Continuity of Formal Power Series Products in Nonlinear Control Theory
aa r X i v : . [ m a t h . O C ] F e b CONTINUITY OF FORMAL POWER SERIES PRODUCTS INNONLINEAR CONTROL THEORY
W. STEVEN GRAY, MATHIAS PALMSTRØM, AND ALEXANDER SCHMEDING
Abstract.
Formal power series products appear in nonlinear control theory when systemsmodeled by Chen-Fliess series are interconnected to form new systems. In fields like adaptivecontrol and learning systems, the coefficients of these formal power series are estimatedsequentially with real-time data. The main goal is to prove the continuity and analyticityof such products with respect to several natural (locally convex) topologies on spaces oflocally convergent formal power series in order to establish foundational properties behindthese technologies. In addition, it is shown that a transformation group central to describingthe output feedback connection is in fact an analytic Lie group in this setting with certainregularity properties.
MSC2020 : 93C10 (primary), 46A04, 46A13, 47N70, 22E65, 46B45, 16T30
Keywords : nonlinear control systems, Chen–Fliess series, system interconnection, Silvaspace, real analytic, locally convex Lie group, regularity of Lie groups
Contents
1. Introduction 12. Preliminaries 32.1. Chen-Fliess series 32.2. Formal power series products induced by system interconnection 53. Continuity of formal power series products 63.1. Continuity of shuffle product and shuffle inverse 73.2. Continuity of the composition product 114. Analyticity of the composition and shuffle product 135. The Lie group ( δ + K mLC hh X ii , ◦ , δ ) 16Appendix A. Infinite-dimensional calculus 23References 241. Introduction
The interconnection of simple input-output systems to form more complex and usefulsystems is commonplace in science and engineering. When each component is a nonlineardynamical system, a weighted infinite sum of iterated integrals known as a
Chen-Fliessseries , F c , provides a convenient way to represent its local behavior [Fli81, Fli83, FLLL83,LL96, Wan90]. When this series converges on some set of admissible inputs, F c defines a socalled Fliess operator . It is uniquely specified by a formal power series c in K hh X ii , known W. STEVEN GRAY, MATHIAS PALMSTRØM, AND ALEXANDER SCHMEDING as its generating series , where X is a finite set of indeterminants, and K is a suitable field.An interconnection of two Chen-Fliess series F c and F d , represented by F c ✷ F d , induces acorresponding algebra ( K hh X ii , ✷ ′ ) so that F c ✷ F d = F c ✷ ′ d [Fer79, Fer80, Fli81, GDE14,GL05]. Algebras defined in this manner provide computational frameworks for explicitlycomputing the generating series of interconnected systems for the purposes of analysis anddesign, especially in the field of nonlinear control theory. Historically, the coefficients of c have been determined by direct calculations using state space models derived from physicallaws and other first principles [Isi95, NS90]. But with the growth of adaptive control andnew types of learning based technologies, there is increasing interest in estimating thesecoefficients using real-time data and numerical methods from the field of system identification[GVD20, PA16]. Assuming that a given sequence of estimates asymptotically approaches itstrue value as more data is collected, a difficult problem in its own right, there is a fundamentalquestion regarding continuity. Consider a sequence of generating series c i , i > F c i , i ≥
1. If c i → c in some manner,is it also true that F c i , i ≥ F c , i.e., doesthe limit point c ensure a convergent Chen-Fliess series? The answer, of course, dependsdirectly on the ambient sets and the assumed topologies. For example, in [DGS21, WA19]the claim is shown to be false on the subset of locally convergent series in K hh X ii , (i.e., a setof generating series under which their corresponding Fliess operators are known to convergeat least locally) endowed with the ultrametric topology and where the operator space hasan L p type topology. As the ultrametric topology mirrors the algebra but provides almostno information on the analytic behavior of the series, this outcome is not surprising. Onthe other hand, in [DGS21] the claim is shown to be true when the ultrametric topology isreplaced with a certain Banach topology on a subspace. Ultimately, the question boils downto identifying topological vectors spaces contained in K hh X ii which ensure that every limitpoint is a generating series with a well defined Fliess operator in some sense.The main goal of this paper is to address a natural follow-up question: Suppose c i , d i ∈ K hh X ii , i ≥ c and d , respectively, suchthat F c i and F d i , i ≥ c ✷ ′ d has the same convergence properties as c and d (this theory is well understood, see[GW02, TG12, WA19]), under what conditions does c i ✷ ′ d i c ✷ ′ d ? Is it even possible toidentify infinite-dimensional spaces with respect to which the products are smooth or evenanalytic?Three formal power series products will be considered: the shuffle product, which mod-els a type of parallel connection [Fli81]; a composition product modeling series connections[Fer79, Fer80, GL05]; and a group product for a transformation group known to model dy-namic output feedback, a central object of study in control theory [GDE14]. In addition,the continuity of the shuffle inverse will be addressed. (A preliminary version of this analysiswas presented in [Pal20].) The shuffle group appears in the context of feedback lineariza-tion [GDET14, GE17]. In each case continuity will be considered in both the Fr´echet andSilva topologies. In addition, analyticity of these product will be characterized. It should ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 3 be noted that the Fr´echet topology was used in [WA19] to show that the shuffle and com-position products preserve a type of global convergence . Continuity issues in this settingare beyond the scope of the present paper. However, the Fr´echet topology is employed asa natural (locally convex) topology on the space of all power series. Convergence in thistopology does not preserve growth bounds. Thus, it is necessary to endow the space oflocally convergent series with the finer Silva topology. Next it will be shown that the out-put feedback transformation group is a locally convex
Lie group (see [Nee06] for a surveyon (infinite-dimensional) Lie theory). This result builds on the development of a pre-Liealgebra presented in [DEG16, Foi15]. Lie groups have a long history in feedback controltheory originating with the work of Brockett in [Bro76]. More recent applications in thiscontext have appeared in [GE17, GE21], albeit only in the formal case where an explicitdifferential structure is not specified. The present work will provide a means to fill this gap.Finally, the regularity of these Lie groups is investigated. Roughly speaking, regularity ofa Lie group asks for the existence and smooth parameter dependence of certain ordinarydifferential equations on the Lie group. Note that since the Lie groups at hand are notmodeled on Banach spaces, the usual theory for existence and uniqueness of ordinary dif-ferential equations does not apply. However, it is shown that the Fr´echet Lie groups areregular. While some progress on the regularity problem is made for the Silva Lie groups,their regularity largely remains an open problem that the authors plan to pursue in futurework.
Acknowledgements
A.S. wishes to thank the University of Bergen, Norway, where hewas employed while most of the present work was carried out.2.
Preliminaries
Throughout this paper let K ∈ { R , C } , namely either the field of real numbers R or thefield of complex numbers C . It will be essential to admit complex coefficients in order todiscuss analyticity of mappings on infinite-dimensional spaces. Note that the continuityresults are unaffected by this choice. Refer to Appendix A for more information regardingcalculus on infinite-dimensional spaces.2.1. Chen-Fliess series. An alphabet X = { x , x , . . . , x m } is any nonempty and finite setof noncommuting symbols referred to as letters . A word η = x i · · · x i k is a finite sequenceof letters from X . The number of letters in a word η , written as | η | , is called its length . Theempty word, ∅ , is taken to have length zero. The collection of all words having length k isdenoted by X k . Define the set of all words X ∗ = S k ≥ X k , which constitutes a monoid underthe concatenation product. Any mapping c : X ∗ → K ℓ is called a formal power series . Often c is written as the formal sum c = P η ∈ X ∗ ( c, η ) η , where the coefficient ( c, η ) is the image of η ∈ X ∗ under c . The support of c , supp( c ), is the set of all words having nonzero coefficients.A series c is said to be proper when ∅ 6∈ supp( c ). The set of all noncommutative formal powerseries over the alphabet X is denoted by K ℓ hh X ii . The subset of series with finite support,i.e., polynomials, is represented by K ℓ h X i . Each set is an associative K -algebra under thecatenation product and an associative and commutative K -algebra under the shuffle product , W. STEVEN GRAY, MATHIAS PALMSTRØM, AND ALEXANDER SCHMEDING that is, the bilinear product uniquely specified by the shuffle product of two words( x i η ) ( x j ξ ) = x i ( η ( x j ξ )) + x j (( x i η ) ξ ) , where x i , x j ∈ X , η, ξ ∈ X ∗ and with η ∅ = ∅ η = η [Fli81]. On K ℓ hh X ii the definition isextended componentwise. K ℓ hh X ii can be viewed as a locally convex space whose topology is briefly described next.First note that identifying a formal power series with the sequence of its coefficients definesan isomorphism of vector spaces K ℓ hh X ii ∼ = Q η ∈ X ∗ K ℓ . The space on the right hand sideis a countable product of Banach spaces, hence a complete metrisable locally convex vectorspace (i.e., a Fr´echet space). Thus, K ℓ hh X ii inherits a canonical Fr´echet space structure.By construction the evaluations a η : K ℓ hh X ii → K ℓ , c ( c, η ) are continuous. Therefore,convergence in this topology is equivalent to separate convergence of all coefficients of a seriestowards the corresponding coefficients of the limit series. Moreover, the Fr´echet topology isinitial with respect to the point evaluations, i.e., a map f to K ℓ hh X ii is continuous if andonly if a η ◦ f is continuous for every word η ∈ X ∗ .Given any c ∈ K ℓ hh X ii one can associate a causal m -input, ℓ -output operator, F c , in thefollowing manner. Let p ≥ t < t be given. For a Lebesgue measurable function u : [ t , t ] → K m , define k u k p = max {k u i k p : 1 ≤ i ≤ m } , where k u i k p is the usual L p -normfor a measurable real-valued function, u i , defined on [ t , t ]. Let L m p [ t , t ] denote the set of allmeasurable functions defined on [ t , t ] having a finite k·k p norm and B m p ( R u )[ t , t ] := { u ∈ L m p [ t , t ] : k u k p ≤ R u } . Assume C [ t , t ] is the subset of continuous functions in L m [ t , t ].Define inductively for each η ∈ X ∗ the map E η : L m [ t , t ] → C [ t , t ] by setting E ∅ [ u ] = 1and letting E x i ¯ η [ u ]( t, t ) = Z tt u i ( τ ) E ¯ η [ u ]( τ, t ) dτ, where x i ∈ X , ¯ η ∈ X ∗ , and u = 1. The Chen-Fliess series corresponding to c is(1) y ( t ) = F c [ u ]( t ) = X η ∈ X ∗ ( c, η ) E η [ u ]( t, t )[Fli81, Fli83]. It can be shown that if there exists real numbers K, M ≥ | ( c, η ) | ≤ KM | η | | η | ! , ∀ η ∈ X ∗ ( | z | := max i | z i | when z ∈ K ℓ ) then the series defining F c converges absolutely and uniformlyfor sufficient small R, T > B m ( R )[ t , t + T ]into B ℓ ∞ ( S )[ t , t + T ] for some S >
0. Any such mapping is called a locally convergent Fliessoperator . Here K ℓLC hh X ii will denote the set of all such locally convergent generating series,i.e., those series satisfying growth condition (2). Given any smooth state space realizationof y = F c [ u ], ˙ z = g ( z ) + m X i =1 g i ( z ) u i , z (0) = z , y = h ( z ) , it is known that the generating series c is determined by(3) ( c j , η ) = L g i · · · L g ik h j ( z ) , η = x i k · · · x i ∈ X ∗ , j = 1 , , . . . , ℓ ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 5 where L g i h j is the Lie derivative of h j with respect to g i .2.2. Formal power series products induced by system interconnection.
Given Fliessoperators F c and F d , where c, d ∈ K ℓLC hh X ii , the parallel and product connections satisfy F c + F d = F c + d and F c F d = F c d , respectively [Fli81]. When Fliess operators F c and F d with c ∈ K ℓLC hh X ii and d ∈ K mLC hh X ii are interconnected in a cascade fashion, the compositesystem F c ◦ F d has the Fliess operator representation F c ◦ d , where the composition product of c and d is given by(4) c ◦ d = X η ∈ X ∗ ( c, η ) ψ d ( η )( )[Fer79, Fer80]. Here denotes the monomial 1 ∅ , and ψ d is the continuous (in the ultrametricsense) algebra homomorphism from K hh X ii to the set of vector space endomorphisms on K hh X ii , End( K hh X ii ), uniquely specified by ψ d ( x i η ) = ψ d ( x i ) ◦ ψ d ( η ) with ψ d ( x i )( e ) = x ( d [ i ] e ) , i = 0 , , . . . , m for any e ∈ K hh X ii , and where d [ i ] is the i -th component seriesof d ( d [0] := ). By definition, ψ d ( ∅ ) is the identity map on K hh X ii .When two Fliess operators F c and F d are interconnected to form a feedback system with F c in the forward path and F d in the feedback path, the generating series of the closed-loopsystem is denoted by the feedback product c @ d . It can be computed explicitly using theHopf algebra of coordinate functions associated with the underlying output feedback group [GDE14]. Specifically, in the single-input, single-output case where X = { x , x } and ℓ = 1,define the set of unital Fliess operators F δ = { I + F c : c ∈ K LC hh X ii} , where I denotes theidentity map. It is convenient to introduce the symbol δ as the (fictitious) generating seriesfor the identity map. That is, F δ := I such that I + F c := F δ + c = F c δ with c δ := δ + c . Theset of all such generating series for F δ will be denoted by δ + K LC hh X ii . The central ideais that ( F δ , ◦ , I ) forms a group of operators under the composition F c δ ◦ F d δ = ( I + F c ) ◦ ( I + F d ) = F c δ ◦ d δ , where c δ ◦ d δ := δ + c ⊚ d , c ⊚ d := d + c ˜ ◦ d δ , and ˜ ◦ denotes the mixed composition product.That is, the product(5) c ˜ ◦ d δ = X η ∈ X ∗ ( c, η ) φ d ( η )( ) , where φ d is analogous to ψ d in (4) except here φ d ( x i )( e ) = x i e + x ( d [ i ] e ) with d [0] := 0[GL05]. The set of unital generating series δ + K hh X ii (not necessarily locally convergent)forms a group ( δ + K hh X ii , ◦ , δ ). The restriction to the set of locally convergent series definesthe subgroup δ + K LC hh X ii . The mixed composition product can be viewed as a right actionof δ + K hh X ii acting freely on K hh X ii [GD13]. The corresponding Hopf algebra H is thefree algebra generated by the coordinate maps a η : δ + K hh X ii → K : c δ ( c, η ) , η ∈ X ∗ under the commutative product µ : a η ⊗ a ξ a η a ξ , W. STEVEN GRAY, MATHIAS PALMSTRØM, AND ALEXANDER SCHMEDING where the unit δ is defined to map every c δ to one. Let V be the K -vector space of coordinatefunctions. If the degree of a η is defined as deg( a η ) = 2 | η | x + | η | x + 1, then both V and thealgebra H are graded and connected with V = L n ≥ V n and H = L n ≥ H n , where V n and H n are sets containing all the degree n elements, and V = H = K δ . The coproduct ∆ isdefined so that ∆ a η ( c δ , d δ ) = a η ( c δ ◦ d δ ) = ( c δ ◦ d δ , η ) . Of primary importance is the following lemma which describes how the group inverse c ◦− δ := δ + c ◦− is computed. Lemma 2.1. [GDE14]
The Hopf algebra ( H, µ, ∆) has an antipode S satisfying a η ( c ◦− δ ) =( Sa η )( c δ ) for all η ∈ X ∗ and c δ ∈ δ + K hh X ii . With this concept, the generating series for the feedback connection, c @ d , can be computedexplicitly as described in the next theorem. It states that feedback in the present context canbe viewed in terms of the group ( δ + K hh X ii , ◦ , δ ) acting on K hh X ii in a specific manner. Theorem 2.1. [GDE14]
For any c, d ∈ K hh X ii it follows that c @ d = c ˜ ◦ ( − d ◦ c ) ◦− δ . In addition to the elementary system interconnections described above, there is the quo-tient connection that is useful in the context of system inversion [GDET14]. This is a type ofparallel connection where the quotient of the subsystems’ outputs is computed. In terms ofgenerating series, the quotient is realized using the shuffle inverse as described next. Divisionby zero is avoided by requiring the divisor series to be non proper.
Theorem 2.2. [GDET14]
The set of non proper series in K hh X ii is a group under theshuffle product. In particular, the shuffle inverse of any such series c is c − = (( c, ∅ )(1 − c ′ )) − = ( c, ∅ ) − ( c ′ ) ∗ , where c ′ := − c/ ( c, ∅ ) is proper, and ( c ′ ) ∗ := P k ≥ ( c ′ ) k . Theorem 2.3. [GDET14]
For c, d ∈ K LC hh X ii , the quotient connection F c /F d has a Fliessoperator representation if and only if d is non proper. In particular, F c /F d = F c/d , where c/d := c d − . Continuity of formal power series products
In this section, the continuity of the various products modeling system interconnectionsdescribed in the previous section is proved. The main goal is to establish continuity onspaces of locally convergent series. In [DGS21] the authors described the space of locallyconvergent Chen-Fliess series as a locally convex space carrying a Silva space topology. Thatconstruction is summarized first, and then the continuity results are presented.Fix
M > k c k ℓ ∞ ,M := sup (cid:26) | ( c, η ) | M | η | | η | ! : η ∈ X ∗ (cid:27) ∈ [0 , ∞ ] ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 7 for each c ∈ K ℓ hh X ii . The set of all c with k c k ℓ ∞ ,M < ∞ is denoted by ℓ ∞ ,M ( X ∗ , K ℓ ). It isstraightforward to check that ℓ ∞ ,M ( X ∗ , K ℓ ) is a vector subspace of K ℓ hh X ii . The function k·k ℓ ∞ ,M is a norm on ℓ ∞ ,M ( X ∗ , K ℓ ). This space is a Banach space as it is isometricallyisomorphic to the Banach space of all bounded functions ℓ ∞ ( X ∗ , K ℓ ) := { c : X ∗ → K ℓ :sup η | ( c, η ) | < ∞} . The Banach space of generating series bounded with respect to theconstant M obviously does not capture all locally convergent series. Indeed for larger M oneobtains series which converge only on a smaller disc. To capture all locally convergent seriesin one space, it is necessary to pass to the limit of these Banach spaces as described next. Definition 3.1 (Locally convergent series as a Silva space) . Consider the union ℓ ∞ , → ( X ∗ , K ℓ ) := [ M> ℓ ∞ ,M ( X ∗ , K ℓ ) . Topologise this space as the locally convex inductive limit of the system (cid:0) ℓ ∞ ,M ( X ∗ , K ℓ ) (cid:1) M> . One can show that the inclusion mappings in this sequence are compact operators, hencethe resulting space is a
Silva space [BS16, DS20]. Since the sequence M k = k , k ∈ N iscofinal, one can always find an M ∈ N for which k c k ℓ ∞ ,M < ∞ . Thus, one could equivalentlywork only with M ∈ N . Though the Silva space topology is more complicated than theBanach spaces from which it was built, some of its properties make it very amenable for theapplications considered here. The most important properties are summarized in the nextlemma. Refer to [Yos57] for proofs and more information about Silva spaces. Lemma 3.1 (Properties of Silva spaces) . (1) A sequence converges in ℓ ∞ , → ( X ∗ , K ℓ ) if and only if there exists M > such that thesequence is contained and converges in the Banach space ℓ ∞ ,M ( X ∗ , K ℓ ) . (2) Silva spaces are sequential, thus a map defined on a Silva space is continuous if andonly if it is sequentially continuous. Moreover, Silva spaces are separable and finiteproducts of Silva spaces are again Silva spaces. (3)
A mapping f : ℓ ∞ , → ( X ∗ , K ℓ ) → E into a locally convex space is continuous (differen-tiable) if and only if for every M > the induced mapping f M := f | ℓ ∞ ,M ( X ∗ , K ℓ ) : ℓ ∞ ,M ( X ∗ , K ℓ ) → E is continuous (differentiable). Perhaps the most striking property of the Silva topology is that one can address continuityand differentiability questions in the Banach spaces from which the Silva space is built. Thiswill be demonstrated in the next section addressing the continuity of formal power seriesproducts.3.1.
Continuity of shuffle product and shuffle inverse.
The following lemma is a pre-requisite for proving continuity of the shuffle product.
Lemma 3.2.
Fix
M > . If c, d ∈ ℓ ∞ ,M ( X ∗ , K ℓ ) , then c d ∈ ℓ ∞ ,M ǫ ( X ∗ , K ℓ ) for any M ǫ = M (1 + ǫ ) , ǫ > and k c d k ℓ ∞ ,M ǫ ≤ K ǫ k c k ℓ ∞ ,M k d k ℓ ∞ ,M , W. STEVEN GRAY, MATHIAS PALMSTRØM, AND ALEXANDER SCHMEDING where K ǫ = sup η ∈ X ∗ ( | η | + 1) / (1 + ǫ ) | η | ≤ ˆ K ǫ := e − (1 + ǫ ) / (log(1 + ǫ )) .Proof: For any η ∈ X ∗ | ( c d, η ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) | η | X k =0 X ν ∈ Xkξ ∈ X | η |− k ( c, ν )( d, ξ )( ν ξ, η ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ | η | X k =0 X ν ∈ Xkξ ∈ X | η |− k k c k ℓ ∞ ,M M k k ! k d k ℓ ∞ ,M M | η |− k ( | η | − k )! ( ν ξ, η )= k c k ℓ ∞ ,M k d k ℓ ∞ ,M M | η | | η | X k =0 k ! ( | η | − k )! (cid:18) | η | k (cid:19) = k c k ℓ ∞ ,M k d k ℓ ∞ ,M M | η | η X k =0 | η | != k c k ℓ ∞ ,M k d k ℓ ∞ ,M M | η | ( | η | + 1)! . Note that this bound is achievable when c = P η ∈ X ∗ K c M | η | | η | ! η and d = P η ∈ X ∗ K d M | η | | η | ! η for any K c , K d ≥
0. Now define M ǫ = M (1 + ǫ ) with ǫ > | ( c d, η ) | M | η | ǫ | η | ! ≤ k c k ℓ ∞ ,M k d k ℓ ∞ ,M | η | + 1(1 + ǫ ) | η | , ∀ η ∈ X ∗ . Taking the supremum over X ∗ gives k c d k ℓ ∞ ,M ǫ ≤ K ǫ k c k ℓ ∞ ,M k d k ℓ ∞ ,M , ∀ η ∈ X ∗ , where K ǫ = sup η ∈ X ∗ ( | η | + 1) / (1 + ǫ ) | η | . The upper bound for K ǫ is found by showing that f ǫ ( x ) = ( x + 1) / (1 + ǫ ) x has a single maximum at x ∗ ǫ = (1 / log(1 + ǫ )) − > < ǫ ≤ e −
1, and ˆ K ǫ = f ǫ ( x ∗ ǫ ) = e − (1 + ǫ ) / (log(1 + ǫ )). In this case, the upper boundis tight (see Figure 1). For ǫ > e − K ǫ = 1 and ˆ K ǫ >
1, and thus this upper bound isconservative.
Theorem 3.1.
The shuffle product is continuous on K hh X ii and K LC hh X ii with respect tothe Fr´echet and the Silva topology, respectively.Proof: Consider the shuffle product on K hh X ii . Since the topology of K hh X ii is initial withrespect to the coordinate functions a η : K hh X ii → K , c ( c, η ), it suffices to prove that a η ◦ is continuous for each η ∈ X ∗ . However, as was seen in the proof of Lemma 3.2 for η ∈ X ∗ , it follows that a η ◦ ( c, d ) = ( c d, η ) = | η | X k =0 X ν ∈ Xkξ ∈ X | η |− k ( c, ν )( d, ξ )( ν ξ, η ) ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 9
Figure 1.
Sample plots of f ǫ ( x ) and ˆ K ǫ in Lemma 3.2.= | η | X k =0 X ν ∈ Xkξ ∈ X | η |− k a ν ( c ) a ξ ( d )( ν ξ, η ) . This shows that a η ◦ ( c, d ) is a polynomial in the variables a ν ( c ) , a ξ ( d ). Since the co-ordinate functions are continuous in the series c, d , it is clear that the shuffle product iscontinuous. Thus, the shuffle product is continuous on K hh X ii . For the correspondingresult on K LC hh X ii , apply Lemma 3.2 for any ǫ > k ( c d ) − ( c j d j ) k ℓ ∞ ,M ǫ = k ( c − c j ) d + c j ( d − d j ) k ℓ ∞ ,M ǫ ≤ k ( c − c j ) d k ℓ ∞ ,M ǫ + k c j ( d − d j ) k ℓ ∞ ,M ǫ ≤ K ǫ k ( c − c j ) k ℓ ∞ ,M k d k ℓ ∞ ,M + K ǫ k c j k ℓ ∞ ,M k ( d − d j ) k ℓ ∞ ,M . Thus, lim j →∞ k ( c d ) − ( c j d j ) k ℓ ∞ ,M ǫ = 0, proving the second part of the theorem.The next lemma will be needed for proving continuity of the shuffle inverse as well as forproving continuity of the composition product in the next section. Lemma 3.3. If c and c j , j ≥ are proper series in ℓ ∞ ,M ( X ∗ , K ) for some M ∈ N , and k c − c j k ℓ ∞ ,M → as j → ∞ , then for N ∈ N sufficiently large it follows that ∞ X n =1 k c n − c nj k ℓ ∞ ,N → as j → ∞ .Proof: It is first shown that the sum is uniformly bounded for some large enough N ∈ N .The fact that c j → c in ℓ ∞ ,M and that k · k ℓ ∞ ,N < k · k ℓ ∞ ,M for nonzero proper elementswhenever N > M , implies that one can choose N ∈ N so that k c k ℓ ∞ ,N + sup j ∈ N k c j k ℓ ∞ ,N ≤ . Define a proper series d ∈ ℓ ∞ ,N ( X ∗ , K ) by( d, η ) := ( k c k ℓ ∞ ,N + sup j ∈ N k c j k ℓ ∞ ,N ) N | η | | η | ! , η = ∅ . Then ( d, η ) ≥ | ( c, η ) | + | ( c j , η ) | for any η ∈ X ∗ and all j ∈ N . In fact, ( d n , η ) ≥ | ( c n , η ) | + | ( c nj , η ) | for any n ≥ j ∈ N by a standard induction argument. In particular, if N ′ ≥ N , then k d n k ℓ ∞ ,N ′ ≥ k c n − c nj k ℓ ∞ ,N ′ . Now observe that( d n , η ) ≤ ( k c k ℓ ∞ ,N + sup j ∈ N k c j k ℓ ∞ ,N ) n N | η | (cid:18) ( n −
1) + | η | n − (cid:19) | η | ! ≤ n N | η | (cid:18) ( n −
1) + | η | n − (cid:19) | η | != 12 n (4 N ) | η | | η | ! (cid:18) ( n −
1) + | η | n − (cid:19) | η | . The first inequality can be shown via induction. Moreover, one can show the existence of apositive constant K for which (cid:18) ( n −
1) + | η | n − (cid:19) | η | ≤ K, ∀ n ∈ N , | η | ≥ n. As d is proper, ( d n , η ) = 0 for all words | η | < n . Therefore,( d n , η ) ≤ n K (4 N ) | η | | η | ! , ∀ η ∈ X ∗ , so that k d n k ℓ ∞ , N ≤ K/ n . Setting N := 4 N gives ∞ X n =1 k c n − c nj k ℓ ∞ ,N ≤ ∞ X n =1 k d n k ℓ ∞ ,N ≤ ∞ X n =1 n K = K < ∞ . Having shown that the sum is uniformly bounded, it is now claimed that for each n ≥ j →∞ k c n − c nj k N = 0. If this holds, thenlim j →∞ ∞ X n =1 k c n − c nj k ℓ ∞ , N = ∞ X n =1 lim j →∞ k c n − c nj k ℓ ∞ , N = 0 . To prove the claim, define N n := N (1 + (1 − n )) for n ≥ N > N n > N n − > · · · >N = N . It is shown by induction on n ≥ k c n − c nj k ℓ ∞ ,Nn → j → ∞ . The case n = 1 follows immediately as N = N and k c − c j k ℓ ∞ ,N ≤ k c − c j k ℓ ∞ ,M . Let n >
1. Usingthe bilinearity of the shuffle product it follows that k c n − c nj k ℓ ∞ ,Nn = k ( c − c j ) c ( n − + c j ( c ( n − − c ( n − j ) k ℓ ∞ ,Nn ≤ K n k c − c j k ℓ ∞ ,Nn − k c ( n − k ℓ ∞ ,Nn − + K n k c j k ℓ ∞ ,Nn − k c ( n − − c ( n − j k ℓ ∞ ,Nn − , where as in Lemma 3.2 the K n > ǫ n > N n = (1 + ǫ n ) N n − . By the induction hypothesis, the latter expression tends to ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 11 zero, implying the same for the former. This proves the claim since k c n − c nj k ℓ ∞ , N ≤k c n − c nj k ℓ ∞ ,Nn , ∀ n ≥ Proposition 3.1.
Denote by ( K LC hh X ii ) × the set of invertible elements of the algebra ( K LC hh X ii , ) . The shuffle inverse − : ( K LC hh X ii ) × → ( K LC hh X ii ) × , c ( c, ∅ ) − X k ≥ ( c ′ ) k , where c ′ = − c/ ( c, ∅ ) , is well defined and continuous.Proof: Well definedness follows from [GDET14, Theorem 5]. To show continuity, firstobserve that ( K LC hh X ii ) × is an open subset of K LC hh X ii . Indeed it is easily verified thatfor any η ∈ X ∗ the evaluation map a η is continuous on the Silva space K LC hh X ii . Inparticular, ( K LC hh X ii ) × = a − ∅ ( K \ { } ) is open. Since K LC hh X ii is sequential, the sameis true for the open subset ( K LC hh X ii ) × , and consequently it suffices to test continuity of − via sequences. With this in mind suppose c j → c for elements c j , c ∈ ( K LC hh X ii ) × , say k c − c j k ℓ ∞ ,M → M >
0. Then also k c ′ − c ′ j k ℓ ∞ ,M →
0. Since c ′ and c ′ j are properseries, applying Lemma 3.3 gives (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =0 ( c ′ ) k − ( c ′ j ) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ ∞ ,N = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ∞ X k =1 ( c ′ ) k − ( c ′ j ) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ ∞ ,N → N > M . Hence, k − ( c ) − − ( c j ) k ℓ ∞ ,N = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( c, ∅ ) − ∞ X k =0 ( c ′ ) k − ( c j , ∅ ) − ∞ X k =0 ( c ′ j ) k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ℓ ∞ ,N → , or in other words, − ( c j ) → − ( c ).3.2. Continuity of the composition product.
In addition to Lemma 3.3 the next resultis needed in order to address the continuity of the composition product.
Lemma 3.4.
Fix
M > . If c ∈ ℓ ∞ ,M ( X ∗ , K ℓ ) and d ∈ ℓ ∞ ,M ( X ∗ , K m ) , then c ◦ d ∈ ℓ ∞ ,M ǫ ( X ∗ , K ℓ ) for any M ǫ = M (1 + ǫ ) , ǫ > φ ( m k d k ℓ ∞ ,M ) with φ ( x ) = x/ p x / x and k c ◦ d k ℓ ∞ ,M ǫ ≤ k c k ℓ ∞ ,M ( K ǫ ◦ φ )( m k d k ℓ ∞ ,M ) , where K ǫ ( a ) = sup η ∈ X ∗ ( | η | + 1)(1 + a ) | η | / (1 + ǫ ) | η | .Proof: It was shown in [GL05] that under the stated conditions | ( c ◦ d, η ) | ≤ k c k ℓ ∞ ,M ((1 + φ ( m k d k ℓ ∞ ,M )) M ) | η | ( | η | + 1)! , ∀ η ∈ X ∗ . Therefore, | ( c ◦ d, η ) | M ǫ | η | ! ≤ k c k ℓ ∞ ,M (1 + φ ( m k d k ℓ ∞ ,M )) | η | ( | η | + 1)(1 + ǫ ) | η | , ∀ η ∈ X ∗ . Taking the supremum over X ∗ proves the lemma. Theorem 3.2.
The composition product on K mLC hh X ii is continuous in the Silva topology.Proof: Left and right continuity of the composition product is first proved, beginning withleft continuity. Let
M > c, d ∈ ℓ ∞ ,M ( X ∗ , K m ), and assume c j , j ≥ ℓ ∞ ,M ( X ∗ , K m ) converging to c . Applying Lemma 3.4 gives k ( c ◦ d ) − ( c j ◦ d ) k ℓ ∞ ,M ǫ = k ( c − c j ) ◦ d k ℓ ∞ ,M ǫ ≤ k c − c j k ℓ ∞ ,M ( K ǫ ◦ φ )( m k d k ℓ ∞ ,M ) . Thus, lim j →∞ k ( c ◦ d ) − ( c j ◦ d ) k ℓ ∞ ,M ǫ = 0.Right continuity is addressed next. It is more complicated given the nonlinearity in theright argument of the product. Let c, d ∈ ℓ ∞ ,M ( X ∗ , K m ) and assume d j , j ≥ ℓ ∞ ,M ( X ∗ , K m ) converging to d . For a fixed ξ ∈ X ∗ , observe that | (( c ◦ d ) − ( c ◦ d j ) , ξ ) | = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X η ∈ X ∗ ( c, η )( η ◦ d − η ◦ d j , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ ∞ X n =0 k c k ℓ ∞ ,M M n n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X η ∈ X n ( η ◦ d − η ◦ d j , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k c k ℓ ∞ ,M ∞ X n =0 M n n ! (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X r ≥ ,...,rm ≥ r ··· + rm = n (( x r · · · x r m m ) ◦ d − ( x r · · · x r m m ) ◦ d j , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Applying the identities x ni = n ! x ni , n ≥ c d ) ◦ e = ( c ◦ e ) ( d ◦ e ) gives | (( c ◦ d ) − ( c ◦ d j ) , ξ ) | ≤ k c k ℓ ∞ ,M ∞ X n =0 M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X r ≥ ,...,rm ≥ r ··· + rm = n (cid:18) nr · · · r m (cid:19) (( x r · · · x r m m ) ◦ d − ( x r · · · x r m m ) ◦ d j , ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . = k c k ℓ ∞ ,M ∞ X n =0 M n (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) m X k =0 x k ◦ d ! n − m X k =0 x k ◦ d j ! n , ξ !(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = k c k ℓ ∞ ,M ∞ X n =0 (cid:12)(cid:12)(cid:0) ¯ d n − ¯ d nj , ξ (cid:1)(cid:12)(cid:12) , where ¯ d := M x P mk =0 d [ k ] and ¯ d j := M x P mk =0 d j [ k ] are proper series in K hh X ii . Here d [ k ] denotes the k -th component series of d . It is clear that ¯ d ∈ ℓ ∞ ,M ( X ∗ , K ), and ¯ d j is a ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 13 sequence in ℓ ∞ ,M ( X ∗ , K ). Furthermore, ¯ d j → ¯ d as j → ∞ since k ¯ d − ¯ d j k ∞ ,M = sup η ∈ X ∗ | ( ¯ d − ¯ d j , η ) | M | η | | η | ! ≤ m X k =1 sup η ∈ X ∗ M | ( x ( d [ l ] − d j [ k ]) , η ) | M | η | | η | != m X k =1 sup x η ∈ X ∗ | ( d [ k ] − d j [ k ] , η ) | M | η | ( | η | + 1)!= m X k =1 sup η ∈ X ∗ | ( d [ k ] − d j [ k ] , η ) | M | η | | η | !( | η | + 1) ≤ m k d − d j k ℓ ∞ ,M . Finally, right continuity follows by applying Lemma 3.3 with
N > M sufficiently large sothat k ( c ◦ d ) − ( c ◦ d j ) k ℓ ∞ ,N ≤ k c k ℓ ∞ ,M ∞ X n =0 k ¯ d n − ¯ d nj k ℓ ∞ ,N . (6)Note that the estimates for left and right continuity imply joint continuity of the compo-sition product due to the following simple observation that for N as above k ( c ◦ d ) − ( c j ◦ d j ) k ℓ ∞ ,N ≤ k ( c ◦ d ) − ( c j ◦ d ) k ℓ ∞ ,N + k ( c j ◦ d ) − ( c j ◦ d j ) k ℓ ∞ ,N ( ) ≤ k ( c − c j ) ◦ d k ℓ ∞ ,N + k c j k ℓ ∞ ,M ∞ X n =0 k ¯ d n − ¯ d nj k ℓ ∞ ,N , where the last inequality is a direct consequence of (6). Hence we see that the productis sequentially continuous (as each sequence convergent in the Silva topology is alreadycontained in one of the Banach steps). By Lemma 3.1 implies that the product is continuousas each of the p M,K is continuous for every
M, K >
Analyticity of the composition and shuffle product
In this section it is proved that the formal power series products and inverse presented inthe previous sections are not only continuous but also analytic. Note that on the infinite-dimensional spaces involved, both complex and real analyticity make sense, cf. Appendix A.For real analyticity one needs only to identify the complexification of the spaces R hh X ii and R LC hh X ii .As locally convex spaces, the complexification of R hh X ii is C hh X ii . This is clear on thelevel of vector spaces, and for the topology simply note that as topological vector spaces C hh X ii = R hh X ii ⊕ i R hh X ii . Similarly, the complexification of the Silva space R LC hh X ii is C LC hh X ii . Again this is clear on the level of vector spaces but more complicated on thelevel of the vector space topologies. However, also the vector space topologies coincide as itis easy to see that for every M > ℓ ∞ ,M ( X ∗ , C ℓ ) is the complexification of ℓ ∞ ,M ( X ∗ , R ℓ ), and the inductive limit of a sequence of compact operators between Banachspaces commutes with the formation of complexifications [HSTH01, Theorem 3.4].Having identified the complexification of the infinite-dimensional spaces, observe that theshuffle product, the composition product and the shuffle inverse are all well defined on boththe complexification and on the real space. Hence, if it can be proved that these mappings areholomorphic on the complexification, then real analyticity is obtained for the correspondingmappings on the real space. Before continuing with the shuffle product and the shuffleinverse, it is helpful to recall a special type of locally convex algebra. Definition 4.1.
Let ( A, β ) be an associative unital locally convex algebra, i.e., A is a locallyconvex space such that the bilinear map β is continuous and admits a unit with β ( , x ) = x = β ( x, ) . Then A is called a continuous inverse algebra (CIA) if the unit group A × is an open subset of A , and inversion ι : A × → A × is continuous. Proposition 4.1.
The algebras ( K hh X ii , ) and ( K LC hh X ii , ) are continuous inversealgebras with respect to their natural topologies.Proof: It was shown in Theorem 3.1 that the bilinear shuffle product is continuous withrespect to the Silva and the Fr´echet topology. Furthermore, the non proper series are preciselythe invertible elements with respect to the shuffle product. By definition of a non properseries it is evident that if A is either the algebra K hh X ii or the algebra K LC hh X ii , then A × = a − ∅ ( C \{ } ) is open as the preimage of an open set under a continuous map. Continuityof the shuffle inverse for the Silva topology on K LC hh X ii was established in Proposition 3.1.To see that the shuffle inverse is also continuous on ( K hh X ii ) × it suffices to test continuityof the composition a η ◦ − for every η ∈ X ∗ . However, due to the definition of the shuffleinverse, it is clear that a η ◦ − ( c ) is a polynomial in finitely many evaluations of the series c .Therefore, a η ◦ − is continuous, and hence the shuffle inverse is continuous in the Fr´echettopology.It is well known that the unit group of a CIA is an infinite-dimensional Lie group. Beforestating the next result, recall the following notion from infinite-dimensional Lie theory. Definition 4.2.
Consider a Lie group G with unit and write L ( G ) for the Lie algebra of G . Let λ g : G → G, λ g ( h ) = gh be the left multiplication with a fixed element g ∈ G . Then G is called C r -regular , r ∈ N ∪ {∞} , if for each C r -curve u : [0 , → L ( G ) the initial valueproblem ( ˙ γ ( t ) = γ ( t ) .u ( t ) := T λ γ ( t ) ( u ( t )) γ (0) = 1 has a (necessarily unique) C r +1 -solution Evol( u ) := γ : [0 , → G and the map evol : C r ([0 , , L ( G )) → G, u Evol( u )(1) is smooth. A C ∞ -regular Lie group G is called regular (in the sense of Milnor ). The function space C r ([0 , , L ( G )) is endowed with the compact open C r -topology (controlling a functionand its derivatives on compact subsets). With this topology and pointwise addition and scalar multiplication ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 15
Every Banach Lie group is C -regular (cf. [Nee06]). Several important results in infinite-dimensional Lie theory are only available for regular Lie groups. For example the interplaybetween Lie algebra and Lie group hinges on regularity as this property guarantees existenceof a smooth Lie group exponential function. Moreover, if one wants to lift morphisms of Liealgebras to the Lie group by integration, this requires the group to be regular, cf. [KM97]. Proposition 4.2.
The group (( K hh X ii ) × , ) with the Fr´echet topology and the group (( K LC hh X ii ) × , ) with the Silva topology are C -regular analytic Lie groups.Proof: It was established that the groups are unit groups of continuous inverse algebras,hence they are infinite-dimensional analytic Lie groups by [Gl¨o02a, Theorem 5.6]. Moreover,since the shuffle product is abelian, and K hh X ii and K LC hh X ii are both complete locallyconvex spaces, an application of [GN12, p.3 Corollary and Proposition 3.4 (a)] shows that theLie groups K hh X ii and K LC hh X ii are C -regular (even with analytic evolution map evol). Remark 4.1. In [GN12, Lemma 2.2] it was proved that the solution to the initial valueproblem for regularity in the unit group of a CIA is given by the Volterra series (7) γ ( t ) = 1 + ∞ X n =1 Z t Z t n − · · · Z t η ( t ) · · · η ( t n )d t . . . d t n . Hence the Volterra series describes both the solution of the initial value problem in K hh X ii and the subgroup δ + K LC hh X ii . Proposition 4.3.
The composition product on K hh X ii and on K LC hh X ii is analytic.Proof: In light of the previous observations regarding the complexifications, it suffices toprove the statement for the case K = C . Fix η ∈ X ∗ . By definition of the compositionproduct, an induction argument shows that a η ( c ◦ d ) is a polynomial in finitely many a γ ( c )and a ρ ( d ) for words such that | γ | , | ρ | ≤ | η | (for a detailed proof see [Pal20, Lemma 83]). Asthe coordinate functions are continuous linear (thus holomorphic) in the Fr´echet topologyon K hh X ii and in the Silva topology on K LC hh X ii , one can deduce the following:(1) the composition product ◦ : K hh X ii → K hh X ii is continuous with respect to theFr´echet topology (which is initial with respect to the a η );(2) for every η ∈ X ∗ the map ( c, d ) a η ( c ◦ d ) is holomorphic both on K hh X ii and on K LC hh X ii .Furthermore, the coordinate functions a η , η ∈ X ∗ separate the points on C hh X ii and on C LC hh X ii . Now apply Lemma A.1. Since the composition product is continuous on C hh X ii and analytic after composition with a η , η ∈ X ∗ , the composition product is analytic as amapping on C hh X ii . A similar argument holds for the composition product on C LC hh X ii as continuity for this product was established in Theorem 3.2. C r ([0 , , L ( G )) is a locally convex space. Thus, it makes sense to define smooth mappings on this space, cf.Appendix A. The Lie group ( δ + K mLC hh X ii , ◦ , δ )A Lie group structure on the group ( δ + K mLC hh X ii , ◦ , δ ) is presented in this section.This group is known to have an associated graded and connected Hopf algebra ( H, µ, ∆)as described in Section 2.2 and more completely in [GDE14, Section 3]. This structurewill play an important role in the proof of the Lie group property. Note, however, that( δ + K m hh X ii , ◦ , δ ) is not the character group of said Hopf algebra, and thus the Lie theoryfor such groups from [BDS16, DS20] is not directly applicable. The main claim, as statedbelow, is established from first principles. Theorem 5.1.
The group ( δ + K mLC hh X ii , ◦ , δ ) is an analytic Lie group under the Silvatopology.Proof: The proof is carried out in four main steps.
Step 1:
The group product is continuous in the Silva topology . Fix M ≥ M ǫ = M (1 + ǫ ). If c, d ∈ ℓ ∞ ,M ( X ∗ , K m ) then the proof of Theorem 3.2 can be easily modifiedto show that c ˜ ◦ d δ is continuous in the Silva topology. Specifically, the only change is inthe definition of ¯ d and ¯ d j . For example, ¯ d = M ( P mk =0 x k + x d [ k ]). In which case, it followsdirectly that c δ ◦ d δ = δ + d + c ˜ ◦ d δ is continuous in both its left and right arguments inthe Banach space ℓ ∞ ,M ǫ ( X ∗ , K m ). Joint continuity follows then verbatim as in the proof ofTheorem 3.2. Step 2:
The group inverse is degreewise a polynomial . Assume without loss of generalitythat m = 1. Let c j → c in ℓ ∞ ,M ( X ∗ , K ). It was shown in [GDE14] that the compositioninverse preserves local convergence. Thus, there exists an M > c ◦− δ ∈ δ + ℓ ∞ ,M ( X ∗ , K ) and ( c δ,j ) ◦− ∈ δ + ℓ ∞ ,M ( X ∗ , K ) for every j ≥
1. Set M = max( M, M ). Since H is graded and connected with respect to the degree grading, it follows from Lemma 2.1(cf. [Man08]) that ( c ◦− δ , η ) = S ( a η )( c ) = − a η ( c ) − X S ( a ′ ( η ) )( c ) a ′ ( η ) ( c )= − a η ( c ) + deg( a η ) X k =1 ( − k +1 µ k ◦ ∆ ′ k ( a η )( c ) , (8)where ∆ ′ a = ∆ a − a ⊗ δ − δ ⊗ a = P a ′ ( η ) ⊗ a ′ ( η ) is the reduced coproduct in the notationof Sweedler , ∆ ′ k = ∆ ′ k − ⊗ id is defined inductively, and µ k is the k -fold multiplication inthe target algebra. In particular, a ′ ( η ) ∈ V n and a ′ ( η ) ∈ H n with n , n < n . As thesummation in (8) is always finite, the η component of c ◦− δ is a polynomial in the variables { a ξ ( c ) : deg( a ξ ) ≤ deg( a η ) } . This implies immediately that inversion is continuous (andanalytic) in the Fr´echet space δ + K mLC hh X ii . However, this does not yet yield continuitywith respect to the Silva space topology on δ + K mLC hh X ii . Given the bijection between δ + K hh X ii and K hh X ii , for brevity a η ( c δ ) will be written as a η ( c ). ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 17
Step 3:
Continuity of the group inverse in the Silva topology . It is first proved thatinversion is continuous at the unit δ . It is again assumed without loss of generality that m = 1. Recalling that c δ := δ + c , the series c δ,j = δ + c j , j ∈ N converges to δ in the Silvatopology if and only if the series c j converges to 0 in ℓ ∞ ,M ( X ∗ , K ) for some M >
0. Fix c ∈ ℓ ∞ ,M ( X ∗ , K ) and define ¯ c = P η ∈ X ∗ KM | η | | η | ! η with K = k c k ℓ ∞ ,M so that | ( c, η ) | ≤ (¯ c, η ), ∀ η ∈ X ∗ . It can be verified directly that y = F ¯ c δ [ u ] = u + F ¯ c [ u ] has the state spacerealization ˙ z = MK (1 + u ) , z (0) = K, y = z + u. Therefore, y = F ¯ c ◦− δ [ u ] = u + F ¯ c ◦− [ u ] has the realization(9) ˙ z = MK ( z − z ) + z u, z (0) = K, y = − z + u. It is shown in [GDE14, Theorem 6] that c ◦− = ( − c )@ δ , where the right-hand side denotesthe generating series for the unity feedback system v y defined by y = F − c [ u ] and u = v + y . Combining this fact with a minor extension of [TG12, Lemma 10], it follows thatthe condition | ( c, η ) | ≤ (¯ c, η ) implies | ( c ◦− , η ) | ≤ | (¯ c ◦− , η ) | , ∀ η ∈ X ∗ . The fastest growingcoefficients of ¯ c ◦− have been shown to be the sequence (¯ c ◦− , x k ), k ≥ η ∈ X ∗ of length k (cid:12)(cid:12) ( c ◦− , η ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) (¯ c ◦− , η ) (cid:12)(cid:12) ≤ (cid:12)(cid:12) (¯ c ◦− , x k ) (cid:12)(cid:12) = (cid:12)(cid:12) L kg h ( z ) (cid:12)(cid:12) , where the right-most inequality follows from (3) with g ( z ) = ( M/K )( z − z ), h ( z ) = − z ,and z = K as derived in (9). A direct calculation gives(10) (¯ c ◦− , x k ) = b k ( K ) KM k k ! , k ≥ , where the first few polynomials b k ( K ) are: b ( K ) = − b ( K ) = − Kb ( K ) = − K − K b ( K ) = − K − K + 15 K b ( K ) = −
24 + 154 K − K + 315 K − K b ( K ) = −
120 + 1044 K − K + 4900 K − K + 945 K b ( K ) = −
720 + 8028 K − K + 70532 K − K + 45045 K − K b ( K ) = − K − K + 1008980 K − K + 1406790 K − K + 135135 K ...When K ≤ b k ( K ) ≤ ¯ b k , where ¯ b k , k ≥ the real analytic function G ( x ) = −
11 + W ( − x − , where W is the Lambert W-function (see [TG12, Example 5]). In which case, there existsgrowth constants ¯ K, ¯ M > b k ≤ ¯ K ¯ M k k !, k ≥
0. Combining this inequality with(10) gives (cid:12)(cid:12) ( c ◦− , η ) (cid:12)(cid:12) ≤ k c k ℓ ∞ ,M ¯ K ( M ¯ M ) | η | | η | ! , ∀ η ∈ X ∗ . Hence, if c δ,j → δ in K × ℓ ∞ ,M ( X ∗ , K ), then c ◦− δ,j → δ in ℓ ∞ ,MM ( X ∗ , K ). Therefore, inver-sion is continuous at the unit with respect to the Silva topology. Exploiting the fact thatinversion is a group antimorphism, this implies that inversion is continuous everywhere on δ + K mLC hh X ii in the Silva topology. Step 4:
Group product and inverse are analytic.
Since the complexification of δ + R mLC hh X ii is δ + C mLC hh X ii , it suffices to consider the complex case. In view of LemmaA.1 and Step 1, all one needs to prove is that for every η ∈ X ∗ the mappings ( c δ , d δ ) a η ( c δ ◦ d δ ) and c δ a η ( c ◦− δ ) are holomorphic. Regarding the composition product recall that( δ + c ) ◦ ( δ + d ) = δ + d + c ˜ ◦ d δ . Now for the mixed composition ˜ ◦ it was shown in the proof ofProposition 4.3 that a η ( c ˜ ◦ d δ ) is given by a polynomial in finitely many of the variables a ξ ( c )and a ν ( d ). Hence, this part of the product is analytic on δ + C mLC hh X ii , and therefore thecomposition product is analytic. Similarly, for the inversion ι , Step 2 shows that a η ◦ ι ( c ) isgiven as a polynomial in finitely many evaluations of c . As before, the coordinate functionsare holomorphic and this implies that a η ◦ ι is holomorphic on δ + C mLC hh X ii . Hence, theinversion is also holomorphic.The argument for the Lie group structure on subsets of locally convergent series can beadapted almost verbatim to the case where no convergence of the series is assumed. Corollary 5.1.
The group ( δ + K m hh X ii , ◦ , δ ) is an analytic Lie group.Proof: Again it suffices to prove the case where K = C . In Step 2 of the proof for Theorem5.1 it was shown that after composition with a coordinate function a η both the compositionand the inversion in the group are given by a polynomial in finitely many coordinate functionsapplied to the arguments. Since the Fr´echet topology is initial with respect to the coordinatefunctions, it follows directly that the group operations are continuous. Applying Lemma A.1gives immediately that the group operations are also analytic.While the Fr´echet Lie group δ + K m hh X ii is much simpler (topologically speaking) thanthe Silva group δ + K mLC hh X ii , it supplies a useful template for the Lie theoretic argumentsconsidered next, namely, identifying the Lie algebra and proving that a Lie group is regularin the sense of Milnor. The first goal is to establish these properties for the simpler Fr´echetLie group. Subsequently, it is shown that these results then imply corresponding propertiesfor the Silva Lie group. However, it is first necessary to introduce a new structure which Alternatively, continuity can be deduced from a more general criterion, see [AR05, Lemma 1.3].
ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 19 will yield a convenient description of the Lie bracket. This structure is the so called pre-Lieproduct , which was developed in [Foi15] for the case where m = 1 and generalized in [DEG16,Section 3.2] for the case where m ≥ Definition 5.1.
Let X = { x , x , . . . , x m } and denote by d [ i ] the i -th component of a series d ∈ K m hh X ii . The pre-Lie product is the bilinear product on K m hh X ii × K m hh X ii c ✁ d = X η ∈ X ∗ ( c, η ) η ✁ d, where η ✁ d is defined inductively by ( x η ) ✁ d = x ( η ✁ d )( x j η ) ✁ d = x j ( η ✁ d ) + x ( η d [ j ]) , j = 1 , , . . . , m and ∅ ✁ d = 0 . This product can be viewed as the linear part of the group product, that is,(11) c δ ◦ d δ = δ + d + c ˜ ◦ d δ = δ + c + d + c ✁ d + O ( c, d ) , where O ( c, d ) denotes all terms depending linearly on c and on higher powers of d . One canshow that the pre-Lie product preserves the length of words in the sense that ( η ✁ ξ, ν ) = 0when | η | + | ξ | 6 = | ν | . Therefore, the product is well defined as it is locally finite. Moreover,defining d [0] = 0, the recursive formulas reduce to a single expression( x j η ) ✁ d = x j ( η ✁ d ) + x ( η d [ j ]) , j ∈ { , , . . . , m } . (12) Example 5.1.
Consider the computation of the pre-Lie product for a few words of shortlength. For example, if c = x n , n ∈ N and d ∈ K m hh X ii , then x n ✁ d = x n ( ∅ ✁ d ) = 0. Forany x k ∈ X with k = 0, x k ✁ d = x k ( ∅ ✁ d ) + x ( ∅ d [ k ]) = x d [ k ] , k ∈ { , , . . . , m } . (13)Observe a x k ( x k ✁ d ) = 0 as every word in the support of x k ✁ d must have the prefix x .Furthermore, it is clear from (13) that the length of the words in supp( x k ✁ d ) coincide withthe length of those in supp( d ) except incremented by one. On the other hand, if d = e k η (where e k ∈ R m is the k -th unit vector), thendeg( a x k ✁ d ) = 2 + deg( a η ) ≥ deg( a x k ) + deg( a η ) , k ∈ { , , . . . , m } . Indeed one always obtains | η ✁ d | + | η ✁ d | x ≥ | η | + | η | x + | d | + | d | x (where the length of asum of words is defined as the maximum of the lenght of the words). Consider next c = x j x k where both j and k are not zero. Applying the definition gives x j x k ✁ d = x j ( x k ✁ d ) + x ( x k d [ j ])= x j ( x k ( ∅ ✁ d ) + x d [ k ]) + x ( x k d [ j ])= x j x d [ k ] + x ( x k d [ j ]) . (14)For comparison, it follows from (5) that x j x k ˜ ◦ d δ = φ d ( x j x k )( ) = φ d ( x j ) ◦ φ ( x k )( )= φ d ( x j )( x k + x d [ k ])= x j ( x k + x d [ k ]) + x ( d [ j ] ( x k + x d [ k ]))= x j x k + x j x d [ k ] + x ( d [ j ] x k ) + x ( d [ j ] ( x d [ k ]))= x j x k + x j x k ✁ d + x ( d [ j ] ( x d [ k ])) , which is consistent with (11). Applying now the coordinate function a x j x k to (14) gives a x j x k ( x j x k ✁ d ) = 0 for any series d . A trivial induction shows that a η ( η ✁ d ) = 0 , ∀ η ∈ X ∗ , d ∈ K m hh X ii . Finally, consider a word η with | η | x = 0. Observe a η ( ρ ✁ d ) = 0 because every word in thesupport of a η ( ρ ✁ d ) must contain at least one x and | η | x = 0. Proposition 5.1.
The Lie algebra of δ + K m hh X ii is the space K m hh X ii with the Lie bracketgiven by the formula [ c, d ] = c ✁ d − d ✁ c. (15) Proof:
The Lie bracket of the Lie algebra associated to the Lie group δ + K m hh X ii isgiven by evaluating the Lie bracket of left invariant vector fields on δ + K m hh X ii at theidentity δ . Note that since δ + K m hh X ii is an affine subspace of δ + K m hh X ii , it is easy tosee that the left-invariant vector field associated to c ∈ K m hh X ii is given by the formula X c ( δ + e ) = c + e ✁ c , hence[ c, d ] = [ X c , X d ]( δ )= ( dX d ◦ X c − dX c ◦ X d )( δ )= lim t → t − ( X d ( δ + tc ) − X c ( δ ) − X c ( δ + td ) + X d ( δ ))= lim t → t − ( d + tc ✁ d − d − c − td ✁ c + c )= lim t → t − t ( c ✁ d − d ✁ c ) = c ✁ d − d ✁ c. Corollary 5.2.
The Lie algebra of ( δ + K mLC hh X ii , ◦ , δ ) is K mLC hh X ii with bracket (15).Proof: The canonical inclusion ι : δ + K mLC hh X ii → δ + K hh X ii is the restriction of acontinuous linear map to a closed (affine linear) subset, whence smooth. Obviously it is aLie group morphism. Derivating the morphism at the identity δ yields a Lie group morphism L ( ι ) := T δ ι : T δ ( δ + K mLC hh X ii ) → T δ ( δ + K hh X ii ) , v ι ( v )(= v ) . ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 21
Observe that the Lie bracket on L ( δ + K mLC hh X ii ) = T δ ( δ + K mLC hh X ii ) ∼ = K mLC hh X ii coincides(pointwise) with the one of L ( δ + K hh X ii ), and the latter is (15).Regularity of the Fr´echet Lie group δ + K hh X ii is investigated next. For a curve γ δ ( t ) =( δ + γ ( t )) ∈ δ + K hh X ii consider the Lie type differential equation ( ˙ γ δ ( t ) = γ δ ( t ) .c ( t ) = c ( t ) + γ ( t ) ✁ c ( t ) γ δ (0) = δ, (16)where c : [0 , → K hh X ii is a continuous curve. For every η ∈ X ∗ observe that ( γ δ ( t ) , η ) =( γ ( t ) , η ). Now since the coordinate functions are continuous linear, a differential equation isobtained for every word η ∈ X ∗ :( ˙ γ δ ( t ) , η ) = ( c ( t ) , η ) + ( γ ( t ) ✁ c ( t ) , η )= ( c ( t ) , η ) + X ρ ∈ X ∗ ( γ ( t ) , ρ )( ρ ✁ c ( t ) , η ) = ( c ( t ) , η ) + X ≤| ρ |≤| η | ( γ ( t ) , ρ )( ρ ✁ c ( t ) , η ) . (17)The computations in Example 5.1 have been used above, and the products of elements in K m are taken as componentwise products. Note now that the sum in (17) only appears if | η | x = 0. Hence, if a word does not contain the letter x , then the differential equation (17)reduces to ( γ ( t ) , η ) = Z t ( c ( s ) , η )d s, ∀ η ∈ X ∗ , | η | x = 0 . (18)Since ( c ( t ) , η ) is a continuous K m -valued curve, one can solve the above equation for all t ∈ [0 , η is a word with | η | x = 0, observe that all elements in (17) appearing ascoefficients of evaluations of γ are continuous K m -valued curves of the form ( c ( t ) , η ) or C ρ,η : [0 , → K m , t C ρ,η ( t ) := ( ρ ✁ c ( t ) , η ) . (19)It is now proved via induction on the length of the words that equation (17) admits asolution on [0 ,
1] for every word. Note first that for any word without an x (such as theempty word, which is the only length zero element), the statement follows directly from theintegral equation (18). If | η | = n > | η | x = 0, the statement follows again from (18). To obtain solutions for thewords of length n containing x , pick an enumeration ( η i ) i ∈ I n of words of length n . Usingthe enumeration and (19), define v n ( t ) := ( γ ( t ) , η )( γ ( t ) , η )...( γ ( t ) , η | I n | ) , C n ( t ) := C η ,η ( t ) · · · C η ,η | In | ( t ) C η ,η ( t ) 0 . . . ...... . . . . . . C η | In |− ,η | In | C η | In | ,η ( t ) · · · C η | In | ,η | In |− , b n ( t ) := X | ρ | 1] (via the usual solution theory for linear differential equations on finite-dimensionalspaces). This completes the induction, and thus, one can iteratively solve the inhomogeneouslinear system (20) for every n ∈ N with a unique solution on [0 , § v n ( t ) = C n ( t ) v n ( t ) + b n ( t ) , n ∈ N . The earlier discussion has shown that this system is lower diagonal, i.e., the right-hand sideof the equation in degree n depends only on the solutions up to degree n . One can now solvethe differential equation on the Fr´echet space by adapting the argument in [Dei77, p. 79-80]:Lower diagonal systems can be solved iteratively component-by-component, if each solutionexists on a time interval [0 , ε ] for some fixed ε > 0. Choosing ε = 1, observe that the Lietype equation (16) admits a unique global solution which can be computed iteratively. Thus,the following result is evident. Proposition 5.2. The Lie group δ + K m hh X ii is C -regular.Proof: It was seen in the discussion above that the Fr´echet Lie group δ + K m hh X ii is C -semiregular. However, due to [Han19, Corollary D] every C -semiregular Lie group modeledon a Fr´echet space is already C -regular.Observe that one can leverage the regularity of the Fr´echet Lie group in the investigationof the regularity for the Silva Lie group δ + K mLC hh X ii . The inclusion ι : δ + K mLC hh X ii → δ + K m hh X ii is a Lie group morphism which relates the solutions of the evolution equationon the Silva and the Fr´echet Lie group. Indeed, [Gl¨o15, 1.16] shows that for a continuouscurve c : [0 , → L ( δ + K mLC hh X ii ) = K mLC hh X ii a solution to the evolution equation (16) in δ + K mLC hh X ii must satisfy ι ◦ Evol δ + K mLC hh X ii ( c ) = Evol δ + K m hh X ii ( L ( ι ) ◦ c ) = Evol δ + K m hh X ii ( c ) , where c is interpreted canonically as a curve into L ( δ + K m hh X ii ) = K m hh X ii via the naturalinclusion. Hence, the Silva Lie group will be C -semiregular if and only if it can be provedthat the solutions to the evolution equation on the Fr´echet Lie group are bounded when thecurve c is bounded. Unfortunately, at present it is not obvious how to bound these solutionsto the evolution equation, which leads to the following. Open problem : Is the Silva Lie group δ + K mLC hh X ii C -semiregular? ONTINUITY OF FORMAL POWER SERIES PRODUCTS IN NONLINEAR CONTROL THEORY 23 Remark 5.1. (1) Note that words which do not contain the letter x do not yield thenecessary bound for the solution of the evolution equation as the differential equationreduces to the integral equation (18) for these words. (2) For words which contain the letter x , the linear system (20) governs the evolutionequation. A natural Ansatz for the problem would thus be to apply a Gronwall typeargument. Looking closer at the pre-Lie product, one easily sees that the top-levelwords (i.e., of length n when dealing with length n -words) only yield an exponentialbound in the Gronwall argument. Unfortunately, there seems to be no clear way tobound the norm of the inhomogeneity b n in (20). (3) Observe that regularity of the Silva Lie group δ + K mLC hh X ii follows almost directlyonce C -semiregularity is known: Having the semiregularity in place, it is assumedthat the estimates will directly yield that for every curve c taking values in L ( δ + K m hh X ii ) ∩ B k·k M (0) = { x ∈ K m hh X ii | k x k M ≤ } , M > , the evolution Evol( c ) is contained in B k·k N K (0) , N, K > fixed (but depending on M ).If this is true, C -regularity of δ + K mLC hh X ii follows from the arguments presentedin the proof of [BS16, Theorem 4.3] . Appendix A. Infinite-dimensional calculus In this appendix we recall some basic definitions concerning the infinite-dimensional cal-culus used throughout the article. For more information we refer to the presentations in[Gl¨o02b, Nee06]. Definition A.1. Let r ∈ N ∪ {∞} and E , F locally convex K -vector spaces and U ⊆ E open. A map f : U → F is called a C r K -map if it is continuous and the iterated directionalderivatives d k f ( x, y , . . . , y k ) := ( D y k · · · D y f )( x ) exist for all k ∈ N with k ≤ r and y , . . . , y k ∈ E and x ∈ U , and the mappings d k f : U × E k → F so obtained are continuous. If f is C ∞ R , it is called smooth . If f is C ∞ C , it is saidto be complex analytic or holomorphic and that f is of class C ω C . Definition A.2 (Complexification of a locally convex space) . Let E be a real locally convextopological vector space. Endow the locally convex product E C := E × E with the followingoperation ( x + iy ) . ( u, v ) := ( xu − yv, xv + yu ) , ∀ x, y ∈ R , u, v ∈ E. The complex vector space E C is called the complexification of E . Identify E with the closedreal subspace E × { } of E C . Definition A.3. Let E , F be real locally convex spaces and f : U → F defined on an opensubset U . f is called real analytic (or C ω R ) if f extends to a C ∞ C -map ˜ f : ˜ U → F C on an openneighborhood ˜ U of U in the complexification E C . Recall from [Dah11, Proposition 1.1.16] that C ∞ C functions are locally given by series of continuoushomogeneous polynomials (cf. [BS71b, BS71a]). This justifies the abuse of notation. For r ∈ N ∪ {∞ , ω } , being of class C r K is a local condition, i.e. if f | U α is C r K for everymember of an open cover ( U α ) α of its domain, then f is C r K . (See [Gl¨o02b, pp. 51-52] for thecase of C ω R , the other cases are clear by definition.) In addition, the composition of C r K -maps(if possible) is again a C r K -map (cf. [Gl¨o02b, Propositions 2.7 and 2.9]). Definition A.4 ( C r K -Manifolds and C r K -mappings between them) . For r ∈ N ∪ {∞ , ω } ,manifolds modeled on a fixed locally convex space can be defined as usual. Direct productsof locally convex manifolds, tangent spaces and tangent bundles as well as C r K -maps betweenmanifolds may be defined as in the finite-dimensional setting.For C r K -manifolds M, N the notation C r K ( M, N ) denotes the set of all C r K -maps from M to N . Furthermore, for s ∈ {∞ , ω } define the locally convex C s K -Lie groups as groups witha C s K -manifold structure turning the group operations into C s K -maps. The following lemma seems to be part of the mathematical folklore, a proof can be foundin [BS16, Lemma A.3]. Lemma A.1. Let U be an open subset of a complex locally convex space E and F be acomplex locally convex space which is sequentially complete. Consider a set Λ ⊆ L ( F, C ) of complex linear functionals which separates the points on F . 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Ph.D. thesis, Rutgers University,New Brunswick, NJ 1990[Yos57] Yoshinaga, K. On a locally convex space introduced by J. S. E. Silva . J. Sci. Hiroshima Univ. Ser.A (1957):89–98 Old Dominion University, Norfolk, Virginia 23529, USA Email address : [email protected] Department of Mathematics, Universitet i Bergen, All´egate 41, 5020 Bergen, Norway Email address : [email protected] FLU, Nord university, Høgskoleveien 27, 7601 Levanger, Norway Email address ::