Persistence of Excitation in Reproducing Kernel Hilbert Spaces, Positive Limit Sets, and Smooth Manifolds
Andrew J. Kurdila Jia Guo Sai Tej Paruchuri Parag Bobade




                                              
                                              
                                              

 
                                                





VTME]Andrew J. Kurdilakurdila@vt.edu,    
VTME]Jia Guojguo18@vt.edu,               
VTME]Sai Tej Paruchurisaitejp@vt.edu,    
UMichAE]Parag Bobadeparagsb@umich.edu

[VTME]Department of Mechanical Engineering, Virginia Tech, Blacksburg, VA 24060, USA
[UMichAE]Department of Aerospace Engineering, University of Michigan, Ann Arbor, MI 48109, USA
                                                    
                                                    
                                                    

          
                           
Adaptive Estimation, Reproducing Kernel, Persistence of Excitation             
                             
                                          
                                          


                          
This paper studies the relationship between the positive limit sets of continuous semiflows and the newly introduced definition of persistently excited (PE) sets and associated subspaces of reproducing kernel Hilbert (RKH) spaces. It is shown that if the RKH space contains a rich collection of cut-off functions, persistently excited sets are contained as subsets of the positive limit set of the semiflow. The paper demonstrates how the new PE condition can be used to guarantee convergence of function estimates in the RKH space embedding method for adaptive estimation. In particular, the paper is applied to uncertain ODE systems with positive limit sets given by certain types of smooth manifolds, and it establishes convergence of adaptive function estimates over the manifolds.









§ INTRODUCTION



 

In this paper we study the method of reproducing kernel Hilbert (RKH) space embedding for adaptive estimation of uncertain, or unknown, dynamic systems that are governed by systems of coupled, nonlinear ordinary differential equations (ODEs). The RKH embedding method for adaptive estimation has been introduced  in <cit.>. This general formulation constructs estimates in ℝ^d of the state of the unknown governing ODEs as well as estimates of an unknown function contained in the RKH space H that characterizes the uncertain governing ODEs.    
This paper investigates several unanswered questions related to the new notion of  persistency of excitation  (PE)  that has been introduced in the latter two of  these three papers. We derive  relationships  between the PE condition over an indexing set Ω that is a subset of the state space X and the positive limit sets of semiflows over X. We also  construct or select good kernels that define the RKH space H in applications where the governing semiflows exhibit certain asymptotic structural properties. These latter properties are expressed in terms of PE  conditions over  some classes of  smooth manifolds.  

The RKH embedding method generates a distributed parameter system, and its associated estimates  evolve in the generally  infinite dimensional space ℝ^d × H.   There are many nontrivial questions about approximations and realizable implementations of the method, and  some study of the convergence of finite dimensional approximations of solutions of the RKH embedding equations is given  in <cit.>. In this short paper,  we only consider the convergence of the estimates generated by the governing  DPS system in the infinite dimensional state space ℝ^d× H that defines the RKH embedding formulation. The current investigation can be viewed as providing needed insight and intuition into the structure of the solutions of the RKH embedding equations, which is much needed for the  effective choice of approximating subspaces in practical implementations. 



 


 §.§ Adaptive Estimation for Uncertain Nonlinear ODEs



 

A common setup for estimation of uncertain nonlinear systems  starts with an ordinary differential equation that can be decomposed into known and unknown parts, 

 

    ẋ(t)=g_0(x(t)) + g(x(t)),    x(0)=x_0



 
with x(t)∈ℝ^d for t∈ℝ^+,  g_0:ℝ^d →ℝ^d a known function,  and g:ℝ^d →ℝ^d an unknown function. One important problem  of adaptive estimation for such a  nonlinear system is to use the full state observations {x(t)}_t∈ℝ^+ to construct an evolution law for a state  estimate x̂(t) that approximates x(t), in the sense that  x̂(t) → x(t) as t→∞. In the language of adaptive estimation this is referred to as convergence of state  estimates.
A canonical model estimator for the original equation might choose the evolution law for the estimate to be 

 

    ẋ̂̇(t)=Ax̂(t)+g_0(x(t))+ĝ(t,x(t)) - Ax(t)



 
for a known matrix A∈ℝ^d, although many alternatives exist of course. Here ĝ(t):=ĝ(t,·) is an estimate of the unknown function g.  On defining  the state error x̃(t)=x(t)-x̂(t) and the function error g̃(t):= g̃(t,·):=g(·)-ĝ(t,·), the associated  error equation is obtained as 

 

    ẋ̃̇(t)=Ax̃(t) + g̃(t,x(t)).



 
At a bare minimum then,   adaptive estimation methods for the above uncertain nonlinear ODEs must guarantee that trajectories of this error equation converge to zero.  
It is usually  considerably more difficult to guarantee that the    time-varying function estimate ĝ(t,·):ℝ^d →ℝ converges in the sense that  ĝ(t,·)→ g as  t→∞.  It is this latter problem that is the primary  concern of this paper. 


 
To gain some appreciation of the issues and nuances arising in the function estimation  problem, we consider two examples. Figures <ref> and  <ref> depict the phase portaits of the uncertain systems studied in Examples <ref> and  <ref>, respectively. These figures also include plots of the error in function estimates obtained by the RKH embedding techniques with the kernel of the RKH space selected as described in <ref>. 

 



 The first example is a case of a supercritical Hopf bifurcation, which can be found in many textbooks on dynamical systems <cit.>. The system equations are given by
 
 

    ẋ_1
    ẋ_2
       =  x_2 + x_1(1 - x_1^2 - x_2^2) 
    
     -x_1 + x_2(1 - x_1^2 - x_2^2)
     .



 
Here we define g_0(x):={x_2+x_1(1-x_1^2-x_2^2),0}^T, g(x):={0,-x_1+x_2(1-x_1^2-x_2^2)} in Equation <ref>. 
The figures below make clear that the positive limit set ω^+(x_0) is the circle S^1 for all x_0∈ℝ^2 for this dynamical system. When the method of RKH embedding is applied to this uncertain nonlinear system, we can obtain estimates ĝ(t) of g whose error is depicted in Figures <ref>(b,c).  These estimates have been constructed from finite dimensional approximations as discussed in <cit.> using basis functions that are a collection of  (extrinsic) Sobolev-Matern kernels discussed in Corollary <ref> centered over the positive limit set. 
























 



 In this example, the dynamical system contains a homoclinic loop. The example is studied in detail in <cit.>. The governing equations are 
 
 

    ẋ_1 
    ẋ_2    =  2x_2 
     
     2x_1 - 3x_1^2 + λ x_2(x_1^3 - x_1^2 + x_2^2)



 
In this example we define g_0(x):={2x_2,0}^T and g(x):={0,2x_1-3x_1^2+λ x_2(x_1^3-x_1^2+x_2^2) }^T in Equation <ref>. Again, application of the RKH embedding method of adaptive estimation to this problem can yield approximations ĝ(t) of g with error depicted in Figure <ref>(b,c). These estimates have been constructed from finite dimensional approximations as discussed in <cit.> using basis functions that are a collection of  (extrinsic) Sobolev-Matern kernels discussed in Corollary <ref> centered over the positive limit set.
























 
Several observations about these two examples of application of the RKH embedding method are noteworthy and motivate this paper. 
In each case, the governing equations have the form of the nonlinear ODEs given in Equations <ref>,  and are have error equations of the form in Equation <ref> that is studied in  detail in this paper. The first important observation to make from  the examples   is to note that the positive 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 some offline, optimization-based estimate of an unknown function, it would come as no surprise that its estimates consist of functions that are supported on or near the prescribed data. We will see that, roughly speaking,  convergence of the RKH embedding method is guaranteed by a newly introduced  PE condition and  estimates of the unknown function  are built over regions of the state space where trajectories are in some sense “concentrated.” Here the notion of concentration is understood in terms of the positive limit set ω^+(x_0), which is known to attract trajectories if the orbit is precompact. <cit.>  

Moreover, both of   the positive limit sets in the examples  are striking and exhibit considerable structure: they can often be interpreted as manifolds. In Example <ref> shown in  Figure <ref>, global  solutions of the governing equations exist for every initial condition in ℝ^d. All trajectories converge to the positive limit set, which happens to be the canonical connected, compact, Riemannian manifold,  S^1. The flows generated for various parameters in λ in Example <ref> exhibit more diverse qualitative limiting behaviors. As shown in Figure <ref>(a) when λ=0, the homoclinic loop encircles a stable region. Trajectories inside this region are all limit cycles. For these initial conditions, the positive limit sets are smooth,  regularly embedded submanifolds of ℝ^d. The form of these embedded manifolds is not as simple as in Example <ref>, that is, they are not one of the  well-known, “iconic” manifolds. When λ<0, the equilibrium (x_e,0) for x_e>0 becomes unstable. It can be shown that the homoclinic loop becomes the ω-limit set of all the trajectories starting from this region <cit.>.  

In either case, the examples illustrate a phenomenon that is common to many uncertain estimation problems. While the observations Γ^+(x_0)=∪_t∈ℝ^+x(t) are contained in ℝ^d, there is an underlying set or manifold that supports, approximately supports, or attracts the observed trajectories. We are interested in this paper in understanding conditions that establish that the RKH embedding method “converges over” these underlying structures. 

To frame our discussion of the RKH embedding method we briefly review the general strategy of “linear-in-parameters” (LIP) methods for adaptive estimation of uncertain nonlinear systems of ODEs. So-called LIP estimation might best be described as a part of the technical folklore for methods in adaptive estimation. 
This approach is ubiquitous in the adaptive estimation literature and is a  well-known tool among researchers who study this topic.
It   is safe to say that the most popular versions of adaptive estimation for the above type of uncertain  nonlinear ODEs choose  the function estimate in terms of a  linear-in-parameters  representation 
ĝ(t,·)=∑_k=1^n
ϕ_k(·)α_k(t)=Φ^T(·)  α(t)
with α_k(t) a time-varying 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 are known as the regressors, a common term arising from applications in  nonlinear regression. 
If the unknown function g has the representation g=∑_k=1^nϕ_k(·)α^*_k=Φ^T(·)α^* for some unknown constants {α^*_k}_k≤ n, then the error in function estimates is g̃(t,·)=Φ(·)α̃(t) with the parameter error  α̃(t):=α^*-α(t)∈ℝ^n.  In this case the error in the function estimates g̃(t)→ 0 if the finite set of parameters errors converge α̃(t)
→ 0∈ℝ^n. It is for this reason that the task of estimating functions in the usual LIP framework reduces to questions of parameter convergence in ℝ^n.    

One of the foundations   of modern adaptive estimation for ODEs has been recognition of the fact that persistency of excitation conditions can be sufficient to guarantee parameter convergence. The notion of persistence of excitation in its conventional form, that is, as it pertains to the the ODE error Equations <ref>, is defined next.



The  regressors Φ  are persistently excited by the positive orbit  Γ^+(x_0) if there are positive constants γ_1,γ_2,T, and Δ such that for each t≥ T,

 

    γ_1 α^2_ℝ^d≤∫_t^t+Δ( α^⊤Φ( x(τ) ) Φ( x(τ)                )^⊤α)   dτ≤γ_2α^2_ℝ^d




 
for all α∈ℝ^N.

 The papers <cit.> and a  number of standard texts <cit.> on adaptive estimation make a careful study of this condition and how it facilitates a proof that the parameter error α̃(t) converges to zero as t→∞. 
In some cases it is too much to hope that all the parameters α̂(t):={α̂_1(t), …,α̂_n(t)} in the approximations ĝ(t,·)=∑_k=1,…,nϕ_k(·)α̂_k(t) converge. A  means of weakening the above PE condition introduces the notion of partial persistency of excitation.  
One version of the definition of a partial PE condition modifies  the inequalities above and  replaces them with the condition that

 

    γ_1P_V v_ℝ^n^2
    ≤α^T ∫_t^t+ΔΦ(x(τ))Φ^T(x(τ)) dτ·α≤γ_2 P_Vv_ℝ^n^2




 
with  P_V:ℝ^n →ℝ^n a projection onto a linear subspace V⊂ℝ^n. This generalization then can be used to guarantee, as a special case, that only certain of the coefficient estimates converge, but not all. 


 
As we will discuss in more detail shortly, the method of RKH embedding recasts the above adaptive estimation problem so that the state errors x̃(t)  and function errors g̃
(t):=g̃(t,·) evolve in a product space having the form ℝ^d ×ℍ with ℍ=H^d a vector-valued RKH space of functions. The space H is known as the hypothesis space and its selection is based on what class of priors or information seems relevant regarding the estimation problem at hand.  
 The precise form of the PE definition in this paper, and the associated theorems that depend on it, are written for a model problem with the vector-valued function g:=Bf
with f:ℝ^d →ℝ a scalar valued function and B∈ℝ^d× 1. This restriction does not seem too severe, simplifies the notation considerably, and conveys the underlying geometric relationships between orbits Γ^+(x_0) of semiflows starting at x_0∈ X, persistency of sets Ω,  and RKH spaces H_Ω. Moreover, the  extension to general vector-valued functions would proceed in principle along the same lines as the strategy in <cit.> used for consensus estimation. 



 


 §.§ Overview of New Results



 

In either of the papers <cit.> some of the standard questions regarding RKH embedding have been discussed such as existence of solutions and well-posedness, continuous dependence on initial conditions, as well as stability and convergence of finite dimensional approximations.  In this paper we focus primarily on building more intuition and insight regarding the newly introduced notion of persistency of excitation in the RKH embedding method. Starting with an RKH space H_X=span{𝔎(x,·)| x∈ X} of functions over X, we then define for some indexing set Ω⊆ X the subspace 
H_Ω:=
span{𝔎(x,·)  |  x∈Ω}. Note carefully that functions in H_Ω are supported on X, which is why Ω is referred to as the indexing set. This space is related to, but distinct from, the space R_Ω(H_X) that are restrictions of functions in H_X to the subset Ω. 
We have the following definition of persistence of excitation for the RKH error Equations  <ref>.

 


The indexing set Ω and RKH space H_Ω  are persistently excited by the orbit Γ^+(x_0) if there are positive constants γ_1,γ_2,T, and Δ such that for each t≥ T,

 

    γ_1f^2_H_Ω≤∫_t^t+Δ( ℰ_x(τ)^*ℰ_x(τ)f,f                    )_H_X  dτ≤γ_2f_H_Ω^2




 
Here ℰ_x:f↦ f(x) is the evaluation functional at x and ℰ_x^* is its adjoint operator. 
The classical definition given above defines persistency of excitation for a specific set of regressors and the trajectory of a semidynamical system. The new PE condition holds for an indexing  set Ω⊆ X, space of functions H_Ω, and a trajectory of a semidynamical system.
  It should be noted that the  PE condition above is over a set Ω⊂ X, which may or may not be the entire state space X. It this sense it bears some resemblance of the  partial  PE conditions that are defined over subspaces of parameters in ℝ^n. The similarity in form is all the more apparent when we note that ·_H_Ω:=P_Ω(·)_H_X with P_Ω the H_X-orthogonal projection onto H_Ω: the closed subspace H_Ω is endowed with the norm it inherits  from H_X. 
It should also be pointed out that both the set Ω and the kernel 𝔎_X (that determines H_X, and therefore determines H_Ω⊆ H_X)  are free to be selected when trying to apply the  above PE condition in the method of RKH embedding. 


 
Intuitively, we expect some kernels are more useful than others in the RKH embedding method, and one of the primary thrusts of this paper is to explore the alternatives.  As we will see, we obtain a strong conclusion about what type of indexing sets Ω are PE when we restrict attention to kernels that define function spaces that are good at separating important subsets of X. There are many ways to think about how well the functions in a space H_X separates points or sets. We find that  one important class of RKH spaces consists of functions that feature a rich set of (possibly smooth) cut-off or bump functions. Our first primary result in Theorem <ref> is that if the RKH space does indeed contain a rich collection of these functions, then we have the following  implication, 

    “Ω and   H_Ω are PE”  ⟹  Ω⊆ω^+(x_0),

with ω^+(x_0) the positive limit set of a trajectory starting at x_0. 
In other words in terms of the new definition of PE,  if a trajectory Γ^+(x_0) persistently excites a  set Ω, the set Ω  is contained in the positive limit set ω^+(x_0).
This result provides novel insight into the structure of this type of persistently excited systems : PE sets are not transient but rather consist of points whose neighborhoods are visited by the trajectory  infinitely often. In fact, a bit more is actually required as illustrated in Theorem <ref>: the “time of visitation” is bounded below in a certain sense. This intuition should be compared with the interpretations of the usual Definition <ref>: a vector signal t↦Φ(x(t))∈ℝ^n is (partially) PE if on average it visits all directions in (a subspace of) ℝ^n. 


 
We should also emphasize at this point that while the intent of this paper is to inform and enhance our understanding of the RKH embedding method, the result in Equation <ref> is not dependent on the fact that the trajectory  under study t↦ x(t)∈ℝ^d happens to be the solution of our model ODE problem in Equation <ref>. We have worked to express the condition in Equation <ref> in very general terms. As long as the PE condition holds under the hypothesis described above, and  Γ^+(x_0) is the forward orbit  of a continuous semiflow on the complete metric space (X,d_X), we conclude that Ω⊆ω^+(Ω).


 
It is  then natural to ask how to choose kernels that exhibit the separation properties that enable the conclusion above.
One approach, which we refer to as an intrinsic method, applies to cases in which  X=Ω is in fact a compact, connected, smooth, Riemannian manifold M.  Here we assume that M is known and that the kernel over M is known. Example <ref> is the type of problem we have in mind here, where the positive limit set is a simple well-known manifold. Numerous intrinsic kernels can be defined over the circle, the sphere, or more generally homogeneous manifolds <cit.>.  In this case we choose kernels that guarantee that the native space H_M is in fact equivalent to a certain Sobolev space W^r,2(M) for r large enough. That such equivalences are possible follows from the Sobolev embedding theorem. <cit.> The Sobolev spaces defined over such a manifold M can be shown to contain a rich family of smooth cutoff functions. In this framework, if the forward orbit Γ^+(x_0) for some t_0∈ℝ of  any continuous flow on M is PE, Corollary <ref> implies that the  manifold is transitive. That is, it supports a flow  that has a dense orbit. The study of when a particular manifold is transitive is of interest in its own right <cit.>, so the new PE condition can be used to study whether a manifold is transitive. 


 
While  this is an interesting result, it is not usually strictly or directly  applicable to understanding the convergence properties of the RKH embedding problem. There are two essential problems here. First, there are many problems where the positive limit set might be a nice smooth, compact, Riemannian  manifold ω^+(x_0)=M, but we do not know the form of the manifold a priori. In such cases defining the kernel in closed form to be used in analysis or approximation is impossible. Example <ref> is of this type: the positive limit set is a smooth manifold, but it is not one over which catalogs of intrinsic kernels are defined. It is also possible,  on the other hand, that we do know the exact form of the manifold, but it is not one of the standard manifolds like the circle, sphere, or torus.  Even if  we in principle can define the kernel  through the fundamental solutions of certain elliptic differential operator on the manifold M as in Corollary <ref>,  it may be intractable to compute this fundamental solution for the manifold at hand. This problem can be as hard, or harder perhaps, that the original estimation problem. 


 
It should be kept in mind that the aim of the RKH embedding method is to carry out adaptive estimation of uncertain nonlinear ODEs. It is typically the case in such situations that the exact form of the positive limit set is unknown. That is, we are more interested in problems like Example <ref>, in contrast to Example <ref>.
In this case,  we assume the M  is an unknown, connected, smooth, 
 (regularly) embedded submanifold of ℝ^d, and  we resort to an extrinsic method. In this technique we build a well-defined kernel on X for a large set X that contains M, and then we define a kernel by restriction on the manifold M ⊆ X. It can be the case that a  plethora of kernels exist for good kernels over the large space X=ℝ^d.  Taking care to choose  the kernel smooth enough, we obtain a kernel on M defined by restriction. The expression for the kernel on M is given in terms of the kernel on the larger space X, which is known. Corollary <ref> then shows that that M=ω^+(x_0) in this case. All of the numerical examples depicted in Figures <ref> and <ref> have been computed using this extrinsic method.






§ NOTATION


 
In this paper the symbols ℕ^+, ℝ,ℝ^+ denote the non-negative integers, real numbers, and non-negative real numbers, respectively. The expression 
a ≲ b 
means that there is a  constant c>0 that does not depend on a,b such that a ≤ c · b. The symbol ≳ is defined similarly.  
The paper makes use of Lebesgue spaces and Sobolev spaces on subsets Ω of ℝ^d,  and it also uses these spaces when they are defined more generally on  measurable subsets Ω of  certain  Riemannian  manifolds M.   The norm on the Banach spaces
  L^p(Ω):=L^p_μ(Ω) of μ-integrable  functions over  Ω⊆ℝ^d take the familiar  form 
f^p_L^p(Ω):=∫_Ω |f(x)|^pdμ
with the measure μ on ℝ^d for  1≤ p<∞, with the usual modification for p=∞. Recall that  when Ω⊆ℝ^d, the Sobolev space W^r,p(Ω) for a positive integer r consists of functions that have weak derivatives of all orders less than or equal to r in L^p(Ω), and the norm on these Banach spaces is usually written 

 

    f_W^r,p(Ω)^p:=∑_0≤ |α| ≤ r∫_Ω |∂^|α|f/∂ x^α |^p dx



 
with the summation taken over all multi-indices α=(α_1,…,α_d),  |α|=∑_i=1,…,dα_i, and here the measure μ is selected to be Lebesgue measure dx.  The Sobolev spaces for non-integer r>0 are defined in terms of interpolation theory as discussed in <cit.>. 
A bit more detail is required to define the spaces L^p(Ω) and W^r,p(Ω) for Ω⊆ M, with M a manifold.  In this paper M is always assumed to be a connected, complete Riemannian manifold with a positive injectivity radius and bounded geometry.  See <cit.>, Chapter 7 or <cit.> for a discussion of these properties.  For purposes in this paper, it suffices to note that  compact, connected Riemannian manifolds and ℝ^d satisfy these conditions. For such  a Riemannian manifold M denote the  metric g and inner product <·,·>_g,p on the tangent space T_pM. We define the associated volume measure dμ on M, and its local representation in terms of the set of coordinates (x^1,…,x^d) is given by dμ(x):=√(det(g))dx^1… dx^d. The norm f_L^p(Ω) has the same expression given above with the measure selected to be the usual volume measure on the manifold M. 
The Banach spaces W^r,p(Ω) for measurable subsets Ω⊆ M are equipped with the norm 

 

    f^p_W^r,p(Ω):=
    ∑_j=0,…,r∫_Ω
    |∇^jf|^p_g,pdμ(p)



 
for 1≤ p<∞ where ∇ is the covariant derivative over (M,g). When applied to a set Ω⊆ M=ℝ^d, the definitions over the manifold M define norms that are equivalent to the usual ones for Sobolev spaces defined on subsets of ℝ^d. As discussed in <cit.> in this case the expression in Equation <ref> amounts to a simple reweighting of the derivative terms in Equation  <ref>. The Sobolev spaces W^r,p(M) for non-integer r>0 are, as in the case above, defined via interpolation theory. <cit.> The non-integer spaces are crucial to the statement of trace theorems for Sobolev spaces, which are used in this paper to study the restrictions of functions that define certain RKH spaces. 

 



§ REPRODUCING KERNEL HILBERT (RKH) SPACES



 


In this paper we make use of several properties of real, scalar-valued, RKH spaces.  Also, the analysis below is readily extended to real, vector-valued RKH spaces ℍ:=H^k for k∈ℕ.  See <cit.> for the case where this is carried out in the context of consensus estimation. 





 


 §.§ Basic Definitions and Constructions


 

An RKH space H_X of functions that map a set X ⊆ℝ^d →ℝ  is defined in terms of a real-valued, continuous, symmetric, and positive type function  𝔎_X:X × X→ℝ that is referred to as the kernel underlying the RKH space. The subscript on 𝔎_X is used to emphasize the set over which the kernel, as well as the functions in H_X are defined.  When we say that 𝔎_X is of positive type, this means that 
∑_i,j≤ 1^N α_i 𝔎_X(x_i,x_j) α_j:=α^T𝕂_X,Nα≥ 0
for all { x_1,…,x_N}⊂ℝ^d 
and   α:={α_1,⋯,α_N}^T∈ℝ^N, with the collocation matrix associated with {x_i}_1≤ i≤ N defined as  𝕂_X,N:=
[𝔎_X(x_i,x_j)]∈ℝ^N× N. So all the collocation matrices of a  kernel of positive type are positive semidefinite. We say that the kernel is of strictly positive type if all of its collocation matrices 𝕂_X,N:=[𝔎_X(x_i,x_j)] for distinct  points {x_k}_1≤ k≤ N  are strictly positive 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 product of two such functions 𝔎_X,x,𝔎_X,y is defined to be (𝔎_X,x,𝔎_X,y)_H_X:=𝔎_X(x,y) for all x,y∈ X.  The RKH space 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→∞∑_i=1^N α_N,i𝔎_X, x_N,i},



 
where α_N,i∈ℝ and x_N,i∈ X.
The closure above is taken with respect to the candidate inner product. The Hilbert space (H_X,(·,·)_H_X) above is also known as the native space induced by the kernel 𝔎_X.  It is well-known <cit.> that with this construction the reproducing property 
(𝔎_X,x,f)_H_X=f(x)
holds for all f∈ H_X and x∈ X. Any Hilbert space H is in fact a RKH space if all of the evaluation functionals that act on H are in fact bounded operators from H→ℝ. If it is further known  that if for some positive  constant 𝔎 we have  sup_x∈ X𝔎_X(x,x)≤𝔎̅_X<∞, then the evaluation operator ℰ_x:H_X →ℝ given by 
ℰ_x:=ℰ_H_X,x: f↦ f(x) 
is a uniformly bounded linear operator since 
|f(x)| = |ℰ_x f| = |(𝔎_X,x,f)_H_X|≤√(𝔎_X(x,x))f_H_X.
This implies that f_C(X)≲f_H_X, and therefore we have the  continuous inclusion H_X ↪ C(X). We will only consider kernels 𝔎_X on X for which such a constant 𝔎_X exists. 

 

Later in the paper we also make extensive use of the closed subspaces 
H_Ω:= span{𝔎_X,x |  x∈Ω}⊆ H_X for some subset Ω⊆ X. These spaces are important in understanding how the new PE condition are applied. One important fact is that we have the H_X-orthogonal decomposition H_X=H_Ω⊕ V_Ω
with V_Ω the kernel of the trace or restriction operator on the set Ω⊆ X,
V_Ω:={ f∈ H_X  |  R_Ω f=. f |_Ω=0}.
That is, f∈ V_Ω if and only if f(x)=0 for all x∈Ω. This fundamental property follows from the analysis in <cit.> and <cit.>. Finally, in some cases when we specifically discuss  spaces derived from restrictions of functions H_X to a subset Ω, we denote these RKH spaces as R_Ω(H_X). 




 


 §.§ Separation of Closed Sets by Reproducing Kernels



 


The current paper is interested in understanding how the use of a RKH space can make precise certain notions of convergence in adaptive estimation. We want to understand the geometric implications of the PE condition, that is, what it implies  about the trajectories of the dynamical system and the PE set. Essentially, we will “test convergence" in X of trajectories x(t)→x̅ by the condition that f(x(t))→ f(x̅) for all f∈ H_X. As we will see, it can be important for understanding persistence that the space H_X contain enough functions to separate, in a certain sense, the points of X.  Here an example can illustrate the the problem.  It is known that it is always possible to induce a metric d_K associated with the kernel 𝔎_X as described in <cit.>.
The problem is, our semiflows will be continuous with respect to some metric d_X, and the topology induced by d_X may not be the same as that generated by d_K.  In fact it is easy to come up with kernels for which this is the case. As noted in Remark 1 of <cit.>, the bilinear kernel k_X(x,y):=x^⊤y for x,y∈ℝ^d induces a RKH space H_X for which the only subsets that can be separated are linear manifolds. 
In this specific case, d_K induces a topology that is strictly coarser than the usual topology on X:=ℝ^d. Specifically, the metric d_K can be used to discriminate convergence to a particular line through the origin, but not convergence to a point on that line. 


 
We will see that some  useful geometric insights regarding the PE condition and positive orbits result if we do not allow the  kernel to induce such a coarse topology.  We would like the metric generated by the kernel to be equivalent with that on the state space. Reference <cit.> gives one example of a useful and simple  separation property.  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 all a∈ A and f(b)≠ 0. This condition can be used to prove that d_K and d_X define the same topology. However, we will employ an even stronger condition, one that is well-suited to the construction of native spaces that contain  well-known  classes of  differentiable functions.   We assume  the existence of a rich family of bump functions in H_X. We say that b_r,x:X→ [0,1] is a bump function on (X,d_X) associated with the open ball B_r,x={y∈ X | d_X(x,y)<r} provided that 1) b_r,x=1 on a neighborhood of x, and 2) b_r,x is zero outside a compact set contained in B_r,x. It is immediate that if for any open set B_r,x, there is an associated bump function b_r,x∈ H_X, then the RKH space H_X separates the d_X-closed subsets of X. We say that the space H_X contains a rich family of bump functions if it contains a bump function b_r,x for each open ball B_r,x.  The construction of smooth bump functions on X=ℝ^d is a classical exercise in analysis on manifolds,  <cit.>  pages 49–51. In practice, the RKH space H_X (even when X≠ℝ^d) will be selected so that it contains them. See the proofs below of Corollaries <ref> and <ref>. 


 



§ THE RKH EMBEDDING METHOD



 


In this paper, we study a model problem of adaptive estimation for uncertain nonlinear systems governed by ordinary differential equations that have the form

 

    ẋ(t) = A x(t) + B f(x(t)),      x(0) = x_0



 
with A ∈ℝ^d × d a Hurwitz matrix, B ∈ℝ^d × 1, and f: ℝ^d →ℝ. This equation is a special case of the general form in Equation <ref>, with g:ℝ^d→ℝ^d:=Bf. Methods for ensuring that this system of ODEs has local or global solutions are well-known, <cit.>, and in this paper we always assume that for each x_0∈ X the equations  have classical solutions on ℝ^+. In this equation, it is assumed that the matrices A and B are known, but the (nonlinear) function f is unknown. The adaptive estimation problem considered in this paper  uses the observations of the full state, x(t) for all t ≥ 0, to construct estimates x̂(t) → x(t) and f̂(t) → f as t →∞. While f is unknown in our adaptive estimation problem, information about this function is reflected in the choice of an hypothesis space H of functions to which f belongs. Perhaps the most familiar choice of hypothesis space H is one that is finite dimensional
H_n :=  { f = ∑_i=1,…,nα_i ϕ_i  }
with {ϕ_i }_i=1^n some fixed set of basis functions and ϕ_i: ℝ^d →ℝ for 1 ≤ i ≤ n. If we suppose for the moment that the unknown function f = ∑_i=1^n α^*_i ϕ_i ∈ H_n, then one canonical choice of an estimator is

 

    ẋ̂̇(t)    = A x̂(t) + B Φ^T(x(t)) α̂(t) 
    α̇̂̇(t)    = - Γ^-1Φ(x(t)) B^T P(x(t) - x̂(t))



 
with P ∈ℝ^d × d the symmetric positive definite solution of Lyapunov's equation PA + A^T P = -Q for a user-designed symmetric positive definite matrix Q ∈ℝ^d × d, and Γ∈ℝ^n × n symmetric and positive definite. When the errors in state x̃(t) = x(t) - x̂(t) and parameter errors  α̃(t) = α^* - α̂(t) are defined, it can be shown directly that the errors satisfy the equations


 

    ẋ̃̇(t) 
    α̇̃̇(t)
        
        =
        [                  A        B Φ^T(x(t)); -Γ^-1Φ(x(t)) B^T P                  0 ]x̃(t) 
    α̃(t)
        
        +
       E(t)





 for E(t):={Be(t),0}^T, with e(t) = 0 if f∈ H_n. If it happens that f ∉ H_n, then e(t):= f(x(t)) - ∑_i=1^n α^*_i ϕ_i(x(t)) := f(x(t))-f_n(x(t)) with f_n a suitable finite dimensional approximation of f.  Precise conditions on the exponential stability of this system are a classical topic in adaptive estimation for uncertain ODEs. See <cit.> when e(t)=0. When e(t)≠ 0, see <cit.> for related  discussions of ultimate boundedness of errors. 

 


In this paper, we are interested in a class of dynamical systems where the unknown function f belongs to the RKH space H. The  generic RKH space H may be the full space H_X or one of its closed subspaces H_Ω described in Section <ref>. The plant, estimator and the learning laws for this case can be expressed as

 

    ẋ(t)    = A x(t) + B ℰ_x(t) f, 
    ẋ̂̇(t)    = A x̂(t) + B ℰ_x(t)f̂(t), 
    ḟ̂̇(t)    = Γ^-1 (B ℰ_x(t))^* P (x(t) - x̂(t)),



 
where x(t), x̂(t), A, B, and P are defined as above. But the (nonlinear) functions f and f̂(t) belong to the RKH space H and ℰ_x: H →ℝ^d is the evaluation functional that is defined as ℰ_x f = f(x) for all x ∈ X and f ∈ H. Furthermore, the term Γ∈ℒ(H,H) in the above equation is a self-adjoint, linear positive definite operator. The error equation analogous to the classical case shown above is given by



    ẋ̃̇(t) 
    ḟ̃̇(t)
           =
        [                    A             B ℰ_x(t); -Γ^-1 (B ℰ_x(t))^* P                    0 ]x̃(t) 
    f̃(t)
        
       = 𝔸(t)
        x̃(t) 
    f̃(t).




Note that the evolution of the above error equation is in  ℝ^d × H as opposed to on ℝ^d ×ℝ^n in the classical adaptive estimator case. Some elementary conditions that guarantee the existence of solutions, as well as their continuous dependence on initial conditions, are given in <cit.>. In this paper we always assume that for each x_0∈ X the equations admit a classical  solution t↦ (x̃(t),f̃(t))∈ℝ^d× H for t∈ℝ^+. The following theorem, which simplifies considerably the analysis in <cit.>, shows that this is reasonable for many common choices of the RKH space H.

 

Suppose that the RKH space H is generated by a kernel 𝔎_X:X× X →ℝ for which sup_x∈ X𝔎_X(x,x) ≤𝔎<∞. Then for each (x̃_0,f̃_0) ∈ℝ^d × H there is a unique solution of Equation <ref> in C^1([0,∞),ℝ^d× H).



 

As discussed in Section <ref>, the hypotheses guarantee that the evaluation operator ℰ_x:H→ H is linear and uniformly bounded in x∈ X. It is immediate that 

    lim_t→∞1/tlog^+(𝔸(t))=0

with log^+(ξ):=max(0,log(ξ)). As discussed on page 211 of <cit.> the governing equations have a unique global solution in time.



 
The definition of the PE condition proves sufficient for convergence of function estimates f̂(t)→ f generated by the RKH embedding method, much as in the conventional, finite dimensional case.  The analysis of convergence of parameters (ie, functions in our case) is notoriously long, so in this short paper we merely outline the proof in a special case. The full and lengthy details (for general P and Hurwitz A) are given in <cit.>. 

We say that a family of functions ℱ over a set S is uniformly equi-continuous if for each ϵ>0, there is a δ_ϵ>0 such that for all f∈ℱ and a,b∈ S, |a-b|<δ_ϵ⇒f(a)-f(b)<ϵ.

 

Suppose that P=I, A is negative definite, ℱ={f(x(·))|f∈ H,f_H=1} is uniformly equi-continuous, and the trajectory Γ^+(x_0) is persistently exciting in the sense of Definition <ref>. Then the solution x̃,f̃ of Equations <ref> satisfy
    lim_t→∞x̃(t)=0 and 
    lim_t→∞f̃(t)=0.



 

The proof that x̃(t)→ 0 follows along lines that are entirely analogous to the classical or finite dimensional case, see <cit.> for the details when arguments are lifted to the infinite-dimensional state space ℝ^d × H. The conclusion that f̃(t)→ 0∈ H follows immediately from Theorem 3.4 of <cit.>, provided that we can prove that there exists constants T,Δ,δ,γ>0 such that for each t≥ T and f∈ H with f_H=1 there is an s∈ [t,t+Δ] such that 

    B_ℝ^d | ∫_s^s+δ E_x(τ)fdτ |=
    ∫_s^s+δ BE_x(τ) f dτ_ℝ^d > γ.

However, the condition above can be shown to be equivalent to the PE Definition <ref> provided that the integrand is smooth enough to eliminate the possibility of certain “rapid switching” behavior. The equivalence of conditions as in Equation <ref> to those similar to  Definition <ref> in the classical, finite dimensional case have been studied in great detail. See <cit.> for a detailed discussion with excellent illustrative examples of pathological rapid switching in the finite dimensional case. In the case at hand, Equation <ref> follows from the fact that f(x(·))∈ℱ, a family of uniformly equi-continuous functions. The lengthy details of the proof can be found in <cit.>. 


 



§ SEMIFLOWS AND PERSISTENCE OF EXCITATION (PE)



 

In this section we recall of few of the basic definitions of dynamical systems theory that will be essential to the analysis of this paper.  The aim is to be able to define persistence of excitation, not only for the model problem in Equations <ref> or <ref>, but for more  general evolutions on metric spaces. In particular we obtain a PE condition that can be applied to flows on Riemannian manifolds, which encompass  a few of our examples.  A continuous semiflow or semidynamical system on the complete metric space (X,d_X)  is  defined in terms of a continuous  semigroup {S(τ)}_τ≥ 0 on X .  The manner in which systems of ODEs can generate such a semigroup, and thereby a semidynamical system is well-studied <cit.>. The positive orbit Γ^+(x_0) starting at x_0  defined to be the set 

 

    Γ^+(x_0):=⋃_τ≥ 0 S(τ)x_0 ⊆ X.



 
The positive limit set ω^+(x_0) associated with the initial condition x_0 is defined to be 
 

    ω^+(x_0):=⋂_t≥ 0⋃_τ≥ tS(τ)x_0,



 
which is equivalently expressed as 

 

    ω^+(x_0)={ 
    y∈ X  |  ∃ t_k →∞ s.t.lim_k→∞ S(t_k)x_0  → y
    }.






 


 §.§ Persistence and Positive Limit Sets



 

The next few results illustrate simple and often  intuitive relationships between persistently excited sets, positive orbits Γ^+(x_0), and the positive limit set ω^+(x_0).  We start with a simple result that illustrates an intuitive notion of what the new PE Definition <ref> entails.

 


Let 𝔎:X→ℝ be a monotone nonincreasing radial basis function and suppose  the associated kernel 𝔎_X(x,y):=𝔎(d_X(x,y)) induces the RKH space H_X↪ C(X) for some fixed ϵ > 0. Define the measurable sets 
I_t,ϵ={τ∈ [t,t+Δ]   |  x(τ)∈ B_ϵ,x_∞}
for each t≥ T_0 with B_ϵ, x_∞ the open ball of radius ϵ centered at x_∞. If the 
the Lebesgue measure μ satisfies
μ(I_t,ϵ) ≥γ_ϵ  >0
for some constant γ_ϵ for all t≥ T_0, then the singleton indexing set Ω:={x_∞} and the closed subspace H_Ω⊂ H_X are  persistently excited in the sense of Definition <ref>. In particular, if x(t)→ x_∞, the set Ω:={x_∞} and closed subspace H_Ω are persistently excited. 

Before proving the above theorem, let us unpack the above definition to understand the relatively straightforward underlying idea. The interval I_t,ϵ is the set of times contained in the interval [t,t+Δ] during which the trajectory t↦ x(t) is within ϵ of the point x_∞. This theorem says that if a trajectory  spends at least γ_ϵ amount of time in each interval [t,t+Δ]  in the ball of radius ϵ centered at x_∞, then  Ω={x_∞} and H_Ω are persistently excited.


 

Without loss of generality, we assume that the kernel 𝔎_X is normalized 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=α𝔎_X,x_∞ we have f_H_Ω=α=f(x_∞). 
Only the lower bound of the persistency definition is problematic, and we compute directly that 

    ∫_t^t+Δ ( ℰ^*_x(τ)ℰ_x(τ) f,f  )_H_X dτ
        = α^2 ∫_t^t+Δ𝔎_X^2(x(τ),x_∞) dτ
    
    
    
    
       ≥α^2  ( min_0≤ξ≤ϵ𝔎^2(ξ)  ) μ(I_t,ϵ) ≥γ_ϵ𝔎_ϵf_H_Ω^2

with 𝔎_ϵ:=min_0≤ξ≤ϵ𝔎^2(ξ). 



 
Theorem <ref> gives a direct interpretation of the persistency condition when we consider a singleton Ω:={x_∞} in terms of visitation to a neighborhood of x_∞. It also suggests that there are many choices of kernels 𝔎_X that induce PE spaces H_Ω for any convergent trajectory x(t)→ x_∞. 
The monotonicity of the kernel in the above theorem is satisfied for a host of common choices of RKH spaces, see Chapter 9 of <cit.> for the definition of completely monotone radial basis functions and kernels. This fact illustrates a significant difference with the conventional PE definition: there are many convergent trajectories that simply are not classically PE for a given set of regressors. We also note that if X=ℝ^d, there is a direct extension of this theorem for the finite set Ω:={x_∞,1,…,x_∞,M}, see Lemma 3.4  in <cit.>. 

We begin our study of the geometric nature of PE sets by noting that the forward orbit is always dense in PE sets.


 


Let 
 H_X   be an  RKH space of functions    over the domain X and suppose that this RKH space includes a rich family of bump functions. If the PE condition in Definition  <ref> holds for a subset  Ω⊆ X, then the forward 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.




 

Suppose to the contrary that there is an y∈Ω for which there is  no such convergent sequence.  This means that there is an  open ball B_r,y such that Γ^+(x_0)⋂ B_r,y=∅. 
But since we have 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∉C_r,y
with C_r,y⊂ B_r,y a compact set. However, 
from Section <ref>, 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)⋂ B_r,x=∅, the integral in Definition <ref> is equal to zero

 

    ∫_t^t+Δ(ℰ^*_x(τ)ℰ_x(τ)b_r,y,b_r,y)_H_Ω dτ
        = ∫_t^t+Δ|b_r,y(x(τ))|^2 dτ=0



 
for each t≥ T. 
Since 

 

    b_r,y_H_Ω:=P_Ω b_r,y_H_X=b_r,y_H_X≳b_r,y_C(X)>0,



 
this is a contradiction of the PE property in Definition <ref> and the theorem is proven.



 
Note that Theorem <ref> does not require that the set of times t_k→∞.  Recall, on the other hand,  that the positive limit set ω^+(x_0)  is contained in the closure of all accumulation points  of the orbit Γ^+(x_0) for sequences of the form {S(τ_k)x_0}_k∈ℕ, as τ_k→∞. 
Next, we discuss a relationship of the positive limit set ω^+(x_0) and a PE space H_Ω over the indexing set Ω⊂ X in Definition <ref>. 


 


Let H_X be the RKH space of functions over X  and  suppose that this RKH space  includes a rich family of bump functions. If the PE condition in Definition  <ref> holds for Ω, then 
Ω⊆ω^+(x_0).



 

The proof of this result is similar to the argument in Theorem <ref>, so we only outline it. For an arbitrary 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 arbitrary y∈Ω and fix r_0>0.  Choose t_0 ∈ (T,T+Δ) such that x(t_0)∈ B_r_0,y.  Such an x(t_0) must exist. 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 b_r_0,y_H_Ω>0,   for which  the integral ∫_T^T+Δ |b_r_0,y(x(τ))|^2 dτ =0 would follow from the condition that B_r_0,y∩{x(τ) | τ∈ (T,T+Δ)}=∅. This is a contradiction of the PE condition. We can then set r_1=r_0/2 and repeat this process seeking a t_1∈ (2T, 2T+Δ) such that x(t_1)∈ B_r_1,y, and so 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 desired conditions above, and we must have y∈ω^+(x_0). 






 


 §.§ Persistence of Excitation for Semiflows on Manifolds



 

A careful reading of the Definition <ref> makes clear that it depends on the orbit Γ^+(x_0) of a continuous semiflow {S(t)}_t≥ 0 on a complete metric space (X,d_X), a subset Ω⊆ X, and an admissible kernel 𝔎_X that defines the RKH space H_X (and therefore also the closed subspace H_Ω). Since it applies to subsets of complete metric spaces, it makes sense to consider much more general systems than the ODEs in the model Equations <ref> or <ref>. For instance, we have the following result for semiflows on manifolds, the case when the state space X=Ω=M in the PE Definition <ref>. Note that below the semigroup S(t) that defines the positive orbit Γ^+(x_0) is defined on all of M=X. 


 


Suppose that M  is a  compact,   connected, d-dimensional  Riemannian manifold, and 𝔎_M is kernel that induces a native space H_M whose norm ·_H_M is equivalent to that of the Sobolev space W^r,2(M). If the orbit Γ^+(x_0) persistently excites H_M, then ω^+(x_0)=M. 




We first show that there are indeed such kernels 𝔎_M that induce a native space H_M ≈ W^r,2(M). The Sobolev embedding theorem  on ℝ^d states that W^r,2(ℝ^d) is continuously embedded in C(ℝ^d), W^r,2(ℝ^d)↪ C(ℝ^d) when r>d/2. As noted on page 1748 of <cit.>, this fact can be used to conclude that W^r,2(M) is continuously embedded in C(M), W^r,2(M)↪ C(M). 
This means that we have  
|ℰ_xf|=|f(x)|≤f_C(Ω)≲f_W^r,2(M)
for each x∈ M and f∈ W^r,2(M). In other words, 
each evaluation functional ℰ_x: W^r,2(M)→ℝ on M  is  bounded.  But  W^r,2(M) is a Hilbert space; boundedness of all its evaluation functionals implies that  W^r,2(M) is a RKH space. 
We define  the  Sobolev-Matern kernel 𝔎_M^r of smoothness r>0 to be  the unique fundamental solution of the elliptic  differential  operator equation  ∑_1≤ℓ≤ r  (∇^ℓ)^*∇^ℓ𝔎_M^r=δ where ∇ is the covariant derivative operator over the manifold M and δ denotes the Dirac distribution. When we define the native space H_M in terms of the Sobolev-Matern kernel 𝔎_M^r, we have  H_M 
≈ W^r,2(M) for the chosen range r>d/2. The details of this analysis are given in   <cit.> for the case when M is a smooth Riemannian manifold that satisfies our standing assumptions on M,  or see reference  <cit.> for the special case M:=ℝ^d.    We next show  that the RKH space H_M  defined in this way contains a rich family of (smooth) cutoff or bubble functions. This proof is not surprising given what we know about Sobolev spaces on subsets of ℝ^d. One way to define W^r,2(Ω) is as the completion of C^∞(Ω) in the Sobolev norm, so  the space C^∞(Ω) is dense in W^r,2(Ω).  It is well-known that for any open ball contained  ℝ^d, there is a smooth cutoff function with compact support contained in that ball. This is a standard result in the study of manifolds and the construction of partitions of unity. <cit.>  It follows that the Sobolev space W^r,2(Ω) contains a rich family of bubble functions. The result extends more generally to Sobolev spaces W^r,2(M) using the exponential map. The details of the proof are rather long, which we simply outline below.  (Particular examples of such a construction can be found in <cit.> on page 1749 and again on page 1751 of the same reference.) If f̂ is a cutoff function on a ball B_x,r⊂ℝ^d, it is possible to construct an associated cutoff function on the image Exp_q(B_x,r)⊂ M under the  exponential map Exp_q:T_qM→ M from f:=f̂∘ Exp_q^-1.  Such an f is always an element of C^∞(M) since f̂ is just a smooth representation of f with respect to a compatible C^∞ chart. The only technical difficulty is showing that f∈ W^r,2(M). But this follows from Lemma 3.2 of <cit.> which states that the exponential operator Exp_q induces a map g→ g∘ Exp_q that is  boundedly invertible from W^r,p(Exp_q(Ω)) to W^r,p(Ω) for any measurable set Ω⊆ℝ^d. Alternatively, we can argue that W^r,2(M) is the completion of C^∞(M) (<cit.>, Section 7.4.5) with respect to the norm in Equation <ref>. We conclude that  W^r,2(M) contains a rich family of (smooth) cutoff functions. If  the motion over the manifold M satisfies the persistency condition in Definition <ref>, then M:=ω^+(x_0). 



 
This example illustrates that the newly introduced persistency condition can be  applicable, in principle, to the study of certain evolutions over smooth   Riemannian manifolds. Still, the analysis in the example above is fairly abstract.  Perhaps more importantly, it is not a simple task to come up with a closed form expression for the Sobolev-Matern kernel. Of course this can be done for some standard manifolds like ℝ^d, the circle, or a torus, since the Sobolev-Matern kernels can be written down for these cases.  But it  is not readily accomplished  for some arbitrary  manifold M. The definition of the space H_M≈ W^r,2(M) is intrinsic here: it depends on the (usually unknown) domain of the manifold  M, the atlas of charts used to define the manifold, and the covariant derivative operator  intrinsic to the manifold. 


 
We next discuss how it is possible to come up with constructions of a kernel for M that is extrinsic in the sense that it is defined by the restriction of some known kernel on a larger domain that contains M. This terminology is used in <cit.> that studies the approximation properties of spaces constructed in such a fashion. This line of attack is particularly  useful to the study of unknown or uncertain  dynamical systems via the RKH embedding method. The persistency of excitation condition is cast in terms of the kernel on the larger space in this case, which is assumed to have a known closed form expression. 
Carefully note that the forward orbit Γ^+(x_0) in the following theorem is defined in terms of a semigroup S(t):M→ M, but M is a proper subset of X. 


 


Let M be an m-dimensional, smooth, compact, (regularly) embedded submanifold of X:=ℝ^d, and suppose that the {S(t)}_t∈ℝ^+ defines a d_M-continuous semiflow on M with d_M the metric on M. Denote by 𝔎_X^r the Sobolev-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 space generated by 𝔎_M. If the orbit Γ^+(x_0) of the semiflow on M persistently excites R_M(H_X), then M=ω^+(x_0). 



 

 The Matern-Sobolev kernels over ℝ^d are given for r>d/2 by
𝔎_X(x,y)=𝔎_X(x-y_ℝ^d)
with
𝔎(ξ)
=2^1-(r-d/2)/Γ(r-d/2)ξ^r-d/2𝔅_r-d/2(ξ)
for all x,y∈ℝ^d and ξ:=x-y_ℝ^d   with 𝔅_r-d/2 the Bessel function of order r-d/2. (<cit.>, page 1771 or <cit.>, page 1957) As in the last example, we have H_X(ℝ^d)≈ W^r,2(ℝ^d) under the condition that r>d/2.
That the   candidate kernel  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 from standard results on RKH spaces,  <cit.> Section 4.2  and <cit.> Sections 2.2.1-2.2.2. 
At this point we do not yet have a rigorous notion of  exactly how smooth the restricted functions in R_M(H_X) are, nor do we know whether the spaces R_M(H_X) contain a rich set of cutoff functions.  
But from Lemma 4 of <cit.>, we know that  R_M(H_X)=T(H_X)=T(W^r,2(ℝ^d)) where T is the trace operator
T: f→ f|_M.
 From Proposition 2 of <cit.>, under the standing assumptions on M, the trace operator T:f→ f |_M is a continuous operator from W^r,2(ℝ^d) onto W^r-(d-m)/2,2(M) for r> (d-m)/2 and 1≤ m≤ d. In summary then, if we choose the kernel 𝔎_X on ℝ^d with a sufficiently large smoothness index r, we have 
 R_M(H_X)=
 T(W^r,2(ℝ^d))
 ≈ W^r-(d-m)/2,2(M).
 This set of equivalencies gives a precise notion of the smoothness or regularity of the restricted functions in the RKH space R_M(H_X): the RKH space over M is equivalent to the Sobolev space having smoothness r-(d-m)/2. 
 The remainder of the proof is now that same as in Corollary <ref>.    



 
Note that the statement of persistence in Definition <ref> is expressed in terms of the kernel 𝔎_M:=
 𝔎_X|_M× M, which can be used for computations since a closed form for 𝔎_X is known.
 
 

§ CONCLUSIONS

 This paper derives sufficient conditions for the convergence of function estimates in the RKH embedding method that are based on the recently introduced notion of persistently excited indexing sets Ω and subspaces H_Ω of an RKH space H_X. The paper establishes that persistently excited subsets are contained  as subsets of  the positive limit sets, if the RKH space has a rich collection of bump functions. We have also introduced both intrinsic and extrinsic methods for defining an appropriate RKH space in the event that the positive limit set is in fact certain types of smooth manifold. The extrinsic method seems particularly well-suited for the estimation of uncertain nonlinear systems since the form of the positive limit set is unknown. 
 
 The theoretical results of this paper establish that a reasonable choice of basis functions for practical finite dimensional approximations include radial basis functions (defined in terms of the kernel of the RKH space) that are centered on or near the positive limit set. It remains an open question as to how to devise versions of the RKH embedding strategy that adaptively selects the basis as estimation is carried out. 
 
 
 
 
 
 
 Andrew J. Kurdila would like to acknowledge the support of the Army Research Office under the award Distributed Consensus Learning for Geometric and Abstract Surfaces, ARO Grant W911NF-13-1-0407.
 

 
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