 2 x 1 γp x 2 γp... xn γp = X n x 1 γp x 2 γp... where he iffereniable curve γ : I R U is assume o be soluion of he former equaion for he iniial value γp 0 = p. Say a smooher S = U T σut R n T K R n T o be compaible wih X provie ha he curve y ρp = K R n T σut γp ϕ is soluion of he equaion y1 ρp = Y 1 y 1 ρp y 2 ρp y2 ρp = Y 2 y 1 ρp y 2 ρp... yn ρp = Y n y 1 ρp y 2 ρp... where q = S ϕ yp 0 = S ϕ p an Y 1 Y 2... Y n = K Rn T σut p ω X Obviously if p is a fixe poin for S ϕ hen yq an xp are soluions of an respecively for he same iniial value p = xp 0 = yp 0. Remark. If p = q ha is o say if p is a fixe-poin for S ϕ hen from Definiion he relaions p U : xp C Nϕ σ UT p an p U : yp C Nϕ σ UT p are rue herefore xp yp max q 1q 2 Cσ UT p ϕq 1 ϕq 2 From he former relaion one can buil some proximiy crieria. If he maximum isance among poins in any se σ UT p is boune ha is o say if here is δ > 0 such ha p U : max q 1q 2 Cσ UT p ϕq 1 ϕq 2 < δ hen > 0 : xp yp < δ

8 SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY 43 Proposiion Le U T σ UT be a co-algebra in calgn an for every poin p of U le µ p : σ UT [0 be a measure for σ UT p such ha he se σ UT p is µ p -measurable. If for every p U he following coniion hol a p σ UT p. b σ UT p is a close subse of U. c µ p σut p = 1 hen he pair U T σ UT V T KVT is a smooher where K VT σ UT p f = σ UT p f µ p Proof. Obviously K VT is linear wih respec o is secon coorinae an by efiniion i saisfies coniion a in Definiion herefore i remains o be prove K VT o saisfy coniion b oo. I is a well-known fac ha each coorinae f j of any measurable funcion f is he i of a sequence { f j n n N } of sep-funcions each of which of he form m fn j = c j in χ E in i=1 such ha each of he E in is µ p -measurable an for every i N c j in = f j α i for some α i E in besies n N : E in E jn = i j an m i=1 E in = σ UT p. In aiion m f j µ p = c j in µ pχ Ein σ UT p Now from saemen c i follows ha n i= n N : m µ p χ Ein = µ p σ UT p = 1 i=1 herefore n N : m c in µ p χ Ein C Nfσ UT p i=1 where c in = c 1 in c2 in. Finally since σ UT p is assume o be close he proposiion follows. 2. A compaibiliy crierion Alhough smoohers can be useful in several areas he aim of his paper is is applicaion in ifferenial equaions in which only hose smoohers being compaible wih he associae vecors are useful. To buil a compaibiliy crierion he following resul is a powerful ool.

9 44 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS Theorem Le M n A n be a smooh manifol an U ϕ a char. Le h : U U san for he one-parameer group associae o a smooh vecor fiel X an S = U T σ UT R n T K R n T a smooher. If he following relaion hols δ > 0 < δ : K Rn T σ UT h p ϕ = σut p ϕ h hen S is compaible wih X. Proof. Firs from we obain ha K Rn T yq = K R n T σut h p ϕ ϕq = yq 0 = K R n T σut p ϕ = ϕ S ϕ p Now i is no ifficul o see ha y q = K Rn T σut h p ϕ K Rn T σut h 0 p ϕ 0 an using he former equaion becomes an by coninuiy y q = K R n T σut p ϕ h K R n T 0 K R n T 0 0 K R n T σut p ϕ = σut p ϕ h ϕ 0 K R n T σ UT p ϕ h ϕ K Rn T = σ UT p ϕ h ϕ = ϕ h ϕ σ UT p 0 herefore aking ino accoun an y q = K R n T σut p ω X 0 accoringly if ϕ h p = x 1 p x 2 p... is soluion of for he iniial value p hen y 1 q y 2 q... = K R n T σut h p ϕ is soluion of he equaion for he iniial value q = S ϕ p being Y 1 Y 2... = K R n T σut p ω X

10 SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY 45 Corollary Wih he same coniions as in he preceing heorem if p is a fixe poin for S ϕ an xp = x 1 p x 2 p... is soluion of he iniial value problem xp = X xp xp 0 = ϕp hen yp = y 1 p y 2 p... = K R n T σut h p ϕ is soluion for he iniial value problem yp = Y yp = K R n T σut p ω X yp 0 = ϕp Remark The smooher efine in saisfies he coniions of he former corollary because each poin x y of R 2 is a fixe-poin. However he smooher S = U T σ UT R n T K R n T such ha he law σ UT sens each poin x 0 y 0 R 2 ino he subse } A x0y 0 = {x y R 2 x 0 x x yy0 0 y e x0 he associae ransform of which is K R 2 T : fx y 2 3y σ UT xy 0 2 3y y fx + u y + v uv = 1 v y 0 fx + u y + v uv is compaible wih he vecor 1 y an sens he poin x y ino x y his is o say S ϕ x y = x y Thus here is no fixe poin for S ϕ. Of course his smooher ransforms he soluion + x 0 y 0 e of he equaion x = 1 y = y for he iniial value x 0 y 0 a = 0 ino he soluion + x y 0e of he same equaion for he iniial value x y 0. Definiion Given a smooher S = U T σ UT R n T K R n T efine over a char U ϕ of a smooh manifol M A n an a smooh vecor fiel X efine he erivaive X S by he following expression X S p = K R n T σut h p ϕ K R n T σut p ϕ h 0

11 46 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS Corollary If X S = 0 hen S is compaible wih X. Proof. Obviously aking ino accoun Definiion from X S = 0 he saemen follows. Definiion Given a smooh vecor fiel X he associae one-parameer group of which is h : U U say a measure-fiel {µ p p U} o be invarian wih respec o X provie ha for every p U an each measurable subse E of σ UT p he following relaion hols R : µ hp Nh E = µ p E accoringly he measure µ p E remains unalere uner he one-parameer group h : U U associae o X. Remark In [4] i is shown ha for a wie class of vecor fiels each iffereniable-map ψ : U C saisfying he equaion n X ψp = X i x i ψx1 x 2... = 0 saisfies also he equaion i=1 ψh p = 0 accoringly ψh p oes no epen upon he parameer ; where for every coninuous funcion f f enoes he maximal exension by coninuiy. Thus an invariance crierion can consis of proving he exisence of a iffereniable map ψ E : U R for each measurable subse E U such ha { ψe p = µ p E X ψ E p = 0 Theorem If a measure fiel {µ p p U} is invarian wih respec o X an for each R he member of corresponing one-parameer group h : U U is a calgn-morphism hen he smooher S = U T σ UT R n T K Rn T such ha K VT σut p ϕ = ϕ µ p is compaible wih X. σ UT p Proof. Firs because for each R he map h : U U is assume o be a calgn-morphism hen by virue of we have ha K VT σut h p ϕ = K VT Nh σut p ϕ = Nh σ UT p ϕ µ p

12 SMOOTHERS AND THEIR APPLICATIONS IN AUTONOMOUS SYSTEM THEORY 47 an because Nh σut p = { h q q σ UT p } hen Nh σ UT p ϕ µ p = σ UT p ϕ h µ hp herefore since he invariance of {µ p p U} wih respec o X is assume hen for every subse E σ UT p he following relaion hols µ hp Nh E = µ p E herefore from i follows ha ϕ h µ hp = ϕ h µ p σ UT p σ UT p consequenly K VT σut h p ϕ = ϕ h µ p = K VT σut p ϕ h σ UT p accoringly he smooher S saisfies he coniions of he preceing heorem. Example Consier he iniial value problem x = y = 0.1 cos x x0 y0 = x0 y 0 he soluion of which is { x = x0 + y = y sin + x 0 sinx 0 where we are assuming he funcion 0.1 cos x in he secon equaion of o be he consequence of a perurbaion working over he vecor fiel 1 0. The map σ sening each x y R 2 ino he close se [x π x + π] [y 1 y + 1] ogeher wih he operaor K efine as follows K : f 1 x y f 2 x y 1 1 π f 1 x + u y + vuv 4π 1 π 1 1 π f 2 x + u y + vuv 4π 1 π

13 48 SOUTHWEST JOURNAL OF PURE AND APPLIED MATHEMATICS form a smooher such ha K ransforms he vecor cosx of he equaion ino he perurbaion-free vecor 1 0 herefore i ransforms also ino he following iniial value problem x = y = 0 x0 y0 = x0 y 0 Now i is no ifficul o see K o be compaible wih he vecor fiel of he equaion herefore K ransforms also he soluion of ino he soluion of as one can see in he following equaliy 1 1 π x 0 + u + : uv = x 0 + 4π 1 π π y0 + v sinx 0 + u + 4π 1 π sinx 0 + u uv = y 0 an { x = x0 + y = y 0 is nohing bu he general soluion of Thus K sens ino an also sens he general soluion of ino he perurbaion-free soluion of References 1. Aámek J. Herrlich an H. Srecker G. E. Absrac an concree caegories. The joy of cas. Pure an Applie Mahemaics A Wiley-Inerscience Publicaion. John Wiley & Sons Inc. New York xiv+482 pp Kobayashi S. an Nomizu K. Founaions of Differenial Geomery Inerscience Publish Köhe G. Topological vecor spaces Springer-Verlag New York Palomar Tarancón J. E. Vecor spaces over funcion fiels. Vecor spaces over analyic funcion fiels being associae o orinary ifferenial equaions Souhwes J. Pure Appl. Mah. no. 2 pp