Regional boundary controllability of time fractional diffusion processes

0.24 MB·

by  Fudong Ge

Regional boundary controllability of time fractional diffusion
processes
Fudong Gea, YangQuan Chenb,∗, Chunhai Kouc
aCollege of Information Science and Technology, Donghua University, Shanghai 201620, PR China
bMechatronics, Embedded Systems and Automation Lab, University of California, Merced, CA 95343, USA
cDepartment of Applied Mathematics, Donghua University, Shanghai 201620, PR China
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n Abstract
a
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In this paper, we are concerned with the regional boundary controllability of the Riemann-
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1
Liouville timefractionaldiffusion systems oforderα ∈ (0,1]. Thecharacterizations ofstrate-

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C gic actuators are established when the systems studied are regionally boundary controllable.
O The determination of control to achieve regional boundary controllability with minimum
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h energy is explored. We also show a connection between the regional internal controllability
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a
and regional boundary controllability. Several useful results for the optimal control from an
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[ implementation point of view are presented in the end.

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Keywords: regional boundary controllability; time fractional diffusion processes; strategic
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4 actuators; minimum energy control
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7 2010 MSC: 35R11, 93B05, 60J60
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1. Introduction
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1 In the past several decades, a lot of work has been carried out to deal with the problem of
:
v
steeringasystemtoatargetstate, especiallyaftertheintroductionofthenotionsofactuators
i
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and sensors [1, 2]. However, in many real-world applications, we are only concerned with
r
a
those cases where the target states of the problem studied are defined in a given subregion of
the whole space domain. Then the regional idea emerges and we refer the reader to [3, 4, 5]
for more information on the concept of regional analysis for the Gaussian diffusion process.
Besides, it should be pointed out that not only does the concept of regional analysis make
sense closer to real-world problems, it also generalizes the results of existence contributions.
In addition, after the introduction of continuous time random walks (CTRWs) by Mon-
trollandWeiss[6], theanomalousdiffusionequationoffractionalorder hasattractedincreas-
ing interest and has been proven to be a useful tool in modeling many real-world problems
∗
Corresponding author
Email addresses: [email protected](Fudong Ge), [email protected](YangQuan Chen),
[email protected](Chunhai Kou)
Preprint submitted to IMA J. Math. Control Info. January 13, 2016
[7, 8, 9, 10, 11]. More precisely, the mean squared displacement (MSD) of anomalous diffu-
sion process is described by a power law of fractional exponent, which is smaller (in the case
of sub-diffusion) or bigger (in the case of super-diffusion) than that of Brownian motion. It
is confirmed that the time fractional diffusion system, where the traditional first order time
derivative is replaced by a Riemann-Liouville time fractional derivative of order α ∈ (0,1],
can be used to well characterize those sub-diffusion process [7, 8]. For example, the flow
through porous media microscopic processes [12], or swarm of robots moving through dense
forest [13] etc. For the fractional calculus, as we all know, it has shown great potential in
science and engineering applications and some phenomena such as self-similarity, nonsta-
tionary, non-Gaussian process and short or long memory process are all closely related to
fractional calculus [14, 15, 16]. It is now widely believed that, using fractional calculus in
modeling can better capture the complex dynamics of natural and man-made systems, and
fractional order controls can offer better performance not achievable before using integer
order controls [17, 18], which in fact raise important theoretical challenges and open new
research opportunities.
Motivated by theargument above, the contribution of this present work is on the regional
boundary controllability of the anomalous transport process described by time fractional dif-
fusion systems. More precisely, for an open bounded subset Ω ⊆ Rn with smooth boundary
∂Ω, we consider:
• A subregion Γ of ∂Ω which may be unconnected.
• Various kinds of actuators (zone, pointwise, internal or boundary) acting as controls.
Therestofthispaperisorganizedasfollows. Themathematicalconceptofregionalboundary
controllability and several preliminaries are presented in the next section, then we present an
example which is regional boundary controllability but not globally boundary controllable.
Section 3 is focused on the characterizations of Γ−strategic actuators and our main result
on regional boundary controllability with minimum energy problem is given in Section 4. In
Section 5, a connection between internal and boundary regional controllability is established
and at last, we work out some useful results for the optimal control from an implementation
point of view.
2. Regional boundary controllability
2.1. Problem statement
In this paper, we consider the following abstract time fractional diffusion system:
Dαz(t) = Az(t)+Bu(t), t ∈ [0,b],
0 t
(2.1)
 lim I1−αz(t) = z ,
0 t 0
 t→0+
 2
where A generates a strongly continuous semigroup {Φ(t)} on the Hilbert space Z :=
t≥0
H1(Ω), z ∈ L2(0,b;Z) and the initial vector z ∈ Z. It is supposed that B : Rp →
0
Z is the control operator and u ∈ L2(0,b;Rp) depends on the number and structure of
actuators. Moreover, the Riemann-Liouville fractional derivative Dα and the Riemann-
0 t
Liouville fractional integral Iα are, respectively, given by [15],[16]
0 t
d 1 t
Dαz(t) = I1−αz(t), α ∈ (0,1] and Iαz(t) = (t−s)α−1z(s)ds, α > 0. (2.2)
0 t dt0 t 0 t Γ(α)
Z0
Definition 2.1. [19] For any given f ∈ L2(0,b;Z), α ∈ (0,1], a function v ∈ L2(0,b;Z) is
said to be a mild solution of the following system
Dαv(t) = Av(t)+f(t), t ∈ [0,b],
0 t
lim I1−αv(t) = v ∈ Z, (2.3)
( t→0+0 t 0
if it satisfies
t
z(t) = K (t)v + (t−s)α−1K (t−s)f(s)ds, (2.4)
α 0 α
Z0
where K (t) = α ∞θφ (θ)Φ(tαθ)dθ, {Φ(t)} is the strongly continuous semigroup gener-
α 0 α t≥0
ated by A, φ (θ) = 1θ−1−1ψ (θ−1) and ψ is a probability density function defined by
α R α α α α α
∞
1 Γ(nα+1)
ψ (θ) = (−1)n−1θ−αn−1 sin(nπα), θ > 0. (2.5)
α
π n!
n=1
X
In addition, we have [20, 21]
∞ ∞ Γ(1+ν)
ψ (θ)dθ = 1 and θνφ (θ)dθ = , ν ≥ 0. (2.6)
α α
Γ(1+αν)
Z0 Z0
By Lemma 2.1, the mild solution z(.,u) of (2.1) can be given by
t
z(t,u) = K (t)z + (t−s)α−1K (t−s)Bu(s)ds. (2.7)
α 0 α
Z0
Let H : L2(0,b;Rp) → Z be
b
Hu = (b−s)α−1K (b−s)Bu(s)ds, ∀u ∈ L2(0,b;Rp) (2.8)
α .
Z0
Supposethat{Φ∗(t)} , generatedbytheadjointoperatorofA, isalsoastronglycontinuous
t≥0
semigroup on the space Z. For any v ∈ Z, it follows from hHu,vi = hu,H∗vi that
H∗v = B∗(b−s)α−1K∗(b−s)v, (2.9)
α
where h·,·i is the duality pairing of space Z, B∗ is the adjoint operator of B and
∞
K∗(t) = α θφ (θ)Φ∗(tαθ)dθ.
α α
Z0
3
Let γ : H1(Ω) → H12(∂Ω) be the trace operator of order zero, which is linear continuous
and surjective, γ∗ denotes the adjoint operator. Moreover, if Γ ⊆ ∂Ω, pΓ : H21(∂Ω) → H12(Γ)
defined by
p z := z| (2.10)
Γ Γ
and for any z¯∈ H21(Γ), the adjoint operator p∗ can be given by
Γ
z¯(x), x ∈ Γ,
p∗z¯(x) := (2.11)
Γ
( 0, x ∈ ∂Ω\Γ.
2.2. Definition and characterizations
Letω ⊆ ΩbeagivenregionofpositiveLebesguemeasure. Denotetheprojectionoperator
on ω by the restriction map
p : H1(Ω) → H1(ω), (2.12)
ω
then we are ready to state the following definitions.
Definition 2.2. The system (2.1) is said to be exactly (respectively, approximately) region-
ally controllable on ω at time b if for any y ∈ H1(ω), given ε > 0, there exists a control
b
u ∈ L2(0,b;Rp) such that
p z(b,u) = y respectively, kp z(b,u)−y k ≤ ε . (2.13)
ω b ω b H1(ω)
Definition 2.3. The system ((cid:0)2.1) is said to be exactly (respectively, (cid:1)approximately) region-
1
ally boundary controllable on Γ ⊆ ∂Ω at time b if for any zb ∈ H2(Γ), given ε > 0, there
exists a control u ∈ L2(0,b;Rp) such that
p (γz(b,u)) = z respectively, kp (γz(b,u))−z k ≤ ε . (2.14)
Γ b Γ b 1
H2(Γ)
(cid:16) (cid:17)
Proposition 2.1. The following properties are equivalent:
(1) The system (2.1) is exactly regionally boundary controllable on Γ at time b;
1
(2) Im(pΓγH) = H2(Γ);
1
(3) Ker(pΓ)+Im(γH) = H2(∂Ω);
1
(4) For any z ∈ H2(Γ), there exists a positive constant c such that
kzk ≤ ckH∗γ∗p∗zk . (2.15)
H12(Γ) Γ L2(0,b;Rp)
Proof. By Definition 2.3, it is not difficult to see that (1) ⇔ (2).
1 1
(2) ⇒ (3) : For any z ∈ H2(Γ),let zˆbe the extension of z to H2(∂Ω). Since Im(pΓγH) =
H21(Γ), there exists u ∈ L2(0,b;Rp), z1 ∈ Ker(pΓ) such that zˆ= z1 +γHu.
1
(3) ⇒ (2) : For any z˜ ∈ H2(∂Ω), z˜ = z1 + z2, where z1 ∈ Ker(pΓ) and z2 ∈ Im(γH).
Then there exists a control u ∈ L2(0,b;Rp) such that γHu = z . Hence, it follows from the
2
1
definition of pΓ that Im(pΓγH) = H2(Γ).
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(1) ⇔ (4) : The equivalence between (1) and (4) can be deduced from the following
general result [22]: Let E,F,G be reflexive Hilbert spaces and f ∈ L(E,G), g ∈ L(F,G).
Then the following two properties are equivalent
(1) Im(f) ⊆ Im(g),
(2) ∃ γ > 0 such that kf∗z∗kE∗ ≤ γkg∗z∗kF∗, ∀z∗ ∈ G.
By choosing E = G = H12(Γ), F = L2(0,b;Rp), f = Id 1 and g = pΓγH, then we
H2(Γ)
complete the proof.
Proposition 2.2. There is an equivalence among the following properties:
h1i The system (2.1) is approximately regionally boundary controllable on Γ at time b;
1
h2i Im(pΓγH) = H2(Γ);
1
h3i Ker(pΓ)+Im(γH) = H2(∂Ω);
h4i The operator p γHH∗γ∗p∗ is positive definite.
Γ Γ
Proof. By Proposition 2.1, h1i ⇔ h2i ⇔ h3i. Finally, we show that h2i ⇔ h4i. In fact, since
Im(pΓγH) = H12(Γ) ⇔ (pΓγHu,z)H1/2(Γ) = 0 for any u ∈ L2(0,b;Rp) implies z = 0,
where (·,·)H1/2(Γ) is the inner product of H21(Γ). Let u = H∗γ∗p∗ωz. Then we see that
Im(pΓγH) = H21(Γ) ⇔ (pΓγHH∗γ∗p∗Γz,z)H1/2(Γ) = 0 implies z = 0, z ∈ H21(Γ),
i.e., the operator p γHH∗γ∗p∗ is positive definite and the proof is complete.
Γ Γ
Remark 2.1. (1) A system which is boundary controllable on Γ is boundary controllable on
Γ for every Γ ⊆ Γ.
1 1
(2) The definitions (2.2) can be applied to the case where Γ = ∂Ω and there exist systems
that are not boundary controllable but which are regionally boundary controllable. This is
illustrated by the following example
2.3. An example
Consider the following two dimension time fractional diffusion equation defined on Ω =
[0,1]×[0,1], which is excited by a zone actuator:
Dαz(x,y,t) = ∂2 z(x,y,t)+ ∂2 z(x,y,t)+p u(t) in Ω×[0,b],
0 t ∂x2 ∂y2 D
 lim I1−αz(x,y,t) = 0 in Ω, (2.16)
0 t
 t→0+

 z(ξ,η,t) = 0 on ∂Ω×[0,b],

where α ∈(0,1], D = {0}×[d ,d ] ⊆ Ω, A = ∂2 + ∂2 with λ = −(i2 +j2)π2, ξ (x,y) =
1 2 ∂x2 ∂y2 ij ij
∞
2aijcos(iπx)cos(jπy), aij = (1−λij)−21, Φ(t)z = exp(λijt)(z,ξij)Zξij and Kα(t)z(x) =
i,j=1
α ∞θφ (θ)Φ(tαθ)z(x)dθ = ∞ E (λ tα)(z,ξ ) Pξ (x). Further, since
0 α α,α ij ij Z ij
i,j=1
R P
∞
(H∗γ∗z)(t) = (b−t)α−1 E (λ (b−t)α)(γ∗z,ξ ) (p ,ξ )
α,α ij ij Z D ij Z
i,j=1
X
5
and (p ,ξ ) = 2aij [sin(jπd )−sin(jπd )+jπ(cos(jπd )−cos(jπd )], there exists d ,d ∈
D ij Z jπ 2 1 2 1 1 2
[0,1] satisfying Ker(H∗) 6= {0} (Im(p H) 6= L2(ω)), i.e., the system (2.16) is not boundary
D
controllable.
Moreover, let d = 0, d = 1, Γ = {0}×[1, 3] and z = ξ (0,y),(i,j = 4k,k = 1,2,3,···).
1 2 2 4 4 ∗ ij
Obviously, z is not reachable on ∂Ω. However, since
∗
2a
ij
E (t) > 0 (t ≥ 0) and (p ,ξ ) = [sin(jπ/2)+jπ(cos(jπ/2)−1)], j = 1,2,···,
α,α D ij Z
jπ
we see that
∞
(H∗γ∗p∗z )(t) = Eα,α(λij(b−t)α)(ξ ,γ∗z ) (p ,ξ )
Γ ∗ (b−t)1−α ij ∗ H1/2(Γ) D ij Z
i,j=1
∞
= P 2aijEα,α(λij(b−t)α)(ξ ,γ∗z )
jπ(b−t)1−α ij ∗ H1/2(Γ) (2.17)
i,j=1,j6=4k
P×[sin(jπ/2)+jπ(cos(jπ/2)−1)]
6= 0.
Hence z is regionally boundary controllable on Γ = {0}×[1, 3].
∗ 4 4
To end this section, we finally recall a necessary lemma to be used afterwards.
Lemma 2.1. [23] Let Ω ⊆ Rn be an open set and C∞(Ω) be the class of infinitely differen-
0
tiable functions on Ω with compact support in Ω and u ∈ L1 (Ω) be such that
loc
u(x)ψ(x)dx = 0, ∀ψ ∈ C∞(Ω). (2.18)
0
ZΩ
Then u = 0 almost everywhere in Ω.
3. Regional strategic actuators
The characteristic of actuators to achieve the regionally approximately boundary con-
trollable of the system (2.1) will be explored in this section.
As citedin[1], aactuator canbeexpressed by a couple(D,g)where D ⊆ Ω isthesupport
of the actuator and g is its spatial distribution. To state our main results, it is supposed that
p
thecontrolaremadebypactuators(D ,g ) andlet Bu = p g (x)u (t), wherep ∈ N,
i i 1≤i≤p Di i i
i=1
gi(x) ∈ Z, u = (u1,u2,···,up) and ui(t) ∈ L2(0,b). As citedPin [24], all these distributed
parameter systems with moving sensors and actuators form the so-called cyber-physical
systems, whicharerichinrealworldapplications. Forinstance, inthepestspreadingprocess,
p is the number the spreading machines and u (·) stands for the control input strategic of
i
every spreading machines with respect to time t [25]. Then the system (2.1) can be rewritten
as
p
Dαz(t,x) = Az(t,x)+ p g (x)u (t), (t,x) ∈ [0,b]×Ω,
0 t Di i i
 i=1 (3.1)
 lim 0It1−αz(t,x) = z0(x)P.
t→0+

 6
Moreover, we suppose that −A is a self-adjoint uniformly elliptic operator, by [26], we get
that there exists a sequence (λ ,ξ ) : k = 1,2,···,r , j = 1,2,··· such that
j jk j
(1) For each j = 1,2,···, λ is the eigenvalue of operator A with multiplicities r and
j j
0 > λ > λ > ··· > λ > ···, lim λ = −∞.
1 2 j j
j→∞
(2) For each j = 1,2,···, ξ (k = 1,2,···,r ) is the orthonormal eigenfunction corre-
jk j
sponding to λ , i.e.,
j
1, k = k ,
m n
(ξ ,ξ ) =
jkm jkn ( 0, km 6= kn,
where 1 ≤ k ,k ≤ r , k ,k ∈ N and (·,·) is the inner product of space Z.
m n j m n
Hence, the sequence {ξ ,k = 1,2,···,r ,j = 1,2,···} is a orthonormal basis in Z, the
jk j
strongly continuous semigroup {Φ(t)} on Z generated by A is
t≥0
∞ rj
Φ(t)z(x) = exp(λ t)(z,ξ )ξ (x), x ∈ Ω (3.2)
j jk jk
j=1 k=1
XX
∞ rj
and for any z(x) ∈ Z, it can be expressed as z(x) = (z,ξ )ξ (x).
jk jk
j=1k=1
P P
Definition 3.1. A actuators (suite of actuators) is said to be Γ−strategic if the system
under consideration is regionally approximately boundary controllable on Γ at time b.
Before to show our main result in this part, by Eq.(3.2), for any z ∈ L2(Ω), we have
∞
K (t)z(x) = α θφ (θ)Φ(tαθ)z(x)dθ
α α
Z0
∞ ∞ rj
= α θφ (θ) exp(λ tαθ)(z,ξ )ξ (x)dθ
α j jk jk
Z0 j=1 k=1
XX
∞ rj ∞ α(λ tα)n ∞
= j (z,ξ )ξ (x) θn+1φ dθ
jk jk α
n!
j=1 k=1 n=0 Z0
XXX
∞ rj ∞ α(n+1)!(λ tα)n
j
= (z,ξ )ξ (x)
jk jk
Γ(αn+α+1)n!
j=1 k=1 n=0
XXX
∞ rj
= αE2 (λ tα)(z,ξ )ξ (x),
α,α+1 j jk jk
j=1 k=1
XX
∞
where Eµ (z) := (µ)n zn, z ∈ C, α,β,µ ∈ C, Reα > 0 is the generalized Mittag-
α,β Γ(αn+β) n!
n=0
Leffler function inPthree parameters and here, (µ)n is the Pochhammer symbol defined by
(see [27], Section 2.1.1)
(µ) = µ(µ+1)···(µ+n−1), n ∈ N. (3.3)
n
7
If α,β ∈ C such that Reα > 0, Reβ > 1, then (see Section 2.3.4, [28] or Section 5.1.1, [29])
αE2 = E −(1+α−β)E . (3.4)
α,β α,β−1 α,β
It then follows that
∞ rj
K (t)z(x) = E (λ tα)(z,ξ )ξ (x) (3.5)
α α,α j jk jk
j=1 k=1
XX
and
t ∞ rj p t
τα−1K (τ)Bu(t−τ)dτ = gi u (t−τ)τα−1E (λ τα)dτξ (x), (3.6)
α jk i α,α j jk
Z0 j=1 k=1 i=1 Z0
XXX
where gi = (p g ,ξ ), j = 1,2,···, k = 1,2,···,r , i = 1,2,···,p and E (z) :=
jk Di i jk j α,β
∞
zi , Reα > 0, β,z ∈ C is known as the generalized Mittag-Leffler function in two
Γ(αi+β)
i=0
parameters.
P
Theorem 3.1. For any j = 1,2,···, define p×r matrices G as
j j
g1 g1 ··· g1
j1 j2 jrj
g2 g2 ··· g2
Gj =  ..j1 ..j2 .. j..rj , (3.7)
. . . .
 p p p 
 g g ··· g 
 j1 j2 jrj 
 
where gi = (p g ,ξ ), j = 1,2,···, k = 1,2,···,r , i = 1,2,···,p. Then the suite of
jk Di i jk j
actuators (D ,g ) is said to be Γ−strategic if and only if
i i 1≤i≤p
p ≥ r = max{r } and rank G = r for j = 1,2,···. (3.8)
j j j
1 1
Proof. For any z∗ ∈ H2(Γ), denote by (·,·)H1/2(Γ) the inner product of space H2(Γ), we
then see that
∞ rj p b
(p γHu,z ) = τα−1E (λ τα)u (b−τ)dτgi z = 0, t ∈ [0,b], (3.9)
Γ ∗ H1/2(Γ) α,α j i jk jk
j=1 k=1 i=1 Z0
XXX
where z = (p γξ ,z ) , j = 1,2,···, k = 1,2,···,r . Further, Lemma 2.1 gives
jk Γ jk ∗ H1/2(Γ) j
∞ rj
tα−1E (λ tα)gi z = 0 := (0,0,···,0) ∈ Rp for t > 0,i = 1,2,···,p. (3.10)
α,α j jk jk p
j=1 k=1
XX
Then we conclude that the suite of actuators (D ,g ) is Γ−strategic if and only if
i i 1≤i≤p
∞
bα−1E (λ bα)G z = 0 ⇒ z = 0, (3.11)
α,α j j j p ∗
j=1
X
8
where z = (z ,z ,···,z )T is a vector in Rrj and j = 1,2,···.
j j1 j2 jrj
(a) If we assume that p ≥ r = max{r } and rank G < r for some j = 1,2,···, there
j j j
exists a nonzero element z˜∈ H12(Γ) with z˜j = z˜j1,z˜j2,···,z˜jrj T ∈ Rrj such that
(cid:0) (cid:1)
G z˜ = 0 . (3.12)
j j p
It then follows from E (λ tα) > 0 (t ≥ 0) that we can find a nonzero vector z˜ satisfying
α,α j
∞
bα−1E (λ bα)G z˜ = 0 . (3.13)
α,α j j j p
j=1
X
This means that the actuators (D ,f ) are not Γ−strategic.
i i 1≤i≤p
(b) However, on the contrary, if the actuators (D ,g ) are not Γ−strategic, i.e.,
i i 1≤i≤p
1
Im(pΓγH) 6= H2(Γ), then there exists a nonzero element z 6= 0n satisfying
(p γHu,z) = 0 for all u ∈ L2(0,b;Rp). (3.14)
Γ H1/2(Γ)
Then we can find a nonzero element zj∗ ∈ Rrj such that
Gj∗zj∗ = 0p. (3.15)
This allows us to complete the conclusion of the theorem.
4. Regional boundary controllability with minimum energy control
In this section, we explore the possibility of finding a minimum energy control when the
system (2.1) can be steered from a given initial vector z to a target function z on the
0 b
boundary subregion Γ. The method used here is an extension of those in [1, 2, 3, 4, 5].
Consider the following minimization problem
infJ(u) = bku(t)k2 dt
u 0 Rp (4.1)
 u ∈ U = {uR ∈ L2(0,b;Rp) : p γz(b,u) = z },
 b Γ b
where, obviously, Uis a closed convex set. We then show a direct approach to the solution
b
of the minimum energy problem (4.1).
Theorem 4.1. If the system (2.1) is regionally approximately boundary controllable on Γ,
1
then for any zb ∈ H2(Γ), the minimum energy problem (4.1) has a unique solution given by
u∗(t) = (p γH)∗R−1(z −p γK (b)z ), (4.2)
Γ Γ b Γ α 0
where R = p γHH∗γ∗p∗ and H∗ is defined in Eq.(2.9).
Γ Γ Γ
9
Proof. To begin with, since the solution of (2.1) excited by the control u∗ is given by
t
z(t,u∗) = K (t)z + (t−s)α−1K (t−s)Bu∗(s)ds, (4.3)
α 0 α
Z0
we get that
b
p γz(b,u∗) = p γ K (b)z + (b−s)α−1K (b−s)Bu∗(s)ds
Γ Γ α 0 α
(cid:20) Z0 (cid:21)
= p γK (b)z +p γH(p γH)∗R−1(z −p γK (b)z )
Γ α 0 Γ Γ Γ b Γ α 0
= z .
b
Next, we show that if the system (2.1) is regionally approximately boundary controllable
1
on Γ at time b, then the operator RΓ is coercive. In fact, for any z1 ∈ H2(Γ), there exists a
control u ∈ L2(0,b,Rp) such that
z = p γ[K (b)z +Hu] (4.4)
1 Γ α 0
and
hR z ,z i = kH∗γ∗p∗z k2
Γ 1 1 H1/2(Γ) Γ 1 L2(0,b,Rp)
= B∗(b−·)α−1K∗(b−·)γ∗p∗z 2
α Γ 1 L2(0,b,Rp)
≥ (cid:13)kz k2 . (cid:13)
(cid:13) 1 H1/2(Γ) (cid:13)
1 1
Moreover, since RΓ ∈ L H2(Γ),H2(Γ) , by the Theorem 1.1 in [30], it follows that RΓ is
an isomorphism. (cid:16) (cid:17)
Finally, we prove that u∗ solves the minimum energy problem (4.1). For this purpose,
since p γz(b,u∗) = z , for any u ∈ L2(0,b,Rp) with p γz(b,u) = z , one has
Γ b Γ b
p γ[z(b,u∗)−z(b,u)] = 0, (4.5)
Γ
which follows that
b
0 = p γ (b−s)α−1K (b−s)B[u∗(s)−u(s)]ds = p γH[u∗ −u].
Γ α Γ
Z0
Thus, by
b
J′(u∗)(u∗ −u) = 2 hu∗(s)−u(s),u∗(s)ids
Z0
b
= 2 u∗(s)−u(s),(p γH)∗R−1(z −p γK (b)z ) ds
Γ Γ b Γ α 0
Z0
b(cid:10) (cid:11)
= 2 p γH[u∗(s)−u(s)],R−1(z −p γK (b)z ) ds
Γ Γ b Γ α 0
Z0
= 0, (cid:10) (cid:11)
it follows that J(u) ≥ J(u∗), i.e., u∗ solves the minimum energy problem (4.1) and the proof
is complete.
10