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Abstract This article provides a dendriform-tree setting for Fliess operators with matrix valued-inputs, a class of analytic nonlinear input-output systems. Such a description is convenient, for example, in quantum control. In particular, a description of such Fliess operators is provided using planar binary trees. Sufficient conditions for convergence are also given. This concept is then applied to solve a bilinear equation and related to more sophisticated combinatoric objects to give a glimpse of its full potential in control applications. 1. Introduction Fliess operators provide a general framework under which nonlinear input–output systems can be studied (Fliess, 1981; Gray & Wang, 2002; Gray & Li, 2005). Let $$X=\{x_0,x_1,\ldots,x_m\}$$ be an alphabet and $$X^{\ast}$$ the free monoid comprised of all words over $$X$$ (including the empty word $$\emptyset$$) under the catenation product. A formal power series $$c$$ in $$X$$ is any mapping of the form $$X^{\ast}\rightarrow \mathbb{R}^{\ell}\!:\eta \mapsto (c,\eta)$$. The set of all such mappings will be denoted by . The support of an arbitrary series $$c$$ is $$\operatorname{supp}(c)=\{\eta\in X^\ast: (c,\eta) \neq 0\}$$. A series having finite support is called a polynomial, and the set of all polynomials is . For a measurable function $$u: [a, b] \rightarrow\mathbb{R}^m$$ define $$\lVert{u}\rVert_{L_p}=\max\{\lVert{u_i}\rVert_{L_p}: \ 1\le i\le m\}$$, where $$\lVert{u_i}\rVert_{L_p}$$ is the usual $$L_p$$-norm for a measurablereal-valued component function $$u_i$$. Define iteratively for each $$\eta\in X^{\ast}$$ the mapping $$E_\eta: L_1^m[t_0, t_0+T]\rightarrow C[t_0, t_0+T]$$ by $$E_\emptyset[u] = 1$$, and Exiη′[u](t,t0)=∫t0tui(τ)Eη′[u](τ,t0)dτ, (1.1) where $$x_i\in X$$, $${\eta'}\in X^{\ast}$$ and $$u_0= 1$$. The input–output operator corresponding to $$c$$ is then Fc[u](t):=∑η∈X∗(c,η)Eη[u](t), which is called a Fliess operator. If the generating series $$c$$ is locally convergent, i.e., if there exist constants $$K,M>0$$ such that |(c,η)|≤KM|η||η|! (1.2) for all $$\eta\in X^{\ast}$$, where $$\lvert{\eta}\rvert$$ denotes the number of letters in $$\eta$$, then $$F_c[u]$$ converges absolutely and uniformly on $$[t_0,t_0+T]$$ if $$T$$ and $$\lVert{u}\rVert_{L_p}$$ are sufficiently small. In general, the input–output map $$F_c:u\rightarrow y$$ needs not to have a state space realization, however, many familiar and relevant examples are obtained from the state space setting (Isidori, 1995). A tacit assumption in the existing theory for Fliess operators is that the inputs are mutually commutative, i.e., the functions associated with the letters of $$X$$ commute for different times and among each other. The proposition here is that this assumption results in a great deal of simplification but also hides certain underlying algebraic structures that are important in applications like control on Lie groups (Brockett, 1973) and quantum control (D’Alessandro, 2007). As a motivating example, consider a bilinear system z˙(t)=Az(t)+B(t)z(t)u(t), (1.3) where $$B$$ is a smooth matrix-valued function on $$[0,T]$$. One can view $$u$$ as the user controlled input and $$B$$ as a disturbance input. Now let $$z_i$$ be the solution of (1.3) when $$z(0)=e_i=[0,\ldots,0,1,0,\ldots,0]^T$$ with the $$1$$ in the $$i$$-th position and define $$Z(t)= [z_1(t), \cdots, z_n(t)]$$, where $$n$$ is the dimension of the system. Then Z˙(t)=(A+B(t)u(t))Z(t)=:U(t)Z(t), (1.4) where in general $$U(t_1)U(t_2) \neq U(t_2)U(t_1)$$. This is, for example, the standard setting for a regulator problem in which the input–output map from disturbance $$B$$ to some output $$y=CZ$$ needs to be determined when $$u(t)=u_0\in \mathbb{R}$$. Equation (1.4) is also the usual starting point for control theory on Lie groups. Systems of the form (1.4) are also ubiquitous in quantum mechanics. Take, for instance, the case of a spin particle in a magnetic field $$B_m$$ whose direction changes in time. The function $$U$$ is proportional to the scalar product $$S \cdot B_m$$, where $$S=(S_x,S_y,S_z)$$ represents the spin vector with spin operators for the $$x$$, $$y$$ and $$z$$ axes as its components. Now suppose the magnetic field at $$t = t_1$$ is parallel to the $$x$$-axis and at $$t=t_2$$ parallel to the $$y$$-axis. Then $$U(t_1) \varpropto \lvert{B_m}\rvert S_x$$, $$U(t_2) \varpropto \lvert{B_m}\rvert S_y$$, and $$[U(t_1), U(t_2)] \varpropto B_m^2 [S_x , S_y ] \varpropto B_m^2 S_z \neq 0$$. Therefore, $$U(t_1)$$ and $$U(t_2)$$ do not commute. Moreover, systems of the form Z˙(t)=U(t)F(Z(t)) can be considered in this class using a coordinate change $$\bar{Z}:=F(Z)$$ that is valid on a neighbourhood of $$Z(0)=I$$. In which case, Z¯˙(t)=(dF−1(Z¯)dZ¯ |Z¯=Z¯(0))−1U(t)Z¯(t)=:W(t)Z¯(t). In general, a series representation of the solution of (1.4) can be obtained by successive iterations. That is, Z(t)=I+∑n=1∞∫0tU(t1)dt1∫0t1U(t2)dt2⋯∫0tn−1U(tn)dtn. (1.5) This series has a representation in terms of the so-called time-ordered exponential (see equation (1.6) below), which is defined using the time-ordering operator T(U(t1)U(t2)⋯U(tn)):= ∑σ∈SnΘnσU(tσ(1))U(tσ(2))⋯U(tσ(n)), where $$\Theta_n^\sigma = \prod_{i=1}^{n-1}\Theta(t_{\sigma(i)}-t_{\sigma(i+1)})$$, $$\Theta$$ is the Heaviside step function, $$\sigma$$ is a permutation in $$S_n$$, the group of permutations of order $$n$$ (Bauer et al., 2013). For example, T(U(t1)U(t2))=Θ(t1−t2)U(t1)U(t2)+Θ(t2−t1)U(t2)U(t1). Because of the symmetry of the simplex consisting of all ordered $$n$$-tuples $$(t_1,t_2,\ldots ,t_n)$$ in the integration limits, this operator provides the following identity: ∫0tdt1∫0t1dt2⋯∫0tn−1dtnU(t1)U(t2)⋯U(tn) =1n!∫0tdt1∫0tdt2⋯∫0tdtnT(U(t1)U(t2)⋯U(tn)). The solution is thus written as the time-ordered exponential Z(t) =I+∑n=1∞1n!∫0t⋯∫0tT(U(t1)⋯U(tn))dt1⋯dtn =:Texp(∫0tU(s)ds). (1.6) Expression (1.6) disregards the underlying algebra provided by the products of non-commutative iterated integrals in (1.5). However, it is known that by systematically keeping track of the non-commutative orderings of these integrals, a proper exponential solution can be derived. That is, Z(t) =exp(Ω(U(t))), where $${\it\Omega}$$ is the Magnus expansion, which is obtained via a recursion (Magnus, 1954). See also (Blanes et al., 2009; Ebrahimi-Fard & Manchon, 2009a, 2014). In the case of commutative inputs, products of iterated integrals are naturally captured by the shuffle algebra (Ree, 1957; Reutenauer, 1993). This algebra is basically the vector space endowed with the shuffle product. This product is an $$\mathbb{R}$$-bilinear mapping uniquely specified by the shuffle product of two words and for all words $$\eta,\xi\in X^\ast$$. It is easy to see that the shuffle product codifies the integration by parts formula ∫0tui(s)ds∫0tuj(s)ds =∫0tui(s)(∫0suj(r)dr)ds+∫0tuj(s)(∫0sui(r)dr)ds as . The non-commutative version of the integration by parts formula is ∫0tui(s)ds∫0tuj(s)ds =∫0tui(s)(∫0suj(r)dr)ds+∫0t(∫0sui(r)dr)uj(s)ds. (1.7) Note that the second summand on the right-hand side above cannot be generated iteratively as in (1.1), and therefore the shuffle product above is not enough to codify the product of non-commutative iterated integrals. In the context of algebraic combinatorics, the algebra of non-commutative iterated integrals corresponds to a Rota-Baxter algebra of weight zero (Ebrahimi-Fard & Guo, 2008b; Ebrahimi-Fard & Patras, 2013). That is, if the Rota-Baxter operator $$R$$ is identified with the Lebesgue integral operator, then (1.7) is equivalent to R(ui)R(uj)=R(uiR(uj))+R(R(ui)uj). Products of iterated integrals naturally appear when a system’s state is filtered by an output function (Fliess, 1981; Wang, 1990), in the computation of bounds for iterated integrals (Duffaut Espinosa et al., 2012), and the characterization of system interconnections such as the product, cascade and feedback connections (Gray & Li, 2005; Gray et al., 2014). The first goal of this article is to provide a fully non-commutative extension of the theory of Fliess operators using planar binary trees together with its underlying dendriform algebra. This dendriform algebra is standard in the field of algebraic combinatorics and serves, for instance, as a tool for keeping track of the non-commutativity of iterated integrals. Fliess operators with non-commutative inputs will be referred to as dendriform Fliess operators. The second goal is to give sufficient conditions under which dendriform Fliess operators converge. The concept will then be employed to solve a non-commutative bilinear equation by re-writing it as a dendriform equation, which can be seen as half of a commutator equation (Ebrahimi-Fard & Manchon, 2009b). This can potentially provide a combinatorial perspective to bracket equations related to isospectral flows in control applications (Brockett, 1991; Helmke, 1991), but this topic is beyond the scope of this article. The article is organized as follows. Section 2 provides a brief tutorial treatment of dendriform algebras. In Section 3, planar binary trees are presented, which allow the extrapolation of combinatorial tools to the realm of Fliess operators. Also, the non-commutative version of the shuffle product is given. These results are then applied in Section 4 to define dendriform Fliess operators. Then the convergence of dendriform Fliess operators is addressed. In Section 5, a closed-form for the solution of a dendriform equation is provided. Finally, conclusions are given in Section 6. 2. Dendriform algebras The goal of this section is to briefly introduce parenthesized words and their relationship to dendriform algebras. More complete treatments of these concepts can be found in Loday & Ronco (1998) and Ebrahimi-Fard & Guo (2008a). Let $$X$$ be a finite alphabet as before and define $$\mathfrak{P}X=X\cup \{\lfloor,\rfloor\}$$. The free semigroup under catenation generated by $$\mathfrak{P}X$$ is denoted $$\mathfrak{P}X'$$. For $$\eta=q_{1}q_{2}\cdots q_{n}\in \mathfrak{P}X'$$, let $$s(\eta)_i$$ denote the number of $$\lfloor$$’s in $$q_{1}\cdots q_{i}$$ minus the number of $$\rfloor$$’s in $$q_{1}\cdots q_{i}$$. Definition 2.1 A word $$\eta=q_{1}q_{2}\cdots q_{n} \in \mathfrak{P}X'$$ is called a parenthesized word if its parenthesization is balanced, i.e., it satisfies: i. $$s(\eta)_i \ge 0$$ for $$i=1,\ldots,n-1$$ and $$s(\eta)_n=0.$$ ii. $$q_{i}q_{i+1}\neq x_{i_1}x_{i_2}$$ for $$x_{i_1},x_{i_2}\in X$$ and $$i=1,\ldots,n-1$$. iii. $$q_{i}q_{i+1}\neq \lfloor\rfloor, \rfloor\lfloor$$ for $$i=1,\ldots,n-1$$. iv. $$q_1=\lfloor$$ and $$q_n=\rfloor$$ does not occur at the same time. v. There are no sub-words in $$\eta$$ of the form $$\xi \lfloor \nu \rfloor \kappa$$ or $$\lfloor \lfloor \xi \rfloor \rfloor$$ for $$\xi,\nu,\kappa\in \mathfrak{P}X'$$. Parenthesized words are such that $$x_i\lfloor x_j \rfloor \neq \lfloor x_i\rfloor x_j$$ for $$x_i,x_j\in X$$. (See Example 2.1 below for a list of such words.) The set of parenthesized words constitutes a free Magma (Bourbaki, 2005) under balanced parenthesization (Holtkamp, 2011; Melançon, 1992; Ebrahimi-Fard & Guo, 2008a,b). The set of parenthesized words including the empty word $$\emptyset$$ is denoted by $$\mathfrak{P}X^\ast$$. In Section 3, the operation of balanced parenthesization is identified with the grafting operation on trees given in Definition 3.2 since parathesized words correspond one-o-one with binary planar rooted trees (Holtkamp, 2011; Melançon, 1992). A formal power series in $$\mathfrak{P}X$$ is any mapping of the form $$\mathfrak{P}X^\ast\rightarrow \mathbb{R}^{\ell\times n}\!:\eta\mapsto (c,\eta)$$. The set of all such mappings will be denoted by , which forms an $$\mathbb{R}$$-vector space. An alternative to the parenthesization of words is to encode the order in which balanced parentheses appear by using two different products, denoted, say $$\prec$$ and $$\succ$$. For example, xi⌊xj⌋≡xi≺xj and ⌊xi⌋xj≡xi≻xj. (2.1) Using these products, the induced algebraic structure on is described next. Definition 2.2 A dendriform algebra$$D$$ is an $$\mathbb{R}$$-vector space endowed with products $$\prec$$ and $$\succ$$ such that for $$a,b,c\in D$$ the following axioms are satisfied: (a≺b)≺c =a≺(b≺c+b≻c), (2.2a) (a≻b)≺c =a≻(b≺c), (2.2b) a≻(b≻c) =(a≺b+a≻b)≻c. (2.2c) For a commutative dendriform algebra the following holds: $$a \succ b = b \prec a$$. This algebra is the so-called Zinbiel algebra. Similar to (2.1), for every $$\eta\in \mathfrak{P}X^\ast$$ there is a corresponding dendriform product defined recursively by the injection δ(η)={xi≺δ(η′),ifη=xi⌊η′⌋,δ(η′)≻xi,ifη=⌊η′⌋xi,δ(η′)≻xi≺δ(η″),ifη=⌊η′⌋xi⌊η″⌋, where $$x_i \in X$$; $$\eta',\eta''\in \mathfrak{P}X^\ast$$; $$\delta(\emptyset)=\emptyset$$; and $$\delta(x_j)=x_j$$ for all $$x_j\in X$$. For example, δ(xi⌊⌊xj⌋xk⌋)=xi≺(δ(⌊xj⌋xk))=xi≺(xj≻xk). Thus, the set of all dendriform products is given by $$\mathfrak{T}X^\ast=\delta(\mathfrak{P}X^\ast)$$, and any element of $$\mathfrak{T}X^\ast$$ is called a dendriform word. The $$\mathbb{R}$$-vector space formed by the span of these words is the dendriform algebra . Similarly to the case of parenthesized words, is generalized to matrices by replacing scalars with $${\ell\times n}$$ real matrices, in which case the set is denoted . Example 2.1 Let $$\eta=x_i x_j x_k \in X^\ast$$. The rules of parenthesization in Definition 2.1 generate the subset of $$\mathfrak{P}X^\ast$$ Lη ={xi⌊xj⌊xk⌋⌋, xi⌊⌊xj⌋xk⌋, ⌊xi⌊xj⌋⌋xk, ⌊⌊xi⌋xj⌋xk, ⌊xi⌋xj⌊xk⌋}. The word $$\lfloor x_ix_j\rfloor x_k \notin \mathfrak{P}X^\ast$$ since it does not satisfy item $$ii.$$ in Definition 2.1. The corresponding set of dendriform products under the mapping $$\delta$$ is the subset of $$\mathfrak{T}X^\ast$$ Lη′={xi≺(xj≺xk), xi≺(xj≻xk), (xi≺xj)≻xk, (xi≻xj)≻xk, xi≻xj≺xk}. Observe, for instance, that $$x_i\succ (x_j \succ x_k)\notin L_\eta'$$. This is due to the fact that it corresponds to $$\lfloor x_i \rfloor \lfloor x_j \rfloor x_k \notin \mathfrak{P}X^\ast$$, which is not a parenthesized word since $$\rfloor \lfloor$$ occurs. However, axiom (2.2c) gives xi≻(xj≻xk)=(xi≺xj)≻xk+(xi≻xj)≻xk, (2.3) where the summands on the right-hand side correspond to parenthesized words in $$L_\eta$$. An element of is a formal power series on dendriform words that can be viewed as a mapping $$c:\mathfrak{T}X^\ast\rightarrow \mathbb{R}^{\ell\times n}\!:\eta\mapsto (c,\eta)$$. The set of all series in having finite support is denoted by . In addition, any dendriform word can be mapped in the obvious way to the underlying word in $$X^\ast$$ by eparenthezation, expressed by the foliage map$$\varphi: \mathfrak{T}X^\ast \rightarrow X^\ast$$, which replaces both $$\prec$$ and $$\succ$$ with the operation of concatenation. For example, $$\varphi(x_i \prec( x_j \prec x_k )) = x_ix_jx_k \in X^\ast$$. Next define the product . This product is the non-commutative counterpart of the shuffle product, and it is extended bilinearly on . Lemma 2.1 (Loday & Ronco, 1998) is an associative $$\mathbb{R}$$-algebra. Note that a similar notion of a non-commutative shuffle product also appeared in control theory (Agrachev & Gamkrelidze, 1991). An important characteristic of the commutative shuffle product is that it can be defined recursively, which is convenient for computer implementations. For the non-commutative shuffle product such a recursive definition is only available when the words to be shuffled have length less or equal than one. In this regards, the notion of planar binary trees plays a key role as described next. 3. Trees, dendriform words and iterated integrals The objective of this section is to describe the one-to-one correspondence between planar binary trees and dendriform words. Then their relationship to non-commutative iterated integrals is described. The majority of concepts presented in this section can be found in Loday & Ronco (1998); Melançon (1992); Ebrahimi-Fard & Manchon (2014) and the references therein. 3.1 Trees and dendriform words A tree is an undirected connected graph made out of vertices and edges. It is without cycles, which amounts to saying that any two vertices can be connected by exactly one simple path. The sets of vertices and edges of a tree are denoted by $$V$$ and $$\Gamma$$, respectively. A planar binary tree is a finite oriented tree that consists of vertices and oriented edges and is given an embedding in the plane such that all vertices have exactly one incoming edge and two outgoing edges. An oriented edge can be internal, i.e., connecting two vertices, or it can be external, having one loose end. The external outgoing edges which do not end in vertices are the leaves. The root is the unique edge not starting in a vertex. The interior edges of a planar binary tree is the set $$\Gamma$$ minus the root and the leaf edges. Due to the embedding, one can label the $$n$$ leaves of a planar tree consecutively from left to right by $$1,2,...,n$$. A planar $$n$$-ary tree is a planar rooted tree where every vertex has exactly $$n$$ outgoing edges. The order of a tree is defined by its number of vertices. The set of all planar binary trees is denoted by $$\mathfrak{T}$$, and $$\mathfrak{T}_n$$ denotes the set of planar binary trees of order $$n$$, i.e., with $$n$$ vertices. The planar binary trees up to order three are: The single edge tree $$|$$ is known as the trivial tree. A mapping $$c: \mathfrak{T}\rightarrow \mathbb{R}^{\ell\times n}: \tau \mapsto (c,\tau)$$ can be written formally as c=∑τ∈T(c,τ)τ, where $$(c,\tau)$$ is the coefficient of $$c$$ at $$\tau$$. The set of all such mappings is denoted as . A well known fact about planar binary trees with $$n$$ vertices is that their cardinality $$\#(\mathfrak{T}_n)=C_n:=\frac{1}{n+1}\binom{2n}{n}$$, which is the $$n$$-th Catalan number. It is also known that the number of ways of associating $$n$$ applications of a binary operator (e.g., balanced parenthesization) is $$C_n$$. If trees are suitably decorated with a set of symbols, then there is a one-to-one correspondence between trees and dendriform words. Definition 3.1 Let $$V$$ be the set of vertices of tree $$\tau \in \mathfrak{T}$$ and $$D$$ a finite set of symbols. A decoration of $$\tau$$ is a map $$\rho: V \rightarrow D$$. An ordering for the vertices of a tree is introduced by letting vertex $$v_i \in V$$ be the vertex where the paths starting from leaves $$i$$ and $$i+1$$ join together. Note that this ordering is inherited from the left to right ordering of the leaves. This vertex ordering is assumed hereafter. Example 3.1 Let with its leaves labeled from $$1$$ to $$4$$, $$D=\{x,y,z\}$$ and $$V=\{v_1,v_2,v_3\}$$. Figure 1 shows one possible decoration of $$\tau$$ by $$\rho$$, where $$\rho$$ is given by $$\rho(v_1)=x$$, $$\rho(v_2)=y$$ and $$\rho(v_3)=z$$. Fig. 1. View largeDownload slide Tree decoration Fig. 1. View largeDownload slide Tree decoration In general, any $$\tau\in \mathfrak{T}_n$$ can be decorated by the letters in the word $$\eta=x_{i_1}\cdots x_{i_n}\in X^n$$ (here $$X^n$$ denotes the set of all words in $$X^\ast$$ of length $$n$$), e.g., $$\rho(v_j)=x_{i_j}$$. The set of all trees decorated by $$X^\ast$$ is denoted by $$\mathfrak{TD}X^\ast$$, $$\vert{\tau}\vert$$ is the order of $$\tau\in \mathfrak{TD}X^\ast$$, and the foliage of $$\tau$$ with vertices $$\{v_1,\ldots, v_n\}$$ is a mapping $$\psi:\mathfrak{TD}X^\ast \rightarrow X^\ast:\tau \mapsto \rho(v_1)\cdots \rho(v_n)$$, which assumes the ordering of the vertices as described previously. In the following, a tree and the word corresponding to its decoration and ordering of its vertices is combined by denoting $$(\cdot;\cdot) : X^\ast \times \mathfrak{T} \rightarrow \mathfrak{TD}X^\ast :(\eta;\tau) \mapsto\tau_\eta$$. The notation $$\tau_\eta$$ makes explicit the fact that a tree $$\tau \in \mathfrak{T}$$ is being decorated by the word $$\eta \in X^\ast$$ according to the order of its vertices. For example, A formal power series on decorated trees is any mapping $$c:\mathfrak{TD}X^\ast\rightarrow \mathbb{R}^{\ell\times n}\!: {{\tau_\eta}} \mapsto (c,\eta)$$. The $$\mathbb{R}$$-vector space of formal power series on decorated trees is denoted by . The subspace of series with finite support is . In this context, the decoration of trees is a bilinear operation. One way of constructing new trees out of a set of trees (usually called a forest) is by the operation of grafting. Definition 3.2 The grafting of trees is an $$n$$-ary operation $$\vee$$ consisting of joining together $$n$$ trees to the same root edge to form a new tree. More precisely, $$ \vee : \underbrace{\mathfrak{T} \times \cdots \times \mathfrak{T}}_{\displaystyle \mbox{$n$ times}} \rightarrow \mathfrak{T}$$ such that Simply put, grafting is for trees what catenation is for the free monoid $$X^\ast$$. For example, Observe that if $$\tau = \vee(\tau^1 \cdots \tau^m) \in\mathfrak{T}_n$$, then each component tree $$\tau^{i}$$ belongs to $$\mathfrak{T}_{m_i}$$, and $$\sum_{i=1}^m m_i=n-1$$. In this article, the focus is on binary grafting, i.e., $$m=2$$. Any planar binary tree can be decomposed uniquely as $$\tau=\tau^1\vee \tau^2$$ at the vertex where the root edge ends. Recall that by definition, any vertex of a planar binary tree is trivalent (one incoming edge and two outgoing ones). Other tree decompositions such as the ones used in Hopf algebras of trees are non-unique (Holtkamp, 2011). The tree $$\tau^1$$ (respectively $$\tau^2$$) is the left part (respectively the right part) of $$\tau$$. Further decompositions allow one to write any planar binary tree in terms of the trivial singe edge tree $$|$$. The grafting operation $$\vee$$ makes $$\mathfrak{T}$$ the free magma algebra with one generator, which is isomorphic to the free magma of parenthesized words and grafting corresponds to the operation of balance parenthesization through the mapping $$({\it\Phi}\circ \delta)^{-1}$$, where $$\delta: \mathfrak{P}X^\ast\rightarrow \mathfrak{T}X^\ast$$ and $${\it\Phi}: \mathfrak{T}X^\ast \rightarrow \mathfrak{TD}X^\ast$$. It is neither commutative nor associative. The order of the grafting of trees is the sum of the orders of all component trees plus one. That is, for two trees $$\tau^1$$, $$\tau^2$$ of order $$n_1$$, $$n_2$$, respectively, the product $$\tau^1 \vee \tau^2$$ is of order $$n_1 + n_2 + 1$$. Particular types of trees that allow an easy decomposition are the so-called right-combs andleft-combs as shown in Fig. 2. Fig. 2. View largeDownload slide $$a)$$ left-comb, $$b)$$ right-comb. Fig. 2. View largeDownload slide $$a)$$ left-comb, $$b)$$ right-comb. Clearly for a right-comb (respectively left-comb) $$\mathfrak{r}_n=\mathfrak{r}_{n-1}\vee |$$ (respectively $$\mathfrak{l}_{n}=|\vee \mathfrak{l}_{n-1}$$), where $$\mathfrak{r}^k$$ denotes the $$k$$-th order right-comb (respectively $$\mathfrak{l}_k$$ denotes the $$k$$-th order left-comb). One way of realizing the decoration of a tree is by attaching a letter from the alphabet $$X$$ to every grafting operation used in the construction, say $$\vee_{x_i}$$. For example, $$ $$ (3.1a) The grafting operation provides an explicit description of the correspondence between the sets $$\mathfrak{T}X^\ast$$ and $$\mathfrak{TD}X^\ast$$, namely, the isomorphism $${\it\Phi}:\mathfrak{T}X^\ast \rightarrow \mathfrak{TD}X^\ast$$ with inductive definition Φ(ητ)={|∨xiΦ(η′),ifητ=xi≺ητ′′,Φ(η′)∨xi|,ifητ=ητ′′≻xi,Φ(η′)∨xiΦ(η″),ifητ=ητ′′≻xi≺ητ″″, where $$x_i\in X$$, $$\eta'_{\tau'},\eta''_{\tau''}\in \mathfrak{T}X^\ast$$, $${\it\Phi}(\emptyset):=|$$, and $${\it\Phi}(x_j):=| \;\vee_{x_j} |$$ for $$x_j\in X$$. This correspondence up to order three is: In Melançon (1992), the free magma $$\mathfrak{T}X^\ast$$ is defined directly as the set of all planar binary trees whose leaves are decorated with letters in $$X$$. Any $$\eta\in\mathfrak{T}X^\ast$$ will be denoted as $$\eta_{\tau}$$, where it is made explicit the fact that for any dendriform word there exist a decorated tree $${\tau}\in\mathfrak{TD}X^\ast$$ providing the order in which the products $$\prec$$ and $$\succ$$ appear. The corresponding tree is then obtained as $${\it\Phi}(\eta_\tau)=\tau_\eta \in \mathfrak{TD}X^\ast$$, and its inverse satisfies $${\it\Phi}^{-1}(\tau_\eta)=\eta_\tau \in \mathfrak{T}X^\ast$$. Moreover, the foliage of $$\tau_\eta$$ can be written in terms of the map $$\varphi$$ of dendriform words as $$\psi(\tau_\eta) = \varphi({\it\Phi}^{-1}(\tau_\eta))=\eta \in X^\ast$$. The isomorphism $${\it\Phi}$$ is extended linearly over . As in Example 2.1, the axioms in (2.2) allow one to write for $$\ell=n=1$$ which follows directly from (2.3). Hereafter, due to the isomorphism between $$\mathfrak{T}X^\ast$$ and $$\mathfrak{TD}X^\ast$$, no notational distinction will be made between products in $$\mathfrak{T}X^\ast$$ and products in $$\mathfrak{TD}X^\ast$$. The grafting operation allows one to define the shuffle product of trees in an inductive manner. Definition 3.3 (Loday & Ronco, 1998) Let $$\tau^1= \tau^{11}\vee_{x_i} \tau^{12}$$ and $$\tau^2=\tau^{21}\vee_{x_j} \tau^{22}$$ with $$\tau^{k1},\tau^{k2} \in \mathfrak{TD}X^\ast$$. The recursive definition of is $$ $$ (3.2) where for any $$\tau\in \mathfrak{TD}X^\ast$$. Example 3.2 The shuffle product Similarly, the shuffle product Lemma 3.1 (Loday & Ronco, 1998) The shuffle product of trees is non-commutative and associative. Proof The non-commutative nature of is obvious from Example 3.2. The associativity follows directly from Lemma 2.1 and by applying the mapping $${\it\Phi}$$. ■ The dendriform products for trees are given next. Definition 3.4 (Loday & Ronco, 1998) For $$\tau^1= \tau^{11}\vee_{x_i} \tau^{12}$$ and $$\tau^2= \tau^{21}\vee_{x_j} \tau^{22}$$, the dendriform products in terms of the grafting product are: $$ $$ (3.3a) $$ $$ (3.3b) The following theorem shows that the dendriform products of trees satisfy (2.2). Theorem 3.3 (Loday & Ronco, 1998) The products $$\prec$$ and $$\succ$$ in (3.3) satisfy the axioms of dendriform algebras given in (2.2). Proof Let $$\tau^1= \tau^{11}\vee_{x_i} \tau^{12}$$, $$\tau^2= \tau^{21}\vee_{x_j} \tau^{22}$$ and $$\tau^3= \tau^{31}\vee_{x_k} \tau^{32}$$. From (3.3a) and since is associative, one has that Applying (3.3a) once more gives (τ1≺τ2)≺τ3=τ1≺(τ2≺τ3)+τ1≺(τ2≻τ3), which is axiom (2.2a). Similarly, applying (3.3a) and (3.3b) provides which is axiom (2.2b). The third axiom follows from Applying (3.3b) again gives τ1≻(τ2≻τ3)=(τ1≺τ2)≻τ3+(τ1≻τ2)≻τ3. ■ From Theorem 3.3, (3.2) and (3.3), it is clear that as defined in Section 2, and therefore, with the help of the mapping $${\it\Phi}$$, the relationship between the dendriform and shuffle products on $$\mathfrak{T}X^\ast$$ and $$\mathfrak{TD}X^\ast$$ is: This subsection ends with three lemmas which are employed in subsection 3.2 and section 5 to characterize the addition of trees related to non-commutative iterated integrals in terms of non-commutative shuffle products. First, the sum $$\mbox{char}(\mathfrak{T}_{n})$$ of all order $$n$$ trees is defined. Definition 3.5 The characteristic series of the set $$\mathfrak{T}_{n}$$ is defined as char(Tn):=∑τ∈Tnτ. Lemma 3.2 For any $$n\ge 0$$, char(Tn+1)=∑i=0nchar(Tn−i)∨char(Ti). (3.4) Proof Recall that the number of planar trees of order $$n$$ is the $$n$$-th Catalan number, i.e., $$\#\mathfrak{T}_{n}=C_n$$. Since the grafting operation is non-commutative and provides a unique decomposition of planar binary trees, one can prove the claim by showing that the right hand side of (3.4) produces a number of summands equal to the $$(n+1)$$-th Catalan number. First, note that the grafting operation does not generate extra trees in the sense that #supp(∑i=1n1τ1,i∨∑j=1n2τ2,j) =n1n2, which is the product of the number of trees in the left-hand side and right-hand side of the grafting operation. It then follows that #supp(char(Tn+1)) =∑i=0n#supp(char(Tn−i)∨char(Ti)) =∑i=0nCn−iCi=Cn+1, which is Segner’s recurrence relation for the $$(n+1)$$-th Catalan number (Segner, 1758). ■ The collection of all trees of a certain order can be described in terms of the non-commutative shuffle product. Lemma 3.3 The sum of all undecorated trees of order $$n\ge 0$$ is given by $$ $$ (3.5) where and . Note that in Lemma 3.3 is defined recursively by noncommutative shuffles always on the left, however the lemma also holds if the noncommutative shuffles are applied solely on the right. Proof The proof is done by induction on the number of shuffles. For $$n=0,1$$, the identity holds trivially. Assume now that (3.5) holds up to some fixed $$n > 1$$. Using Lemma 3.2 and the associativity of , it follows that Given that , and using the induction hypothesis, the last summand above is (∑i=0n−1char(Ti)∨char(Tn−1−i))∨| t=char(Tn)∨char(T0). Thus, which completes the proof. ■ The definition of a tree descent is needed for the last lemma. Definition 3.6 (Chapoton, 2009) A leaf of $$\tau\in \mathfrak{T}$$ is a descent if it is not the leftmost leaf and if it is pointing to the left. The descent mapping $$\mathrm{des}: \mathfrak{T} \rightarrow \mathbb{N}$$ gives the number of descents in $$\tau$$. Note that in the following example a leaf which is a descent is indicated by a thickened leaf edge. Example 3.4 The numbers of descents for some trees are: Lemma 3.4 (Chapoton, 2009; Ebrahimi-Fard & Manchon, 2014) For the proper left-comb characteristic series $$c = \sum_{n =1}^\infty \mathfrak{l}_n$$, it follows that where $$ $$ (3.6) and (c,τ)=(−1)des(τ)n(n−1des(τ))forτ∈Tn. The reader is referred to (Ebrahimi-Fard & Manchon, 2014) for a concise proof of the previous lemma in terms of rooted trees. Rooted trees and planar binary trees are related via Knuth’s correspondence, and the number of descents corresponds to the number of leaves in the corresponding rooted tree minus one. 3.2 Non-commutative iterated integrals For a matrix-valued measurable function $$u : [0,T] \rightarrow \mathbb{R}^{n\times q}$$, define $$\Vert{u}\Vert_{L_1} = \int_{0}^{T} \Vert{u(s)}\Vert_1 \,ds$$, where $$\Vert{u(s)}\Vert_1:=\max_j \{\sum_{i} \vert{u(s)_{ij}}\}\vert$$. If instead $$u=(u_1,\ldots,u_m)$$, where each $$u_i:[0, T] \rightarrow \mathbb{R}^{n\times q}$$, then $$\Vert{u}\Vert:= \max_i \Vert{u_i}\Vert_{L_1}$$. The set $$L^{m\times(n\times q)}_1[0,T]$$ contains all measurable functions defined on $$[0,T]$$ having finite $$\Vert{\cdot}\Vert$$ norm, and $$B^{m\times (n\times q)}_1(R)[0,T]:= \{u\in L^{m\times (n\times q)}_1[0,T], \Vert{u}\Vert\le R \}$$. In addition define $$u_0=I$$ with $$I$$ denoting the identity matrix. Definition 3.7 Let $$u\in B^{m\times (n\times n)}_1(R)[0,T]$$. The non-commutative iterated integral corresponding to $$\eta_\tau \in \mathfrak{T}X^\ast$$ for $$t\in [0,T]$$ is defined inductively by $$E_{\emptyset}[u]=I$$, and Eητ[u](t)=∫0tEξτ1[u](s)ui(s)Eντ2[u](s)ds, where $$\eta_\tau={\xi}_{\tau^1}\vee_{x_i}{\nu}_{\tau^2}$$, and for any $$x_i\in X$$, $${\xi}_{\tau^1},{\nu}_{\tau^2}\in \mathfrak{T}X^\ast$$, and $$\tau^1, \tau^2 \in \mathfrak{T}$$. The mapping $$E_{\eta_{\tau}}$$ is extended linearly on in the natural way. Example 3.5 From (3.1) and the mapping $${\it\Phi}$$, the iterated integrals up to order three are: Exi[u](t)=∫0tui(s)dsExi≺xj[u](t)=∫0tui(s1)∫0s1uj(s2)ds2ds1Exi≻xj[u](t)=∫0t(∫0s1ui(s2)ds2)uj(s1)ds1Exi≺(xj≺xk)[u](t)=∫0tui(s1)∫0s1uj(s2)∫0s2uk(s3)ds3ds2ds1Exi≻(xj≺xk)[u](t)=E(xi≻xj)≺xk[u](t)=∫0t(∫0sui(s1)ds1)uj(s)(∫0suk(s2)ds2)dsExi≺(xj≻xk)[u](t)=∫0tui(s1)∫0s1(∫0s2uj(s3)ds3)uk(s2)ds2ds1E(xi≺xj)≻xk[u](t)=∫0t(∫0s1ui(s2)∫0s2uj(s3)ds3ds2)uk(s1)ds1E(xi≻xj)≻xk[u](t)=∫0t(∫0s1(∫0s2ui(s3)ds3)uj(s2)ds2)uk(s1)ds1. For a planar binary tree $$\tau=\tau^1 \vee \tau^2 \in \mathfrak{T}$$, the tree factorial is defined as γ(τ)=(|τ1|+|τ2|+1)γ(τ1)γ(τ2), where $$\gamma(|)=1$$ (Butcher, 2008). For instance, the tree factorial of the $$n$$-th order left-comb $$\mathfrak{l}_{n}$$ is γ(ln)=γ(|∨ln−1)=nγ(ln−1). Repeating the procedure $$n$$ times one arrives at $$\gamma(\mathfrak{l}_{n})=n!$$. Thus, the standard factorial is a special case of the tree factorial. An analogous procedure applies for right-comb trees. The next three lemmas and theorem are of central importance to the main results of the article given in Section 4. The first lemma provides bounds for particular types of non-commutative iterated integrals. Hereafter consider $$\bar{u}_i(s):=\Vert{u_i(s)}\Vert_1$$ for $$u_i$$ the $$i$$th component of $$u\in B^{m\times (n\times n)}_1(R)[0,T]$$. Lemma 3.5 Let $$\tau$$ be an arbitrary tree in $$\mathfrak{T}_n$$, $$\mathfrak{l}_{n}$$ the left-comb tree in $$\mathfrak{T}_n$$, $$x_i\in X$$ and $$\eta\in X^n$$. Then the following bounds apply: $$i$$. $$\Vert{E_{(x_i^n)_{\tau}}[u](t)}\Vert_1 \le\frac{\displaystyle \bar{U}_i^{\vert{\tau}\vert}(t)}{\displaystyle \gamma(\tau)}$$, $$ii$$. $$\displaystyle \Vert{E_{\eta_{\mathfrak{l}_{n}}}[u](t)}\Vert_1 \le \prod_{j=1}^n \frac{\displaystyle \bar{U}_j^{n_j}(t)}{\displaystyle n_j !}$$, where $$\bar{U}_j(t) := \int_0^t\bar{u}_j(s) \,ds$$, $$n_j=\vert{\eta}\vert_{x_j}$$ for $$j=0,\ldots,m$$, and $$\vert{\eta}\vert_{x_j}$$ denotes the number of times the letter $$x_j$$ appears in the word $$\eta\in X^\ast$$. Proof The bound in item $$i.$$ is proved by induction on $$n$$. The $$n=0, 1$$ cases are straightforward. Assume the bound holds up to some $$n-1\ge 0$$. Let $$\tau = \tau^1 \vee \tau^2$$ with $$\tau^1\in \mathfrak{T}_k$$ and $$\tau^2 \in \mathfrak{T}_{n-k-1}$$ for $$0\le k \le n-1$$. Then ‖E(xin)‖τ[u](t)‖1 ≤∫0t‖E(xi|τ1|)τ1[u](s)‖1‖ui(s)‖1‖E(xi|τ2|)τ2[u](s)‖1ds ≤∫0t‖ui(s)1U¯‖i|τ1|γ(τ1)U¯i|τ2|γ(τ2)ds =∫0t‖ui(s)‖1U¯i|τ1|+|τ2|γ(τ1)γ(τ2)ds =U¯i|τ1|+|τ2|+1(|τ1|+|τ2|+1)γ(τ1)γ(τ2)=U¯i|τ|γ(τ). Thus, the bound in $$i.$$ holds for all $$n\ge 0$$. The bound in $$ii.$$ is also proved by induction on $$\vert{\mathfrak{l}_{n}}\vert=n$$. The $$n=0,1$$ cases are also evident. Let $$\eta=x_i\eta'$$ with $$\eta'\in X^{n-1}$$, and recall $$\mathfrak{l}_{n}= | \vee \mathfrak{l}_{n-1}$$ with $$\mathfrak{l}_{n-1}$$ the $$(n-1)$$-th left-comb. If the bound in $$ii.$$ holds for $$n-1$$, then the bound for the iterated integral $$E_{\eta_{\mathfrak{l}_{n}}}[u](t)$$ is computed as ‖Eηln[u](t)‖1 ≤∫0t‖ui(s)‖1‖Eηln−1′[u](s)‖1ds ≤∫0t‖ui(s)‖1U¯1n1′(s)⋯U¯mnm′(s)n1′!⋯nm′!ds ≤∏j=1j≠imU¯jnj′(t)nj′!∫0t‖ui(s)‖1U¯ini′(s)ni′!ds =U¯ini′+1(ni′+1)!∏j=1j≠imU¯jnj′(t)nj′!=∏j=1mU¯jnj(t)nj!, where $$n_i=n'_i+1$$ and $$n_j=n'_j$$ for $$j\neq i$$. So the bound in $$ii.$$ applies for all $$n\ge 0$$. ■ The following lemma provides a relationship between the norms of commutative and non-commutative iterated integrals. It plays a key role in the convergence of dendriform Fliess operators. Lemma 3.6 Let $$\eta_\tau\in \mathfrak{T}X^\ast$$, and $$u\in B^{m\times(n\times n)}_1(R)[0,T]$$. The iterated integral corresponding to $$\eta_\tau$$ satisfies ‖Eητ[u](t)‖1≤Eητ[u¯](t),∀t∈[0,T], where $$\bar{u}=(\bar{u}_1, \ldots, \bar{u}_m)^T$$. Proof The lemma is proved by induction on $$\vert{\eta_\tau}\vert=k$$. The result is trivial for $$k=0$$. For $$k=1$$, ‖Exi[u](t)‖1 =maxj∑l=1n|∫0t(ui)lj(s)ds| ≤maxj∑l=1n∫0t|(ui)lj(s)|ds ≤∫0tmaxj∑l=1n|(ui)lj(s)|ds =∫0t‖ui(s)‖1⏟u¯i(s)ds=Exi[u¯](t), where $$\bar{u}_{i}(t) \ge 0$$ is now scalar-valued, i.e., $$\bar{u}$$ is commutative. If the claim holds up to $$|{\tau}|=k$$ for $$\eta_\tau= {\it\Phi}^{-1}(\tau^1_\xi \vee_{x_i} \tau^2_{\nu})$$ with $$|\tau^1_\xi|=k_1$$, $$|\tau^2_\nu|=k_2$$, and $$k_1+k_2=k-1$$, then ‖Eητ[u](t)‖1 ≤∫0t‖Eξτ1[u](s)‖1‖ui(s)‖1‖Eντ2[u](s)‖1ds ≤∫0tEξτ1[u¯](s)u¯i(s)Eντ2[u¯](s)ds =Eητ[u¯](t). Thus the bound holds for all $$k\ge 0$$. ■ It is important to note that even though the components of $$\bar{u}$$ are mutually commutative, the corresponding iterated integrals do not coincide with the fully commutative case where one removes all the ordering provided by trees. Example 3.6 Let $$\eta=x_ix_jx_k$$ for $$i\neq j \neq k$$ pairwise distinct and . Then it follows that E(xixjxk)τ[u¯](t) =∫0t(∫0su¯i(r)dr)u¯j(s)(∫0su¯k(r)dr)ds =∫0tu¯j(s)(∫0su¯i(r)dr)(∫0su¯k(r)dr)ds =Exjxixk[u¯](t)+Exjxkxi[u¯](t), where $$E_{x_jx_ix_k}[\bar{u}](t)$$ and $$E_{x_jx_kx_i}[\bar{u}](t)$$ are commutative iterated integrals distinct from $$E_\eta[\bar{u}](t)$$. The isomorphism between the commutative shuffle product, denoted , and the product of commutative iterated integrals generalizes in the non-commutative setting as follows. Theorem 3.7 Let $$u\in B^{m\times(n\times n)}_1(R)[0,T]$$ and $${\eta}_{\tau^1},{\xi}_{\tau^2}\in \mathfrak{T}X^\ast$$. Then $$ $$ (3.7) Proof Recall that the decorated tree corresponding to $$\eta_{\tau^i}$$ is $$\tau^i_{\eta}={\it\Phi}(\eta_{\tau^i})$$. Identity (3.7) is proved by induction on $$|\tau^1_\eta|+|\tau^2_\xi|=n$$. The claim is trivial for $$n=0,1$$ since $$E_{\emptyset}[u]=I$$ and by definition . Assume (3.7) holds up to some fixed $$n\ge 1$$. If $$\tau^{1}_\eta=\tau^{11}_{\eta_1}\vee_{x_i} \tau^{12}_{\eta_2}$$ and $$\tau^{2}_{\xi}=\tau^{21}_{\xi_1}\vee_{x_j}\tau^{22}_{\xi_2}$$ with $$|\tau^1_\eta|+|\tau^2_{\xi}|=n+1$$, then which completes the proof. ■ The final two lemmas in this section come from grouping trees conveniently. Lemma 3.7 Let and $$\eta=x_{i_1}\cdots x_{i_n}\in X^\ast$$. It then follows that $$ $$ (3.8) Proof Given that $$\eta$$ is fixed, the decoration of all trees on the left hand side of (3.8) is the same. Therefore, applying Lemma 3.3, it follows that Note that the ordering of the interior vertices follows the order in which the non-commutative shuffle products are taken. One can confirm this claim using the non-commutative shuffle product definition in terms of the grafting operation, and the fact that the grafting operation always produces a vertex at the point where the paths starting from leaves $$i$$ and $$i+1$$ join together. Hence, which completes the proof. ■ Lemma 3.8 Let and $$\bar{u}(s)=\Vert{u(s)}\Vert_1$$. The following identity holds $$ $$ (3.9) Proof For brevity define and recall that . From Lemma (3.3a), one has that From Theorem 3.7 and the fact that $$\bar{u}_i$$ commutes with $$\bar{u}_j$$ for any $$i$$ and $$j$$, one can replace the non-commutative shuffle product by its commutative counterpart. Hence, Finally, one only needs to apply the well-known identity to obtain (3.9). This completes the proof. ■ Note that the iterated integral on the right-hand side of (3.9) is commutative. Lemma 3.8 also implies that if the inputs are assumed to commute then produces $$n!$$ identical commutative iterated integrals. For example, if $$n=3$$ then the set of order $$3$$ trees is which obviously has $$C_3=5$$ elements. Recall that a decorated tree can be denoted as $$\tau_\eta $$ and that the corresponding dendriform word is $${\it\Phi}^{-1}(\tau_{\eta})=\eta_\tau$$. Therefore, decorating all elements of $$\mathfrak{T}_3$$ with $$\eta=x_i^3$$ corresponds to dendriform words $$(x_i^3)_{\tau^j}$$ for $$j=1,\ldots,5$$. Now observe that $$(x_i^3)_{\tau^3}$$ identifies the iterated integral E(xi3)τ3[u](t)=∫0t(∫0sui(r)dr)ui(s)(∫0sui(r)dr)ds. After applying Lemma 3.6, one notes that the outermost integral contains a product of single integrals. That is, E(xi3)τ3[u¯](t) =∫0tu¯i(s)(∫0su¯i(r)dr)(∫0su¯i(r)dr)ds =2∫0tu¯i(s)∫0su¯i(r)∫0ru¯i(p)dpdrds=2Exi3[u¯](t). (3.10) The latter iterated integral in (3.10) agrees with the commutative iterated integrals in (1.1). The iterated integrals associated to the dendriform words $$(x_i^3)_{\tau^1},(x_i^3)_{\tau^2},(x_i^3)_{\tau^4}$$ and $$(x_i^3)_{\tau^5}$$ do not contain products of integrals once $$u_i$$ has been replaced with $$\bar{u}_i$$ using Lemma 3.8. Thus, they correspond to four identical commutative} iterated integrals, which when added to (3.10) gives For $$n=4 (C_4=14)$$, there are six iterated integrals containing products of integrals, and 8 iterated integrals that do not contain products. After applying Lemma 3.6 to the 6 iterated integrals containing products, they produce a total of 16 commutative iterated integrals that together with the other 8 iterated integrals amounts to $$4!$$ identical commutative iterated integrals. Recall Lemma 3.6 provides a way to relate certain bounds of non-commutative Fliess operators to those for associated commutative Fliess operators via the functions $$\bar{u}_i(s):=\Vert{u_i(s)}\Vert_1$$, where $$u_i$$ is defined as the $$i$$th component of $$u\in B^{m\times (n\times n)}_1(R)[0,T]$$. Since in the commutative setting the condition $$x_i \prec x_j = x_j \succ x_i$$ holds, this greatly simplifies the computation of bound estimates for noncommutative Fliess operators. 4. Dendriform Fliess operators and their convergence In this section, dendriform Fliess operators are defined, and sufficient conditions for their convergence are provided. 4.1 Dendriform Fliess operators The definition of a dendriform Fliess operator is given first. Definition 4.1 Let $$u\in B^{m\times (n \times n)}_1(R)[0,T]$$ and . A dendriform Fliess operator with generating series $$c$$ is defined by the following summation Fc[u](t)=∑ητ∈TX∗(c,ητ)Eητ[u](t). The operator in (1.5) is a special case of a dendriform Fliess operator. The support of its generating series contains only left-comb trees. This is purely a consequence of the iterative procedure used to derive it. However, defining Fliess operators as a summation comprised of only left-comb trees limits its application as shown in the next example. Example 4.1 Suppose two Fliess operators $$F_c$$ and $$F_d$$ have generating series containing only terms corresponding to left-comb trees. Assume $$c=c'I$$ and $$d=d'I$$ in with $$c',d'$$ being scalar-valued series, and $$u\in B^{1\times (n \times n)}_1(R)[0,T]$$ for some $$R,T > 0$$. Since $$\sum_{\eta_\tau\in \mathfrak{T}X^\ast} \eta_\tau = \sum_{\eta\in X^\ast} \sum_{\tau\in \mathfrak{T}} \eta_\tau $$, their product connection as shown in Fig. 3 is described by Fc[u]Fd[u]=∑n1,n2=0∞∑η∈X n2,ξ∈X n1(c,ηln1)(d,ξln2)Eηln1[u]Eξln2[u]. Recall from Theorem 3.7 that where generates more than just left-comb trees as shown in Example 3.2. At this stage characterizing system interconnections remains an open problem due to its highly non-commutative nature. One possible approach is to introduce an explicit decoupling of the system parameters and corresponding input. That is, Fc[u](t):=∑ητ∈TX∗(c,ητ)⊗Eητ[u](t), where $$\otimes$$ denotes a bilinear decoupling of $$(c,\eta_{\tau})$$ and $$E_{\eta_\tau}[u](t)$$. This decoupling occurs, for instance, in the case of systems evolving in the quantum stochastic process setting. Namely, the system and its inputs evolve in a tensor product space in which the inputs commute with the system’s vector fields under the condition of being affiliated (Parthasarathy, 1992), which in simple words is the quantum counterpart of the standard property of stochastic processes being adapted to an underlying $$\sigma$$-algebra. Fig. 3. View largeDownload slide Product connection of Fliess operators Fig. 3. View largeDownload slide Product connection of Fliess operators 4.2 Convergence of dendriform Fliess operators The next theorem addresses the convergence of dendriform Fliess operators by considering bounds on the coefficients of the corresponding generating series. The results in Section 3 were specifically developed for proving this theorem. Theorem 4.2 Let with coefficients satisfying the growth condition ‖(c,ητ)‖1≤KM|τ|,∀ητ∈TX∗. (4.1) Then there exist $$R,T>0$$ such that for each $$u \in B^{m\times(n\times n)}_1(R)[0,T]$$ the series y(t)=Fc[u](t)=∑ητ∈TX∗(c,ητ)Eητ[u](t) converges absolutely and uniformly on $$[0,T]$$. Proof: Fix some $$T>0$$. Pick $$u\in L^{m\times(n\times n)}_1[0,T]$$ and let $$R:= \max\{\Vert{u},T\}\Vert$$. Since the summation over dendriform words can be decomposed into the summations over words in $$X^\ast$$ (decorations) and the summation over trees, define for any $$k \ge 0$$ the finite sum ak(t)=∑η∈Xk∑τ∈Tk(c,ητ)Eητ[u](t). Using (4.1) and Lemma 3.6, an upper bound for $$a_k(t)$$ is computed as ‖ak(t)‖1 =‖∑η∈Xk∑τ∈Tk(c,ητ)Eητ[u](t)‖ ≤∑η∈Xk‖(c,ητ)‖1∑τ∈Tk‖Eητ[u](t)‖ ≤KMk∑η∈Xk∑τ∈TkEητ[u¯](t). From Theorem 3.7, Lemma 3.3 and the commutativity of $$\bar{u}$$, one has that Lemma 3.8 in tree terminology amounts to This is also equivalent to Continuing the analysis, where $$E_{x_i}[\bar{u}](t) = \bar{U}_i(t)\le \Vert{u}\Vert\le R$$. It is now clear that ∑k=0∞‖ak(t)‖≤∑k=0∞K(MR(m+1))k. Therefore, $$F_c[u](t)$$ converges absolutely and uniformly on $$[0,T]$$ when R<1M(m+1). (4.2) ■ Coefficients bounded as in (4.1) ensure convergence of a local nature since (4.2) conveys a restriction on both $$T$$ and the norm of $$u$$. In the commutative case, however, bound (4.1) provides global convergence in the sense that $$T$$ and $$u$$ can be arbitrarily chosen (Gray & Wang, 2002). The reason for this discrepancy is that in addition to summing over all possible permutations of letters in $$X$$, as in the commutative case, the non-commutative case also requires one to sum over all trees. When applying Lemma 3.6, summing over all trees with the same decoration yields a factorial factor as shown in Lemma 3.8. This is consistent with the bound (1.2) required for the local convergence in the commutative case. A left-comb dendriform Fliess operator is a dendriform Fliess operator whose generating series has support containing only dendriform words corresponding to left-combs. The convergence of such operators is addressed in the next theorem. Theorem 4.3 Let with coefficients satisfying the growth condition ‖(c,ητ)‖1≤KM|τ||τ|!,∀ητ∈TX∗ for some constants $$K,M>0$$ and $$\mathrm{supp}(c)=\{\eta_\tau \in \mathfrak{T}X^\ast, \tau = \tau^k_l,k>0\}$$. Then there exist $$R,T>0$$ such that for each $$u \in B^{m\times(n\times n)}_1(R)[0,T]$$ the series y(t)=Fc[u](t)=∑k=0∞∑η∈Xk(c,ητk)Eητk[u](t) (4.3) converges absolutely and uniformly on $$[0,T]$$. Proof: Fix some $$T>0$$. Pick $$u\in L^{m\times(n\times n)}_1[0,T]$$ and let $$R:= \max\{\Vert{u}\Vert,T\}$$. Define ak(t)=∑η∈Xk(c,ηlk)Eηlk[u](t). Using (4.1) and Lemma 3.6, a bound for $$a_k(t)$$ is computed as ‖ak(t)‖1=‖∑η∈Xk(c,ηlk)Eηlk[u](t)‖1≤∑η∈Xk‖(c,ηlk)‖1‖Eηlk[u](t)‖1≤KMkk!∑η∈XkEηlk[u¯](t). Since $$\mathfrak{l}_k$$ is a left-comb, observe that $$E_{\eta_{\mathfrak{l}_k}}[\bar{u}](t) = E_{\eta}[\bar{u}](t)$$ with the right-hand side being a commutative iterated integral. In which case, This means that one can proceed analogously as in the commutative case. Specifically, where the identity has been used. It then follows directly that ∑k=0∞‖ak(t)‖ ≤K∑k=0∞(MR)k∑α0+⋯+αm=kk!α0!⋯αm! =K∑k=0∞(MR(m+1))k. Therefore, $$F_c[u](t)$$ converges absolutely and uniformly on $$[0,T]$$ when $$R < \frac{1}{M(m+1)}$$. ■ 5. A solution for system $$\dot{Z}=u Z$$ in the dendriform Fliess operator setting Consider with $$(x_1^k)_{\mathfrak{l}_k} := {\it\Phi}((x_1^k;\mathfrak{l}_k)) \in \mathfrak{T}X^k$$. This series is the generating series corresponding to (1.5), the solution of (1.4). Recall that (1.4) can represent the evolution of a closed quantum system (with all quantum constants normalized to $$1$$). In the commutative case, it is known that $$Z(t)=\exp({\it\Omega}(t))$$, where $${\it\Omega}(t)=\int_0^t u(s)\,ds$$. From the Fliess operator point of view, Z(t)=Fc[u](t)=∑k=0∞Ex1k[u](t)=∑k=0∞(Ex1[u](t))kk!=exp(Ex1[u](t)), (5.1) where obviously $$ E_{x_1}[u](t)={\it\Omega}(t)$$. Suppose now that $$u$$ is non-commutative. Then the analogous expression is Fc[u](t)=∑k=0∞E(x1k)lk[u](t), (5.2) which by Theorem 4.3 with $$K=M=1$$ is a well defined operator. Next assume that $$F_c[u](t)$$ has an exponential representation similar to the commutative case. That is, $$F_c[u](t)=\exp({\it\Omega}(t))$$ with $${\it\Omega}(t)= F_d[u](t)$$ for some . Unfortunately, the identities used to obtain (5.1) cannot be used to find an expression for $$d$$. But Lemma 3.3 provides an inductive way for its computation. Starting with the ansatz $${\it\Omega}(t) := {\it\Omega}_1(t):=E_{(x_1)_{\mathfrak{l}_1}}[u](t)$$, and expanding $$\exp({\it\Omega}_1(t))$$ gives Observe that the expansion produces more terms than needed since the result does not coincide with (5.2). The use of a correction term is introduced in order to cancel the unwanted second order terms, i.e., the term that is not a left comb. Therefore, redefining $${\it\Omega}(t)$$ by modifying it as It follows then that the first and second order terms are The dendriform approach allows one to explicitly write the second-order correction term of $${\it\Omega}(t)$$ as In fact, define the product so that Ex1▹x1[u](t) =Ex1≺x1[u](t)−Ex1≻x1[u](t) =∫0t[u(s),∫0su(r)dr]ds, where $$[\cdot,\cdot]$$ denotes the usual commutator. The non-associative product $$\triangleright$$ is an example of a so-called pre-Lie product (Ebrahimi-Fard & Manchon, 2009a). It is also known as the chronological product in control theory (Agrachev & Gamkrelidze, 1978) and is characterized by the pre-Lie identity, which implies, among others, that the Lie bracket $$[x,y]_\triangleright := x \triangleright y - y \triangleright x$$ satisfies the Jacobi identity. (For more details the reader is referred to (Manchon, 2011).) Using this pre-Lie product, the correction procedure can be applied successively at every order. For instance, at order three the correction terms are Ω3(t) :=14E(x1▹x1)▹x1[u](t)+112Ex1▹(x1▹x1)[u](t), which gives $${\it\Omega}(t)={\it\Omega}_1(t) + {\it\Omega}_2(t) + {\it\Omega}_3(t)$$ such that exp(Ω(t)) =I+E(x1)l1[u](t)+E(x12)l2[u](t)+E(x13)l3[u](t)+⋯. It turns out that the generating series $$d$$ of $${\it\Omega}(t):=\sum_{n>0} {\it\Omega}_n(t)$$ satisfies the recursion d[k]=∑n=0∞Bnn!Ld[k−1]▹(n)(x1), with $$d^{[1]}=x_1$$, $$\lim_{k \rightarrow \infty} d^{[k]}= d$$, $$L^{(n)}_{d \triangleright}(x) := d \triangleright (L^{(n-1)}_{d \triangleright}(x))$$, $$L^{(0)}_{d \triangleright}(x)=d$$, and $$B_n$$ denotes the $$n$$-th Bernoulli number, namely, B0=1, B1=−12, B2=16, B4=−130,… andB2k+1=0 for k≥1. Thus, it can be shown that the limit of $$\exp(F_d^{[k]}[u](t))$$ when $$k\rightarrow \infty$$ agrees with (5.2). This is the well-known Magnus expansion – in its pre-Lie form. The more familiar expression for the Magnus expansion is obtained by noting that ELd▹(n)(x1)[u](t) =∫0tadΩ(s)(n)(u(s))ds, and $$ad^{(n)}_{{\it\Omega}}(u):=[{\it\Omega},ad^{(n-1)}_{{\it\Omega}}(u)]$$ with $$ad^{(0)}_{{\it\Omega}}(u)=u$$. Compared to the ordered exponential presented in Section 1, this is the standard exponential function of $${\it\Omega}(t)$$. Recall that in his seminal 1954 article (Magnus, 1954) Wilhelm Magnus proposed a particular differential equation for the matrix-valued function $${\it\Omega}(s;A)$$ such that the solution of the non-autonomous initial value problem, $$\dot{Y}(t)=A(t)Y(t)$$, $$Y(0)=Y_0$$, is given by $$X(s)=\exp(\int_0^s \dot{{\it\Omega}}(x;A)dx)Y_0$$, $${\it\Omega}(0;A)=0$$: Ω˙(s;A)=A(s)+∑n=0∞Bnn!ad∫0sΩ˙(x;A)dx(n)(A(s))=adΩ(s;A)exp(adΩ(s;A))−1(A(s)). By comparison, the Fliess operator $$F_c[u](t)$$ in (5.2) provides an input–output map that encodes in the iterated integrals the underlying algebraic structure of the system $$\dot{Z}=u Z$$. It is also possible to give a closed-form for $${\it\Omega}(t)$$ that does not involve any recursion. It is expressed in terms of the natural dendriform operations, which encode essentially on which side to put integrals in non-commuting expressions. The starting point, algebraically speaking, is associating (1.4) with the linear dendriform equation c=I+u≺c, whose formal solution is $$c = \sum_{k= 0}^\infty {x_1^k}_{\mathfrak{l}_k}$$ (Ebrahimi-Fard & Manchon, 2009b). Since the decorated trees in $$c$$ are only in terms of $$x_1$$, one can focus on the series c′=Φ−1(c)=∑k=0∞lk. In the commutative case, the grafting of trees becomes simply the catenation of words, and $$c'$$ reduces to . Denote $$\bar{c}= c'-I$$, then Thus, one can compute the generating series Note that was replaced by in (3.6), and that the grouping is possible due to the bilinearity and commutativity of . However, in the non-commutative case, trees have to be taken into consideration and needs to be computed in closed form. Therefore, From Lemma 3.4, it follows that (d′,τ)=(−1)des(τ)n(n−1des(τ)). Recall that the map $${\it\Phi}^{-1}$$ and the decoration map allow one to recover the original dendriform generating series $$d$$ of $${\it\Omega}$$. That is, since and $$E_{{\it\Phi}^{-1}(x_1^n,\tau)}[u](t) = E_{{x_1^n}_\tau}[u](t)$$, the element $${\it\Omega}(U)(t) = \log\left(F_c[u](t)\right)$$ is given by the explicit formula Ω(t)=∑n=1∞∑τ∈Tn(−1)des(τ)n(n−1des(τ))Ex1nτ[u](t), (5.3) which is equivalent to the celebrated Mielnik–Plebański–Strichartz formula for the Magnus expansion (Mielnik & Plebański, 1970; Strichartz, 1987). Thus, the dendriform Fliess operator corresponding to the solution of the non-commutative equation $$\dot{Z}=u Z$$ is Z(t)=Fc[u](t)=eΩ(t) with $${\it\Omega}(t)$$ given by (5.3). 6. Conclusions and future research The concept of dendriform Fliess operators was introduced and developed. The algebraic structure basically considers interactions between words in $$X^\ast$$ and planar binary decorated trees. Sufficient conditions for the convergence of such Fliess operators were given for the general case (4.1) and for operators indexed only by left-comb trees (4.3). As an application, the solution of the equation $$\dot{Z} = uZ$$ was given the form of a dendriform Fliess operator with output $$y=Z$$. This output was then written in a closed-form and related to the notion of pre-Lie algebras using the underlying dendriform algebra of these operators. This expression for the output was shown to be consistent with respect to the Magnus expansion and the Mielnik–Plebański–Strichartz formula. Future work includes expanding the theory to handle the non-commutative stochastic and quantum cases, in which the integrators generate extra terms in addition of being intrinsically non-commutative (Parthasarathy, 1992). In that context, the connection to Rota-Baxter algebras will be made clear. Also, the interconnection of systems with non-commutative inputs is currently under study by defining the non-commutative counterparts to the formal power series products in (Gray & Li, 2005). Funding This work was done in part while the first author was affiliated and funded by The University of New South Wales (ADFA) in Canberra, Australia. The second author was supported, while on leave from Old Dominion University in Norfolk, Virginia, USA, by grant SEV-2011-0087 from the Severo Ochoa Excellence Program at the Instituto de Ciencias Matemáticas in Madrid, Spain. The third author was supported by Ramón y Cajal research grant RYC-2010-06995 from the Spanish government and acknowledges support from the Spanish government under project MTM2013-46553-C3-2-P. This research was also funded by a grant from the BBVA Foundation. References Agrachev A. & Gamkrelidze R. ( 1978 ) Exponential representation of flows and chronological calculus. Math. USSR Sbornik ( English translation ), 35 , 727 – 786 . Google Scholar CrossRef Search ADS Agrachev A. & Gamkrelidze R. ( 1991 ) The shuffle product and symmetric groups. Differential equations, dynamical systems, and control science , ( Elworthy K. D. ed.). Bauer M. Chetrite R. Ebrahimi-Fard K. & Patras F. ( 2013 ) Time-ordering and a generalized Magnus expansion. Lett. Math. Phys. , 103 , 331 – 350 . Google Scholar CrossRef Search ADS Blanes S. Casas F. Oteo J. A. & Ros J. ( 2009 ) Magnus expansion: mathematical study and physical applications. Phys. Rep. , 470 , 151 – 238 . 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