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THE FORMAL LAPLACE-BOREL TRANSFORM OF FLIESSOPERATORS 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 Fliessoperator, is defined and developed. Then, in conjunction with the composition productover 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 andthe 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 settingto explicitly derive the relationship between the formal Laplace transform of the inputand 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 nonlinearsystem.
Copyright © 2006 Hindawi Publishing Corporation. All rights reserved.
1. Introduction
Let u : R→ R be a function which is real analytic at a point t0 ∈ R. Its Taylor series ex-pansion is denoted by
u(t)=∞∑
n=0
cu(n)
(t− t0
)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 uby introducing an abstract symbol set X0 = {x0}, namely,
cu =∞∑
n=0
cu(n)xn0 . (1.2)
The formal Laplace transform in this setting is the mapping
� f : u �−→ cu. (1.3)
Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2006, Article ID 34217, Pages 1–14DOI 10.1155/IJMMS/2006/34217
2 The formal Laplace-Borel transform of Fliess operators
Its inverse is the formal Borel transform. When u is entire, its one-sided integral Laplacetransform is often well defined. Setting t0 = 0, it can be written in the form
�[u](s) :=∫∞
0u(t)e−stdt =
∞∑
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�⇐⇒ n!
(s−1)n+1
, n≥ 0. (1.5)
Clearly, the two transforms � f and � are related via
�[u](s)= x0� f [u] |x0→s−1 . (1.6)
An alternative notion of the formal Laplace-Borel transform appears in [23] as a mappingbetween 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 eachinstance, an algebraic representation of an analytic function was used to produce a typeof operational calculus for computing the output responses of a nonlinear system givenvarious analytic input functions. In [9–11] Fliess also applied the formal Laplace-Boreltransform via a tensor product to examine some classical problems regarding transfermatrix representations of linear time-varying systems and generalized state space systems.Later in [3], Delaleau and Rudolph characterized the properness of linear time-varyingsystems using this approach. In [18], Hautus used the formal Laplace-Borel transform tocharacterize 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 mappingfor 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 ofcomputing the formal Laplace-Borel transform of an input-output operator, an idea thatis familiar and useful in linear systems theory. In this setting, noncommutative formalpower series arise as a natural generalization.
The main purpose of this paper is to define the formal Laplace-Borel transform ofa 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 formalpower series [4, 5, 13–16, 20]. The specific class of operators considered is known asFliess operators. Let X = {x0,x1, . . . ,xm} denote an arbitrary alphabet of noncommutativeletters, and X∗ the set of all words over X , including the empty word ∅. For each c inthe set of all formal power series R�〈〈X〉〉, one can formally associate a correspondingm-input, �-output operator Fc in the following manner. Let p ≥ 1 and a < b be given.
Y. Li and W. S. Gray 3
For a measurable function u : [a,b]→Rm, define ‖u‖p =max{‖ui‖p : 1≤ i≤m}, where‖ui‖p is the usual Lp-norm for a measurable real-valued function, ui, defined on [a,b]when p ∈ [1,∞), and ‖ui‖∞ = supt∈[a,b]{|ui(t)| : 1≤ i≤m}. For any fixed p ∈ [1,∞], letLmp [a,b] denote the set of all measurable functions defined on [a,b] having a finite ‖ · ‖pnorm and Bm
p (R)[a,b] := {u∈ Lmp [a,b] : ‖u‖p ≤ R}. With t0,T ∈R fixed, and T > 0, de-fine inductively for each η = xiη′ ∈ X∗ the mapping Eη : Lm1 [t0, t0 +T]→�[t0, t0 +T] by
Eη[u](t, t0
):=∫ t
t0ui(τ)Eη′[u]
(τ, t0
)dτ, (1.7)
where E∅ ≡ 1 and u0 ≡ 1. The input-output operator corresponding to c is the Fliessoperator
Fc[u](t) :=∑
η∈X∗(c,η)Eη[u]
(t, t0
). (1.8)
All Volterra operators with analytic kernels, for example, are Fliess operators. When thereexist 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 byR�
LC〈〈X〉〉. When c ∈ R�LC〈〈X〉〉, it is known that Fc constitutes a well-defined opera-
tor from Bmp (R)[t0, t0 +T] into B�
q(S)[t0, t0 +T] for sufficiently small R,S,T > 0, where thenumbers p,q ∈ [1,∞] are conjugate exponents, that is, 1/p+ 1/q = 1 with (1,∞) being aconjugate pair by convention [17]. Therefore, the specific operator class of interest in thispaper is the set of Fliess operators � := {Fc : c ∈R�
LC〈〈X〉〉}. When � =m, this set forms asemigroup under operator composition, as does the setRm
LC〈〈X〉〉 under the compositionproduct [4, 20]. It will be shown here that the formal Laplace-Borel transform providesan isomorphism between these two semigroups.
The paper is organized as follows. In Section 2, the notion of a formal Laplace-Boreltransform of a Fliess operator is defined. Then its basic properties are developed and a setof examples is given. In Section 3, the composition product is introduced and its relation-ship to the formal Laplace-Borel transform is described. Using the concepts developed inSection 3, the formal Laplace-Borel transform is shown to provide a compact interpreta-tion of the operational calculus of Fliess in Section 4. Examples are given to illustrate thecomputation of the output response of an analytic system. Section 5 concludes the paperwith 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
t0h(t− τ)u(τ)dτ, (2.1)
4 The formal Laplace-Borel transform of Fliess operators
where the kernel function, h, is analytic at t = 0. The operator is completely characterizedby the Laplace transform of the kernel function, namely,H(s) :=�[h](s)=∑k>0 ch(k)s−k,when it exists. Therefore, the usual definition (1.3) of the formal Laplace-Borel transformover a single symbol applies directly to this case. In the more general context of Fliessoperators, a generalization is required. The following preliminaries are needed to ensurethe new definition is well-posed.
Theorem 2.1 [24, Corollary 2.2.4]. Suppose c,d ∈R�LC〈〈X〉〉. If Fc = Fd on Bm∞(1)[t0, t0 +
T] for some finite T > 0, then c = d.
Given the nested nature of the set of spaces Lp[t0, t0 +T] for p = 1,2, . . . ,∞, the follow-ing result is immediate.
Corollary 2.2. Suppose c,d ∈ R�LC〈〈X〉〉. If Fc = Fd on Bm
p (R)[t0, t0 + T] for some p ∈{1,2, . . . ,∞} and real numbers R,T > 0, then c = d.
In light of this uniqueness property, there exists a one-to-one mapping between the setof well-defined Fliess operators, �, and the set of locally convergent formal power series,R�
LC〈〈X〉〉. This guarantees that the following definition is well-posed.
Definition 2.3. Let X = {x0,x1, . . . ,xm}. The formal Laplace transform is defined as
� f : �−→R�LC
⟨〈X〉⟩
: Fc �−→ c.(2.2)
The corresponding inverse transform, the formal Borel transform, is
� f :R�LC
⟨〈X〉⟩−→�
: c �−→ Fc.(2.3)
Note that when m= 0, this definition is consistent with that given in (1.3) in the sensethat 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 � f [u]= cu =� f [Fcu].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 areneeded.
Definition 2.4. For any xi ∈ X , the left-shift operator, x−1i :R�〈〈X〉〉 →R�〈〈X〉〉, is defined
as
x−1i (c)=
∑
η∈X∗(c,η)x−1
i (η), (2.4)
where
x−1i (η)=
⎧⎪⎨⎪⎩
η′ : η = xiη′, η′ ∈ X∗,
0 : otherwise.(2.5)
Y. Li and W. S. Gray 5
Definition 2.5. A Dirac series, δi, is a generalized series with the defining property thatFδi[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�LC〈〈X〉〉 and scalars α,β ∈ R, the following identities
hold.(1) Linearity:
� f[αFc +βFd
]= α� f[Fc]
+β� f[Fd],
� f[αc+βd
]= α� f [c] +β� f [d].(2.6)
(2) Integration:
� f[InFc
]= xn0c,
� f[xn0c
]= InFc,(2.7)
where In(·) is the nth order integration operator so that
InFc[u](t)=∫ t
0
∫ τ1
0···
∫ τn−1
0Fc[u]
(τn)dτn ···dτ2dτ1. (2.8)
(3) Differentiation:
� f[DFc
]= x−10 (c) +
m∑
i=1
δi��(x−1i (c)
),
� f
[x−1
0 (c) +m∑
i=1
δi��(x−1i (c)
)]=DFc.
(2.9)
If xn0 is a left factor of c, that is, c = xn0c′ for some c′ ∈R�
LC〈〈X〉〉, then
� f[DnFc
]= x−n0 (c),
� f[x−n0 (c)
]=DnFc,(2.10)
whereDn(·) is the nth-order differentiation operator so thatDnFc[u](t)=dnFc[u](t)/dtn, and �� denotes the shuffle product (see, e.g., [21]).
(4) Multiplication:
� f[Fc ·Fd
]=� f[Fc]��� f
[Fd],
� f [c��d]=� f [c] ·� f [d].(2.11)
Proof. The properties of linearity and integration are trivial. The multiplication propertyfollows from results in the literature concerning the shuffle product [22, 24]. Only thedifferentiation property remains to be justified. It is shown in [24] that the derivative of
6 The formal Laplace-Borel transform of Fliess operators
a Fliess operator is
d
dtFc[u](t)= Fx−1
0 (c)[u](t) +m∑
i=1
ui(t)Fx−1i (c)[u](t). (2.12)
Applying the formal Laplace transform to this equality gives the first pair of equationsin part (3). Now if x0 is a left factor of c, then Fx−1
i (c)[u](t) = 0 for i = 1,2, . . . ,m. In thiscase, dFc[u](t)/dt = Fx−1
0 (c)[u](t). Proceeding inductively, the second pair of equationsfollows. �
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 operatorapplied 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′). Inwhich case, there exists a c−1 ∈R〈〈X〉〉 such that under the catenation product cc−1 = 1and c−1c = 1. Specifically,
c−1 = 1(c,∅)
(1− c′)−1 = 1(c,∅)
(c′)�. (2.14)
Example 2.8. Let X = {x0,x1,x2} and Fc[u](t)=exp(∫ t
0 u1(t)+u2(t)dt). Observe that Fc[u]can be expanded as
Fc[u](t)=∑
n≥0
1n!
(∫ t
0u1(t) +u2(t)dt
)n
=∑
n≥0
∫ t
0
[u1(τ1)
+u2(τ1)]∫ τ1
0
[u1(τ2)
+u2(τ2)]
×···∫ τn−1
0
[u1(τn)
+u2(τn)]dτn ···dτ2dτ1.
(2.15)
Therefore,
� f[Fc]=
∑
n≥0
(x1 + x2
)n = (x1 + x2)�
. (2.16)
This result can be viewed as an operator generalization of the integral transform pair
et�⇐⇒ (1− s)−1. (2.17)
Other formal Laplace-Borel transform pairs are given in Table 2.1. (See [24, Example2.3.9] for additional discussion related to this example.)
Y. Li and W. S. Gray 7
Table 2.1. Some formal Laplace-Borel transform pairs.
Fc � f
[Fc]
Fc : u �−→ 1 1
Fc : u �−→ tn n!xn0
Fc : u �−→(n−1∑
i=0
(i
n−1
)
i!aiti)eat
(1− ax0
)−n
Fc : u �−→ 1n!
(∫ t
t0
k∑
j=1
uij (τ)dτ
)n(xi1 + xi2 + ···+ xik
)n
Fc : u �−→∑
n≥0
ann!
(∫ t
t0
k∑
j=1
uij (τ)dτ
)n ∑
n≥0
an(xi1 + xi2 + ···+ xik
)n
Fc : u �−→ exp
(∫ t
t0
k∑
j=1
uij (τ)dτ
)(xi1 + xi2 + ···+ xik
)�
Fc : u �−→∫ t
t0
k∑
j=1
uij (τ)dτ exp
(∫ t
t0
k∑
j=1
uij (τ)dτ
)xi1 + xi2 + ···+ xik[
1− (xi1 + xi2 + ···+ xik)]2
Fc : u �−→ cos
(∫ t
t0
k∑
j=1
uij (τ)dτ
)1
1 +(xi1 + xi2 + ···+ xik
)2
Fc : u �−→ sin
(∫ t
t0
k∑
j=1
uij (τ)dτ
)xi1 + xi2 + ···+ xik
1 +(xi1 + xi2 + ···+ xik
)2
Example 2.9. Let X = {x0,x1, . . . ,xm}. Suppose Fc has the generating series c =∑η∈X∗ η,and Fξ is given for some fixed word ξ ∈ X∗. Then
� f[Fc ·Fξ
]=� f[Fc]��� f
[Fξ]= c��ξ
=∑
ν∈X∗
(νξ
)ν,
(2.18)
where ( νξ ) 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
In this section, the formal Laplace-Borel transform of the composition of two Fliess op-erators is considered. This leads to an important semigroup isomorphism between theset of all Fliess operators and the set of all locally convergent formal power series withcompatible dimensions. In system theory applications, this analysis is useful for modelsconsisting of cascaded nonlinear input-output systems. The definition of the compositionproduct first appeared in [4, 5]. Its set of known properties was significantly expanded in[13–17, 20]. The definition is constructed recursively in terms of the shuffle product.
8 The formal Laplace-Borel transform of Fliess operators
Definition 3.1. For any c ∈R�〈〈X〉〉 and d ∈Rm〈〈X〉〉, the composition product is definedas
c ◦d =∑
η∈X∗(c,η)η ◦d, (3.1)
where
η ◦d =⎧⎪⎨⎪⎩
η : |η|xi = 0, ∀i �= 0,
xn+10
[di��(η′ ◦d)
]: η = xn0xiη
′, 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, theith component of (d,ξ).)
Consequently, if
η = xnk0 xikxnk−10 xik−1 ···xn1
0 xi1xn00 , (3.3)
where i j �= 0 for j = 1, . . . ,k, it follows that
η ◦d = xnk+10
[dik��xnk−1+1
0
[dik−1��···xn1+1
0
[di1��xn0
0
]···]]. (3.4)
Alternatively, for any η ∈ X∗ one can uniquely define a set of right factors {η0,η1, . . . ,ηk}of η by the iteration
ηj+1 = xnj+1
0 xij+1ηj , η0 = xn00 , i j+1 �= 0, (3.5)
so that η = ηk with k = |η| − |η|x0 . In which case, η ◦ d = ηk ◦ d, where ηj+1 ◦ d =xnj+1+10 [dij+1��(ηj ◦ d)] and η0 ◦ d = xn0
0 . It was shown in [15, 16] that the compositionproduct of two series is always well defined since the family of series {η ◦ d : η ∈ X∗} islocally finite for any fixed d ∈Rm〈〈X〉〉.
It is easily verified that for any series c,d,e ∈Rm〈〈X〉〉, the composition product is leftdistributive 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 linearseries. A series c is linear if
supp(c)⊆ {η ∈ X∗ : η = xn10 xix
n00 , i∈ {1,2, . . . ,m}, n1,n0 ≥ 0
}. (3.7)
The composition product is associative, that is, (c ◦d)◦ e= c ◦ (d ◦ e); hence (Rm〈〈X〉〉,◦)forms a semigroup [4, 20]. In [16], it is shown that the composition product of two locallyconvergent formal power series is always locally convergent; therefore the setRm
LC〈〈X〉〉 isclosed under composition, and (Rm
LC〈〈X〉〉,◦) 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�
LC〈〈X〉〉 and d ∈ RmLC〈〈X〉〉, the cascade connection of two
Y. Li and W. S. Gray 9
(Fc, Fd) Fc Æ Fd = FcÆd
(c, d) c Æ d
Æ
Æ
� f � f � f� f
Figure 3.1. The isomorphism between the semigroups (�,◦) and (RmLC〈〈X〉〉,◦).
Fliess operators is always another Fliess operator with generating series c ◦d, that is,
Fc ◦Fd = Fc◦d. (3.8)
Therefore, � forms a semigroup under operator composition when � =m. The followingtheorem shows that composition is preserved in a natural sense under the formal Laplace-Borel transform.
Theorem 3.2. Let X = {x0,x1, . . . ,xm}. For any c ∈R�LC〈〈X〉〉 and d ∈Rm
LC〈〈X〉〉,
� f(Fc ◦Fd
)=� f(Fc)◦� f
(Fd),
� f (c ◦d)=� f (c)◦� f (d).(3.9)
Proof. The proof follows directly from the definitions. For any well-defined Fc and Fd,
� f(Fc ◦Fd
)=� f(Fc◦d
)= c ◦d =� f(Fc)◦� f
(Fd). (3.10)
Similarly, for any locally convergent formal power series c and d,
� f (c ◦d)= Fc◦d = Fc ◦Fd =� f (c)◦� f (d). (3.11)�
From Theorem 3.2, when � =m, the formal Laplace-Borel transform provides an iso-morphism between the two semigroups (�,◦) and (Rm
LC〈〈X〉〉,◦), as shown in Figure 3.1.
Example 3.3. Let X = {x0,x1,x2}, Fc[u](t) = cos(∫ t
0 u1(t) + u2(t)dt), and d = (d1 d2)T ∈R2
LC〈〈X〉〉. Define
Fe[u](t)= (Fc ◦Fd)[u](t)= cos
(∫ t
0Fd1 [u](t) +Fd2 [u](t)dt
). (3.12)
The formal Laplace transform of Fe is then
� f[Fe]= c ◦d =
∑
i≥0
(−1)i(x1 + x2
)2i ◦d
= [− (x1 + x2)2]� ◦d = 1
1 +(x1 + x2
)2 ◦d.(3.13)
10 The formal Laplace-Borel transform of Fliess operators
Example 3.4. Let X = {x0,x1, . . . ,xm} and c ∈RmLC〈〈X〉〉. It is easily verified by induction
that for n≥ 1,
xni ◦ c =1n!
(x0ci
)��n, i= 1,2, . . . ,m, (3.14)
where (·)��n denotes the shuffle power.Applying the formal Borel transform to both sides of this identity gives
� f[xni ◦ c
]=� f
[ 1n!
(x0ci
)��n]
= 1n!
[� f
[x0ci
]]n
= 1n!
[∫ t
0Fci[u](τ)dτ
]n.
(3.15)
Example 3.5. First consider the linear ordinary differential equation
dny(t)dtn
+n−1∑
i=0
aidi y(t)dti
=n−1∑
i=0
bidiu(t)dti
(3.16)
with initial conditions y(i)(0) = 0, u(i)(0) = 0, and where ai,bi ∈ R for i = 0,1, . . . ,n− 1.Integrate both sides of the equation n times and assume there exists a c ∈RLC〈〈X〉〉 suchthat y = Fc[u]. Applying the formal Laplace transform to both sides of the equation gives
(δ +
n−1∑
i=0
aixn−1−i0 x1
)◦ c =
n−1∑
i=0
bixn−1−i0 x1,
(1 +
n−1∑
i=0
aixn−i0
)c =
n−1∑
i=0
bixn−1−i0 x1.
(3.17)
Therefore,
c =(
1 +n−1∑
i=0
aixn−i0
)−1 n−1∑
i=0
bixn−1−i0 x1. (3.18)
Rephrased in the language of the integral Laplace transform, this is equivalent to
Y(s)=(
1 +n−1∑
i=0
ai1sn−i
)−1(n−1∑
i=0
bi1sn−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
dny(t)dtn
+n−1∑
i=0
aidi y(t)dti
+k∑
j=2
pju(t)y j(t)=n−1∑
i=0
bidiu(t)dti
(3.20)
Y. Li and W. S. Gray 11
with y(i)(0)= 0, u(i)(0)= 0, and where ai,bi, pj ∈R for i= 0,1, . . . ,n− 1 and j = 2, . . . ,k.Again integrate both sides of the equation n times and assume y = Fc[u]. Applying theformal Laplace transform gives
(1 +
n−1∑
i=0
aixn−i0
)c+
k∑
j=2
pjxn−10 x1
(c �� j
)=n−1∑
i=0
bixn−1−i0 x1. (3.21)
As in [12], a recursive procedure can be applied to solve the algebraic equation iterativelyso that
c = c1 + c2 + ··· (3.22)
with
c1 =(
1 +n−1∑
i=0
aixn−i0
)−1 n−1∑
i=0
bixn−1−i0 x1 (3.23)
and for n≥ 2
cn =(
1 +n−1∑
i=0
aixn−i0
)−1
xn−10 x1
k∑
j=2
pj
∑
ν1≥1,...,ν j≥1ν1+ν2+···+ν j=n
cν1��cν2��···��cν j . (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 ofan analytic nonlinear system represented by a Fliess operator. In this section, the formalLaplace-Borel transform pair in conjunction with the composition product are employedto produce a clear and compact interpretation of this methodology.
Theorem 4.1. Let Fc ∈� and let u be an analytic function with formal Laplace transformcu ∈ Rm
LC〈〈X0〉〉. Then the output function y = Fc[u] is analytic and has formal Laplacetransform cy = c ◦ cu ∈R�
LC〈〈X0〉〉.Proof. The analyticity of y follows from [24, Lemma 2.3.8]. Let cy denote its formalLaplace transform. Local convergence is preserved under the composition product; there-fore the formal power series c ◦ cu is locally convergent, and, for any admissible inputv ∈ 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 computingthe output response of an analytic input-output system.
Example 4.2. Consider the linear time-invariant system y(t) = ∫ t0 h(t− τ)u(τ)dτ, whereh is analytic at t = 0. Then y = Fc[u] with (c,xk0x1) = h(k)(0), k ≥ 0, and zero otherwise.
12 The formal Laplace-Borel transform of Fliess operators
uz y
∫1
1� z
Figure 4.1. A simple Wiener system.
If u(t)=∑k≥0(cu,xk0)tk/k!, then it follows that y(t)=∑n≥0(cy ,xn0 )tn/n!, where
cy = c ◦ cu=∑
k≥0
(c,xk0x1
)xk0x1 ◦ cu
=∑
k≥0
(c,xk0x1
)xk+1
0 cu.
(4.2)
Therefore,
(cy ,xn0
)=n−1∑
k=0
(c,xk0x1
)(cu,xn−1−k
0
), n≥ 1, (4.3)
which is the conventional convolution sum.
Example 4.3. Consider the simple Wiener system shown in Figure 4.1 where z(0) = 0.The mapping u �→ y can be written as
y(t)=∞∑
n=0
(Ex1 [u](t)
)n =∞∑
n=0
Fx �� n1
[u](t)=∞∑
n=0
n!Exn1 [u](t). (4.4)
Therefore y = Fc[u] with c = ∑n≥0n!xn1 . When u(t) = tk/k!, for example, the formalLaplace transform of u is cu = xk0 . From Theorem 4.1 and (3.14) it follows that
cy =∞∑
n=0
n!xn1 ◦ xk0 =∞∑
n=0
(xk+1
0
)��n =∞∑
n=0
((k+ 1)n
)!
((k+ 1)!
)n x(k+1)n0 . (4.5)
Consequently, the output response is
y(t)=∞∑
n=0
((k+ 1)n
)!
((k+ 1)!
)nt(k+1)n
((k+ 1)n
)!=
∞∑
n=0
t(k+1)n
((k+ 1)!
)n = 11− tk+1/(k+ 1)!
. (4.6)
5. Conclusion
In this paper, the formal Laplace-Borel transform of a Fliess operator is defined and de-veloped. This concept provides an isomorphism between the semigroup of all Fliess oper-ators under composition and the semigroup of all locally convergent formal power seriesunder the composition product. As an application in system theory, an explicit relation-ship is derived between the formal Laplace-Borel transforms of the input and outputfunctions of a Fliess operator, which is a compact interpretation of the operational calcu-lus of Fliess.
Y. Li and W. S. Gray 13
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Yaqin Li: Department of Electrical and Computer Engineering, University of Memphis, Memphis,TN 38152, USAE-mail address: [email protected]
W. Steven Gray: Department of Electrical and Computer Engineering, Old Dominion University,Norfolk, VA 23529, USAE-mail address: [email protected]
