# Approximate Controllability of Fractional Sobolev-Type Evolution Equations in Banach Spaces

Authors: N. I. Mahmudov
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. The results of the paper are generalization and continuation of the recent results on this issue.


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## Body

## 1. Introduction

Many social, physical, biological, and engineering problems can be described by fractional partial differential equations. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. In the last two decades, fractional differential equations (see Samko et al. [1] and the references therein) have attracted many scientists, and notable contributions have been made to both theory and applications of fractional differential equations.Recently, the existence of mild solutions and stability and (approximate) controllability of (fractional) semilinear evolution system in Banach spaces have been reported by many researchers; see [2–36]. We refer the reader to El-Borai [3, 4], Balachandran and Park [5], Zhou and Jiao [6, 7] Hernández et al. [8], Wang and Zhou [9], Sakthivel et al. [12, 13], Debbouche and Baleanu [14], Wang et al. [15–21], Kumar and Sukavanam [22], Li and Yong [37], Dauer and Mahmudov [28], Mahmudov [27, 29], and the references therein. Complete controllability of evolution systems of Sobolev type in Banach spaces has been studied by Balachandran and Dauer [23], Ahmed [24], and Feckan et al. [2]. However, the approximate controllability of fractional evolution equations of Sobolev type has not been studied.Motivated by the above-mentioned papers, we study the approximate controllability of a class of fractional evolution equations of Sobolev type:(1)cDtα(Ex(t))=Ax(t)+Bu(t)+f(t,x(t)),t∈[0,b],x(0)=x0,
where A:D(A)⊂X→X and E:D(E)⊂X→X are linear operators from a Banach space X to X. The control function u takes values in a Hilbert space U and u∈L2([0,b],U). B:U→X is a linear bounded operator. The function f∈C([0,b]×X,X) will be specified in the sequel. The fractional derivative cDtα, 0<α<1, is understood in the Caputo sense.Our aim in this paper is to provide a sufficient condition for the approximate controllability for a class of fractional evolution equations of Sobolev type. It is assumed thatE-1 is compact, and, consequently, the associated linear control system (35) is not exactly controllable. Therefore, our approximate controllability results have no analogue for the concept of complete controllability. In Section 5, we give an example of the system which is not completely controllable, but approximately controllable.


## 2. Preliminaries

Throughout this paper, unless otherwise specified, the following notations will be used. LetX be a separable reflexive Banach space and let X* stand for its dual space with respect to the continuous pairing 〈·,·〉. We may assume, without loss of generality, that X and X* are smooth and strictly convex, by virtue of a renorming theorem (see, e.g., [37, 38]). In particular, this implies that the duality mapping J of X into X* given by the relations
(2)∥J(z)∥*=∥z∥,〈J(z),z〉=∥z∥2,∀z∈X
is bijective, homogeneous, demicontinuous, that is, continuous from X with a strong topology, into X* with weak topology, and strictly monotonic. Moreover, J-1:X*→X is also duality mapping.The operatorsA:D(A)⊂X→X and E:D(E)⊂X→X satisfy the following hypotheses:(S1)
A and E are linear operators, and A is closed;(S2)
D(E)⊂D(A) and E is bijective;(S3)
E-1:X→D(E) is compact.The hypotheses (S1)–(S3) and the closed graph theorem imply the boundedness of the linear operatorAE-1:X→X. Consequently, -AE-1 generates a semigroup {S(t);t≥0} in X. Assume that max0≤t≤b∥S(t)∥=:M.Let us recall the following known definitions in fractional calculus. For more details, see [1].Definition 1.
The fractional integral of orderα>0 with the lower limit 0 for a function f is defined as
(3)Iαf(t)=1Γ(α)∫0t‍f(s)(t-s)1-αds,t>0,α>0,
provided the right-hand side is pointwise defined on [0,∞), where Γ is the gamma function.Definition 2.
The Caputo derivative of orderα for a function f can be written as
(4)cDαf(t)=1Γ(n-α)∫0t‍f(n)(s)(t-s)1+α-nds=In-αf(n)(t),t>0,n-1≤α<n.
Iff is an abstract function with values in X, then integrals which appear in the above definitions are taken in Bochner’s sense.
Forx∈X and 0<α<1, we define two families {𝒮E(t):t≥0} and {𝒯E(t):t≥0} of operators by
(5)𝒮α(t)=∫0∞‍Ψα(θ)S(tαθ)dθ,𝒯α(t)=α∫0∞‍θΨα(θ)S(tαθ)dθ,𝒮E(t)=E-1𝒮α(t),𝒯E(t)=E-1𝒯α(t),
where
(6)Ψα(θ)=1πα∑n=1∞‍(-1)n-1Γ(nα+1)n!sin(nπα),θ∈(0,∞),
is the function of Wright type defined on (0,∞), which satisfies
(7)Ψα(θ)≥0,∫0∞‍Ψα(θ)dθ=1,∫0∞‍θζΨα(θ)dθ=Γ(1+ζ)Γ(1+αζ),ζ∈(-1,∞).Lemma 3 (see [2]).
The operators𝒮E and 𝒯E have the following properties. (i)
For any fixedt≥0, SE(t) and TE(t) are linear and bounded operators, and
(8)∥𝒮E(t)x∥≤M∥E-1∥∥x∥,∥𝒯E(t)x∥≤M∥E-1∥Γ(α)∥x∥.(ii)
{𝒮E(t):t≥0} and {𝒯E(t):t≥0} are compact.In this paper, we adopt the following definition of mild solution of (1).Definition 4.
A solutionx(·;u)∈C([0,b],X) is said to be a mild solution of (1) if for any u∈L2([0,b],U), the integral equation
(9)x(t)=𝒮E(t)Ex0n+∫0t‍(t-s)α-1𝒯E(t-s)[Bu(s)+f(s,x(s))]ds
is satisfied.Letx(b;u) be the state value of (9) at terminal time b corresponding to the control u. Introduce the set ℜ(b)={x(b;u):u∈L2([0,b],U)}, which is called the reachable set of system (9) at terminal time b; its closure in X is denoted by ℜ(b)¯.Definition 5.
System (1) is said to be approximately controllable on [0,b] if ℜ(b)¯=X; that is, given an arbitrary ε>0, it is possible to steer from the point x0 to within a distance ε from all points in the state space X at time b.To investigate the approximate controllability of system (9), we assume the following conditions.(H4)
The functionf:[0,b]×X→X satisfies the following:
(a)
f(t,·):X→X is continuous for each t∈[0,b] and for each x∈X,  f(·,x):[0,b]→X is strongly measurable;(b)
there is a positive integrable functionn∈L1([0,b],[0,+∞)) and a continuous nondecreasing function Λf:[0,∞)→(0,∞) such that for every (t,x)∈[0,b]×X, we have
(10)∥f(t,x)∥≤n(t)Λf(∥x∥),liminfr→∞Λf(r)r=σf<∞.(H5)
The following relationship holds:(11)(1+1εMB2M𝒯2b2α-12α-1)M∥E-1∥Γ(α)bααnnnnnnnnnnn×σfsups∈[0,b]n(s)<1,
here MB∶=∥B∥,M𝒯∶=∥𝒯E∥.(H6)
For everyh∈X,zα(h)=ε(εI+Γ0bJ)-1(h) converges to zero as ε→0+ in strong topology, where
(12)Γ0b∶=∫0b‍(b-s)2(α-1)𝒯E(b-s)BB*𝒯E*(b-s)ds,
and zε(h) is a solution of
(13)εzε+Γ0bJ(zε)=εh.


## 3. Existence Theorem

In order to formulate the controllability problem in the form suitable for application of fixed point theorem, it is assumed that the corresponding linear system is approximately controllable. Then it will be shown that system (1) is approximately controllable if for all ε>0, there exists a continuous function x∈C([0,b],X) such that
(14)uε(t,x)=(b-t)α-1B*𝒯E*(b-t)J((εI+Γ0bJ)-1p(x)),nnnnnnnnnnnnnnnnnnnnnnnn0≤t<b,x(t)=𝒮E(t)Ex0+∫0t‍(t-s)α-1𝒯E(t-s)[Bu(s)+f(s,x(s))]ds,
where
(15)p(x)=h-𝒮E(b)Ex0-∫0b‍(b-s)α-1𝒯E(b-s)f(s,x(s))ds.
Having noticed this fact, our goal, in this section, is to find conditions for the solvability of (14). It will be shown that the control in (14) drives system (1) from x0 to
(16)h-εJ((εI+Γ0bJ)-1p(x)),
provided that system (14) has a solution.Theorem 6.
Assume that assumptions (S1)–(S3), (H4), (H5) hold and1/2<α≤1. Then there exists a solution to (14).Proof.
The proof of Theorem6 follows from Lemmas 7–9, infinite dimensional analogue of Arzela-Ascoli theorem, and the Schauder fixed point theorem.For allε>0, consider the operator Φε:C([0,b],X)→C([0,b],X) defined as follows:
(17)(Φεx)(t)n∶=𝒮E(t)Ex0+∫0t‍(t-s)α-1𝒯E(t-s)vvvvvvvvvvvvvvnnnn×[Buε(s,x)+f(s,x(s))]ds,
where
(18)uε(t,x)=(b-t)α-1B*𝒯E*(b-t)J((εI+Γ0bJ)-1p(x)),p(x)=h-𝒮E(b)Ex0-∫0b‍(b-s)α-1𝒯E(b-s)f(s,x(s))ds.It will be shown that for allε>0, the operator Φε:C([0,b],X)→C([0,b],X) has a fixed point. To prove this we will employ the Schauder fixed point theorem.Lemma 7.
Under assumptions (S1)–(S3), (H4), (H5), for anyε>0 there exists a positive number r∶=r(ε) such that Φε(Br)⊂Br.Proof.
Letε>0 be fixed. If it is not true, then for each r>0, there exists a function zr∈Br, but Φε(zr)∉Br. So for some t=t(r)∈[0,b], one can show that
(19)r≤∥(Φεzr)(t)∥≤∥𝒮E(t)Ex0∥+∥∫0t‍(t-s)α-1𝒯E(t-s)f(s,x(s))ds∥+∥∫0t‍(t-s)α-1𝒯E(t-s)Buε(s,x)ds∥=:I1+I2+I3.
Let us estimate Ii,i=1,2,3. By the assumption (H4), we have
(20)I1≤∥𝒮E(t)Ex0∥≤M∥E-1∥∥Ex0∥,(21)I2≤∫0t‍∥(t-s)α-1𝒯E(t-s)(t-s)f(s,x(s))∥ds≤M∥E-1∥Γ(α)∫0t‍(t-s)α-1∥f(s,x(s))∥ds≤M∥E-1∥Γ(α)∫0t‍(t-s)α-1n(s)Λf(∥x(s)∥)ds≤M∥E-1∥Γ(α)bααΛf(r)sups∈Jn(s).
Combining the estimates (19)–(21) yields
(22)I1+I2<M∥E-1∥∥Ex0∥+M∥E-1∥Γ(α)bααΛf(r)sups∈[0,b]n(s)∶=Δ.
On the other hand,
(23)I3≤∫0t‍∥(t-s)α-1𝒯E(t-s)Buε(s,x)∥ds=∫0t‍∥((εI+Γ0bJ)-1p(x))(t-s)α-1(b-s)α-1𝒯E(t-s)nn×BB*𝒯E*(b-t)J((εI+Γ0bJ)-1p(x))∥ds≤∫0t‍∥(t-s)α-1(b-s)α-1𝒯E(t-s)BB*𝒯E*(b-t)∥dsnn×∥J((εI+Γ0bJ)-1p(x))∥≤MB2M𝒯2b2α-12α-1∥J((εI+Γ0bJ)-1p(x))∥=MB2M𝒯2b2α-12α-1∥(εI+Γ0bJ)-1p(x)∥≤1εMB2M𝒯2b2α-12α-1∥p(x)∥≤1εMB2M𝒯2b2α-12α-1Δ.
Thus,
(24)r≤∥(Φεzr)(t)∥≤Δ+1εMB2M𝒯2b2α-12α-1Δ=(1+1εMB2M𝒯2b2α-12α-1)Δ.
Dividing both sides by r and taking r→∞, we obtain that
(25)(1+1εMB2M𝒯2b2α-12α-1)M∥E-1∥Γ(α)bαασfsups∈[0,b]n(s)≥1,
which is a contradiction by assumption (H5). Thus, Φε(Br)⊂Br for some r>0.Lemma 8.
Let assumptions (S1)–(S3), (H4), (H5) hold. Then the set{Φεz:z∈Br} is an equicontinuous family of functions on [0,b].Proof.
Let0<η<t<b and δ>0 such that
(26)∥𝒯E(s1)-𝒯E(s2)∥<η
for every s1,s2∈[0,b] with |s1-s2|<δ. For z∈Br, 0<|h|<δ,t+h∈[0,b], we have
(27)∥(Φεz)(t+h)-(Φεz)(t)∥n≤∥∫0t‍((t+h-s)α-1-(t-s)α-1)nnnnn×𝒯E(t+h-s)[Buε(s,z)+f(s,z(s))]ds∫0t‍∥nn+∥∫tt+h‍(t+h-s)α-1𝒯E(t+h-s)nnnnn×[Buε(s,z)+f(s,z(s))]ds∫tt+h‍∥nn+∥∫0t‍(t-s)α-1(𝒯E(t+h-s)-𝒯E(t-s))nnnnn×[Buε(s,z)+f(s,z(s))]ds∫0t‍∥.
Applying Lemma 3 and the Holder inequality, we obtain
(28)∥(Φεz)(t+h)-(Φεz)(t)∥n≤M∥E-1∥Γ(α)Λf(r)nn×∫0t‍((t+h-s)α-1-(t-s)α-1)n(s)dsnn+M∥E-1∥Γ(α)1εMBM𝒯Δnn×∫0t‍((t+h-s)α-1-(t-s)α-1)(b-s)α-1dsnn+M∥E-1∥Γ(α)Λf(r)∫tt+h‍(t+h-s)α-1n(s)dsnn+M∥E-1∥Γ(α)1εMBM𝒯Δnn×∫tt+h‍(t+h-s)α-1(b-s)α-1dsnn+ηTααΛf(r)∫0t‍(t-s)α-1n(s)ds+ηTαα1εMBM𝒯Δn+ηTααΛf(r)∫0t‍n×∫0t‍(t-s)α-1(b-s)α-1ds.
Therefore, for ε sufficiently small, the right-hand side of (28) tends to zero as h→0. On the other hand, the compactness of 𝒯E(t),t>0, implies the continuity in the uniform operator topology. Thus, the set {Φεz:z∈Br} is equicontinuous.Lemma 9.
Let assumptions (S1)–(S3), (H4), (H5) hold. ThenΦε maps Br onto a precompact set in Br.Proof.
Let0<t≤b be fixed and let λ be a real number satisfying 0<λ<t. For δ>0, define an operator Φελ,δ on Br by
(29)(Φελ,δz)(t)n=α∫0t-λ‍E-1∫δ∞‍θ(t-s)α-1Ψα(θ)S((t-s)αθ)nnnnnnnnnnnn×[Buε(s,z)+f(s,z(s))]dsn=αE-1S(λαδ)nn×∫0t-λ‍∫δ∞‍θ(t-s)α-1Ψα(θ)S((t-s)αθ-λαδ)mmmmmnnn×[Buε(s,z)+f(s,z(s))]ds.
Since E-1 is a compact operator, the set {(Φελ,δz)(t):z∈Br} is precompact in X for every 0<λ<t, δ>0. Moreover, for each z∈Br, we have
(30)∥(Φεz)(t)-(Φελ,δz)(t)∥n≤α∥∫0t‍∫0δ‍θ(t-s)α-1Ψα(θ)S((t-s)αθ)nnnnnnnnn×[Buε(s,z)+f(s,z(s))]dθds∫0t‍∫0δ‍∥nn+α∥∫t-λt‍∫δ∞‍θ(t-s)α-1Ψα(θ)S((t-s)αθ)nnnnnnnnnnn×[Buε(s,z)+f(s,z(s))]dθds∫t-λt‍∫δ∞‍∥n=:J1+J2.
A similar argument, as before, is as follows:
(31)J1≤αM∫0t‍(t-s)α-1(∥Buε(s,z)∥+∥f(s,z(s))∥)dsnnnnn×(∫0δ‍θΨα(θ)dθ)≤αM(1εMBM𝒯Δ∫0t‍(t-s)α-1(b-s)α-1dsnnnnn∫0t‍1ε+Λf(r)∫0t‍(t-s)α-1n(s)ds1ε)(∫0δ‍θΨα(θ)dθ),(32)J2≤αM∫t-λt‍(t-s)α-1(∥Buε(s,z)∥+∥f(s,z(s))∥)dsnnnn×(∫δ∞‍θηα(θ)dθ)≤αMΓ(1+α)(1εMBM𝒯Δ∫t-λt‍(t-s)α-1(b-s)α-1dsnnnnnnnn+Λf(r)∫t-λt‍(t-s)α-1n(s)ds1ε),
where we have used the equality
(33)∫0∞‍θαβΨα(θ)dθ=Γ(1+β)Γ(1+αβ).
From (30)–(32), one can see that for each z∈Br,
(34)∥(Φεz)(t)-(Φελ,δz)(t)∥⟶0asλ⟶0+,δ⟶0+.
Therefore, there are relatively compact sets arbitrary close to the set {(Φεz)(t):z∈Br}; hence, the set {(Φεz)(t):z∈Br} is also precompact in X.


## 4. Main Results

Consider the following linear fractional differential system:(35)DtαEx(t)=Ax(t)+Bu(t),t∈(0,b],(36)x(0)=x0.
It is convenient at this point to introduce the controllability and resolvent operators associated with (35) as
(37)L0b=∫0b‍(b-s)α-1𝒯E(b-s)Bu(s)ds:L2([0,b],U)⟶X,Γ0b=L0b(L0b)*=∫0b‍(b-s)2(α-1)𝒯E(b-s)nnnnnnnn×BB*𝒯E*(b-s)ds:X⟶X,
respectively, where B* denotes the adjoint of B and 𝒯α*(t) is the adjoint of 𝒯α(t). It is straightforward that the operator L0b is a linear bounded operator for 1/2<α≤1.Theorem 10 (see [27]).
The following three conditions are equivalent.(i)
Γ is positive; that is, 〈z*,Γz*〉>0 for all nonzero z*∈X*.(ii)
For allh∈X,J(zε(h)) converges to zero as ε→0+ in the weak topology, where zε(h)=ε(εI+ΓJ)-1(h) is a solution of (13).(iii)
For allh∈X,zε(h)=ε(εI+ΓJ)-1(h) converges to zero as ε→0+ in the strong topology.Remark 11.
It is known that Theorem10 (i) holds if and only if ImL0b¯=X. In other words, Theorem 10 (i) holds if and only if the corresponding linear system is approximately controllable on [0,b]. Consequently, assumption (H6) is equivalent to the approximate controllability of the linear system (35).Theorem 12 (see [27]).
Letp:X→X  be a nonlinear operator. Assume that zε is a solution of the following equation:
(38)εzε+Γ0TJ(zε)=εp(zε),∥p(zε)-q∥⟶0asε⟶0+,q∈X.
Then there exists a subsequence of the sequence {zε} strongly converging to zero as ε→0+.We are now in a position to state and prove the main result of the paper.Theorem 13.
Let1/2<α≤1. Suppose that conditions (S1)–(S3), (H4)–(H5) are satisfied. Besides, assume additionally that there exists N∈L∞([0,b],[0,+∞)) such that
(39)supx∈X∥f(t,x)∥≤N(t),fora.e.t∈[0,b].
Then system (1) is approximately controllable on [0,b].Proof.
Letxε be a fixed point of Φε in Br(ε). Then xε is a mild solution of (1) on [0,b] under the control
(40)uε(t,xε)=(b-t)α-1B*𝒯E*(b-t)J((εI+Γ0bJ)-1p(xε)),p(xε)=h-𝒮E(b)x0-∫0b‍(b-s)α-1𝒯E(b-s)f(s,xε(s))ds
and satisfies the following equality:
(41)xε(b)=𝒮E(b)x0+∫0b‍(b-s)α-1𝒯E(b-s)nnnnnnnnnnn×[Buε(s,xε)+f(s,xε(s))]ds=𝒮E(b)x0+(-εI+εI+Γ0bJ)((εI+Γ0bJ)-1p(xε))+∫0b‍(b-s)α-1𝒯E(b-s)f(s,xε(s))ds=h-ε(εI+Γ0bJ)-1p(xε).
In other words, zε=h-xε(b) is a solution of
(42)(εI+Γ0bJ)(zε)=εp(xε).
By our assumption,
(43)∫0b‍∥f(s,xε(s))∥2ds≤∫0T‍N2(s)ds.
Consequently, the sequence {f(·,xε(·))} is bounded. Then there is a subsequence still denoted by {f(·,xε(·))} and weakly converges to, say, f(·) in L2([0,b],X). Then
(44)∥p(xε)-q∥n=∥∫0b‍(b-s)α-1𝒯E(b-s)[f(s,xε(s))-f(s)]ds∥n≤sup0≤t≤b∥∫0t‍(t-s)α-1𝒯E(t-s)nnnnnnnnnn×[f(s,xε(s))-f(s)]ds∫0t‍∥⟶0,
where
(45)q=h-𝒮EEx0-∫0b‍(b-s)α-1𝒯E(b-s)f(s)ds
as ε→0+ because of the compactness of an operator f(·)  →  ∫0·‍(·-s)α-1𝒯E(·-s)f(s)ds:L2([0,b],X)  →  C([0,b],X). Then by Theorem 12 for any h∈X,
(46)∥xε(b)-h∥=∥zε∥⟶0
as ε→0+. This gives the approximate controllability. The theorem is proved.Remark 14.
Theorem13 assumes that the operator E-1 is compact and, consequently, the associated linear control system (35) is not exactly controllable. Therefore, Theorem 13 has no analogue for the concept of exact controllability.Remark 15.
In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, fractional impulsive differential equations have been used for the system model. Our result can be extended to study the complete and approximate controllability of nonlinear fractional impulsive differential equations of Sobolev type; see [35, 36].


## 5. Applications

Example 16.
LetX=U=L2[0,π]. Consider the following fractional partial differential equation with control:
(47)cDt3/4(x(t,θ)-xθθ(t,θ))=xθθ(t,θ)+g(t,x(t,θ))+u(t,θ),x(t,0)=x(t,π)=0,x(0,θ)=ϕ(θ),:0≤t≤b,:0≤θ≤π.
DefineA:D(A)⊂X→X by A∶=xθθ and E:D(E)⊂X→X by Ex:=x-xθθ, where each domain, D(A)andD(E), is given by
(48){x∈X:x,xθareabsolutelycontinuous,nxθθ∈X,x(t,0)=x(t,π)=0}.A and E can be written as follows:
(49)Ax∶=∑n=1∞‍-n2〈x,en〉en,x∈D(A),Ex=∑n=1∞‍(1+n2)〈x,en〉en,x∈D(E),
respectively, where en(θ)∶=2/πsinnθ, n=1,2,…, is the orthonormal set of eigenvalues of A. Moreover, for any x∈X, we have
(50)E-1x=∑n=1∞‍11+n2〈x,en〉en,AE-1x=∑n=1∞‍-n21+n2〈x,en〉en,S(t)x=∑n=1∞‍exp(-n21+n2t)〈x,en〉en,𝒯E(t)=34∫0∞‍E-1θξ3/4(θ)S(t3/4θ)dθ,𝒯E(t)x=34∑n=1∞‍11+n2×∫0∞‍θξ3/4(θ)exp(-n21+n2t3/4θ)dθ×〈x,en〉en=-∑n=1∞‍1n2t1/4×∫0∞‍ξ3/4(θ)ddtexp(-n21+n2t3/4θ)dθ×〈x,en〉en,Γ0b=∫0b‍(b-s)2(α-1)𝒯(b-s)𝒯(b-s)ds=∫0b‍s2(α-1)𝒯(s)𝒯(s)ds=∫0b‍s2(α-1)34∫0∞‍E-1θξ3/4(θ)S(s3/4θ)dθ×34∫0∞‍E-1θξ3/4(θ)S(s3/4θ)dθds.
It is clear that E-1 is compact. The linear system corresponding to (47) is completely controllable if and only if there exists γ>0 such that 〈Γ0bx,x〉≥γ∥x∥2 for all x∈X. Assume
(51)〈Γ0bx,x〉=916∑n=1∞‍1(1+n2)2nnnn×∫0b‍s2(α-1)(∫0∞‍θξ3/4(θ)exp(-n21+n2s3/4θ)dθ)2dsnnnn×〈x,en〉2⩾γ∑n=1∞‍〈x,en〉2.
Then
(52)9161(1+n2)2nnn×∫0b‍s2(α-1)(∫0∞‍θξ3/4(θ)exp(-n21+n2s3/4θ)dθ)2ds⩾γ,γ≤9161(1+n2)2nnn×∫0b‍s-1/2(∫0∞‍θξ3/4(θ)exp(-n21+n2s3/4θ)dθ)2dsn≤9161(1+n2)2nnn×∫0b‍s-1/2(∫0∞‍θξ3/4(θ)dθ)2ds⟶0,nnnnasn⟶∞,⟹γ=0(contradiction),
and no such γ>0 exists which satisfies (51), and hence the linear system corresponding to (47) is never completely controllable. We show that the associated linear system is approximately controllable on [0,b]. We need to show that (b-s)α-1B*𝒯E*(b-s)x=0,0≤s<b⇒x=0. Indeed,
(53)(b-s)α-1B*𝒯E*(b-s)xn=(b-s)α-134∫0∞‍θξ3/4(θ)∑n=1∞‍11+n2exp(-n21+n2s3/4θ)nnnnnnnnnnnnnnnnnnnnnnnnnn×〈x,en〉en=0,34∫0∞‍θξ3/4(θ)∑n=1∞‍11+n2exp(-n21+n2s3/4θ)〈x,en〉en=0,〈x,en〉=0⟹x=0.
Next, we suppose
(H6)g:[0,b]×R→R. For each x∈R, g(·,x) is measurable and for each t∈[0,b], g(t,·) is continuous. Moreover, supx∈R∥g(t,x)∥≤N(t), for a.e. t∈[0,b].
Definef:[0,b]×X→X by f(t,x)(θ)=g(t,x(t,θ)). Now, system (47) can be written in the abstract form (1). Clearly, all the assumptions in Theorem 13 are satisfied if (H6) holds. Then system (47) is approximately controllable on [0,b].



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*Source: 502839-2013-03-23.xml*