Persistence of Excitation in Reproducing Kernel Hilbert Spaces, Positive Limit Sets, and Smooth Manifolds

P ersiste nce of Excitation in Repro ducing Ker nel Hil b er t Spaces, P ositi v e Limit S ets, an d Smo oth Manifold s Andrew J. Kur dila a , Jia Guo a , Sa i T ej P a ruc huri a , P a rag Bobad e b a Dep artment of Me chanic al Engine ering, Vir gi nia T e ch, Blacksbur g, V A 24060, USA b Dep artment of A er osp ac e Engine ering, University of Michigan, An n Arb or, MI 48109, USA Abstract This paper studies the relationship b etw een th e p ositiv e limit sets of conti nuous semiflo ws and th e newly introdu ced definition of p ersisten tly excited (PE) sets and associated sub spaces of repro ducing kernel Hilb ert (RKH) spaces. It is shown t hat if the RKH space contains a ric h collection of cut-off functions, p ersistently excited sets are contained as subsets of the p ositive limit set of th e semiflo w. The p ap er d emonstrates how the new PE condition can b e used to guarantee converg ence of function estimates in the RKH space embedd ing metho d for adaptive estimation. In particular, the pap er is applied to un certain O DE systems with positive limit sets giv en by certai n types of smooth manifolds, and it establishes con vergence of adaptiv e function estimates over the manifolds. Key wor ds: Adaptive Estimation, R eprod ucing Kernel, Persistence of Excitation 1 In tro duction In this pap er w e study the metho d of repro ducing ker- nel Hilber t (RKH) spac e embedding for adaptive es- timation of uncertain, or unknown, dynamic systems that are gov erned by systems of coupled, no nlinear or - dinary differential equations (ODEs). The RKH embed- ding metho d for ada ptive estimation has been intro- duced in [1,2,3]. This g e neral formulation constructs es- timates in R d of the state o f the unkno wn governing ODEs as well as estimates of an unknown function c o n- tained in the RKH space H that characterizes the un- certain gov er ning ODEs. This pap e r investigates several unanswered questio ns r elated to the new no tion of p er- sistency of exc itation (PE) that has b een introduced in the latter tw o o f these three pap ers. W e derive rela tion- ships betw een the PE co ndition over an indexing set Ω that is a subset of the state spa ce X and the p o sitive limit sets of semiflo ws ov er X . W e also construct or se- lect go o d kernels that define the RKH s pace H in ap- plications where the governing semiflows exhibit certain asymptotic str uctural pro per ties. These latter pr op er- ties are expressed in ter ms of PE conditions ov e r some classes o f smo o th manifolds . Email addr esses: kurdila@vt.edu (Andrew J. Kurdila), jguo18@vt. edu (Jia Guo), saitejp@vt. edu (Sai T ej P aruchuri), paragsb@umi ch.edu (Pa rag Bobade). The RKH em b edding method g enerates a distributed parameter system, a nd its asso ciated estimates evolve in the g enerally infinite dimensional space R d × H . Ther e are many nontrivial questions ab out appr oximations and realizable implemen tatio ns of the metho d, and some study of the c o nv ergence of finite dimensional approxi- mations of so lutions o f the RKH embedding equatio ns is given in [2,3]. In this shor t pap er, we only consider the conv erg e nce of the es timates ge ne r ated by the gov ern- ing DPS system in the infinite dimensional state spa ce R d × H that defines the RKH embedding formulation. The curre n t inv es tigation can b e viewed as providing needed insight and intu itio n into the structur e o f the solutions of the RKH em b edding equatio ns, whic h is m uch needed for the effective choice of approximating subspaces in pra c tica l implementations. 1.1 A daptive Estimation for Unc ertain Nonline ar ODEs A common setup for es tima tion o f uncertain nonlin- ear s y stems starts with an ordinary differential equation that can b e decomp ose d in to known and unkno wn par ts, ˙ x ( t ) = g 0 ( x ( t )) + g ( x ( t )) , x (0) = x 0 (1) with x ( t ) ∈ R d for t ∈ R + , g 0 : R d → R d a known func- tion, and g : R d → R d an unknown function. One imp or- tant problem of adaptive e s timation for such a nonlinear system is to use the full state observ ations { x ( t ) } t ∈ R + to co nstruct a n evolution law for a state estimate ˆ x ( t ) Preprint subm itted t o Au tomatica 27 Sept em b er 2019 that approximates x ( t ), in the sense that ˆ x ( t ) → x ( t ) as t → ∞ . In the lang uage of ada ptive estimation this is re- ferred to a s convergence o f sta te es timates. A ca nonical mo del es timator for the or iginal equation migh t choose the evolution law for the estima te to b e ˙ ˆ x ( t ) = A ˆ x ( t ) + g 0 ( x ( t )) + ˆ g ( t, x ( t )) − Ax ( t ) (2) for a known matrix A ∈ R d , althoug h many alter natives exist of course. Here ˆ g ( t ) := ˆ g ( t, · ) is an estimate o f the unknown function g . O n defining the state er ror ˜ x ( t ) = x ( t ) − ˆ x ( t ) and the function error ˜ g ( t ) := ˜ g ( t, · ) := g ( · ) − ˆ g ( t, · ), the asso cia ted erro r eq uation is obta ined a s ˙ ˜ x ( t ) = A ˜ x ( t ) + ˜ g ( t, x ( t )) . (3) A t a bar e minimum then, adaptiv e es timation metho ds for the ab ove uncertain nonlinear ODEs must g uarantee that tra jectories of this err or equation conv erg e to zero. It is usually c o nsiderably more difficult to guara n tee tha t the time-v arying function estimate ˆ g ( t, · ) : R d → R con- verges in the sense that ˆ g ( t, · ) → g as t → ∞ . It is this latter pro blem that is the pr imary co ncern of this pap er. T o gain some apprecia tion of the issues a nd nuances aris- ing in the function estimation problem, we consider tw o examples. Figures 1 and 2 depict the phase p ortaits of the uncertain systems studied in Examples 1 and 2, re- sp ectively . These figures also include plots of the er ror in function estimates obtained b y the RKH embedding techn iques with the kernel of the RKH space selected as describ ed in 11 . Example 1 The firs t example is a c ase of a sup er critic al Hopf bifur c ation, whi ch c an b e found in many tex tb o oks on dynamic al syst ems [4,5]. The syst em e quations ar e given by ( ˙ x 1 ˙ x 2 ) = ( x 2 + x 1 (1 − x 2 1 − x 2 2 ) − x 1 + x 2 (1 − x 2 1 − x 2 2 ) ) . (4) Her e we define g 0 ( x ) := { x 2 + x 1 (1 − x 2 1 − x 2 2 ) , 0 } T , g ( x ) := { 0 , − x 1 + x 2 (1 − x 2 1 − x 2 2 ) } in Equation 1. The figur es b elow make cle ar that t he p ositive limit set ω + ( x 0 ) is the cir cle S 1 for al l x 0 ∈ R 2 for t his dynamic al system. When the metho d of RKH emb e dding is applie d to this unc ertain nonline ar syst em, we c an obtain est imates ˆ g ( t ) of g whose err or is depicte d in Figur es 1(b, c). These estimates have b e en c onstru cte d fr om finite dimensional appr oximations as discusse d in [2] using b asis functions that ar e a c ol le ction of (extr insic) Sob olev-Matern kernels discusse d in Cor ol lary 11 c enter e d over the p ositive limit set. Example 2 In this example, the dynamic al system c on- tains a homo clinic lo op. The ex ample is studie d in detail in [4 ]. The governing e quations ar e ( ˙ x 1 ˙ x 2 ) = ( 2 x 2 2 x 1 − 3 x 2 1 + λx 2 ( x 3 1 − x 2 1 + x 2 2 ) ) (5) In this example we define g 0 ( x ) := { 2 x 2 , 0 } T and g ( x ) := { 0 , 2 x 1 − 3 x 2 1 + λx 2 ( x 3 1 − x 2 1 + x 2 2 ) } T in Equation 1. Aga in, applic at ion of the R KH emb e dding metho d of adaptive estimation to this pr oblem c an yield appr oximations ˆ g ( t ) of g with err or depicte d in Figur e 2(b,c). These estimates have b e en c onstructe d fr om finite dimensional appr oxi- mations as discusse d in [2] using b asis fun ctions that ar e a c ol le ction of (extrinsic) Sob olev - Matern kernels dis- cusse d in Cor ol lary 11 c enter e d over the p ositive limit set. Several observ ations ab out these t wo e xamples of ap- plication of the RKH e mbedding metho d are notewor- th y and motiv ate this pap er. In each case, the g ov erning equations hav e the form o f the no nlinear O DEs g iven in Equations 1, a nd are have error equa tio ns o f the fo rm in Equation 14 tha t is studied in deta il in this pap er. The first important o bs erv ation to make fro m the examples is to note that the p ositive orbit Γ + ( x 0 ) := ∪ t ∈ R + x ( t ) is the only data that is used to construct estimates of the unknown function. If we were interested in so me offline, optimization-based estimate of an unknown function, it would come as no sur prise that its estimates cons ist of functions that ar e supp orted o n or nea r the prescrib ed data. W e will see that, ro ug hly s pea king, conv erge nc e of the RKH em b edding metho d is guaranteed by a newly int r o duced PE condition and estimates of the unknown function a re built ov er r egions of the state spa c e where tra jectories are in some sense “concentrated.” Here the notion of concent ration is understo o d in terms of the po sitive limit set ω + ( x 0 ), w hich is known to attract tra - jectories if the or bit is precompa ct. [6] Moreov e r, b oth of the po sitive limit sets in the exa mples are s tr iking and e x hibit conside r able struc tur e: they c an often b e interpreted a s manifolds. In Example 1 shown in Figure 1, glo bal solutions of the gov erning equations exist for every initial condition in R d . All tra jectories conv erg e to the po sitive limit set, which happ ens to be the canonica l connected, c ompact, Riemannian mani- fold, S 1 . The flows gener ated for v arious par a meters in λ in Example 2 exhibit more diverse qualitative limit- ing b ehaviors. As shown in Figure 2 (a) when λ = 0 , the homo clinic loo p encir cles a stable r egion. T ra jectories inside this r egion are all limit cycles. F or these initial conditions, the p ositive limit sets ar e smo oth, regular ly embedded submanifolds of R d . The form of these embed- ded ma nifolds is not as s imple as in Exa mple 1, that is, they are not o ne of the well-known, “iconic” manifolds. When λ < 0, the equilibrium ( x e , 0) for x e > 0 b ecomes unstable. It can b e shown that the ho mo c linic lo op b e- comes the ω -limit set of a ll the tr a jectories star ting from this r egion [4]. In either cas e , the examples illustrate a phenomenon that is common to many uncer tain estimation problems. While the observ ations Γ + ( x 0 ) = ∪ t ∈ R + x ( t ) are c o n- tained in R d , there is an under lying s et or manifold that suppo rts, approximately supp orts, or attracts the ob- served tra jectories. W e are interested in this pap er in understanding co nditions that e s tablish that the RK H embedding metho d “c o nv erges over” these under lying structures. 2 (a) Ph ase p ortrait (b) Error in F unction Estimate (c) Error Conto u r Fig. 1. Example 1 (a) Ph ase p ortrait (b) Error in F unction Estimate (c) Error Conto u r Fig. 2. Example 2 T o fr a me our discus sion of the RKH embedding metho d we briefly review the g eneral strategy of “linear-in- parameters ” (LIP) metho ds for adaptive estimation of uncertain nonlinear s ystems of ODEs. So-called LIP estimation might b est b e describ ed as a part of the tec h- nical folklor e for metho ds in adaptive estimation. This approach is ubiquitous in the a daptive estimation liter- ature and is a well-known too l a mo ng resea rchers who study this to pic . It is safe to say that the most p opula r versions o f ada ptiv e estimatio n for the ab ov e t y p e of un- certain nonlinear ODEs choo se the function estimate in terms of a linear- in- parameters representation ˆ g ( t, · ) = P n k =1 φ k ( · ) α k ( t ) = Φ T ( · ) α ( t ) with α k ( t ) a time-v arying parameter, φ k ( · ) : R d → R d a function for k = 1 , . . . , n , the vector α ( t ) := { α 1 ( t ) , . . . , α n ( t ) } T ∈ R n , and the matrix of functions Φ T ( · ) = [ φ 1 ( · ) , . . . , φ n ( · )]. Here the functions in Φ a re known as the re g ressor s, a co mmon term arising from applications in nonlinear re gres- sion. If the unknown function g has the r e pr esenta- tion g = P n k =1 φ k ( · ) α ∗ k = Φ T ( · ) α ∗ for some unknown constants { α ∗ k } k ≤ n , then the erro r in function esti- mates is ˜ g ( t, · ) = Φ( · ) ˜ α ( t ) with the pa rameter error ˜ α ( t ) := α ∗ − α ( t ) ∈ R n . In this case the err or in the func- tion estimates ˜ g ( t ) → 0 if the finite set of parameters error s conv erg e ˜ α ( t ) → 0 ∈ R n . It is fo r this rea son that the task o f e s timating functions in the usual LIP fra me- work reduces to questions of par ameter conv erg ence in R n . One o f the foundations of mo dern adaptive estimation for ODEs has b een r ecognition of the fact that p e rsis- tency of e xcitation conditions can b e sufficien t to guar- antee parameter conv er gence. The no tion of p ersis tence of excitation in its conven tional for m, tha t is, as it p e r - tains to the the ODE er ror Equations 14, is defined nex t. Definition 3 The r e gr essors Φ ar e p ersistently excite d by the p ositive orbit Γ + ( x 0 ) if ther e ar e p ositive c onst ant s γ 1 , γ 2 , T , and ∆ such that for e ach t ≥ T , γ 1 k α k 2 R d ≤ Z t +∆ t  α ⊤ Φ  x ( τ )  Φ  x ( τ )  ⊤ α  dτ ≤ γ 2 k α k 2 R d (6) for al l α ∈ R N . The pap ers [7,8,9,10,11] and a num b er o f standard texts [12,13,14,15] on ada ptive estimation make a careful study of this conditio n a nd how it facilitates a pro of that the para meter e r ror ˜ α ( t ) conv erges to z e ro as t → ∞ . In some cases it is to o muc h to ho p e that all the pa - rameters ˆ α ( t ) := { ˆ α 1 ( t ) , . . . , ˆ α n ( t ) } in the approxima- tions ˆ g ( t, · ) = P k =1 ,..., n φ k ( · ) ˆ α k ( t ) co nv erge. A means of weakening the ab ov e PE condition intro duces the notion of p artial p ersistency o f excitatio n. One version of the definition of a partia l P E condition mo difies the inequalities ab ov e and replaces them with the co ndition that γ 1 k P V v k 2 R n ≤ α T Z t +∆ t Φ( x ( τ ))Φ T ( x ( τ )) d τ · α ≤ γ 2 k P V v k 2 R n 3 with P V : R n → R n a pro jection onto a linear subspac e V ⊂ R n . This generaliza tion then can be used to guar an- tee, as a sp ecial case, that o nly certain of the co efficient estimates co n verge, but not all. As we will discuss in mor e detail sho r tly , the metho d of RKH embedding recasts the ab ov e a daptive estimation problem so that the state er rors ˜ x ( t ) a nd function er- rors ˜ g ( t ) := ˜ g ( t, · ) evolv e in a pro duct space having the form R d × H with H = H d a vector-v alued RK H space o f functions. The spa ce H is known as the hypothesis spa ce and its selec tio n is based on what cla ss of prior s or infor- mation seems r elev an t r e g arding the estimation proble m at hand. The precise form of the PE definition in this pap er, and the a sso ciated theor ems that dep end on it, are written for a mo del pr oblem with the vector-v alued function g := B f with f : R d → R a scalar v alued func- tion and B ∈ R d × 1 . This restriction does not seem to o severe, simplifies the notation co nsiderably , and conv e ys the underlying geometric r elationships be tween orbits Γ + ( x 0 ) of s emiflows star ting at x 0 ∈ X , pers is tency of sets Ω, and RKH spaces H Ω . Mor eov er, the extension to general vector-v alued functions w o uld pro ceed in pr inci- ple along the sa me lines a s the strategy in [16] used for consensus estimation. 1.2 Overview of New Results In either of the pap ers [2,3] some o f the standard ques- tions regarding RKH em b edding ha ve b een discussed such as e x istence of solutions and well-pos e dness, con- tin uo us dep endence on initial co nditions, as well as stability and conv erge nce o f finite dimensional approx- imations. In this pap er we fo cus primar ily on build- ing mor e intuition and insight regar ding the newly int r o duced notion of p ersistency of excitatio n in the RKH e mbedding metho d. Starting with an RKH space H X = span { K ( x, · ) | x ∈ X } of functions ov er X , w e then define for some indexing set Ω ⊆ X the s ubspace H Ω := spa n { K ( x, · ) | x ∈ Ω } . Note ca r efully that func- tions in H Ω are supp o rted on X , which is wh y Ω is referred to as the indexing set. This space is r elated to, but distinct fro m, the space R Ω ( H X ) tha t a re r estric- tions of functions in H X to the s ubset Ω. W e hav e the following definition of p ersistence of excita tio n for the RKH error Equations 1 5. Definition 4 The indexing set Ω and RKH sp ac e H Ω ar e p ersisten t ly excite d by the orbit Γ + ( x 0 ) if ther e ar e p ositive c onstants γ 1 , γ 2 , T , and ∆ such that for e ach t ≥ T , γ 1 k f k 2 H Ω ≤ Z t +∆ t  E ∗ x ( τ ) E x ( τ ) f , f  H X dτ ≤ γ 2 k f k 2 H Ω (7) Here E x : f 7→ f ( x ) is the ev alua tion functional at x and E ∗ x is its adjoint o per ator. The clas sical definition given above defines per sistency o f excitation for a sp ecific set o f regresso rs a nd the tra jectory of a semidynamical system. The new PE condition holds for an indexing set Ω ⊆ X , space of functions H Ω , and a tra jecto r y of a semidynamical sys tem. It should b e noted tha t the PE co ndition a bove is over a set Ω ⊂ X , which may or may no t b e the entire state space X . It this s e ns e it bear s s ome resem bla nce o f the pa rtial PE conditions that a re defined ov er s ubspaces o f para meters in R n . The similarity in form is all the mo re apparent when we no te that k · k H Ω := k P Ω ( · ) k H X with P Ω the H X -orthogo nal pro jection onto H Ω : the closed subspace H Ω is endow ed with the norm it inherits fro m H X . It should also be po int ed out that b oth the set Ω and the kernel K X (that determines H X , and ther e fore determines H Ω ⊆ H X ) are free to b e selected when tr ying to apply the ab ove PE condition in the metho d of RKH embedding. Int uitively , we exp ect so me kernels a re more useful than others in the RKH embedding metho d, and one o f the primary thrusts of this pap er is to explore the alterna- tives. As we will see, we obtain a strong conclusion ab out what type o f indexing sets Ω are PE when we r estrict attent ion to kernels that define function spaces that are go o d at separating imp ortant subsets of X . There ar e many w ays to think ab out how well the functions in a space H X separates po int s or sets. W e find tha t o ne im- po rtant class o f RKH spac e s consists of functions that feature a rich set of (pos s ibly smo oth) cut-off or bump functions. Our first prima ry result in T heo rem 9 is that if the RKH space do es indeed co n ta in a r ich co llection of these functions, then we have the following implication, “Ω and H Ω are PE ” = ⇒ Ω ⊆ ω + ( x 0 ) , (8) with ω + ( x 0 ) the positive limit set o f a tra jectory start- ing at x 0 . In o ther w or ds in ter ms of the new definition of PE, if a tra jectory Γ + ( x 0 ) p ersis ten tly excites a set Ω, the set Ω is contained in the p ositive limit set ω + ( x 0 ). This r esult provides nov el insig ht int o the s tr ucture o f this t yp e of p e r sistently excited systems : PE sets a re not transient but rather consist of p oints whose neigh- bo rho o ds are visited b y the tr a jectory infinitely o ften. In fact, a bit more is actually required as illustr ated in Theorem 7: the “time of visitation” is b ounded b elow in a cer tain sense. This intuition should b e compa red with the in ter pretations o f the usual Definition 3: a v e ctor signal t 7→ Φ( x ( t )) ∈ R n is (partially) P E if on a verage it vis its all directio ns in (a subspace of ) R n . W e should a lso emphasize at this p oint that while the int ent of this pap er is to inform a nd enhance o ur under - standing of the RKH embedding metho d, the result in Equation 8 is no t dep endent on the fact tha t the tra - jectory under study t 7→ x ( t ) ∈ R d happ ens to b e the solution o f our model ODE pro blem in Equa tion 2. W e hav e worked to express the co ndition in E quation 8 in very general ter ms. As lo ng a s the P E co ndition holds under the hypothesis des crib ed ab ov e, a nd Γ + ( x 0 ) is the forward orbit of a contin uous semiflow on the complete metric spa c e ( X , d X ), we conclude that Ω ⊆ ω + (Ω). It is then natura l to ask ho w to choose kernels that ex- hibit the separ ation pro per ties that enable the conclu- sion ab ov e. One a pproach, which we refer to as an in- 4 trinsic metho d, applies to cas es in which X = Ω is in fact a compact, co nnected, smoo th, Riemannian mani- fold M . Here we assume that M is known and that the kernel ov er M is known. Example 1 is the type of pr o b- lem we hav e in mind here, wher e the p ositive limit set is a simple well-kno wn manifold. Numer ous intrinsic ker- nels can be defined ov er the circle, the sphere, or more generally homogeneous manifolds [17]. In this case we choose kernels that gua rantee that the na tiv e space H M is in fact equiv alent to a cer tain So bo lev space W r, 2 ( M ) for r lar ge enough. That s uch eq uiv alences are p ossible follows from the Sob olev embedding theorem. [18] The Sob olev spaces defined over such a manifo ld M can b e shown to co nt a in a rich family o f smo oth cutoff func- tions. In this framework, if the forward or bit Γ + ( x 0 ) fo r some t 0 ∈ R of any c o nt inuous flow o n M is PE , Corol- lary 10 implies that the manifold is transitive. That is, it supp orts a flow that has a dense orbit. The study of when a particular manifold is tr ansitive is of interest in its own right [19], so the new PE co ndition can b e us ed to study whether a ma nifold is transitive. While this is an in ter esting result, it is not usually strictly or directly applica ble to understanding the conv erg e nce pr op erties of the RKH embedding prob- lem. There a re t wo essential pro blems her e . First, there are many pro blems wher e the p ositive limit set might b e a nice smo o th, compact, Riemannian mani- fold ω + ( x 0 ) = M , but w e do not know the form of the manifold a priori . In such ca ses defining the kernel in closed fo r m to be us e d in analysis or approximation is impossible. Example 2 is of this t yp e: the p ositive limit set is a smo oth ma nifold, but it is not one ov er which catalogs of intrinsic kernels a re defined. It is a lso po ssible, on the other ha nd, that w e do kno w the exact form of the ma nifold, but it is not o ne of the standa rd manifolds like the circle, sphere, or torus. E ven if we in principle ca n define the kernel thro ugh the fundamental solutions of c ertain elliptic differential op era tor on the manifold M as in Cor o llary 10, it may b e intractable to compute this fundamen ta l solution for the ma nifold at hand. This pr oblem can b e a s hard, or har de r pe r haps, that the o r iginal estimation pr o blem. It should b e kept in mind that the aim of the RKH embedding metho d is to car ry o ut adaptive es timation of unc ertain nonlinear ODEs. It is typically the cas e in s uc h situations that the exact form o f the p ositive limit set is unknown. That is, we are more interested in pro blems like Exa mple 2, in contrast to E xample 1. In this case, we assume the M is a n unknown, connected, smo o th, (regularly ) embedded submanifold of R d , a nd we resort to a n extrinsic metho d. In this tec hnique w e build a well-defined kernel on X for a larg e set X that co n ta ins M , and then w e define a kernel by restriction o n the manifold M ⊆ X . It can be the case that a plethora of kernels exist for go o d kernels ov er the large spac e X = R d . T aking car e to choose the kernel smo oth eno ugh, we obtain a kernel on M defined by restrictio n. The expression for the kernel on M is g iven in terms of the kernel on the large r space X , which is known. Coro llary 11 then shows that that M = ω + ( x 0 ) in this case. All of the numerical examples depicted in Figures 1 and 2 hav e b een computed using this extrinsic metho d. 2 Notation In this pap er the sy mbo ls N + , R , R + denote the non- negative integers, real num b ers, and non-negative real nu mber s, r e spe ctively . The expression a . b means that there is a constant c > 0 that do es not dep end o n a, b such that a ≤ c · b . The symbol & is defined similarly . The pap er ma kes use of Leb esg ue spac e s and Sob olev spaces on subsets Ω of R d , and it also uses these s pa ces when they are defined more gener a lly on measura ble s ubsets Ω of ce r tain Riemannian manifolds M . The norm on the Ba nach spaces L p (Ω) := L p µ (Ω) of µ -integrable func- tions ov er Ω ⊆ R d take the familiar form k f k p L p (Ω) := R Ω | f ( x ) | p dµ with the measure µ on R d for 1 ≤ p < ∞ , with the us ual mo dification for p = ∞ . Recall that when Ω ⊆ R d , the Sob olev space W r,p (Ω) for a pos itive inte- ger r consists of functions that hav e weak deriv atives of all order s le s s than o r equal to r in L p (Ω), and the nor m on these Ba nach spa ces is usually written k f k p W r,p (Ω) := X 0 ≤| α |≤ r Z Ω     ∂ | α | f ∂ x α     p dx (9) with the summation taken over all mult i- indices α = ( α 1 , . . . , α d ), | α | = P i =1 ,...,d α i , and her e the meas ure µ is selected to b e Le b esg ue measure dx . The So bo lev spaces for non-in teg e r r > 0 are defined in terms of in- terp olation theor y a s discus sed in [1 8]. A bit more de- tail is required to define the spaces L p (Ω) and W r,p (Ω) for Ω ⊆ M , with M a manifold. In this pap er M is a l- wa ys ass umed to b e a connected, complete Riemannian manifold with a p ositive injectivity ra dius and b ounded geometry . See [20], Chapter 7 or [17,21] for a discus- sion of these prop erties. F or purp oses in this pap er, it suffices to note that compact, connected Riemannian manifolds and R d satisfy these conditions. F or such a Riemannian manifold M denote the metric g a nd inner pro duct < · , · > g,p on the tangent space T p M . W e de- fine the asso ciated volume measure dµ on M , and its lo cal represe ntation in terms of the set of co or dinates ( x 1 , . . . , x d ) is given b y dµ ( x ) := p det ( g ) dx 1 . . . dx d . The norm k f k L p (Ω) has the same e x pression given ab ove with the measure selected to b e the us ual volume mea- sure on the manifold M . The Banach spaces W r,p (Ω) for measurable subsets Ω ⊆ M are equipp ed with the no r m k f k p W r,p (Ω) := X j =0 ,...,r Z Ω |∇ j f | p g,p dµ ( p ) (10) for 1 ≤ p < ∞ where ∇ is the cov ariant deriv ative ov er ( M , g ). When a pplied to a s et Ω ⊆ M = R d , the defini- tions ov er the manifold M define nor ms that are equiv a- lent to the usual ones fo r Sob olev spaces defined on sub- sets of R d . As discussed in [17,21] in this cas e the expr e s- sion in Equation 10 amo unts to a simple reweigh ting of 5 the deriv ative terms in E q uation 9. The Sob olev space s W r,p ( M ) for no n-int e g er r > 0 are, as in the case ab ov e, defined via interpolation theory . [20,22] The non-integer spaces are c r ucial to the sta temen t o f trace theor e ms for Sob olev s pa ces, which a r e used in this pap er to study the restrictions o f functions that define certain RK H spa ces. 3 Repro ducing Kernel Hilb ert (RKH) Spaces In this pap er we make use of s everal prop erties of r e al , scalar- v alued, RKH spaces. Also, the analys is b elow is readily extended to real, vector-v alued RKH spaces H := H k for k ∈ N . See [16] for the cas e whe r e this is car ried out in the co n tex t of consensus estimation. 3.1 Basic Defi nitions and Constructions An RKH space H X of functions that map a set X ⊆ R d → R is defined in terms of a r eal-v alued, contin uous , symmetric, and p ositive type function K X : X × X → R that is r eferred to as the kernel underlying the RKH space. The subscript on K X is used to emphasize the set ov er which the kernel, as well as the functions in H X are defined. When we sa y that K X is of po sitive t yp e, this means that P N i,j ≤ 1 α i K X ( x i , x j ) α j := α T K X,N α ≥ 0 for all { x 1 , . . . , x N } ⊂ R d and α := { α 1 , · · · , α N } T ∈ R N , with the collo catio n matrix asso ciated with { x i } 1 ≤ i ≤ N defined as K X,N := [ K X ( x i , x j )] ∈ R N × N . So a ll the collo cation ma tr ices o f a kernel of p ositive type ar e p os- itive semidefinite. W e say that the kernel is of str ictly po sitive t y p e if a ll of its collo ca tion matr ices K X,N := [ K X ( x i , x j )] for distinct p oints { x k } 1 ≤ k ≤ N are strictly po sitive definite. The function K X,x := K X ( x, · ) is known as the kernel function centered at x ∈ X , and a candidate for the inner pro duct of tw o such functions K X,x , K X,y is defined to b e ( K X,x , K X,y ) H X := K X ( x, y ) for all x, y ∈ X . The RKH s pa ce H X is the closed finite span of the set of functions { K X,x | x ∈ X } , that is, H X : = span { K X,x | x ∈ X } = ( f : X → R     f = lim N →∞ N X i =1 α N ,i K X,x N ,i ) , where α N ,i ∈ R and x N ,i ∈ X . The closur e a bove is taken with respe c t to the ca ndida te inner product. The Hilber t space ( H X , ( · , · ) H X ) above is also known a s the native space induced by the kernel K X . It is well-known [23,24,25] that with this constructio n the r epro ducing prop erty ( K X,x , f ) H X = f ( x ) ho lds fo r all f ∈ H X and x ∈ X . An y Hilb ert s pace H is in fac t a RK H spac e if all o f the ev aluation functionals that act o n H a r e in fact bounded op erators from H → R . If it is fur- ther known that if for s ome p ositive constant K we hav e sup x ∈ X K X ( x, x ) ≤ ¯ K X < ∞ , then the ev aluation op er- ator E x : H X → R given by E x := E H X ,x : f 7→ f ( x ) is a uniformly b ounded linear op era tor s ince | f ( x ) | = | E x f | = | ( K X,x , f ) H X | ≤ p K X ( x, x ) k f k H X . This im- plies that k f k C ( X ) . k f k H X , and therefor e we hav e the contin uous inclusion H X ֒ → C ( X ). W e will only consider kernels K X on X for whic h such a co nstant K X exists. Later in the pap er we als o make extensive use of the closed subspa ces H Ω := span { K X,x | x ∈ Ω } ⊆ H X for some subset Ω ⊆ X . These spaces are imp ortant in understanding how the new PE condition a re applied. One imp ortant fact is that we hav e the H X − orthog onal decomp osition H X = H Ω ⊕ V Ω with V Ω the kernel of the trace or restric tion opera tor on the set Ω ⊆ X , V Ω := { f ∈ H X | R Ω f = f | Ω = 0 } . Tha t is, f ∈ V Ω if and only if f ( x ) = 0 for all x ∈ Ω. This fundamental prop erty follows from the analy s is in [24] and [23]. Fi- nally , in some cases when we sp ecifically discuss spaces derived fro m r estrictions o f functions H X to a s ubset Ω, we denote these RK H spaces as R Ω ( H X ). 3.2 Sep ar ation of Close d S et s by R epr o ducing Kernels The curre nt pap er is in ter ested in understanding how the use of a RKH spa ce can make precis e certain no- tions of conv erge nce in a daptive estimation. W e wan t to understand the ge ometric implica tio ns of the PE c ondi- tion, that is, what it implies ab out the tra jectories of the dynamical sys tem and the PE set. E ssentially , we will “test conv er gence” in X of tr a jectories x ( t ) → ¯ x by the condition that f  x ( t )  → f ( ¯ x ) for all f ∈ H X . As we will s ee, it can b e imp orta n t for understanding per- sistence that the spac e H X contain enough functions to separate, in a certa in sense, the p oints o f X . Here an ex- ample can illustra te the the pro blem. It is known that it is alwa ys p oss ible to induce a metric d K asso ciated with the kernel K X as desc r ibe d in [26]. The problem is , o ur semiflows will be con tinuous with resp e c t to so me met- ric d X , and the top ology induced by d X may not b e the same a s that genera ted by d K . In fact it is e asy to come up with kernels fo r which this is the case . As noted in Remark 1 o f [26], the bilinear kernel k X ( x, y ) := x ⊤ y for x, y ∈ R d induces a RKH spa ce H X for whic h the only subsets that ca n b e separa ted are linear ma nifolds. In this s pecific case, d K induces a top olo gy that is strictly coarser than the usual top olog y on X := R d . Sp ecifi- cally , the metric d K can b e used to discriminate con- vergence to a pa rticular line through the origin, but not conv erg e nce to a p oint on that line. W e will see that some useful geo metric insights r egard- ing the P E condition a nd p ositive or bits result if w e do not allow the kernel to induce such a coarse top olog y . W e would lik e the metric genera ted by the kernel to b e equiv- alent with that on the state spac e. Refer ence [2 6] giv es one ex ample of a useful and simple sepa r ation prop er t y . An RKH space H X is said to separate a subset A ⊂ X if for each b / ∈ A there is a function f ∈ H X such that f ( a ) = 0 for a ll a ∈ A and f ( b ) 6 = 0 . This co ndition can be us ed to prov e tha t d K and d X define the same top ol- ogy . Howev er, we will emplo y a n even stro nger condi- tion, o ne that is well-suited to the construction of native spaces that contain well-known c la sses of differen tiable functions. W e a ssume the ex is tence of a rich family of 6 bump functions in H X . W e say that b r,x : X → [0 , 1] is a bump function on ( X , d X ) asso ciated with the op en ball B r,x = { y ∈ X | d X ( x, y ) < r } provided that 1) b r,x = 1 on a neighborho o d o f x , and 2) b r,x is z e ro outside a compact set contained in B r,x . It is immediate that if for any op en set B r,x , there is a n a sso ciated bump function b r,x ∈ H X , then the RKH space H X separates the d X - closed subsets o f X . W e say that the spa c e H X contains a rich family of bump functions if it co nt a ins a bump function b r,x for each op en ball B r,x . The c onstruction of s mo o th bump functions o n X = R d is a cla ssical ex er- cise in analy sis on manifo lds , [27] pages 4 9 –51. In prac- tice, the RKH space H X (even when X 6 = R d ) will b e selected so that it contains them. See the pro ofs b elow of Corollar ies 1 0 and 11. 4 The R KH Emb edding M etho d In this pap er, we study a mo del pr oblem of a daptive estimation for uncerta in nonlinear systems g ov erned by ordinary differential eq uations that ha ve the fo r m ˙ x ( t ) = Ax ( t ) + B f ( x ( t )) , x (0) = x 0 (11) with A ∈ R d × d a Hurwitz matrix, B ∈ R d × 1 , and f : R d → R . This equa tion is a sp ecial case of the general form in Equation 2, with g : R d → R d := B f . Meth- o ds for ensuring that this system of ODEs has lo cal or global solutions a re well-kno wn, [5], a nd in this pa p er we alwa ys a ssume that for each x 0 ∈ X the eq uations have classical solutions on R + . In this equation, it is a ssumed that the matric e s A and B are known, but the (non- linear) function f is unknown. The ada ptive estimation problem considered in this pap er uses the observ atio ns of the full state, x ( t ) for all t ≥ 0 , to co nstruct estimates ˆ x ( t ) → x ( t ) and ˆ f ( t ) → f a s t → ∞ . While f is unknown in o ur adaptiv e estimation problem, information ab out this function is r e fle c ted in the choice of an hypothesis space H of functions to which f b elongs. Perhaps the most familiar choice of hypothesis space H is one that is finite dimensio nal H n := { f = P i =1 ,...,n α i φ i } with { φ i } n i =1 some fixe d s e t of basis functions and φ i : R d → R for 1 ≤ i ≤ n . If we suppo se for the moment that the un- known function f = P n i =1 α ∗ i φ i ∈ H n , then one canoni- cal choice of an estimator is ˙ ˆ x ( t ) = A ˆ x ( t ) + B Φ T ( x ( t )) ˆ α ( t ) ˙ ˆ α ( t ) = − Γ − 1 Φ( x ( t )) B T P ( x ( t ) − ˆ x ( t )) with P ∈ R d × d the symmetric p ositive definite solution of Lyapuno v’s eq uation P A + A T P = − Q for a user- designed sy mmetric p os itiv e definite matrix Q ∈ R d × d , and Γ ∈ R n × n symmetric and p ositive definite. When the e r rors in state ˜ x ( t ) = x ( t ) − ˆ x ( t ) and par ameter erro rs ˜ α ( t ) = α ∗ − ˆ α ( t ) ar e defined, it can b e shown dir ectly that the er rors sa tisfy the equations    ˙ ˜ x ( t ) ˙ ˜ α ( t )    =   A B Φ T ( x ( t )) − Γ − 1 Φ( x ( t )) B T P 0      ˜ x ( t ) ˜ α ( t )    + E ( t ) for E ( t ) := { B e ( t ) , 0 } T , with e ( t ) = 0 if f ∈ H n . If it happens that f / ∈ H n , then e ( t ) := f ( x ( t )) − P n i =1 α ∗ i φ i ( x ( t )) := f ( x ( t )) − f n ( x ( t )) with f n a suit- able finite dimensio nal appr oximation of f . Pr e c ise con- ditions on the ex po nen tia l stability of this system ar e a classical topic in ada ptiv e estimation for uncertain ODEs. See [13,28] when e ( t ) = 0. When e ( t ) 6 = 0, s ee [15,14,29] for r e la ted discussions of ultimate b ounded- ness o f er rors. In this paper, we ar e interested in a class of dynamical systems where the unknown function f b elongs to the RKH space H . The generic RKH space H may b e the full space H X or one of its closed subspa ces H Ω describ ed in Section 3. T he pla nt , e s timator and the lear ning laws for this ca s e can b e expres sed as ˙ x ( t ) = Ax ( t ) + B E x ( t ) f , (12) ˙ ˆ x ( t ) = A ˆ x ( t ) + B E x ( t ) ˆ f ( t ) , (13) ˙ ˆ f ( t ) = Γ − 1 ( B E x ( t ) ) ∗ P ( x ( t ) − ˆ x ( t )) , (14) where x ( t ), ˆ x ( t ), A , B , and P are defined a s ab ov e. But the (nonlinear ) functions f and ˆ f ( t ) b elong to the RKH space H and E x : H → R d is the ev aluation functional that is defined as E x f = f ( x ) for a ll x ∈ X and f ∈ H . F urthermo re, the term Γ ∈ L ( H , H ) in the ab ov e equa- tion is a self-a djoin t, linear p ositive definite op erator . The erro r equa tion analo gous to the clas sical case shown ab ov e is given by ( ˙ ˜ x ( t ) ˙ ˜ f ( t ) ) = " A B E x ( t ) − Γ − 1 ( B E x ( t ) ) ∗ P 0 # ( ˜ x ( t ) ˜ f ( t ) ) = A ( t ) ( ˜ x ( t ) ˜ f ( t ) ) . (15) Note that the evolution of the ab ov e error eq uation is in R d × H as opp ose d to o n R d × R n in the cla ssical ada ptive estimator case. Some elementary conditions that gua r- antee the existence of solutions, a s well as their co n tinu- ous dep endence on initial co nditions, are given in [2,3]. In this pap er w e alwa ys a ssume that for ea ch x 0 ∈ X the equations admit a classical s olution t 7→ ( ˜ x ( t ) , ˜ f ( t )) ∈ R d × H for t ∈ R + . The following theor em, which simpli- fies considera bly the analys is in [2 ,3], shows that this is reasona ble for many co mmon choices of the RK H space H . Theorem 5 Supp ose that t he RKH sp ac e H is gen- er ate d by a kernel K X : X × X → R for which sup x ∈ X K X ( x, x ) ≤ K < ∞ . Then for e ach ( ˜ x 0 , ˜ f 0 ) ∈ R d × H t her e is a unique solution of Equation 15 in C 1 ([0 , ∞ ) , R d × H ) . PR OOF. As discussed in Section 3, the hypotheses guarantee that the ev aluatio n oper ator E x : H → H is linear and unifor mly b ounded in x ∈ X . It is immediate that lim t →∞ 1 t log + ( k A ( t ) k ) = 0 with log + ( ξ ) := max(0 , log( ξ )). As discuss e d on pag e 211 of [30] the governing equations hav e a unique global solution in time. 7 The definition of the PE condition prov es sufficient for conv erg e nce of function estimates ˆ f ( t ) → f gener ated by the RKH embedding metho d, mu ch a s in the conv en- tional, finite dimensio nal case. The analys is o f c o nv er- gence of para meters (ie, functions in our case) is no tori- ously long, so in this shor t pa per we merely o utline the pro of in a sp ecia l case. The full and lengthy details (for general P and Hurwitz A ) are given in [3 1]. W e s ay that a family of functions F over a set S is uni- formly equi-contin uous if for each ǫ > 0, ther e is a δ ǫ > 0 such that for all f ∈ F and a, b ∈ S , | a − b | < δ ǫ ⇒ k f ( a ) − f ( b ) k < ǫ . Theorem 6 Su pp ose that P = I , A is ne gative def- inite, F = { f ( x ( · )) | f ∈ H , k f k H = 1 } is uniformly e qui-c ontinuous, and the tr aje ctory Γ + ( x 0 ) is p ersistently exciting in the sense of Defin ition 4. Then the solu- tion ˜ x, ˜ f of Equations 15 s atisfy lim t →∞ ˜ x ( t ) = 0 and lim t →∞ ˜ f ( t ) = 0 . PR OOF. The pr o of that ˜ x ( t ) → 0 follows along lines that are en tir ely analo gous to the cla ssical or finite di- mensional case, see [3] for the details when ar gument s are lifted to the infinite-dimensiona l sta te spa ce R d × H . The conclusio n that ˜ f ( t ) → 0 ∈ H follows immediately from Theo rem 3 .4 of [32], provided that we ca n prove that there exists constants T , ∆ , δ, γ > 0 such that for each t ≥ T and f ∈ H with k f k H = 1 there is an s ∈ [ t, t + ∆] such that k B k R d      Z s + δ s E x ( τ ) f dτ      =      Z s + δ s B E x ( τ ) f dτ      R d > γ . (16) How ever, the condition ab ove can b e s hown to b e equiv- alent to the PE Definition 4 pr ovided that the integrand is smo oth enoug h to eliminate the possibility o f certain “rapid switching” behavior. The equiv alence of condi- tions as in Equation 16 to thos e similar to Definition 4 in the classic al, finite dimensiona l case have been studied in great detail. See [9] for a detailed discussion with excel- lent illustrative examples of pathological rapid switching in the finite dimensional ca se. In the case at ha nd, Equa- tion 1 6 follows from the fa c t that f ( x ( · )) ∈ F , a family of uniformly equi-co n tinuous functions. The lengthy de- tails of the pro o f ca n b e found in [31]. 5 Semiflows and Persistence of Excitation (PE) In this section we recall o f few o f the bas ic definitions o f dynamical systems theor y that will b e essential to the analysis of this pa pe r . The a im is to b e able to define per sistence of excitation, not only for the mo del proble m in Equa tions 2 or 11 , but for mo re g eneral evolutions on metric spaces. In particular we obtain a P E co ndition that can b e applied to flows on Riemannian manifolds, which encompass a few of o ur ex a mples. A contin uous semiflow o r semidynamical system o n the complete met- ric space ( X , d X ) is defined in terms of a co ntin uo us semi- group { S ( τ ) } τ ≥ 0 on X . The manner in which systems of ODEs can g enerate s uch a semigr oup, and thereb y a semidynamical sys tem is well-studied [6,33]. The p osi- tive or bit Γ + ( x 0 ) sta rting a t x 0 defined to b e the set Γ + ( x 0 ) := [ τ ≥ 0 S ( τ ) x 0 ⊆ X . The p ositive limit se t ω + ( x 0 ) a sso ciated with the initial condition x 0 is defined to b e ω + ( x 0 ) := \ t ≥ 0 [ τ ≥ t S ( τ ) x 0 , which is equiv alently ex pr essed as ω + ( x 0 ) =  y ∈ X | ∃ t k → ∞ s .t. lim k →∞ S ( t k ) x 0 → y  . 5.1 Persistenc e and Posi t ive Limit S et s The next few results illus trate simple and often intuitiv e relationships b etw een p ersis tently excited s ets, p ositive orbits Γ + ( x 0 ), and the p ositive limit set ω + ( x 0 ). W e s tart with a simple result that illustrates an in tuitive no tion of what the new PE Definition 4 entails. Theorem 7 L et K : X → R b e a monotone nonin- cr e asing r adial b asis funct ion and supp ose the asso ciate d kernel K X ( x, y ) := K ( d X ( x, y )) induc es the R KH sp ac e H X ֒ → C ( X ) for some fix e d ǫ > 0 . Define the me asur- able s et s I t,ǫ =  τ ∈ [ t, t + ∆]     x ( τ ) ∈ B ǫ,x ∞  for e ach t ≥ T 0 with B ǫ,x ∞ the op en b al l of r adius ǫ c ent er e d at x ∞ . If the t he Le b esgu e me asur e µ satisfies µ ( I t,ǫ ) ≥ γ ǫ > 0 for some c onstant γ ǫ for al l t ≥ T 0 , then the singleton in- dexing set Ω := { x ∞ } and the close d subsp ac e H Ω ⊂ H X ar e p ersisten t ly excite d in t he sense of Definition 4. In p articular, if x ( t ) → x ∞ , the set Ω := { x ∞ } and close d subsp ac e H Ω ar e p ersistently excite d. Before proving the above theore m, let us unpack the ab ov e definition to understand the relatively str aightfor- ward underlying idea. The interv al I t,ǫ is the set of times contained in the interv al [ t, t + ∆] dur ing which the tra- jectory t 7→ x ( t ) is within ǫ of the point x ∞ . This theo - rem says that if a tra jectory spends at lea st γ ǫ amount of time in each interv al [ t, t + ∆] in the ba ll of radius ǫ centered a t x ∞ , then Ω = { x ∞ } and H Ω are p ers istent ly excited. PR OOF. Without lo ss of g enerality , we assume that the kernel K X is no rmalized so that K X ( x ∞ , x ∞ ) = 1. By definition H Ω := span { K X,x ∞ } when Ω = { x ∞ } , and for each f ∈ H Ω with f = α K X,x ∞ we hav e k f k H Ω = α = f ( x ∞ ). Only the low er b ound of the p ers istency 8 definition is problematic, and we co mpute dir e c tly that Z t +∆ t  E ∗ x ( τ ) E x ( τ ) f , f  H X dτ = α 2 Z t +∆ t K 2 X ( x ( τ ) , x ∞ ) dτ ≥ α 2  min 0 ≤ ξ ≤ ǫ K 2 ( ξ )  µ ( I t,ǫ ) ≥ γ ǫ K ǫ k f k 2 H Ω with K ǫ := min 0 ≤ ξ ≤ ǫ K 2 ( ξ ). Theorem 7 gives a direct interpretation of the p ersis- tency condition when we consider a sing leton Ω := { x ∞ } in terms of visitation to a neighborho o d of x ∞ . It a lso suggests that there are many choices of kernels K X that induce PE spaces H Ω for any co nv ergent tra jec- tory x ( t ) → x ∞ . The mono to nicit y of the kernel in the ab ov e theorem is s a tisfied for a host of commo n choices of RKH space s, see Chapter 9 o f [34] for the definition of c o mpletely monotone radial bas is functions and ker- nels. This fact illustrates a significant difference with the con ventional PE definition: there are man y con ver- gent tra jectories that s imply are no t classically P E for a given set of r egresso rs. W e also note that if X = R d , there is a direc t extensio n of this theorem for the finite set Ω := { x ∞ , 1 , . . . , x ∞ ,M } , see Le mma 3.4 in [3 5]. W e b egin our study of the geometric nature o f PE sets by noting that the forw a rd or bit is alw ays dense in PE sets. Theorem 8 L et H X b e an RKH sp ac e of functions over the domain X and supp ose t hat this RKH sp ac e includes a rich family of bump functions. If the PE c ondition in Definition 4 holds for a subset Ω ⊆ X , then the forwar d orbit Γ + ( x 0 ) is dense in Ω , Ω ⊆ Γ + ( x 0 ) . That is, we have y ∈ Ω = ⇒ ∃{ t k } k ∈ N with lim k →∞ S ( t k ) x 0 → y . PR OOF. Supp ose to the contrary that there is an y ∈ Ω for which there is no such c onv ergent s e - quence. This mea ns that there is a n op en ball B r,y such that Γ + ( x 0 ) T B r,y = ∅ . But since we hav e assumed there is a rich collection of bump functions, there is a bump function b r,y ∈ H X that satisfies b r,y ( y ) = 1, b r,y ( x ) = 0 ∀ x 6∈ C r,y with C r,y ⊂ B r,y a compact set. How ever, from Section 3, H X = H Ω ⊕ V Ω with V Ω = { f ∈ H X | f ( x ) = 0 , ∀ x ∈ Ω } . It follows that b r,y ∈ H Ω . Since Γ + ( x 0 ) T B r,x = ∅ , the integral in Definition 4 is equal to z e r o Z t +∆ t  E ∗ x ( τ ) E x ( τ ) b r,y , b r,y  H Ω dτ = Z t +∆ t | b r,y ( x ( τ )) | 2 dτ = 0 for ea ch t ≥ T . Since k b r,y k H Ω := k P Ω b r,y k H X = k b r,y k H X & k b r,y k C ( X ) > 0 , this is a contradiction of the PE pr op erty in Definition 4 and the theor em is proven. Note that Theorem 8 do es not r equire tha t the set of times t k → ∞ . Reca ll, o n the other hand, that the pos- itive limit set ω + ( x 0 ) is contained in the closure of all accumulation p oints of the o rbit Γ + ( x 0 ) for sequences of the form { S ( τ k ) x 0 } k ∈ N , as τ k → ∞ . Next, we discuss a re la tionship of the p ositive limit set ω + ( x 0 ) a nd a PE space H Ω ov er the indexing se t Ω ⊂ X in Definition 4 . Theorem 9 L et H X b e the R KH sp ac e of functions over X and su pp ose that t his RKH sp ac e includes a rich family of bump fun ct ions. If the PE c ondition in Definition 4 holds for Ω , then Ω ⊆ ω + ( x 0 ) . PR OOF. The pro o f of this result is similar to the argu- men t in Theo rem 8 , so we only outline it. F or an a r bitrary y ∈ Ω we build a sequence { x ( t k ) } k ∈ N := { S ( t k ) x 0 } k ∈ N such that lim k →∞ t k = ∞ , lim k →∞ S ( t k ) x 0 = y . Pick the ar bitrary y ∈ Ω a nd fix r 0 > 0. Cho os e t 0 ∈ ( T , T + ∆) such that x ( t 0 ) ∈ B r 0 ,y . Suc h an x ( t 0 ) m us t exis t. If such a time does not exist, we could choose a bubble function b r 0 ,y on B r 0 ,y as in the last example such that k b r 0 ,y k H Ω > 0, for which the int e- gral R T +∆ T | b r 0 ,y ( x ( τ )) | 2 dτ = 0 would follow fro m the condition that B r 0 ,y ∩ { x ( τ ) | τ ∈ ( T , T + ∆) } = ∅ . This is a contradiction of the PE condition. W e can then set r 1 = r 0 / 2 and rep eat this pro cess seeking a t 1 ∈ (2 T , 2 T + ∆) s uch that x ( t 1 ) ∈ B r 1 ,y , and s o forth to generate { t k } k ∈ N 0 with t k → ∞ and { x ( t k ) } k ∈ N 0 with x ( t k ) → y . These sequences satisfy the desir ed conditions a bove, and we must have y ∈ ω + ( x 0 ). 5.2 Persistenc e of Excitation for Semiflows on Mani- folds A careful r e ading o f the Definition 4 makes cle ar that it depe nds o n the orbit Γ + ( x 0 ) o f a contin uous semiflow { S ( t ) } t ≥ 0 on a complete metric space ( X , d X ), a sub- set Ω ⊆ X , and an admiss ible kernel K X that defines the RKH space H X (and therefore also the clo sed sub- space H Ω ). Since it applies to subsets of complete met- ric spaces, it ma kes sense to consider muc h more general systems than the ODEs in the mo de l Equa tions 2 or 11. F or instance, we hav e the follo wing result for semiflows on manifolds, the case when the state space X = Ω = M in the PE Definition 4. No te that below the semigroup S ( t ) that defines the p ositive or bit Γ + ( x 0 ) is de fined on all of M = X . Corollary 10 Supp ose that M is a c omp act, c onne cte d, d -dimensional Riema n nian manifold , and K M is kernel that induc es a n ative sp ac e H M whose norm k · k H M is e quivalent to that of the Sob olev sp ac e W r, 2 ( M ) . If the orbit Γ + ( x 0 ) p ersistently excites H M , then ω + ( x 0 ) = M . PR OOF. W e first show that there a re indeed such ker- nels K M that induce a native space H M ≈ W r, 2 ( M ). The 9 Sob olev embedding theorem on R d states that W r, 2 ( R d ) is contin uously embedded in C ( R d ), W r, 2 ( R d ) ֒ → C ( R d ) when r > d/ 2. As noted on page 17 4 8 of [1 7], this fact can b e used to conclude that W r, 2 ( M ) is contin uously embedded in C ( M ), W r, 2 ( M ) ֒ → C ( M ). This means that we have | E x f | = | f ( x ) | ≤ k f k C ( Ω) . k f k W r, 2 ( M ) for eac h x ∈ M a nd f ∈ W r, 2 ( M ). In other words, each ev aluation functional E x : W r, 2 ( M ) → R on M is bo unded. But W r, 2 ( M ) is a Hilbe r t space; b oundedness of a ll its ev aluation functionals implies that W r, 2 ( M ) is a RKH space. W e define the Sob olev-Mater n kernel K r M of smo othness r > 0 to b e the unique fundamen- tal solutio n of the elliptic differential ope r ator equa - tion P 1 ≤ ℓ ≤ r ( ∇ ℓ ) ∗ ∇ ℓ K r M = δ w he r e ∇ is the cov ariant deriv ative o per ator ov e r the manifold M a nd δ denotes the Dirac distribution. When we define the na tiv e space H M in terms o f the Sob olev- Matern kernel K r M , we hav e H M ≈ W r, 2 ( M ) for the chosen rang e r > d/ 2. The de- tails of this analysis are given in [17] for the case when M is a smo oth Riemannian ma nifold that satisfies our standing ass umptions on M , or see refer e nc e [36] for the sp ecial case M := R d . W e next show that the RKH space H M defined in this wa y contains a rich fa mily o f (smo oth) cutoff or bubble functions. This pro of is not surprising g iven wha t we know ab out So bo lev space s on s ubsets of R d . One wa y to define W r, 2 (Ω) is a s the completion o f C ∞ (Ω) in the Sob olev norm, so the space C ∞ (Ω) is dense in W r, 2 (Ω). It is w ell-known that for any op en ball contained R d , ther e is a smo oth cutoff function with compact supp or t con taine d in that ball. This is a standard result in the study of manifolds and the con- struction of pa r titions o f unity . [27] It follows that the Sob olev space W r, 2 (Ω) contains a rich family of bubble functions. The re sult extends more genera lly to So bo lev spaces W r, 2 ( M ) using the exp onential map. The details of the pro of are rather long, which we simply o utline be - low. (Particular examples of such a constructio n can b e found in [17] o n pag e 1749 a nd ag ain on pa ge 175 1 o f the same reference.) If ˆ f is a cutoff function o n a ball B x,r ⊂ R d , it is p ossible to cons truct an as so ciated cuto ff function on the image E xp q ( B x,r ) ⊂ M under the ex- po nent ia l map E xp q : T q M → M fro m f := ˆ f ◦ E xp − 1 q . Such a n f is alwa ys an element of C ∞ ( M ) since ˆ f is just a smo o th re pr esentation of f with res pec t to a compat- ible C ∞ chart. The only techn ic a l difficulty is showing that f ∈ W r, 2 ( M ). But this follows from Lemma 3.2 of [17] which states that the ex po nen tia l op er ator E xp q in- duces a map g → g ◦ E xp q that is b oundedly in vertible from W r,p ( E xp q (Ω)) to W r,p (Ω) for any mea surable set Ω ⊆ R d . Alternatively , we can argue that W r, 2 ( M ) is the completion o f C ∞ ( M ) ([20 ], Section 7.4.5) with r esp ect to the norm in Equation 1 0. W e conclude that W r, 2 ( M ) contains a rich family of (smo oth) cutoff functions. If the motio n ov er the manifold M sa tisfies the p ersistency condition in Definition 4, then M := ω + ( x 0 ). This example illus trates that the newly introduced p er- sistency co ndition can b e applicable, in principle , to the study o f certain ev olutions ov er smooth Rieman- nian manifolds. Still, the a nalysis in the exa mple a bove is fairly abstract. Perhaps more imp ortantly , it is not a simple task to come up with a closed form expres sion for the Sobo le v-Matern kernel. Of course this c a n b e done for s ome standa rd manifolds like R d , the cir cle, or a torus, since the Sob olev-Matern kernels can b e wr it- ten down for these case s . But it is not readily a ccom- plished for so me arbitrary manifold M . The definition of the space H M ≈ W r, 2 ( M ) is intrinsic here: it dep ends on the (usually unknown) domain of the manifold M , the a tlas of c harts used to define the manifold, a nd the cov a riant deriv ative o pe rator intrinsic to the ma nifold. W e nex t discuss how it is p os sible to co me up with con- structions of a kernel for M that is extrins ic in the sense that it is defined by the restriction of so me known kernel on a larg er domain that contains M . This termino logy is used in [22] that studies the approximation prop erties of spaces constructed in such a fashion. This line of at- tack is particula rly useful to the study o f unknown or uncertain dy namical s ystems via the RKH e mbedding metho d. The pe rsistency of exc ita tion condition is cast in terms of the kernel o n the lar ger spa ce in this case, which is assumed to hav e a known closed form expres- sion. Carefully note that the fo r ward orbit Γ + ( x 0 ) in the following theo rem is defined in terms of a semigr oup S ( t ) : M → M , but M is a prop er subset of X . Corollary 11 L et M b e an m -dimensional, smo oth, c omp act, (r e gularly) emb e dde d submanifold of X := R d , and supp ose t hat the { S ( t ) } t ∈ R + defines a d M -c ontinuous semiflow on M with d M the metric on M . D enote by K r X the Sob olev-Matern kernel on X = R d for some r > d/ 2 , define the kernel K M ( · , · ) = K r X | M ( · , · ) , and denote by R M ( H X ) the RKH sp ac e gener ate d by K M . If the orbit Γ + ( x 0 ) of the semiflow on M p ersistently excites R M ( H X ) , then M = ω + ( x 0 ) . PR OOF. The Matern-Sob olev kernels ov er R d are given for r > d/ 2 by K X ( x, y ) = K X ( k x − y k R d ) with K ( ξ ) = 2 1 − ( r − d/ 2) Γ( r − d/ 2) ξ r − d/ 2 B r − d/ 2 ( ξ ) for all x, y ∈ R d and ξ := k x − y k R d with B r − d/ 2 the Bessel function of or der r − d/ 2. ([22], page 17 71 or [21], page 195 7) As in the last example, we hav e H X ( R d ) ≈ W r, 2 ( R d ) under the condition that r > d/ 2 . That the ca ndidate k er nel K M defined the restriction K M ( x, y ) := K | M × M ( x, y ) for all x, y ∈ M is in fact an admissible kernel for a RKH space follows fr om standa rd results o n RKH spa ces, [23] Section 4 .2 and [2 4] Sections 2.2.1-2 .2.2. At this po in t we do not y et ha ve a rigorous notion of exactly how smo oth the restricted functions in R M ( H X ) a re, nor do we k now whether the spaces R M ( H X ) contain a rich set of cutoff functions. But fr o m Lemma 4 of [22], we know tha t R M ( H X ) = T ( H X ) = T ( W r, 2 ( R d )) wher e T is the trace opera tor T : f → f | M . F rom P r op osition 2 of [22], under the s tanding a s sumptions on M , the trace op erator T : f → f | M is a con tinuous op era tor from W r, 2 ( R d ) onto W r − ( d − m ) / 2 , 2 ( M ) for r > ( d − m ) / 2 and 10 1 ≤ m ≤ d . In summar y then, if we c ho o se the kernel K X on R d with a sufficien tly large smoo thness index r , we hav e R M ( H X ) = T ( W r, 2 ( R d )) ≈ W r − ( d − m ) / 2 , 2 ( M ) . This s et of equiv alencies gives a precis e notion of the smo othness or regular it y of the res tricted functions in the RKH space R M ( H X ): the RKH space ov er M is e q uiv alen t to the Sob olev space having smo o thness r − ( d − m ) / 2 . The remainder of the pro o f is now that same a s in Coro llary 10 . Note that the statement of p ersistence in Definition 4 is expressed in terms of the kernel K M := K X | M × M , which can be used for computatio ns since a closed fo rm fo r K X is known. 6 Conclusions This pap er derives sufficient conditions for the co n- vergence of function estimates in the RK H e m b edding metho d that ar e based o n the rece n tly introduced notion of p ersistently excited indexing sets Ω and subspa ces H Ω of an RKH spac e H X . The pa pe r establis hes that per sistently excited subsets are contained as subsets of the p ositive limit sets, if the RKH space has a rich collection of bump functions. W e hav e also introduced bo th intrinsic a nd extr insic methods for defining a n appropria te RKH space in the event that the po sitive limit set is in fact certain t yp es of smo o th manifold. The extrinsic metho d seems particula rly well-suited for the estimation of unce rtain nonlinear sy s tems s ince the form of the p ositive limit set is unknown. The theoretica l results of this pap er establish that a r ea- sonable choice of bas is functions for pra ctical finite di- mensional a pproximations include radial ba sis functions (defined in terms of the kernel of the RK H space) that are centered on or near the p ositive limit set. It rema ins an op en ques tio n as to how to devis e versions of the RKH embedding stra tegy that adaptiv e ly selects the basis a s estimation is ca rried out. 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