Some Conclusions about α-times Integrated C Semigroups

Chunjing Liu and Xiaoqiu SongMathematic Department of College of ScienceChina University of Mining and Technology

Jiangsu, Xuzhou,China, 221008

Tao WangDepartment of Mathematics

Jiangsu Houji Senior High SchoolJiangsu, Xuzhou,China, 221008

Abstract— This paper gets two main conclusions about α-times integrated C semigroups based on the Hille exponentialformulas of the other kinds of semigroups, the generationtheorem in terms of the Laplace transform and the propertiesof exponential bounded α-times integrated C semigroups. Oneis the representation theorem, the other is the Laplace inversetransform as well as two deductions.

Index Terms— Intelligent Control; α-times integrated C semi-groups; Representation theorem; Exponential formulas; Laplaceinverse transform

I. INTRODUCTION AND PRELIMINARIES

Functional analysis is one of the mathematics foundations

that control theory and system science develope on. The basic

concepts of functional analysis form at the end of the 19th

century and the beginning of the 20th century, after decades

of development, as an independent branch of mathematics,

it becomes mature enough and seepage in various fields

gradually, including mechanical, electromagnetic field theory,

control theory and system science, etc. Operator theory,

especially the linear operator theory, is the main theory of

the functional analysis, the solving of operator equations

and the reaserch on the solvability of linear operators, have

provided theory basis for the solving of all sorts of algebraic

equations, the differential equations and the comprehensive

control system. Almost all the problems in control theory

can be described in terms of space and operator involved in

functional analysis. For example, the theory of the dual space

and the adjoint operator can explain almost all the duality

theorems in control theory, and the discovery of all these

theorems is the result of the direct deduction of mathematics.

Semigroup of operator, as an important branch of the

operator theory, has an abroad application in partial differ-

ential equations system theory, distributed parameter control

theory, the approximation and quantum mechanics theory.

For example, the solution of the control system model that

represented in developing equation in Banach space mainly

depends on Hille Yosida semigroup and Goldstein semigroup

in distributed parameter control system.

∗This work is partially supported by The Fundamental Research Fundsfor the Central Universities(2010LKSXO8)

The classical theory of C0 semigroups of operators [1] is a

powerful method in studying the first order Cauchy problem.

Later, two kinds of generalizations of C0 semigroups were

developed by some authors. One is C semigroups, it was

studied first by Da Prato [2] and later by Davies and Pang

[3]. The other is n-times integrated semigroups, which was

introduce by Arendt [4] firstly, and further developed by

Kellerman and Hieber [5], further extension to fractionally

integrated semigroups were carried out by Hieber [6]. Re-

cently, Li Yuanchuan and Shaw SenYen [7] have studied

a nature generalization of above two notations to a wider

class of operator families, called n- times integrated Csemigroups, and then it was generalized to α- times integrated

C semigroups by Kuo Chungcheng and Shaw SenYen [8].Based on these results, this paper aims to study the

representation theorem and the Laplace inverse transform of

α-times integrated C semigroups which have considerable

practical significance.

Next, we give the notations that will be used the in the

following:

let X be a Banach space and B(X) be the set of all

bounded linear operators from X to itself, ρ(A) is the

resolvent set of A.

Let j−1 := δ0, the Dirac measure at 0, and for r > −1,

letjr(t) = tr

Γ(r+1) , where Γ(·) is the Gamma function.

Let f(·) be a continuous function, β ≥ −1, the convolution

is defined as follows:

(jβ ∗ f)(t) =

{ ∫ t

0

(t−s)β

Γ(β+1) f(s)ds, β>−1∫ t

0f(t−s)dδ0(s)=f(t), β=−1

,

For α > 0, let [α], (α) be the integer part, the decimal

part of α, and Cα([0, T ], X) be the functions that are αorder continuously differentiable on [0, T ]. Now we define

the fractional differential and integral of functions:

The α times repeated integral (Iαf)(t) = (jα−1 ∗ f)(t) (1)The α order derivative

(Dαf)(t) = (d

dt)[α]+1(j−(α) ∗ [f − f(0)])(t) (2)

DEFINITION1.1. let C ∈B(X) and α ≥ 0. A family

{S(t) : t ≥ 0} in B(X) is called an α-times integrated

C semigroup(see [7] for the case α = n ∈ N) if

2011 International Conference on Internet Computing and Information Services

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DOI 10.1109/ICICIS.2011.22

61

1) S(·)x :[0,∞)→X is continuous for each x∈X;

2) S(t)S(s)x = 1Γ(α) [

∫ s+t

s− ∫ t

0](s + t− r)α−1S(r)Cxdr

for x∈X and s, t ≥ 0, and CS(0) = S(0)C.

It is called a C semigroup(see [2][3]) if

2′) S(0) = C and S(t)S(s) = S(t + s)C for all s, t ≥ 0.

S(·) is said to be nondegenerate if

3) S(t)x = 0 for all t > 0 implies x = 0.

S(·) is said to be exponentially bounded if

4) there are constants M, ω > 0 such that ‖S(t) ‖≤ Meωt

for all t ≥ 0When C = I , an α-times integrated C semigroup reduce to

an α-times integrated semigroup.

Let α ≥ 0, the generator A of a nondegenerate α-times

integrated C semigroup S(·) is defined as

5) x ∈ D(A) and for all t ≥ 0

Ax = y ⇔ S(t)x − jα(t)Cx =∫ t

0

S(s)yds

LEMMA1.2. Let C∈B(X) be an injection, α ≥ 0, a

strong continuous family {S(t) : t ≥ 0} in B(X) is called

an exponentially bounded α-times integrated C semigroup

with generator A, if and only if {S(t) : t ≥ 0} satisfies 4),S(0) = 0, CS(·) = S(·)C, and for each λ > ω, x ∈ X

R(λ,A)Cx = λα

∫ ∞

0

e−λtS(t)xdt

LEMMA1.3. Let A be the generator of an nondegenerate

α-times integrated C semigroup {S(t) : t ≥ 0}, then

1) S(t)x ∈ D(A) and AS(t)x = S(t)Ax for all x ∈ D(A)and t ≥ 0;

2)∫ t

0S(s)xds ∈ D(A) and A

∫ t

0S(s)xds = S(t)x −

jα(t)Cx for all x ∈ X and t ≥ 0.

II. REPRESENTATION THEOREM

In this section, first we get some properties about α-times

integrated C semigroup: then based on the Hille exponential

formulas of the other kinds of semigroups ([1], [9] − [11]),we deduce the representation theorem.

THEOREM2.1. Let [0,∞) ⊂ ρ(A), α, β ≥ 0, t ≥0, {S(t)} is an α-times integrated C semigroup, then T (t) =(jβ ∗ S)(t) is an α + β + 1-times integrated C semigroup.

Proof. When S(t) is an α-times integrated C semigroup,

R(λ,A)Cx = λα

∫ ∞

0

e−λtS(t)xdt for each λ > ω

so

∫ ∞

0

e−λtT (t)xdt =∫ ∞

0

e−λt(jβ ∗ S)(t)xdt

=1

λβ+1

∫ ∞

0

e−λtS(t)xdt

=1

λα+β+1R(λ,A)Cx

that is proved by lemma1.2.

THEOREM2.2. Let A be the generator of an exponentially

bounded α-times integrated C semigroup {S(t) : t ≥ 0},

then

1) S(t)x ∈ Ck(R+, X), and Sk(t)x = S(t)Akx +∑ki=1 jα−i(t)CAk−ix for all x ∈ D(Ak), k = 1, 2, · · · [α];2) S(t)x ∈ Cθ(R+, X), and (DθS)(t)x =∑[θ]+1j=1 jα−(θ)+1−j(t)CA[θ]+1−jx+(j−(θ)∗S)(t)A[θ]+1x for

all x ∈ D(A[θ]+1), 0 ≤ θ < α;

3) (DθS)(0)x = 0 for all x ∈ D(A[θ]+1), 0 ≤ θ < α,

and (DαS)(0)x = x when x ∈ D(A[α]+1)Proof. S(t)x ∈ Ck(R+, X) is obvious from [6], now

differentiate the equation 2) in lemma1.3 with respect to t,we obtain the following equalities:

S′(t)x =αtα−1

Γ(α + 1)Cx + S(t)Ax

= jα−1(t)Cx + S(t)Ax

S′′(t)x = jα−2(t)Cx + S′(t)Ax

= jα−2(t)Cx + jα−1(t)CAx + S(t)A2x

...

S(k)(t)x = S(t)Akx +k∑

i=1

jα−i(t)CAk−ix

Thus 1) is proved.

To prove 2), by equation (2) and S(0) = 0, we have

(DθS)(t)x =d[θ]+1

dt(j−(θ) ∗ [S − S(0)])(t)x

=d[θ]+1

dt(j−(θ) ∗ S)(t)x

yet (j−(θ) ∗ S)(t) is an α − (θ) + 1-times integrated Csemigroup by theorem 2.1, so 2) is obtained from 1)by taking S(t) = (j−(θ)∗S)(t), k = [θ]+1, α = α−(θ)+1.

3) is easily got by taking t = 0 in 2).LEMMA2.3. Let A be a closed operator, {S(t) : t ≥ 0}

is an n-times integrated C semigroup with generator A, and

there exist constants M ′ ≥ 0, ω′ ∈ R, such that ‖ S(n)(t) ‖≤M ′eω′t, then for any x ∈ X, and t ≥ 0, we have

S(n)(t)x = limm→∞(I − t

mA)−mCx

= limm→∞(

m

t)m(

m

t− A)−mCx

and the limit is uniform in t on any bounded interval.

THEOREM2.4. Let A be a closed operator, {S(t) : t ≥0} is an α-times integrated C semigroup with generator

A and there exist constants M ′ ≥ 0, ω′ ∈ R, such that

‖ (DαS)(t) ‖≤ M ′eω′t, then for any x ∈ X, and t ≥ 0, we

have

(DαS)(t)x = limm→∞(I − t

mA)−mCx

= limm→∞(

m

t)m(

m

t− A)−mCx

62

and the limit is uniform in t on any bounded interval.

Proof. Theorem2.1 and 2.2 yields that, for any x ∈X, and t ≥ 0

(DαS)(t)x =d[α]+1

dt(j−(α) ∗ S)(t)x

and (j−(α) ∗ S)(t) is [α] + 1-times integrated C semigroup.

According to lemma 2.3, we completes the proof by

replacing n with [α] + 1 and S(t) with (j−(α) ∗ S)(t).COROLLARY2.5. The condition is the same as theo-

rem2.4, then for any x ∈ D(A[α]+1) and t ≥ 0, we have

1) (j−(α) ∗ S)(t)x

= limm→∞J [α]+1((I − t

mA)−mCx)

= limm→∞J [α]+1((

m

t)m(

m

t− A)−mCx)

which J(f(t)) =∫ t

0f(r)dr, f : [0,∞) → X is continuous.

and 2) S(t)x

= limm→∞

1Γ(α)

∫ t

0

(t − s)α−1(I − s

mA)−mCxds

= limm→∞

1Γ(α)

∫ t

0

(t − s)α−1(m

s)m(

m

s− A)−mCxds

and the limit is uniform in t on any bounded interval.

Proof. In order to prove these two equalities, we only have

to repeatedly integrate the equality in theorem2.4 by [α] +1 and α times respectively, now we only prove 2):

S(t)x = (Iα(DαS))(t)x

= (jα−1 ∗ (DαS))(t)x = (tα−1

Γ(α)∗ (DαS)(t)x

= limm→∞

1Γ(α)

∫ t

0

(t − s)α−1(I − s

mA)−mCxds

= limm→∞

1Γ(α)

∫ t

0

(t − s)α−1(m

s)m(

m

s− A)−mCxds

III. LAPLACE INVERSE TRANSFORM

Shi Deming and Yang Deshan[12] have given the Laplace

inverse transform of C semigroups in 1944, then Laplace

inverse transform of n times integrated semigroups was

introduced by two different approaches, Cao Dexia[13] get

it based on the relations among these semigroups, recently

Rong Rong[14] obtain it with the results that Hening B. and

Neubrander[15] have got. In this section, we try to study the

two kinds of the Laplace inverse transform as well as two

deductions of exponentially bounded α times integrated Csemigroups, based on the generation theorem in terms of the

Laplace transform, the properties and the Gamma function.

LEMMA3.1. Let A be the infinitesimal generator of an

exponentially bounded C semigroup, ‖S(t)‖ ≤ Meωt, γ >max{0, ω}, then for all x ∈ D(A), we have∫ t

0

S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt(λ − A)−1Cx

dλ

λ

and the integral on the right hand converges uniformly in tfor t in bounded intervals.

LEMMA3.2. Let α ≥ 0, C ∈ B(X) be an injection, then

the following are equivalent:

1) A generates an exponentially bounded α-times inte-

grated C semigroup {S(t) : t ≥ 0.

2) There exist ω > 0, that (ω,∞) ⊂ ρ(A), and for all

μ > ω, A generates an exponentially bounded (μ − A)−αCsemigroup {T (t) : t ≥ 0, and

T (t) = (μ − A)−α(d

dt)[α]+1(j−(α) ∗ S)(t)

LEMMA3.3. ω ≥ 0, F [λ] : (ω,∞) → X, let F [λ]satisfies: F (λ) = λ

∫ ∞0

e−λtα(t)dt, α(t) = 0, and ‖α(t +h) − α(t)‖ ≤ Mheω(t+h), t, h ≥ 0, then

α(t) =i

2πi

∫ γ+i∞

γ−i∞eλtF (λ)

dλ

λ(γ > ∞)

and the integral on the right hand converges uniformly in tfor t in bounded intervals.

THEOREM3.4. Let A be a closed operator, C ∈ B(X)is injective, ρ(A) �= ∅, λ ∈ ρ(A), and A is the gen-

erator of an exponentially bounded α-times integrated Csemigroup,‖S(t)‖ ≤ Meωt(ω ≥ 0), γ > ω, then for all

x ∈ D(A), we obtain∫ t

0

S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λα

dλ

λ

=1

2πi

∫ γ+i∞

γ−i∞Iα(eλtR(λ,A)Cx)

dλ

λ

and the integral on the right hand converges uniformly in tfor t in bounded intervals.

Proof. For ‖S(t)‖ ≤ Meωt(ω ≥ 0), we let α(t) =∫ t

0S(s)xds, F (λ) = (λ−A)−1Cx

λα ,∀x ∈ D(A), obviously,

α(t) satisfies the conditions of Lemma3.3, and

F (λ) =(λ − A)−1Cx

λα=

∫ ∞

0

e−λtS(t)xdt

=∫ ∞

0

e−λtxd

∫ t

0

S(s)ds

= λ

∫ ∞

0

e−λt(∫ t

0

S(s)ds)xdt (λ > ω)

also satisfies the conditions of Lemma3.3, so we have∫ t

0

S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λα

dλ

λ(γ > ω)

63

the former part is proved now.

On the other hand, Lemma3.2 implies that A also generates

an exponentially bounded (μ−A)−αC semigroup {T (t) : t ≥0}, together with Lemma3.1, we get∫ t

0

T (s)xds

=1

2πi

∫ γ+i∞

γ−i∞eλt(λ − A)−1(μ − A)−αCx

dλ

λ

=1

2πi

∫ γ+i∞

γ−i∞eλt(λ − A)−1R(μ,A)αCx

dλ

λ

=R(μ,A)α

2πi

∫ γ+i∞

γ−i∞eλt(λ − A)−1Cx

dλ

λ

It follows from Lemma3.2 that

S(t) = (μ − A)α(IαT )(t) (3)

so∫ t

0S(s)xds

= (μ − A)α

∫ t

0

(IαT )(s)xds

= (μ − A)αIα(∫ t

0

T (s)xds)

= (μ − A)α R(μ,A)α

2πi

∫ γ+i∞

γ−i∞Iα(eλt(λ − A)−1Cx)

dλ

λ

=1

2πi

∫ γ+i∞

γ−i∞Iα(eλtR(λ,A)Cx)

dλ

λ

that is proved.

COROLLARY3.5. The condition is the same as theo-

rem3.4, then for all x ∈ X

S(t)x =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λαdλ

=1

2πi

∫ γ+i∞

γ−i∞Iα(eλtR(λ,A)Cx)dλ

and the integral on the right hand converges uniformly in tfor t in bounded intervals.

Proof. By theorem3.4,∫ t

0

S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λα

dλ

λ

multiply this equation by A that

A

∫ t

0

S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)ACx

λα

dλ

λ

then it is seen from Lemma1.3 and

tα

Γ(α + 1)=

12πi

∫ γ+i∞

γ−i∞eλtλ−α−1dλ

that S(t)x

=tα

Γ(α + 1)Cx +

12πi

∫ γ+i∞

γ−i∞eλt R(λ,A)ACx

λα

dλ

λ

=1

2πi

∫ γ+i∞

γ−i∞eλtλ−α−1Cxdλ

+1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)ACx

λα

dλ

λ

=1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λα(λ − A + A)

dλ

λ

=1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λαdλ

this completes the former part of the corollary, together with

theorem(3.4) and (3), the other part can be proved in the

same way.

COROLLARY3.6. The condition is the same as theo-

rem3.4, then all x ∈ X∫ t

0

(t − s)S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λα

dλ

λ2

=1

2πi

∫ γ+i∞

γ−i∞Iα(eλtR(λ,A)Cx)

dλ

λ2

and the integral on the right hand converges uniformly in tfor t in bounded intervals.

Proof. Theorem3.4 shows∫ t

0

S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λα

dλ

λ

integrating the equation from 0 to t, we get∫ t

0

(t − s)S(s)xds =1

2πi

∫ t

0

∫ γ+i∞

γ−i∞eλs R(λ,A)Cx

λα

dλ

λds

=1

2πi

∫ γ+i∞

γ−i∞(eλt − 1)

R(λ,A)Cx

λα

dλ

λ2

and for1

2πi

∫ γ+i∞

γ−i∞

R(λ,A)Cx

λα

dλ

λ= 0

so∫ t

0

(t − s)S(s)xds =1

2πi

∫ γ+i∞

γ−i∞eλt R(λ,A)Cx

λα

dλ

λ2

the other part of this corollary can be got similarly from

theorem(3.4) and (3).

ACKNOWLEDGMENT

The authors would like to thank the referees for their

valuable suggestions, this paper is supported in part by

The Fundamental Research Funds for the Central Univer-

sities(2010LKSXO8).

64

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