**OPERATORS AND THE COMPOSITION PRODUCT**

YAQIN LI AND W. STEVEN GRAY
*Received 28 July 2005; Revised 14 April 2006; Accepted 25 April 2006*

The formal Laplace-Borel transform of an analytic integral operator, known as a Fliess operator, is defined and developed. Then, in conjunction with the composition product over formal power series, the formal Laplace-Borel transform is shown to provide an iso-morphism between the semigroup of all Fliess operators under operator composition and the semigroup of all locally convergent formal power series under the composition prod-uct. Finally, the formal Laplace-Borel transform is applied in a systems theory setting to explicitly derive the relationship between the formal Laplace transform of the input and output functions of a Fliess operator. This gives a compact interpretation of the op-erational calculus of Fliess for computing the output response of an analytic nonlinear system.

Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.

**1. Introduction**

Let*u*:R*→*Rbe a function which is real analytic at a point*t*0*∈*R. Its Taylor series

ex-pansion is denoted by

*u(t)=∞*

*n=*0

*cu*(n)

*t−t*0

*n*

*n!* *.* (1.1)

Within its radius of convergence,*u*is completely characterized by the sequence of coeﬃ
-cients*{cu*(n)}*∞n=*0. From this, one can construct a formal power series representation of*u*

by introducing an abstract symbol set*X*0*= {x*0*}, namely,*

*cu=*

*∞*

*n=*0

*cu*(n)x*n*0*.* (1.2)

*The formal Laplace transform in this setting is the mapping*

ᏸ*f*:*u−→cu.* (1.3)

Hindawi Publishing Corporation

International Journal of Mathematics and Mathematical Sciences Volume 2006, Article ID 34217, Pages1–14

*Its inverse is the formal Borel transform. Whenu*is entire, its one-sided integral Laplace
transform is often well defined. Setting*t*0*=*0, it can be written in the form

ᏸ[u](s) :*=*
_{∞}

0 *u(t)e*

*−st _{dt}_{=}∞*

*n=*0

*cu*(n)

_{∞}

0

*tn*

*n!e−stdt*

*=s−*1
*∞*

*n=*0

*cu*(n)

*s−*1*n*,

(1.4)

using the Laplace transformation pair

*tn _{⇐⇒}*ᏸ

*1*

_{n!}_{s}−*n*+1

_{,}

_{n}_{≥}_{0.}

_{(1.5)}

Clearly, the two transformsᏸ*f* andᏸare related via

ᏸ[u](s)*=x*0ᏸ*f*[u]*|x*0*→s−*1*.* (1.6)

An alternative notion of the formal Laplace-Borel transform appears in [23] as a mapping between two formal power series.

The formal Laplace-Borel transform was first used by Fliess et al. for nonlinear
sys-tems analysis in [6–8,12,19]. A related approach was later given by Minh in [22]. In each
instance, an algebraic representation of an analytic function was used to produce a type
of operational calculus for computing the output responses of a nonlinear system given
various analytic input functions. In [9–11] Fliess also applied the formal Laplace-Borel
transform via a tensor product to examine some classical problems regarding transfer
matrix representations of linear time-varying systems and generalized state space systems.
Later in [3], Delaleau and Rudolph characterized the properness of linear time-varying
systems using this approach. In [18], Hautus used the formal Laplace-Borel transform to
characterize the invertibility properties of so-called smooth linear systems. More recently,
in [2] it was applied to obtain conditions for well-posedness of the input-output mapping
for several classes of boundary control systems. The formal Laplace-Borel transform
ap-plied in the analysis of linear systems, as with the analysis of input and output functions,
mostly involves formal power series over a single-letter alphabet, and thus is
commuta-tive in nature. What is absent in the nonlinear systems literature is an explicit notion of
*computing the formal Laplace-Borel transform of an input-output operator, an idea that*
is familiar and useful in linear systems theory. In this setting, noncommutative formal
power series arise as a natural generalization.

The main purpose of this paper is to define the formal Laplace-Borel transform of
a class of analytic input-output operators using noncommutative formal power series,
and then to show how this notion can provide an alternative interpretation of the
*op-erational calculus of Fliess when combined with the composition product of two formal*
power series [4,5, 13–16,20]. The specific class of operators considered is known as
*Fliess operators. LetX= {x*0,x1,...,x*m}*denote an arbitrary alphabet of noncommutative

letters, and*X∗*the set of all words over*X, including the empty word∅. For eachc*in
the set of all formal power seriesR_{X, one can formally associate a corresponding}

For a measurable function*u*: [a,b]*→*R*m*_{, define}_{u}

*p=*max{u*ip*: 1*≤i≤m}, where*

*uip*is the usual*Lp*-norm for a measurable real-valued function,*ui*, defined on [a,*b]*

when*p∈*[1,∞), and*ui∞=*sup* _{t}∈*[

*a*,

*b*]

*{|ui*(t)|: 1

*≤i≤m}. For any fixedp∈*[1,∞], let

*Lm*

*p*[a,b] denote the set of all measurable functions defined on [a,*b] having a finite · p*

norm and*Bm*

*p*(R)[a,b] := {u*∈Lmp*[a,b] :*up≤R}. Witht*0,T*∈*Rfixed, and*T >*0,

de-fine inductively for each*η=xiη∈X∗*the mapping*Eη*:*Lm*1[t0,t0+*T]→*Ꮿ[t0,t0+*T] by*

*Eη*[u]*t,t*0:=

_{t}

*t*0

*ui*(τ)E*η*[u]*τ,t*0*dτ,* (1.7)

where*E∅≡*1 and *u*0*≡*1. The input-output operator corresponding to *c*is the Fliess

operator

*Fc*[u](t) :=

*η∈X∗*

(c,η)E*η*[u]*t,t*0*.* (1.8)

All Volterra operators with analytic kernels, for example, are Fliess operators. When there
exist real numbers*K*,M >0 such that

_{(c,η)}_{:=}_{max}_{(c,η)}_{j}_{: 0}_{≤}_{j}_{≤}_{}

*≤KM|η||η|!,* (1.9)

the formal power series*c* *is said to be locally convergent. (Here|η|* denotes the
num-ber of symbols in *η∈X∗*.) The subset of all locally convergent series is denoted by

R

*LCX. When* *c∈*R*LCX, it is known that* *Fc* constitutes a well-defined

opera-tor from*Bm*

*p*(R)[t0,*t*0+*T*] into*B _{q}*(S)[t0,t0+

*T*] for suﬃciently small

*R,S,T >*0, where the

numbers*p,q∈*[1,*∞] are conjugate exponents, that is, 1/ p*+ 1/q*=*1 with (1,∞) being a
conjugate pair by convention [17]. Therefore, the specific operator class of interest in this
paper is the set of Fliess operatorsᏲ:= {F*c*:*c∈*R*LCX}. When=m, this set forms a*

semigroup under operator composition, as does the setR*m*

*LCX*under the composition

product [4,20]. It will be shown here that the formal Laplace-Borel transform provides an isomorphism between these two semigroups.

The paper is organized as follows. InSection 2, the notion of a formal Laplace-Borel transform of a Fliess operator is defined. Then its basic properties are developed and a set of examples is given. InSection 3, the composition product is introduced and its relation-ship to the formal Laplace-Borel transform is described. Using the concepts developed in Section 3, the formal Laplace-Borel transform is shown to provide a compact interpreta-tion of the operainterpreta-tional calculus of Fliess inSection 4. Examples are given to illustrate the computation of the output response of an analytic system.Section 5concludes the paper with a brief summary of the main results.

**2. The formal Laplace-Borel transform of a Fliess operator**
Consider a causal linear integral operator

*y(t)=*
_{t}

*t*0

where the kernel function,*h, is analytic att=*0. The operator is completely characterized
by the Laplace transform of the kernel function, namely,*H(s) :=*ᏸ[h](s)*=* *k>*0*ch*(k)s*−k*,

when it exists. Therefore, the usual definition (1.3) of the formal Laplace-Borel transform over a single symbol applies directly to this case. In the more general context of Fliess operators, a generalization is required. The following preliminaries are needed to ensure the new definition is well-posed.

Theorem 2.1 [24*, Corollary 2.2.4]. Supposec,d∈*R

*LCX. IfFc=FdonBm∞*(1)[t0,*t*0+

*T] for some finiteT >0, thenc=d.*

Given the nested nature of the set of spaces*Lp*[t0,t0+*T] forp=*1, 2,...,∞, the

follow-ing result is immediate.

*Corollary 2.2. Supposec,d∈*R

*LCX. IfFc=Fd* *onBmp*(R)[t0,*t*0+*T] for some* *p∈*

*{1, 2,...*,∞}*and real numbersR,T >0, thenc=d.*

In light of this uniqueness property, there exists a one-to-one mapping between the set of well-defined Fliess operators,Ᏺ, and the set of locally convergent formal power series,

R

*LCX. This guarantees that the following definition is well-posed.*

*Definition 2.3. LetX= {x*0,x1,...,x*m}. The formal Laplace transform is defined as*

ᏸ*f* :Ᏺ*−→*R*LC*

*X*

:*Fc−→c.* (2.2)

*The corresponding inverse transform, the formal Borel transform, is*

Ꮾ*f* :R*LC*

*X−→*Ᏺ

:*c−→Fc.*

(2.3)

Note that when*m=*0, this definition is consistent with that given in (1.3) in the sense
that a fixed function*u*can be represented as the constant operator*Fcu, that is,u=Fcu[v]*

for all*v∈Bm*

*p*(R)[t0,t0+*T] and*ᏸ*f*[u]*=cu=*ᏸ*f*[F*cu*].

It is next shown that many of the familiar properties of the integral Laplace transform also have counterparts in the present context. To facilitate the analysis, two concepts are needed.

*Definition 2.4. For anyxi∈X, the left-shift operator,x−i*1:R*X →*R*X, is defined*

as

*x−*1
*i* (c)*=*

*η∈X∗*

(c,η)x*−*1

*i* (η), (2.4)

where

*x−i*1(η)*=*

⎧ ⎪ ⎨ ⎪ ⎩

*η*:*η=xiη*, *η∈X∗*,

*Definition 2.5. A Dirac series,δi*, is a generalized series with the defining property that

*Fδi*[u]*=ui*(t) for any 1*≤i≤m.*

The main result concerning elementary properties of the formal Laplace-Borel trans-form is stated below.

*Theorem 2.6. Given any* *c,d∈*R

*LCXand scalarsα,β∈*R*, the following identities*

*hold.*

*(1) Linearity:*

ᏸ*f*

*αFc*+*βFd*

*=α*ᏸ*f*

*Fc*

+*β*ᏸ*f*

*Fd*
,
Ꮾ*f*

*αc*+*βd=α*Ꮾ*f*[c] +*β*Ꮾ*f*[d].

(2.6)

*(2) Integration:*

ᏸ*f*

*In _{F}*

*c*

*=xn*

0*c,*
Ꮾ*f*
*xn*
0*c*
*=In _{F}*

*c*,

(2.7)

*whereIn*_{(·) is the}_{n}_{th order integration operator so that}

*InFc*[u](t)*=*

_{t}

0

* _{τ}*1

0 *···*

* _{τ}_{n}_{−}*1

0 *Fc*[u]

*τndτn···dτ*2*dτ*1*.* (2.8)

*(3) Diﬀerentiation:*

ᏸ*f*

*DFc*

*=x−*1

0 (c) +
*m*

*i=*1

*δi*

*x−*1

*i* (c)

,

Ꮾ*f*

*x−*1

0 (c) +
*m*

*i=*1

*δi*

*x−*1

*i* (c)

*=DFc.*

(2.9)

*Ifxn*0*is a left factor ofc, that is,c=x*0*ncfor somec∈*R*LCX, then*

ᏸ*f*

*Dn _{F}*

*c*

*=x−n*

0 (c),

Ꮾ*f*

*x−n*

0 (c)

*=Dn _{F}*

*c*,

(2.10)

*whereDn*_{(·) is the}_{n}_{th-order di}_{ﬀ}_{erentiation operator so that}_{D}n_{F}

*c*[u](t)=*dnFc*[u](t)/

*dtn _{, and}_{}_{denotes the shu}_{ﬄ}_{e product (see, e.g., [}_{21}_{]).}*

*(4) Multiplication:*

ᏸ*f*

*Fc·Fd*

*=*ᏸ*f*

*Fc*
ᏸ*f*
*Fd*
,

Ꮾ*f*[cd]*=*Ꮾ*f*[c]*·*Ꮾ*f*[d].

(2.11)

a Fliess operator is

*d*

*dtFc*[u](t)*=Fx−*01(*c*)[u](t) +
*m*

*i=*1

*ui*(t)F*x−*1

*i* (*c*)[u](t). (2.12)

Applying the formal Laplace transform to this equality gives the first pair of equations
in part (3). Now if*x*0is a left factor of*c, thenF _{x}−*1

*i* (*c*)[u](t)*=*0 for*i=*1, 2,...,m. In this
case,*dFc*[u](t)/dt*=Fx−*1

0 (*c*)[u](t). Proceeding inductively, the second pair of equations

follows.

The following definition is utilized in the examples which follow.

*Definition 2.7 [*1]. Let *c∈*R*X* be proper (i.e., (c,∅)*=0). Then the star operator*
applied to*c*is defined as

*c*:=

*n≥*0

*cn*_{:=}_{(1}* _{−}_{c)}−*1

_{,}

_{(2.13)}

where*cn*_{denotes the catenation power.}

Observe that when*c*is not proper, it is always possible to write*c=*(c,∅)(1*−c*). In
which case, there exists a*c−*1_{∈}_{R}_{X}_{such that under the catenation product}* _{cc}−*1

_{=}_{1}

and*c−*1_{c}_{=}_{1. Specifically,}

*c−*1* _{=}* 1

(c,∅)(1*−c*)*−*1*=*
1

(c,∅)(c)*.* (2.14)

*Example 2.8. LetX= {x*0,x1,*x*2*}*and*Fc*[u](t)=exp(

_{t}

0*u*1(t)+u2(t)dt). Observe that*Fc*[u]

can be expanded as

*Fc*[u](t)*=*

*n≥*0

1
*n!*

_{t}

0*u*1(t) +*u*2(t)dt

*n*

*=*

*n≥*0

_{t}

0

*u*1

*τ*1

+*u*2

*τ*1

*τ*1

0

*u*1

*τ*2

+*u*2

*τ*2

*× ···*
* _{τ}_{n}_{−}*1

0

*u*1*τn*+*u*2*τndτn···dτ*2*dτ*1*.*

(2.15)

Therefore,

ᏸ*f*

*Fc*

*=*

*n≥*0

*x*1+*x*2

*n _{=}*

*x*1+*x*2

*.* (2.16)

This result can be viewed as an operator generalization of the integral transform pair

*et _{⇐⇒}*ᏸ

_{(1}

*1*

_{−}_{s)}−

_{.}_{(2.17)}

Table 2.1. Some formal Laplace-Borel transform pairs.

*Fc* ᏸ*f*

*Fc*

*Fc*:*u−→*1 1

*Fc*:*u−→tn* *n*!*xn*0

*Fc*:*u−→*

_{n}_{}*−*1

*i=*0

*i*
*n−*1

*i*! *aiti*

*eat* _{1}_{−}_{ax}

0

−*n*

*Fc*:*u−→* 1

*n*!

*t*

*t*0
*k*

*j=*1
*uij*(*τ*)*dτ*

*n*

*xi*1+*xi*2+*···*+*xik*
*n*

*Fc*:*u−→*

*n≥*0
*an*
*n*!
*t*
*t*0
*k*

*j=*1
*uij*(*τ*)*dτ*

*n* _{}

*n≥*0
*an*

*xi*1+*xi*2+*···*+*xik*
*n*

*Fc*:*u−→*exp

*t*

*t*0
*k*

*j=*1
*uij*(*τ*)*dτ*

*xi*1+*xi*2+*···*+*xik*

*Fc*:*u−→*

*t*

*t*0
*k*

*j=*1

*uij*(*τ*)*dτ*exp
*t*

*t*0
*k*

*j=*1
*uij*(*τ*)*dτ*

*xi*1+*xi*2+*···*+*xik*

1*−xi*1+*xi*2+*···*+*xik*
2

*Fc*:*u−→*cos

_{t}

*t*0
*k*

*j=*1
*uij*(*τ*)*dτ*

1

1 +*xi*1+*xi*2+*···*+*xik*
2

*Fc*:*u−→*sin

*t*

*t*0
*k*

*j=*1
*uij*(*τ*)*dτ*

*xi*1+*xi*2+*···*+*xik*

1 +*xi*1+*xi*2+*···*+*xik*
2

*Example 2.9. LetX= {x*0,*x*1,...,x*m}. SupposeFc*has the generating series*c=* *η∈X∗η,*
and*Fξ*is given for some fixed word*ξ∈X∗*. Then

ᏸ*f*

*Fc·Fξ*

*=*ᏸ*f*

*Fc*
ᏸ*f*
*Fξ*
*=cξ*
*=*
*ν∈X∗*

*ν*
*ξ*

*ν,* (2.18)

where (* _{ξ}ν*) denotes the number of subwords of

*ν*which are equal to

*ξ*(see [21, page 127]).

**3. The formal Laplace-Borel transform and the composition product**

*Definition 3.1. For anyc∈*R_{X}_{and}_{d}_{∈}_{R}*m _{X, the composition product is defined}*

as

*c◦d=*

*η∈X∗*

(c,η)η*◦d,* (3.1)

where

*η◦d=*
⎧
⎪
⎨
⎪
⎩

*η*:*|η|xi=*0, *∀i=*0,

*xn*+1
0

*di(η◦d)*:*η=x*0*nxiη*, *n≥*0, *i=*0.

(3.2)

(Here*|η|xi* denotes the number of symbols in*η*equivalent to*xi*and*di*:*ξ→*(d,ξ)*i*, the
*ith component of (d,ξ).)*

Consequently, if

*η=xnk*

0 *xikx*0*nk−*1*xik−*1*···x*
*n*1

0 *xi*1*x*
*n*0

0 , (3.3)

where*ij=*0 for*j=*1,*...,k, it follows that*

*η◦d=xnk*+1

0

*dikxn*0*k−*1+1

*dik−*1*···x*
*n*1+1

0

*di*1*x*

*n*0

0

*···.* (3.4)

Alternatively, for any*η∈X∗*one can uniquely define a set of right factors*{η*0,*η*1,...,η*k}*

of*η*by the iteration

*ηj*+1*=xnj*

+1

0 *xij*+1*ηj*, *η*0*=x*
*n*0

0 , *ij*+1*=*0, (3.5)

so that *η=ηk* with *k= |η| − |η|x*0. In which case, *η◦d=ηk◦d, where* *ηj*+1*◦d=*

*xnj*+1+1

0 [d*ij*+1*(ηj◦d)] andη*0*◦d=x*
*n*0

0 . It was shown in [15,16] that the composition

product of two series is always well defined since the family of series*{η◦d*:*η∈X∗}*is
locally finite for any fixed*d∈*R*m _{X.}*

It is easily verified that for any series*c,d,e∈*R*m _{X, the composition product is left}*

distributive over addition. That is,

(c+*d)◦e=c◦e*+*d◦e,* (3.6)

but in general*c◦*(d+*e)=c◦d*+*c◦e. An exception is the class of series called linear*
*series. A seriesc*is linear if

supp(c)*⊆η∈X∗*:*η=xn*1

0 *xixn*00,*i∈ {1, 2,...,m},n*1,n0*≥*0

*.* (3.7)

The composition product is associative, that is, (c*◦d)◦e=c◦*(d*◦e); hence (*R*m _{X,◦)}*

forms a semigroup [4,20]. In [16], it is shown that the composition product of two locally
convergent formal power series is always locally convergent; therefore the setR*m*

*LCX*is

closed under composition, and (R*m*

*LCX,◦) also forms a semigroup. The central *

prop-erty of the composition product is its relationship to the composition of two Fliess
oper-ators. Namely, for any*c∈*R

(*Fc, Fd*) *Fc*Æ*Fd=Fc*
Æ*d*

(*c, d*) *c*Æ*d*
Æ

Æ

[image:9.468.171.298.72.134.2]Ꮾ*f* ᏸ*f* ᏸ*f* Ꮾ*f*

Figure 3.1. The isomorphism between the semigroups (Ᏺ,*◦*) and (R*m*

*LCX*,*◦*).

Fliess operators is always another Fliess operator with generating series*c◦d, that is,*

*Fc◦Fd=Fc◦d.* (3.8)

Therefore,Ᏺforms a semigroup under operator composition when*=m. The following*
theorem shows that composition is preserved in a natural sense under the formal
Laplace-Borel transform.

*Theorem 3.2. LetX= {x*0,x1,...,x*m}. For anyc∈*R*LCXandd∈*R*mLCX,*

ᏸ*fFc◦Fd=*ᏸ*fFc◦*ᏸ*fFd*,

Ꮾ*f*(c*◦d)=*Ꮾ*f*(c)*◦*Ꮾ*f*(d).

(3.9)

*Proof. The proof follows directly from the definitions. For any well-definedFc*and*Fd*,

ᏸ*f*

*Fc◦Fd*

*=*ᏸ*f*

*Fc◦d*

*=c◦d=*ᏸ*f*

*Fc*

*◦*ᏸ*f*

*Fd*

*.* (3.10)

Similarly, for any locally convergent formal power series*c*and*d,*

Ꮾ*f*(c*◦d)=Fc◦d=Fc◦Fd=*Ꮾ*f*(c)*◦*Ꮾ*f*(d). (3.11)
FromTheorem 3.2, when*=m, the formal Laplace-Borel transform provides an *
iso-morphism between the two semigroups (Ᏺ,*◦) and (*R*m*

*LCX,◦), as shown in*Figure 3.1.

*Example 3.3. LetX= {x*0,*x*1,x2*},Fc*[u](t)*=*cos(

_{t}

0*u*1(t) +*u*2(t)dt), and*d=*(d1*d*2)*T∈*
R2

*LCX. Define*

*Fe*[u](t)*=*

*Fc◦Fd*

[u](t)*=*cos
*t*

0*Fd*1[u](t) +*Fd*2[u](t)dt

*.* (3.12)

The formal Laplace transform of*Fe*is then

ᏸ*f*

*Fe*

*=c◦d=*

*i≥*0

(−1)*i _{x}*
1+

*x*2

2*i _{◦}*

*d*

*=−x*1+*x*22*◦d=* 1

1 +*x*1+*x*22

*◦d.*

*Example 3.4. LetX= {x*0,x1,...,x*m}*and*c∈*R*mLCX. It is easily verified by induction*

that for*n≥*1,

*xn*
*i* *◦c=*

1
*n!*

*x*0*ci*

*n*

, *i=*1, 2,...,*m,* (3.14)

where (·)*n*_{denotes the shu}_{ﬄ}_{e power.}

Applying the formal Borel transform to both sides of this identity gives

Ꮾ*f*

*xn*

*i* *◦c*

*=*Ꮾ*f*

_{1}
*n!*

*x*0*ci*

*n*
*=* 1
*n!*
Ꮾ*f*
*x*0*ci*

_{n}

*=* 1
*n!*

_{t}

0*Fci*[u](τ)dτ

*n*

*.*

(3.15)

*Example 3.5. First consider the linear ordinary di*ﬀerential equation

*dn _{y(t)}*

*dtn* +
*n*_{}*−*1

*i=*0

*aid*
*i _{y(t)}*

*dti* *=*
*n*_{}*−*1

*i=*0

*bid*
*i _{u(t)}*

*dti* (3.16)

with initial conditions*y*(*i*)_{(0)}_{=}_{0,}* _{u}*(

*i*)

_{(0)}

_{=}_{0, and where}

_{a}*i*,*bi∈*Rfor*i=*0, 1,...,*n−*1.

Integrate both sides of the equation*n*times and assume there exists a*c∈*R*LCX*such

that*y=Fc*[u]. Applying the formal Laplace transform to both sides of the equation gives

*δ*+

*n*_{}*−*1

*i=*0

*aixn*0*−*1*−ix*1

*◦c=n−*

1

*i=*0

*bixn*0*−*1*−ix*1,

1 +

*n*_{}*−*1

*i=*0

*aixn*0*−i*

*c=n*

*−*1

*i=*0

*bixn*0*−*1*−ix*1*.*

(3.17)

Therefore,

*c=*

1 +

*n*_{}*−*1

*i=*0

*aixn*0*−i*

−1_{n}_{−}_{1}

*i=*0

*bix*0*n−*1*−ix*1*.* (3.18)

Rephrased in the language of the integral Laplace transform, this is equivalent to

*Y*(s)*=*

1 +

*n*_{}*−*1

*i=*0

*ai _{s}_{n}*1

_{−}_{i}−1_{n}_{−}_{1}

*i=*0

*bi _{s}_{n}*1

_{−}_{i}
*U(s)*

*=*

*sn*_{+}*n−*
1

*i=*0

*aisi*

−1_{n}_{−}_{1}

*i=*0

*bisi*

*U(s).*

(3.19)

Now consider the nonlinear diﬀerential equation

*dn _{y(t)}*

*dtn* +
*n*_{}*−*1

*i=*0

*aid*
*i _{y(t)}*

*dti* +
*k*

*j=*2

*pju(t)yj*(t)*=*
*n*_{}*−*1

*i=*0

*bid*
*i _{u(t)}*

with*y*(*i*)_{(0)}_{=}_{0,}* _{u}*(

*i*)

_{(0)}

_{=}_{0, and where}

_{a}*i*,*bi*,*pj∈*Rfor*i=*0, 1,*...,n−*1 and *j=*2,...,*k.*

Again integrate both sides of the equation*n*times and assume *y=Fc*[u]. Applying the

formal Laplace transform gives

1 +

*n*_{}*−*1

*i=*0

*aixn*0*−i*

*c*+

*k*

*j=*2

*pjxn*0*−*1*x*1

*cj _{=}n−*
1

*i=*0

*bixn*0*−*1*−ix*1*.* (3.21)

As in [12], a recursive procedure can be applied to solve the algebraic equation iteratively so that

*c=c*1+*c*2+*···* (3.22)

with

*c*1*=*

1 +

*n*_{}*−*1

*i=*0

*aix*0*n−i*

−1_{n}_{−}_{1}

*i=*0

*bixn*0*−*1*−ix*1 (3.23)

and for*n≥*2

*cn=*

1 +

*n*_{}*−*1

*i=*0

*aixn*0*−i*

−1

*xn−*1
0 *x*1

*k*

*j=*2

*pj*

*ν*1*≥*1,*...*,*νj≥*1
*ν*1+*ν*2+*···*+*νj=n*

*cν*1*cν*2*···cνj.* (3.24)

**4. Operational calculus for the output response of Fliess operators**

In [7,12,19], an operational calculus was developed to compute the output response of an analytic nonlinear system represented by a Fliess operator. In this section, the formal Laplace-Borel transform pair in conjunction with the composition product are employed to produce a clear and compact interpretation of this methodology.

*Theorem 4.1. LetFc∈*Ᏺ*and letube an analytic function with formal Laplace transform*

*cu∈*R*mLCX*0*. Then the output function* *y=Fc*[u] is analytic and has formal Laplace

*transformcy=c◦cu∈*R*LCX*0*.*

*Proof. The analyticity of* *y* follows from [24, Lemma 2.3.8]. Let *cy* denote its formal

Laplace transform. Local convergence is preserved under the composition product;
there-fore the formal power series*c◦cu*is locally convergent, and, for any admissible input

*v∈Bm*

*p*(R)[t0,t0+*T*],

*y=Fcy*[v]*=Fc*

*Fcu[v]*

*=Fc◦cu*[v]. (4.1)

Applying the formal Laplace transform to both sides of the equation gives*cy=c◦cu*.

The following examples further illustrate the applications of this result for computing the output response of an analytic input-output system.

*Example 4.2. Consider the linear time-invariant systemy(t)=*0*th(t−τ)u(τ)dτ, where*

*u* *z* 1 *y*

1 *z*

Figure 4.1. A simple Wiener system.

If*u(t)=* *k≥*0(c*u*,x*k*0)t*k/k!, then it follows thaty(t)=* *n≥*0(c*y*,x0*n*)t*n/n!, where*

*cy=c◦cu*

*=*

*k≥*0

*c,x*0*kx*1

*xk*0*x*1*◦cu*

*=*

*k≥*0

*c,x*0*kx*1

*xk*0+1*cu.*

(4.2)

Therefore,

*cy*,x0*n*

*=n−*

1

*k=*0

*c,xk*

0*x*1

*cu*,x0*n−*1*−k*

, *n≥*1, (4.3)

which is the conventional convolution sum.

*Example 4.3. Consider the simple Wiener system shown in*Figure 4.1where*z(0)=*0.
The mapping*u→y*can be written as

*y(t)=*
*∞*

*n=*0

*Ex*1[u](t)

*n _{=}∞*

*n=*0

*Fxn*

1 [u](t)*=*

*∞*

*n=*0

*n!Exn*

1[u](t). (4.4)

Therefore *y=Fc*[u] with *c=* *n≥*0*n!x*1*n*. When *u(t)=tk/k!, for example, the formal*

Laplace transform of*u*is*cu=x*0*k*. FromTheorem 4.1and (3.14) it follows that

*cy=*

*∞*

*n=*0

*n!xn*1*◦xk*0*=*

*∞*

*n=*0

*x*0*k*+1

_{n}*=*

*∞*

*n=*0

(k+ 1)n!

(k+ 1)!*nx*

(*k*+1)*n*

0 *.* (4.5)

Consequently, the output response is

*y(t)=*
*∞*

*n=*0

(k+ 1)n!

(k+ 1)!*n*

*t*(*k*+1)*n*

(k+ 1)n!*=*
*∞*

*n=*0

*t*(*k*+1)*n*

(k+ 1)!*n=*

1

1*−tk*+1_{/(k}_{+ 1)!}*.* (4.6)

**5. Conclusion**

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Yaqin Li: Department of Electrical and Computer Engineering, University of Memphis, Memphis, TN 38152, USA

*E-mail address:*[email protected]

W. Steven Gray: Department of Electrical and Computer Engineering, Old Dominion University, Norfolk, VA 23529, USA