Chronological algebras combinatorics and control

This article investigates the geometric and algebraic foundations of exponential product expansions in nonlinear control. A survey of historic developments in geometric control theory on one side, and algebraic combinatorics on the other side exhibits parallel developments and demonstrates how respective ndings translate into powerful tools on the other side. Chronological algebras are shown to provide the fundamental structure that uni es geometric, algebraic and combinatorial theories. 1 Introduction Asymptotic expansions are a fundamental tool for the analysis of nonlinear dynamical systems since closed-form solutions are rarely practical, if at all available. In nonlinear control theory, Lie series expansions, which date back to the work of M. Fliess [11, 12] in the 1970s, have been the most widely used tool for the analysis of optimality and controllability, for realization theory, and for obtaining stabilization results. However, a closer look at the Lie series from a geometric point of view exhibits substantial shortcomings. For example, the series do not immediately exhibit their character as exponential Lie series, and truncations of the series do not correspond to series of any approximating systems. Considering noncommuting ows as the heart of nonlinear control, it is more natural to work with exponential product expansions. Starting in the late 1970s A. Agrachev and R. Gamkrelidze [1, 2] began to develop a comprehensive theory from this point of view { coining the term chronological calculus. While providing the geometric and analytic framework, their work did not yet provide the explicit formulae that allow for e cient calculations in applications. In the mid 1980s H. Sussmann independently managed to derive explicit formulae for in nite product expansions of the Chen-Fliess series [51] by utilizing explicit formulae for Hall bases of free Lie algebras. This work used basic techniques from di erential equations (variation of parameters) combined with structural properties of Lie algebras, speci cally Lazard elimination. The second section of this article surveys the main strands of this control-theoretic background. The third section exhibits parallel developments in algebraic combinatorics. Again, the focal points are explicit formulae for in nite exponential product expansions, which in this combinatorial setting have been obtained by Melan con and Reutenauer[37, 38] using rewriting systems (rather than techniques from di erential equations). Based on ndings by Viennot [55] these explicit formulae are valid for a considerably larger class of bases than the original Hall bases (as de ned e.g. in Bourbaki [5]). However, these formulae exhibit a features that seems to be more than just a cosmetic aw, as they mix the concatenation and shu e products, which do belong into di erent algebras! The fourth section introduces chronological products as the natural product of control theory. It is shown how these products simplify the exponential product expansions, and how they naturally arise when solving di erential equations by iteration. As a powerful corollary one obtains explicit formulae (in coordinates) for normal forms of free nilpotent control systems which are widely used as approximating systems, in particular, in path planning algorithms. 1 The nal section provides a more formal description of chronological algebras. Extending recent work by M. Kawski and H. Sussmann [22], abstract chronological algebras are presented as iterated integral functionals. As a highlight the Chen-Fliess series is identi ed as a resolution of the identity map on an algebra of functions. A few concluding remarks point to related ongoing work in homological algebra which investigates Leibniz algebras which are dual to the chronological algebras that are the foundations of nonlinear control theory. 2 Series and product expansions in control This work is concerned with nite dimensional control systems on a manifold M _ x = f0(x) + m Xi=1 uifi(x): (1) de ned by a set of real-analytic vector elds fi. The controls ui are measurable and take values in compact intervals, typically containing zero in their interiors. A special case are systems without drift, that is systems for which f0 0. Looking for geometric invariants K. T. Chen [6] associated in 1957 to every smooth path : [a; b] 7! IRm the formal power series ( ) = 1 + 1 X p=1 X (i1;:::ip) Z dxi1 : : : dxip Xi1 : : :Xip (2) in noncommuting indeterminates X1; : : :Xm. The integrals are de ned recursively by Z dxi1 : : : dxip = Z b a Z t dxi1 : : : dxip 1 d ip(t) (3) where t denotes the restriction j[a;t] of the path : [a; b] 7! IRm to the interval [a; t]. In the 1970 M. Fliess [12, 11] recognized the utility of the Chen series for investigations of nonlinear control systems. Following the notation of Sussmann [49] consider a formal control system _ S = S X0 + m X 1=1 uiXi! (4) initialized at the identity S(0) = I in the associative algebra Â(X0; : : :Xm) of formal power series (with real coe cients) in the noncommuting indeterminates Xi. For controls ui as above this system has a unique solution. As can be easily veri ed by direct 2 substitution, this solution is explicitly given by the Chen-Fliess series: Ser(T; u) =XI I(T; u)XI (5) where the sum ranges over all multi-indices I = (i1; : : : ip) with p 0 and 1 ij m. Here XI = Xi1Xi2 : : :Xip, and I(T; u) = Z T 0 Z t1 0 Z tp 1 0 uip(tp)uip 1(tp 1) : : : ui1(t1) dt1dt2 : : : dtp (6) For a particular control system (1) initialized at a point p 2 IRn, together with an analytic (\output") function : IRm 7! IR, substitute the system vector elds fj for the indeterminates Xj. Interpreting the products fI as compositions of rst order partial di erential operators, one obtains an \asymptotic series for the propagation of along trajectories of (1)" Serf(T; u) =XI I(T; u)(fI )(p) (7) Sussmann [49]] showed that this series converges uniformly, (on su ciently small time intervals) to along solution curves of the system (1) that are initialized and remain in any given compact subset of M . Detailed analysis and estimates of the iterated integrals in this series are at heart of the proofs of many su cient and necessary high-order conditions for small-time local controllability and optimality, compare e.g. [23, 25, 47, 52]. A typical such result by Stefani [47] considers the case of m = 1, ju( )j 1, f0(p) = 0. Write (adk+1v; w) = [v; (adkv; w)] to denote iterated Lie brackets of vector elds. Let S1 = f(adkf0; f1): k 0g, and Sk+1 = Sk[f[v; w]: v 2 S1; w 2 Skg. It is shown that if (ad2kf1; f0)(p) is linearly independent from the set S2k 1 evaluated at p for some k 1 then the system is not small-time controllable about p. This means that for su ciently small times T 0, the initial point p does not lie in the interior of the reachable set Rp(T ) (the set of all points that can be reached from p in time T by solution curves of the system (1)). The proof uses the linear independence of (ad2kf1; f0)(p) to nd a function such that (p) = 0, (f )(p) = 0 for all f 2 S2k 1 and ((ad2kf1; f0) )(p) > 0. After elaborate manipulations of the series (7) Stefani shows that essentially (x(T; u)) = ((2k)!) 1 R T 0 (R s 0 u(s) ds)2kdt+ o(jT j2k+1) 0. Aside from technical estimates on the size of the iterated integrals, these manipulations involve the rewriting of linear combinations of higher order partial di erential 3 operators as rst order operators (Lie-brackets) and combining large numbers of iterated integrals over simplices f(t1; : : : ; tp): 0 tp : : : t2 t1 Tg to obtain the structurally much simpler integral R T 0 (R s 0 u(s) ds)2kdt. For illustration, in the classical case of k = 1 (Clebsch-Legendre condition for optimality) typical such steps involve the identities [f1[f1; f0]] = f1f1f0 2fff0f1 + f0f1f1 and R T 0 (R t 0 u1(s))2u0(t) dt = 2 R T 0 R t1 0 R t2 0 u0(t1)u1(t2)u1(t3) dt3dt2dt1. The book-keeping issues become horrendous, but it also becomes clear that there are some hidden algebraic and combinatorial structures that make these manipulations possible. These algebraic structures were partially utilized for a partial factorization of the Chen-Fliess series (5) (factorization of the in nite partial sum corresponding to terms involving at most three factors f1) in the proof [23] of a necessary condition involving the iterated Lie bracket [[f0; f1]; [[f0; f1]; f0]]]. A complete factorization of the Chen-Fliess series (5) was achieved by Sussmann [51], but it necessitated the additional structural properties of Lie algebras and Hall sets, see below. As successful as the Chen-Fliess series was for deriving the early conditions for controllability and optimality, it exhibits some serious drawbacks. First of all the number of terms is very large { more speci cally, the number of terms of total degree less or equal to N is of the order of (m+1)N 1. For practical purposes it is cumbersome that truncations of the Chen-Fliess series, which one might want to use for approximations of solutions, do not correspond to series expansions for any approximate systems. Closely related is that intricate cancellations among the terms of the Chen-Fliess series make it virtually impossible to decide accessibility, i.e to decide whether all solution curves of a system (starting from a xed initial point) lie within a nontrivial submanifold of the state-space. (The standard in nitesimal criterion for accessibility is the Lie algebra rank condition: The system (1) is accessible from p if and only if the set of all iterated Lie brackets of the vector elds f0; : : : fm spans the tangent space TpM .) But most importantly, it is clear from elementary considerations that the Chen-Fliess series (5) is an exponential Lie series, i.e. there exists a Lie series (T; u) (whose coe cients depend on the controls) such that Ser(T; u) = exp( (T; u)). This may be seen using di erential equations arguments related to the Campbell-Baker-Hausdor formula [48, 51], or via the combinatorial argument (Ree's theorem [42, 43]) that the iterated integrals I of the Chen-Fliess series (5) satisfy the shu e relations I J = Ix J (8) for all multi-indices I and J . (The next section provides a detailed discussion of the shu e product and its algebra.) 4 For practical purposes one may consider any ordered basis B = fB1; B2; : : :g for the free Lie algebra L(X0; : : :Xm), and ask for explicit formulae for the coe cients k(T; u) in the series log(Ser(T; u)) = (T; u) = P1k=1 k(T; u)Bk as functionals on the space of controls u. (Unless otherwise speci ed all algebras are assumed to have coe cients in an in nite eld k of characteristic zero.) A general formula for such coordinates of the rst kind has been provided in [51] in terms of a spanning set of the free Lie algebra L(X0; : : :Xm). This provides a signi cant reduction in the number of terms (as compared to the original Chen-Fliess series (5) as the dimension of subspace of the free Lie algebra on (m + 1) generators that is spanned by all iterated products of the generators of length less than N is equal to 1 N PkjN (k)(m + 1)N=k (compare e.g. [5] x3.3) where is the Moebius function ( (n) = 0 if there exists a prime p such that p2jn, and (n) = ( 1)s if n is the product of s distinct primes). Alternatively, one may ask for explicit formulae for coordinates of the second kind, i.e. explicit formulae for the coe cients B(T; u) in an in nite directed product expansion Ser(T; u) = Y1B2B exp( B(T; u)B) (9) where B Lk(X ) is any suitable basis of the free Lie algebra. This is the route successfully taken by Sussmann [51], and also by Melan con and Reutenauer [37, 38] in the combinatorial setting. This product expansion relies on the basic di erential equations technique of variation of parameters combined with structural properties of Hall sets and Lazard elimination [5]: Suppose k a eld of scalars, X is a set and c 2 X . Then the free Lie algebra Lk(X ) over k generated by X is the direct sum of the one-dimensional subspace f c: 2 kg and of a Lie-subalgebra of Lk(X ) that is freely generated by the set f(adjc; b): b 2 X n fcg; j 0g. The origins of the rst bases for free Lie algebras go back to the work of Ph. Hall [17] who investigated commutator groups, and they were formalized in a Lie algebra setting by M. Hall [15, 16]. Several other bases were proposed in subsequent years, most notably Lyndon bases [7] and Sir sov bases [46], which initially were regarded as structurally di erent. However, Viennot [55] showed that all these bases arose from the same fundamental principles (31). In the following we will refer to these as generalized Hall bases, or as Hall-Viennot bases. To characterize these bases start with a set X . The free magma M(X ) over X is the set of all parenthesized words, i.e. the union M(X ) = [1n=1Mn(X ) where the sets Mn(X ) are de ned inductively by M1(X ) = X and Mn(X ) = [i+j=nMi(X ) Mj(X ). A generalized Hall set over a set X is any strictly ordered subset ~ H M(X ) that 5 satis es [55] (in the literature one frequently nds de nitions which reverse the ordering <, and/or which consider the words read from right to left) X ~ H Suppose a 2 X . Then (w; a) 2 ~ H i w 2 H, w < a and a < (w; a). (10) Suppose u; v; w; (u; v) 2 ~ H. Then (u; (v; w)) 2 ~ H i v u (v; w) and u < (u; (v; w)). The original Hall elements (in a narrow sense) as de ned e.g. in Bourbaki [5] require that the ordering of the Hall elements be compatible with the length of the word, i.e. u; v 2 ~ H and u < v imply that juj jvj. Viennot showed that one can omit this condition, and still obtain bases for free Lie algebras. The importance of Hall sets for our purposes arises from the fact that their images under the canonical map ':M(X ) 7! Lk(X ) from the free magma into the free Lie algebra are bases for Lk(X ) [5, 16, 55]. Here, '(a) = a for a 2 X , and '((w; z)) = ['(w); '(z)] for w; z 2 M(X ): (11) The restriction of the map ' to any generalized Hall-set ~ H M(X ) is one-to-one by construction [55]. Hence the inverse image ' 1(H) 2 ~ H M(X ) of an element H 2 '( ~ H) Lk(X ) is well-de ned, and we denote it by ~ H for notational convenience. Caution is advised with this notation as, for example, one may have fx; y; (x; y); (y; (x; (x; y)))g ~ H M(fx; yg). Then '((x; y)) = [x; y] = [y; x], and '((y; (x; (x; y)))) = [y; [x; [x; y]]] = [x; [y; [x; y]]] (due to the anti-commutativity and Jacobi-identity in Lk(X )). Consequently, ' 1([x; [y; [x; y]]]) = ' 1([y; [x; [x; y]]]) = (y; (x; (x; y))) 6= (x; (y; (x; y))) inMX (unless x = y). This distinction between formal brackets, or parenthesized words in M(X ) on one side, and elements of the free Lie algebra Lk(X ) on the other side is necessary as Hall-sets, and the iterated integrals depend critically on the factorization of the Hall words. However, since [x; y] = [ y; x] in any Lie algebra it is impossible to de ne left and right factors of Lie-brackets. Sussmann's product expansion [51] of the Chen-Fliess series (5) originally used Hallsets in the narrower sense as found Bourbaki [5]. But it can be shown that with some minor modi cations the original proof also holds for the more general Hall-Viennot bases. Speci cally, consider a generalized Hall set ~ H = f ~ H1 ~ H2; : : :g M(X ) (with X = f0; 1; : : : ; mg). De ne two sets of iterated integrals that are related via C ~ H(T; u) = R T 0 c ~ H(t; u)dt for ~ H 2 ~ H, and de ned recursively by ca(t; u) = ua(t) for a 2 X and c ~ H(t; u) = 1 m! C ~ H1(t; u) m c ~ H2(t; u) if ~ H = ( ~ H1; ( ~ H1; : : : ; ( ~ H1 | {z } m-times ; ~ H2)):::) 2 ~ H (12) 6 and either ~ H2 2 X or ~ H2 = ( ~ H21; ~ H22) with ~ H1 6= ~ H21 and ~ H21; ~ H22 2 ~ H. The main theorem of [51] states that these iterated integrals are the coe cients of the exponential product expansion (9) of the Chen-Fliess-series { very carefully written as C ~ H(T; u) = '( ~ H)(T; u) for ~ H 2 ~ H M(X ). It is informative to revisit some key-steps of Sussmann's original proof [51] as they reveal very close links to key structural properties of both chronological algebras and Hall-sets, linking combinatorics and di erential equations. For the sake of notational convenience consider the system without drift _ S = S m Xi=1 uiXi! (13) initialized at the identity S(0) = I in the associative algebra Âk(X1; : : :Xm) of formal power series in the noncommuting indeterminates Xi (with coe cients in a ed k of characteristic zero). (The system (4) is obtained via renumbering and setting u0 1.) Choose an ordered Hall set ~ H. For any formal bracket ~ H 2 ~ H denote the image '( ~ H) in the free Lie algebra Lk(X1; : : :Xm) by H. Iteratively using the technique of variation of parameters the solution may be expressed in the form S(t) = Sj(t) Y H2Fj e H(t;u)H (14) where Fj H is an increasing family of subsets constructed iteratively, starting with F0 = ;. At each stage the varied parameter Sj, starting with S0 = S, satis es its own di erential equation _ Sj(t) = Sj(t)0@ X H2Gj _ H(t; u)H1A (15) where the subsets Gj H are constructed iteratively, starting with G0 = fX1; : : :Xmg the original set of generators. At each stage j 1 select the least element H j 2 Gj 1 (with respect to the ordering induced from ~ H), and de ne Fj = Fj 1 [ fH j g. By di erentiating (14) (using the new value for j) and substituting into (13) one obtains, after some basic manipulations [51], the new di erential equation (15) where the next set Gj must be chosen as Gj = f(ad H j ; H):H 2 Gj 1; H 6= H j g (16) Note that by choice of H j 2 Gj 1 all elements of Gj are again Hall-elements, i.e. Gj 2 H. Using Lazard's elimination theorem one observes that the free Lie algebra 7 L(fX1; : : :Xmg) is the direct sum of the one-dimensional subspaces spanned by the elements of Fj and a subalgebra generated freely by the new set Gj. The key step to obtain a formula for the iterated integral coe cients H(t; u) repeatedly uses the basic fact that for any two elements X, and Y of a Lie algebra etXY e tX = X 0 1 ! t (ad X; Y ): (17) More speci cally, the technical manipulations leading to the di erential equation (15) for the next value of j involve the critical step: e j (t;u) 0B@ X H2Gj 1nfH j g d dt H(t; u) H1CA e j (t;u) = X 0 X H2Gj 1nfH j g 1 ! H j (t; u) d dt H(t; u) (ad H j ; H) (18) From the coe cients of the iterated brackets (ad H j ; H) in this expression one directly obtains the formula (12) for the iterated integrals (ad H j ;H)(t; u). For technical details, and precise discussion of convergence etc. see the original reference [51]. This exponential product expansion has found numerous applications. The proofs of the known high-order su cient and necessary conditions simplify considerably as now much fewer iterated integrals need to be analyzed, and these iterated integrals have very special structures, allowing one to more quickly nd upper and lower bounds. For an e cient calculation related to the classical Clebsch-Legendre condition see [22]. In a similar style the necessary conditions of Stefani [47] and Kawski [23] may also be derived much more e ciently [26]. The most successful applications of the product expansion may lie in many path planning algorithms, e.g. [18, 19, 27, 40, 41, 53]. These are closely related to normal forms for free nilpotent control systems, which may be immediately read o from the explicit formula for the iterated integrals H . With a slight abuse of notation use deparenthesized Hall words (compare the next section) to index the coordinates xH and consider the control system _ xa = ua if a 2 X _ xHK = xH _ xK if H;K;HK 2 H jcalW (X ) (19) The suggestive notation employed here is to imply that H and K are the uniquely determined left and right factors of the Hall word HK (compare the next section for 8 details). For any control U(t) one directly reads o the value of the coordinate functions along the solution curve (starting at x = 0 xH(t) = mH H(t; u) (20) where mH is a multi-factorial as apparent from the recursive formulas for H(t; u). Essentially the same normal forms for free nilpotent systems have been obtained by Grayson and Grossman [14] who veri ed, essentially by direct calculation, that the vector elds de ning the system (19) generate a free Lie algebra. Independently of these developments, Agrachev and Gamkrelidze [1, 2] developed a completely geometric formalism for dealing with compositions of ows of time-varying vector elds. This chronological calculus avoided the pitfalls of the Chen Fliess series, but it lacked the explicit formulae of Sussmann's exponential product expansion that were made possible by utilizing speci c bases for free Lie algebras. The later sections of this article will revisit the chronological formalism in detail and further develop the algebraic background. 3 Combinatorics of exponential products The applications of the product expansion to control make it natural to use di erential equations techniques, as in the preceding section. However, it also is clear that underlying the manipulations of the iterated integrals there are some basic algebraic and combinatorial structures. The combinatorial objects contain just the essential information needed for the manipulations and avoid all the excessive notational burden of the iterated integrals. The goal of this section is to exhibit the (combinatorial) analogue of the exponential product expansion of the Chen-Fliess series as a factorization of the identity map on the space of noncommuting formal power series. To this end we review the properties of some basic algebraic and combinatorial objects. Throughout we will largely follow the terminology of Lothaire [34]. The set of indeterminates X = fX1; : : :Xmg is commonly called an alphabet. (In general the alphabet may be in nite, but for our purposes it su ces to consider nite alphabets.) We will use the variables a; b; c; : : : to denote its elements, which are also referred to as letters. The free monoid over X , denoted W (X ) is the set of all words (or noncommuting monomials) endowed with the noncommutative, associative concatenation product (w; z) 7! wz. The empty word is denoted by 1 and satis es 1w = w1 = w for all w 2 W (X ). Write W0(X ) = W (X ) n f1g for the set of nonempty 9 words. Throughout, k denotes a eld of scalars of characteristic zero. The associative algebras of noncommutative polynomials noncommutative formal power series (with coe cients in k) are denoted by Ak(X ) and Âk(X ), respectively. Associated with the basis W (X ) is the pairing < ; > : Âk(X ) Ak(X ) 7! k de ned by < ; > def = X w2W (X ) w w for = X w2W (X ) ww 2 Âk(X ); = X w2W (X ) ww 2 Ak(X ) (21) With this pairing Âk(X ) is the algebraic dual of Ak(X ), and Ak(X ) is the topological dual of Âk(X ) with the usual topologies, compare [22] for details. Use Ak;0(X ) and Âk;0(X ) to denote the sets of noncommutative polynomials and power series with zero constant term, respectively, i.e. Ak;0(X ) = f 2 Ak(X ): <1; >= 0g Âk;0(X ) = f 2 Âk(X ): < ; 1>= 0g (22) The algebra Âk(X ) is equipped with a natural co-product: the diagonal map : Âk(X ) Âk(X ) 7! Âk(X ) (23) is the k-algebra homomorphism that is de ned on generators a 2 X by (a) = a 1+ 1 a. Its transpose is the bilinear map X:Ak(X ) Ak(X ) 7! Ak(X ) de ned by =< (u); w z> for all u 2 Âk(X ); w; z 2 Ak(X ) (24) For a detailed discussion of the Hopf-algebra structures compare Reutenauer [44, 54]. Alternatively, one may characterize the shu e product combinatorially by de ning 1Xw = wX1 = w for all w 2 W (X ) and, for a; b 2 X and w; z 2 W (X ), (wa)X(zb) = ((wa)Xz)b + (wX(zb))a: (25) This recursive characterization is useful to clarify how the shu e product is related to the iterated integrals. Using Ree's theorem [42, 43, 54] one may establish that the Chen-Fliess series is an exponential Lie series (compare the previous section) by showing that the coe cients I(T; u) satisfy the shu e relations (8). More precisely, on a suitable space U of controls, let IIF(X ) be the algebra of iterated integral functionals considered as a subalgebra of all mappings from UX into U (compare the next sections, or [22], for technical details). Using the terminology introduced 10 in this section the linear map :A(X ) 7! IIF(X ) of (6) may be de ned recursively by (a) : (T; u) 7! R T 0 ua(t)dt for a 2 X (wa) : (T; u) 7! R T 0 w(t; u)ua(t)dt for a 2 X ; 1 6= w 2 W (X ) (26) We use the symbols (wa) and wa interchangeably. To satisfy the shu e relation means that the map is an associative algebra homomorphism from the algebra A(X ) equipped with the shu e product to the algebra of iterated integral functionals under pointwise multiplication. It is instructive to take a closer look at a direct proof by induction on the combined length of the words (wa) and (zb). The empty word 1 is mapped to the iterated integral functional 1(t; u) 1. For any letter a 2 X it is clear that ax 1(t; u) = a(t; u) = a(t; u) 1 = a(t; u) 1(t; u). The induction step uses the recursive de nitions of the shu e product (25) and of the map . The calculation shows clearly how shu e product of the indices corresponds to the pointwise multiplication of the iterated integrals, and how the recursive characterization (25) of the shu e product encodes integration by parts. (wa)x (zb)(u; T ) = = ((wa)x z)b+(wx (zb))a(u; T ) (using (25)) = ((wa)x z)b(u; T ) + (wx (zb))a(u; T ) (linearity) = R T 0 (wa)x z(t; u) ub(t)dt+ R T 0 wx (zb)(t; u) ua(t)dt (using (26)) = R T 0 wa(t; u) z(t; u) ub(t) + w(t; u) zb(t; u) ua(t) dt (by induction hypothesis) = R T 0 wa(t; u) d dt zb(t; u) + d dt ( wa(t; u)) zb(t; u) dt(using (26)) = wa(T; u) zb(T; u) (integration by parts) (27) In the literature, one also nds partially commutative shu e products [10], and a noncommutative shu e product [3] on spaces of permutations { this latter is closely related to the product expansion and the iterated integrals. The following illustrations serve to suggestively demonstrate how permutations may be mapped to the iterated integrals over simplices as they occur in the Chen-Fliess series, and how the shu e product of permutations corresponds to the Cartesian product of simplices, which in turn may again be decomposed into sums of simplices. 11 For a word w = (i1; : : : im) containing each of the letters 1; 2; : : : n at most once de ne w = f(t1; : : : ; tn) : 0 ti1 ti2 : : : tim 1; and tk = 0 if ij 6= k for 1 j mg, and extend linearly to the group algebra. For example, if n = 2 then (1) = [0; 1] f0g, (1) = f0g [0; 1], (21) = f(t1; t2) : 0 t2 t1 1g, and the shu e product (1x 2) = 12 + 21 maps to the Cartesian product 1x 2 = [0; 1] [0; 1]. 6 6 6 1x 2 1 2 = 21 [ 12 The following is typical example in the case of three letters f1; 2; 3g = [ [ (12)x 3 = 312 [ 132 [ 123 0 t1 t2 1; 0 t3 1 For multiplicative integrands f(x; y; z) = f1(x) f2(y) f3(z) this example leads to the following suggestive formula (using x; y; z, instead of t1; t2; t3 for better readability): Z 1 0Z y 0 ( )dx dy Z 1 0 ( )dz =Z 1 0Z y 0Z x 0 ( ) dz dx dy+Z 1 0Z y 0Z z 0 ( ) dx dz dy+Z 1 0Z z 0Z y 0 ( ) dx dy dz The partial factorization of the Chen-Fliess series as in [23] made repeated use of such identities (going from the right to the left side). Thus the composition and products of the iterated integral functionals are combinatorially encoded using the shu e product of the algebra Ak(X ). The transpose of the shu e product is the diagonal map on the algebra Âk(X ) (23). Inside this algebra Âk(X ) of formal power series the noncommutative polynomials (or nite linear combinations of words) correspond to partial di erential operators (of nite order) that are obtained as compositions of the vector elds de ning the original control 12 system (1). Among the in nite linear combinations in Âk(X ) there are also some distinguished elements which may be regarded as points (or translations, or evaluation at points) on a formal group. Here we review the combinatorial properties of these distinguished elements of Âkk(X ). De ne Lie polynomials to be the elements of the smallest subspace of Â(X ) that contains X and that is closed under the Lie bracket [ ; ]: (w; z) 7! [w; z] = wz zw. Note that this subspace may be identi ed with the free Lie algebra L(X ) over X . Furthermore denote by L̂(X ) the set of Lie series, that is the set of all elements s = P1n=1 sn 2 Â(X ) all of whose homogeneous components sn = Pjwj=n w 2 L(X ) are Lie polynomials. (Here jwj denotes the length of the word w 2 W (X ).) For letters a; b 2 X , one easily sees that (ab) = (ab) 1 + a b + b a + 1 (ab), and hence ([a; b]) = (ab) (ba) = ((ab) 1 + 1 (ab)) ((ba) 1 + 1 (ba)) = [a; b] 1+ [b; a] 1. Repeating this argument inductively, one obtains the Friederich's criterion that an element s 2 Â(X ) is a Lie series if an only if (s) = s 1 + 1 s. Using the de nition of the shu e as the transpose of the diagonal map one obtains as a direct consequence that the ~ L(X ) is orthogonal to all nontrivial shu es. Suppose ` 2 L(X ). Then<`; w X z> = < (`); w z> = <` 1; w z> + <1 `; w z> 6= 0 only if w = 1 or z = 1: (28) From this criterion for Lie series one easily obtains a similar criterion for a series 2 Â(X ) to be an exponential Lie series [44], i.e. that there exists a Lie series L̂(X ) (denoted log( )) such that = exp( ) def = P1n=0 1 n! n. ( ) = () log( ) 2 L̂(X ) (29) In terms of the shu e product this immediately translates into Ree's theorem [42, 43] that a series 2 Â(X ) is an exponential Lie series if and only if its coe cients < ; w> , w 2 W (X ), satisfy the shu e relations: Indeed, since < ; wXz>=< ( ); w z> the condition ( ) = is equivalent to < ; w X z>=< ; w> < ; z> (30) This property justi es the name group-like elements for the exponential Lie series. Before we can return to the exponential product expansion we need to revisit Hall bases. Bases for free Lie algebras proposed by M. Hall [15, 16], Lyndon [7], and Sir sov [46] 13 were originally considered to be distinct, until Viennot [55] showed that they all arise from a fundamental unique factorization principle, leading to the de nition we gave above (10). A consequence of the unique factorization principle is that the restriction of the canonical map :M(X ) 7! W (X ) (from the free magma into the free monoid, (a) = a for a 2 X , and ( ~ w; ~ z) = ( ~ w) (~ z) for ~ w; ~ z 2 M(X )) to any generalized Hall-set ~ H M(X ) is one-to-one. This e ectively allows us to conveniently \drop the parentheses", identifying unparenthesized Hall words H with parenthesized elements ~ Hof the Hall-set. The unique factorization principle [55] that underlies the construction of these bases asserts that for any such xed generalized Hall-set ~ H, now considered as a subset ofW (X ), every word w 2 W (X ) has a unique factorization into a nonincreasing product of Hall words, i.e. there exist unique Hj 2 H, such that w = H1H2 : : :Hs and H1 H2 : : : Hs (31) In particular, this allows one to quickly insert brackets or parenthesis into a Hallword to nd the images under the inverse map 1: ( ~ H) 7! M(X ): Any Hall-word H 2 ( ~ H n X ) W (X ) n X , that is not a generator, may be written in the form H = Ka with a 2 X and w 2 W (X ). Then w uniquely factorizes as above and the parenthesation map 1 gives 1(H) = 1(H1H2 : : : Hsa) = 1(H1) 1(H2) : : : 1(Hs)a ::: (32) This special structure is nicely illustrated pictorially by the structure of the binary tree: QQQQQQQQQQ H1 H2 H3 : : : Hs 2 Hs 1 Hs< a In complete analogy, the bracketing is the linear map [ : ]: (Ak(X ) 7! Ak(X ) de ned for generators a; b 2 X with a < b by [a] = a and [a; b] = ab ba, and recursively on words w 2 W (X ) with unique factorization w = H1H2 : : :Hs by [H1H2 : : :Hs 1Hs] = [[H1]; [[H2]; [: : : ; [[Hs 1]; [Hs]]] :::]] (33) 14 Note that for a < b as above this implies [b; a] = ba (whereas [a; b] = ab ba). For a Hall set ~ H M(X ) the map [ : ] maps the image ( ~ H) W (X ) to a subset of the free Lie algebra, in particular, to a basis of Lk(X ). In terms of the map ', compare (11), one may consider [ : ] an extension of = ' 1 from ( ~ H) to AkX . For any ordered basis B of the free Lie algebra Lk(X ), the set P = fBi1Bi2 : : : Bis: s > 0; Bij 2 B; Bi1 Bi2 : : : Bisg is a basis, called the Poincar e-Birckho -Witt basis, of the universal enveloping algebra of Lk(X ) which in turn may be identi ed with Ak(X ). In a combinatorial setting the quest for formulas for the iterated integrals in the exponential product expansion amounts to searching for explicit formulae for the dual bases of such Poincar e-Birckho -Witt bases. In Melan con and Reutenauer [37] provided such an explicit formula for the dual basis for the PBW basis constructed on a Lyndon basis H for the free Lie algebra L(X ). They recognized that the formula was virtually identical to the one obtained by Sch utzenberger [45] in 1958 for Hall-bases in the original narrow sense. Later work [38] showed that these formulae do indeed hold for the general Hall-Viennot bases H as in (10). Modulo some minor notational changes, such as reading words from the left or from the right, the Melan con and Reutenauer's result is expressed in the following formula a la Hopf algebra [37] X w2W (X )w w = Y H2H exp([H] H) (34) where a = a if a 2 X ; Ma = Ma if H =Ma 2 H; a 2 X ;M 2 W (X ) and w = 1 n1! 1 ns! H1 xn1X : : :X H1 xn1 if w = Hn1 1 : : : Hns s ; with Hj 2 H; and H1 > : : : > Hs: (35) In the special case of a Hall word H = H1 : : : Hsa with a 2 X , H1 > : : : > Hs < a with Hi 2 H one obtains H = H1X H2X : : :X Hs a (36) which is a remarkable mix of the shu e product and one instance of the concatenation product. These two products are really at home in two di erent algebras which are their respective (algebraic and/or topological) duals (for details see [22, 44, 54]). The next section introduces an entirely di erent product which will beautifully resolve this aw. 15 4 Chronological products Chronological products go back at least as far as the work of Sch utzenberger [45]. In 1958 he used a product > in the algebra A(X ) that satis es the identity (42) and thus is a chronological product. Incidentally, while never giving a name to this product, Sch utzenberger de ned the shu e product, which he denoted by >, as the symmetrization w>z = w>z + z>w of the chronological product. The term chronological product was coined later and introduced into nonlinear control theory by Agrachev and Gamkrelidze who developed a comprehensive chronological calculus starting in the late 1970s [1, 2]. The chronological product also was implicitly a key ingredient in Sussmann's exponential product expansion (15) (18). In this section we shall make the case that the chronological product is the most natural product of nonlinear control. The subsequent section will give a more abstract foundation and more theoretical properties of the chronological algebra. Probably the most basic, best known, and best studied example, e.g. [4], to illustrate the fundamental concepts of geometric nonlinear control is the system _ x1 = u1 ju1j 1 _ x2 = u2 ju2j 1 _ x3 = x1u2 x2u1 : (37) This system is completely controllable (every point can be reached from every other point), since the distribution spanned by the vector elds f1 = @ @x1 x2 @ @x3 and f2 = @ @x2 + x1 @ @x3 is nonintegrable due to the nonvanishing commutator [f1; f2] = 2 @ @x3 . To more clearly see the structure perform the global coordinate change y1 = x1, y2 = x2, and y3 = (x3 + x1x2)=2 to rewrite the system in the less symmetric, but simpler form _ y1 = u1 ju1j 1 _ y2 = u2 ju2j 1 _ y3 = y1u2 : (38) It is convenient to consider the function u3(t) = (u1 ? u2)(t) def = Z t 0 u1(s) ds u2(t) (39) a virtual third control which allows one to steer the system into an independent third direction corresponding to the commutator [f1; f2] = 2 @ @x3 . Via highly oscillatory controls this concept gives rise to path-planning algorithms [28, 29]. Such virtual control 16 also lend themselves to be used in cost-functionals to arrive at nonlinear optimal feedback stabilization schemes. This concept of the virtual controls may be considered a starting point to motivate a possible chronological product (u1 ? u2) of two control functions u1 and u2. If the functions u1 and u2 are both locally integrable, then so is the product (u1?u2). Also note that the symmetrization (u1?u2)+(u2?u1) corresponds to the derivative of the pointwise product of the integrated controls (u1 ? u2 + u2 ? u1)(t) = d dt Z t 0 u1(s) ds Z t 0 u2(s) ds (40) Finally, one easily veri es that this product satis es a three term identity (u1 ? (u2 ? u3)) (t) = R t 0 u1(s); ds R t 0 u2(s)ds u3(t) = R t 0 (R s 0 u1( ) d ) u2(s)ds u3(t) + R t 0 (R s 0 u2( ) d ) u1(s)ds u3(t) = ((u1 ? u2) ? u3) (t) + ((u2 ? u1) ? u3) (t) (41) In general de ne a (right) chronological algebra to be a linear space C (over a eld k) that is endowed with a bilinear product :C C 7! C which satis es the (right) chronological identity v (w z) = (v w) z + (w v) z for all v; w; z 2 C (42) Thus the space Lloc([0;1)) of locally integrable functions de ned on the interval [0;1) is a (right) chronological algebra with the product ? de ned as above. In the following it turns out to often be more convenient to work with the integrated controls which live in the space ACloc([0;1)) of locally absolutely continuous functions. On this space de ne closely related product by (U1 U2)(t) def = Z t 0 U1(s)U 0 2(s) ds (43) where U 0(s) = d dsU(s) denotes the derivative. One immediately veri es that (F (G H))(t) = R t 0 F (s) d ds R s 0 G( )H 0( ) d ds = R t 0 F (s)G(s) H 0(s) ds = R t 0 (R s 0 F ( )G0( ) d + R s 0 G( )F 0( ) d ) H 0(s) ds = ((F G) H)(t) + ((F G) H)(t) (44) 17 is true for all F 2 AC loc and G;H 2 AC loc;0. Hence the space AC loc;0([0;1)) of locally absolutely continuous functions that vanish at zero is (right) chronological algebra with the product . It is the requirement R t 0(FG)0 = (FG)(t) that stands in the way of extending to AC loc AC loc without giving up the chronological identity. (This is intimately related to the impossibility to equip any chronological algebra with an identity that satis es 1 1 = 1, compare (58)). However, it is convenient to still use that U 1 = 0 for any U , and 1 U = U for U 2 AC loc;0. The symmetrization of the product of (43) yields the (associative) pointwise product on AC loc;0 (which in turn corresponds to the shu e product in the combinatorial setting). (U1 U2) + (U2 U1) = (U1 U2) for U1; U2 2 AC loc;0 (45) There are many familiar chronological subalgebras of the ones de ned above, most notably those of polynomial and of (real, or complex) exponential functions. In either case the multiplication rules are xn ? xm = 1 n+1xn+m+1 ent ? emt = 1 ne(n+m)t xn xm = m n+mxn+m ent emt = m n+me(n+m)t (46) The standard method of solving time-varying linear di erential equations by iteration is e ciently encoded by the chronological product: The integrated form of the initial value problem, the universal control system (13) _ S = S , (0) = 1 with = Pmi=1 uiXi may be very compactly written using chronological products S = 1 + S (47) Iteration immediately yields an explicit series expansion (respecting the noncommutative character of the variables Xj inside ) S = 1 + (1 + S ) = 1 + + ((1 + S ) ) = 1 + + ( ) + ((1 + S ) ) ) ... = 1 + + ( ) + (( ) ) + ((( ) ) ) : : : (48) Upon expansion this immediately yields a suggestive formula for the iterated integrals (6) of the Chen Fliess series. For a word w 2 W (X ) and a letter a 2 X wa(t; u) = Z T 0 w(s; u) d ds a(s; u) ds = ( w( ; u) a( ; u)) (t) (49) 18 Similarly, the formula for the iterated integrals (12) in the exponential product expansion (9) simpli es dramatically when written in terms of chronological products CHK(t; u) = CH(t; u) CK(t; u) if H;K;HK 2 H (50) Recall that the original formula (12) contains factorials that correspond to stationary subsequences of the nonincreasing sequences of Hall words in the unique factorization (31). It is by virtue of the identity (65) that the chronological product absorbs these factorials that are necessary when using the shu e product (which corresponds to pointwise multiplication of the iterated integrals (27)). Chronological products also provide the most compact way for specifying a normal form for free nilpotent control systems of rank r. The key is to index the coordinates by Hall words H 2 H(r) def = fH 2 H: jhj rg (rather than by the integers): _ xa = ua if a 2 X xHK = xH xK if H;K;HK 2 H(r)(X ) (51) For a typical Hall set on the alphabet X = f0; 1g and r = 5 the normal from for a free nilpotent system is readily obtained from (51) as _ x0 = u0 _ x1 = u1 _ x01 = x0 _ x1 = x0 u1 _ x001 = x0 _ x01 = x20 u1 using 1(001) = (0(01)) _ x101 = x1 _ x01 = x1x0 u1 using 1(101) = (1(01)) _ x0001 = x0 _ x001 = x30 u1 using 1(0001) = (0(0(01))) _ x1001 = x1 _ x001 = x1x20 u1 using 1(1001) = (1(0(01))) _ x1101 = x1 _ x101 = x21x0 u1 using 1(1101) = (1(1(01))) _ x00001 = x0 _ x001 = x40 u1 using 1(00001) = (0(0(0(01)))) _ x10001 = x1 _ x0001 = x1x30 u1 using 1(10001) = (1(0(0(01)))) _ x11001 = x1 _ x1001 = x21x20 u1 using 1(11001) = (1(1(0(01)))) _ x01001 = x01 _ x001 = x01x30 u1 using 1(01001) = ((01)(0(01))) _ x01101 = x01 _ x101 = x01x21x0 u1 using 1(01101) = ((01)(1(01))) (52) The coordinates in this normal form are chosen such that there are no factorials in this system description consequently the values of xH(t; u) and the iterated integrals H(t; u) of the exponential product expansion will disagree by multi-factorials. It is straightforward to extract from this formula (51) the components of the system vector elds fa, a 2 X in these coordinates. Speci cally, rewrite the system 19 (51) in the form _ x = X a2X uafa(x): (53) and nd that fa = X H2H(r) xHa 1 @ @xH (54) where (wb)a 1 = w if a = b and (wb)a 1 = 0 else (for any w 2 W (X ) and b 2 X ). Moreover, if w = H1H2 : : : Hs is the unique factorization (31) of a word w 2 W (X ) into a nonincreasing product of Hall words, then de ne xw = xH1xH2 : : : xH1 . The Lie algebra generated by the system vector elds fa, a 2 X is, by construction, free nilpotent of rank r. This also may be veri ed by direct calculation. Incidentally, Grayson and Grossman [14] rst proposed essentially the formula (54) and then veri ed by direct calculation that the associated Lie algebra was free { thereby obtaining essentially the formulas for the iterated integrals of the exponential product expression (9) through another independent route. 5 The free chronological algebra This section provides a more formal treatment of chronological algebras and derives some simple, but remarkable identities. Following [22] the free chronological algebra is presented as a chronological algebra of iterated integral functionals, and the Chen Fliess series is recognized as the image of the resolution of the identity map under a natural isomorphism. On the spaces Ak;0(X ) and Âk;0(X ) of noncommutative polynomials and power series with zero constant term, de ne a bilinear, noncommutative, nonassociative product by identifying w a = wa for any nonempty word w 2 W0(X ) and any letter a 2 X , and using the chronological identity (42) to de ne w (z a) = (w z) a + (z w) a for a 2 X ; w; z 2 W0(X ) (55) One easily sees that with this product Ck(X ) def = (Ak;0(X ); ) is a chronological algebra that is free in the usual sense: If C is any chronological algebra then any :X 7! C extends uniquely to a chronological algebra homomorphism ̂ : Ck(X ) 7! C. Utilizing the recursive de nition of the shu e product (25), one recognizes that the shu e product is (the extension to Ak(X ) of) the symmetrization of the chronological product: w z + z w = w X z for w; z 2 W0(X ) (56) 20 which implies in particular that w (z a) = (w X z)a for a 2 X ; w; z 2 W0(X ) (57) Note that the special case of wXw = 2w w clearly indicates that one cannot extend the chronological product to the empty word without giving up some fundamental property. For notational convenience one may extend to Ak(X ) Ak;0(X )[Ak;0(X ) Ak(X ) by declaring 1 w = w for w 2 W0(X ) and w 1 = 0 for w 2 W (X ): (58) One readily veri es that the identities (42), (56), and (57) are still satis ed. As an example consider the chronological product of two two-letter words (ab) (cd) = ((a b) c) d+ (c (a b)) d = ((a b) c) d+ ((c a) b) d+ ((a c) b) d: (59) which includes the special case (ab) (ab) = abab+2aabb = 12(ab)X(ab). This example leads to a number of identities that relate the di erent products (concatenation, shu e and chronological) on the set Ak;0(X ). Due to the lack of associativity one needs to distinguish between left and right chronological powers: For w 2 Ak;0(X ) de ne w 1 = 1(w) = wx 1 = w, and inductively for n 1 n+1(w) = w n(w) (60) w (n+1) = w n w (61) wx (n+1) = w X wxn = wxn X w (62) (allowing w 2 Ak(X ) for the shu e powers). One obtains the useful identities for w 2 Ak;0(X ) for n > 1w w (n 1) = (n 1) w n (63) n(w) = (n 1)! w n (64) wxn = n! w n (= n n(w)) (65) In the case of n = 2 note that wx 2 = wXw = w w+w w = 2w 2 = 2 2(w). The case n = 3 is just the chronological identity (42) with v = w = z: 3(w) = w w 2 = 2w 3. In general, for n 3, one obtains by induction w w (n 1) = w (w (n 2) w) = (w w (n 2)) w + (w (n 2) w) w = (n 2) w (n 1) w + w n = (n 1) w n (66) 21 and as corollaries for n 3 n(w) = w ( n 1(w)) = (n 2)! w (w (n 1)) = (n 1)! w (n 1) (67) and wx n = wX wx (n 1) = w X (n 1)! w (n 1) = (n 1)! w w (n 1) + w (n 1) w = (n 1)! ((n 1) + 1) w n = n! w n (68) Returning to the exponential product expansion (9) recall that the multi-factorials disappear when the iterated integrals are written in terms of chronological powers (50). The multi factorials are ever present when using pointwise multiplication of the iterated integrals (12), when using shu e products for the dual PBW bases (36), or when di erentiating the normal forms of free nilpotent systems (20). Speci cally, rewrite the formulae (36) for the dual PBW bases of Melan con and Reutenauer [37]using chronological powers: If H = Hn1 1 : : :Hns s a is the unique standard factorization of a Hall word then the corresponding dual basis element is with factorials H = 1 n1! 1 ns! H1 : : : H1 | {z } n1 times : : : Hs : : : Hs | {z } ns times a )) :::)) (69) and without factorials H = H1 n1 H2 n2 : : : Hs ns a ::: (70) In either case the uniform chronological products replace the mix of shu e and concatenation products. However, the disappearance of the multi-factorials comes at the cost of a reversal of the bracketing of repeat factors, illustrated for the case ofH = wwwzzza with w = H1 > z = H2 2 H, a 2 X . In this case H = 1 36( w ( w ( w ( z ( z ( z a)))))) = ((( w w) w) (((( z z) z) a) = (( w) 3 (( w) 3 a)) (71) Pictorially: 22 QQQQQ QQQQQ QQQQQQQQQQ (( w w) w) ((( z z) z) a) However, for the dual PBW bases elements of words w 2 W (X ) n H that are not Hall words, the noncommutative chronological product requires an explicit summation that is implicit in the formula (35) using shu e products. Speci cally, if w = Hn1 1 : : : Hnr r is the standard factorization of w 2 W (X ) with Hi 2 H, Hi > Hi+1 then w = X 2Sr H (1) n (1) : : : H (r 1) n (r 1) H (r) n (r) ::: (72) where the sum is taken over all permutations of the set f1; 2; : : : rg. For letters a 2 X de ne the maps &a; and a on Ak(X ) as transposes of the left and right translations w 7! aw and w 7! wa, i.e. &(1) = (1) = 0 and for w 2 W (X ) and b 2 X by &a;(bw) = w if a = b; and &a;(bw) = 0 else a(wb) = w if a = b; and a(wb) = 0 else (73) Using the recursive de nition (25) and the commutativity of the shu e product one easily sees that both &a; and a are derivations on (Âk(X );X) for a 2 X . However, using (55), one sees that on the chronological algebra Ck(X ) = (Ak(X ); ) only &a; is a derivation (for a 2 X ) (where de ned, and using the extended convention (58) to resolve the special cases). Speci cally, for w; z 2 Ak;0(X ), and a; b 2 X &a;(w (zb)) = &a;((w z + z w)b) = (&a;(wXz))b = (&a;(w)) (zb) + w (&a;(zb)) (74) while the right truncation a is not a derivation since, e.g. for distinct letters a; b; c; d 2 X , b(ab cd) = 0 (compare (59)), whereas ( b(ab)) (cd)+(ab) ( b(cd)) = acd+cad 6= 0. Instead one has (for w; z 2 Ak;0(X ), and a; b 2 X ) a(w (zb)) = w ( a(zb)): (75) This property of being only a one-sided derivation plays a fundamental role in the theoretical development of presentations of the free chronological algebra [22]. Note, that 23 in general the product (composition) of two derivations is not a derivation, but their commutator always is a derivation. Consequently, one obtains well-de ned derivations &` and ` on (Ak(X );X) for ` 2 L̂k(X ). All these formal objects have natural analytic interpretations, compare [22] for details: Recall that by virtue of Ree's theorem (30) the set Ĝk(X ) Âk(X ) of exponential Lie series has a natural Lie group structure (when endowed with the usual topology). Considering the exponential Lie series as formal points, the algebra (Ak(X );X) is interpreted as an algebra of functions on Ĝk(X ) with pointwise multiplication by virtue of (30). Speci cally, an element w 2 Ak(X ) is identi ed with the linear functional w: 7!< ; w> on Ĝk(X ). The formal tangent vectors to Ĝk(X ) at a point 2 Ĝk(X ) are the linear functionals 2 Âk(X ) that satisfy (wXz) = ( (w)) < ; z > + < ; w> ( (z)) for all w; z 2 Ak(X ). One easily veri es, compare [22], that any such is of the form = ` for some element ` 2 L̂k(X ). A simple calculation makes the connection between formal tangent vectors and the right truncations ` < `; (wXz)> = < ( `); (w z)> = <( )(1 `+ ` 1); (w z)> = < ); ( `(w) z) + (w `(z))> = < ; ( `(w)Xz) + (wX `(z))> (76) Finally, any ` 2 L̂k(X ) may also be identi ed with the map `: 7! ` ( 2 Ĝk(X )) which is interpreted as a left invariant tangent vector eld on Ĝk(X ). Similarly one may consider right invariant vector elds which are related to the left truncations &` with ` 2 L̂k(X ). Following [22] one may present the free chronological algebra Ck(X ) as an chronological algebra of dynamic functionals. This is in complete analogy to the presentation of the associative polynomial algebra k[X1; : : : :Xn] in commuting variables Xi by k[x1; ::xn] of polynomial functions as the subalgebra of k(km) that is generated by the projections xi: (p1; : : : ; pn) 7! pi, i = 1; : : : n. In the chronological case, one may for example, choose U = AC loc;0([0;1)) (the algebra of locally absolutely continuous functions that vanish at zero) as ring of coe cients. Then U (UX ) is a chronological algebra under pointwise multiplication. De ne IIFk(X ) as the chronological subalgebra of U (UX ) that is generated by the projections a:UX 7! U de ned by a(U) = Ua for a 2 X . Denote by the map Ak(X ) 7! IIFk(X ) that sends the generators a 2 X to (a) = a. As is shown in [22] the map is not only surjective, but also a 24 chronological algebra isomorphism The proof relies on the choice of ACloc;0([0;1))which is a su ciently large set of dynamic scalars, and on the fact that a is not aderivation on the chronological algebra. Speci cally, for any given set ffa 2 Ak(X ): a 2Xg there is a unique g 2 Ak;0(X ), (namely g =Pa2X faa) such that a(g) = fa foreach a 2 X .Finally we return to the Chen Fliess series which appears in a new light in this advancedcontext. Recall the in nite product expansion a la Hopf algebra (34) of [37]. On the lefthand side we nd thesumPw2W (X )w w which may be interpreted as the identity mapon Ak(X ), under the standard identi cation of the the space of linear maps Hom(V; V )on a vector space V with the tensor product V V (here V = Ak(X ) and V = Âk(X )is its algebraic dual). Thus the Chen-Fliess-series (5) is nothing else than the image ofthe identity map on Ak(X ), identi edwithPw2W (X ) w w 2 Âk(X ) Ak(X ) under themap idÂk(X ). On either side one may go from the in nite series to the exponentialproduct expansion. The two product expansions are mapped into each other by thesame map idÂk(X )[22].6 Concluding remarksThis article has laid out how the algebraic structure of chronological algebras perfectlymatches the geometric features of noncommuting ows that are at the heart of a necontrol systems. This chronological structure is useful for describing geometric fea-tures, for analysis (controllability, optimality, approximating systems), and for design(including stabilization and path planning). The parallels between the combinatorialsetting and the dynamical systems point of view may be exploited in several ways:For proofs, calculations, and for obtaining algebraic and geometric insight into speci cfeatures and properties, one has the choice between combinatorial and di erential equa-tions approaches. The realization of the the abstract chronological algebra in termsof integral functionals and dynamic variables makes the abstract algebra tangible. Inthe other direction, the combinatorial language allows one to dispense of the excessivenotation that used to burden manipulations of iterated integrals: The isomorphismallows one to instead only manipulate the indices.The chronological algebra structure has recently found substantial attention also out-side nonlinear control theory: Following the pioneering work of Crouch and Grossman[9] who saw the applicability of the framework of noncommuting ows to the numericalsolution of systems of di erential equations under state constraints, several numerical25 analysts have been exploring a very closely related formalism, compare e.g. [39].In the completely di erent arena of homological algebra, in particular Loday [30, 31, 32,33], has analyzed the algebraic structure of Leibniz algebras. These are characterizedby the three-term identity[x; [y; z]] = [[x; y]; z] [[x; z]; y] for all x; y; z 2 g(77)As such a Leibniz algebra is like a Lie algebra but without the requirement of anticom-mutativity. As spelled out in [31], the study of Leibniz algebras may be motivated bythe observation that the identity (77) alone is su cient to prove as basic properties asd2 = 0 for the boundary map d of the complexC (g) = : : : g n d! g n 1 ! : : : ! g ! k(78)for a Lie algebra g. The Leibniz algebra, as a nonabelian version of a Lie algebra, is inti-mately related to the chronological algebra which may be regarded as a nonassociativepre-shu e algebra. Technically the Leibniz algebra is a dual algebra structure to thatof the chronological algebra [33]. For a comprehensive introduction to the notion ofduality, speci cally Koszul duality of quadratic operads, we refer the interested readerto [13]. 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