Hindawi Publishing Corporation
Discrete Dynamics in Nature and Society
Volume 2012, Article ID 481295, 24 pages
doi:10.1155/2012/481295
Research Article
A Posteriori Error Estimates for
a Semidiscrete Parabolic Integrodifferential
Control on Multimeshes
Wanfang Shen
School of Mathematic and Quantitative Economics, Shandong University of Finance and Economics,
Jinan 250014, China
Correspondence should be addressed to Wanfang Shen, wanfangshen@gmail.com
Received 18 August 2012; Accepted 26 September 2012
Academic Editor: Hua Su
Copyright q 2012 Wanfang Shen. This is an open access article distributed under the Creative
Commons Attribution License, which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.
We extend the existing techniques to study semidiscrete adaptive finite element approximation
schemes for a constrained optimal control problem governed by parabolic integrodifferential
equations. The control problem involves time accumulation and the control constrain is given
in an integral obstacle sense. We first prove the uniqueness and existence of the solution of this
optimal control problem. We then derive the upper a posteriori error estimators for both the state
and the control approximation, which are useful indicators in adaptive multimesh finite element
approximation schemes.
1. Introduction
For the last twenty years great progress has been made on standard finite element
approximation of optimal control problems governed by elliptic or parabolic equations; see,
for example, 1–9, although it is impossible to give even a very brief review here. More
specifically, research on finite element approximation of parabolic optimal control problems
can be found in, for example, 10, 11. Systematic introduction of the finite element method
for PDEs and optimal control problems can be found in, for example, 5, 12, 13.
In many real modeling applications it is important to consider time accumulation.
Parabolic integrodifferential equations and their control of this nature appear in heat
conduction in materials with memory, population dynamics, and viscoelasticity; see, for
example, Friedman and Shinbrot 14, Heard 15, and Renardy et al. 16. This calls for more
studies on the finite element approximation on integrodifferential equations. For instance,
finite element methods for parabolic integrodifferential equations with a smooth kernel have
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Discrete Dynamics in Nature and Society
been discussed in, for example, Cannon et al. 17, Sloan and Thomée 18, and Yanik and
Fairweather 19.
Recently adaptive finite element method has been widely studied and applied in
practical control computations. In order to obtain a numerical solution of acceptable accuracy
for the optimal control problem, the finite element meshes have to be refined according to a
mesh refinement scheme. Adaptive finite element approximation uses a posteriori indicators
to guide the mesh refinement procedure. Only the area where the error indicator is larger will
be refined so that a higher density of nodes is distributed over the area where the solution
is difficult to approximate. In this sense efficiency and reliability of adaptive finite element
approximation rely very much on the error indicator used.
Adaptive finite element schemes have been widely studied for some optimal control
problems governed by elliptic and parabolic PDEs as well. Some of recent accounts
of progress can be found, for example, in 11, 20, 21. Although there exists so much
progress in adaptive finite element elliptic and parabolic control problems, it is much more
complicated to study and implement adaptive multimeshes computational schemes for
parabolic integrodifferential control problems. More specificity, there has been a lack of a
posteriori error estimates for any parabolic integrodifferential control problem, in spite of the
fact that such control problems are widely encountered in practical engineering applications
and scientific computations as we discussed above.
In this work, we extend the results and the techniques used in 20–22 to a quadratic
optimal control problem governed by a linear parabolic integrodifferential equation, which
can be generalized to the control problems with more general objective functions. The
semidiscrete finite element scheme for this problem is presented. We derive the upper a
posteriori error estimates for the semidiscrete finite element approximation
for the case where
the control constraints are given in an integral sense: Uad {v ∈ X; ΩU v 0, t ∈ 0, T }.
We are interested in the following optimal control problems:
min J u, yu T
u∈Uad ⊂X
g y hu dt
1.1
0
subject to
yt αy t
γt, τyτdτ f Bu,
in Ω × 0, T ,
0
y 0,
on ∂Ω × 0, T ,
y|t0 y0 ,
1.2
in Ω,
where Ω and ΩU are bounded open sets in Rn n 2 with the Lipschitz boundary ∂Ω
and ∂ΩU , y0 ∈ H01 Ω, f ∈ L2 0, T ; L2 Ω, U L2 ΩU , X L2 0, T ; U, Uad ⊆ X is a
closed convex subset; α is a linear strongly elliptic self-adjoint partial differential operator of
second order with coefficients depending smoothly on spacial variables, γt, τ is an arbitrary
second-order linear partial differential operator, with coefficients depending smoothly on
both of time and spacial variables in the closure of their respective domains, and B is a
suitable linear and continuous operator.
Here we assume g· is a convex functional which is continuously differential on
L2 Ω, and h· is a strictly convex continuously differential functional on U. We further
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3
assume that hu → ∞ as uU → ∞ and that g· is bounded below. Details will be
specified later.
The plan of the paper is as follows: in Section 2, we give the weak formulation and
prove the existence and uniqueness of the solution for this optimal control problem. In
Section 3, we will give a brief review on the finite element methods and optimality conditions
and construct the semidiscrete finite element approximation schemes for the optimal control
problem. In Section 4, the upper a posteriori error bounds in L2 0, T ; H 1 Ω-norm are
derived for the control problem. In Section 5, we obtain the a posteriori error bounds in
L2 0, T ; L2 Ω-norm for the control problem.
2. Existence and Uniqueness of the Solution of the Model Problem
In this paper, we adopt the standard notation W m,q Ω for Sobolev spaces on Ω with norm
m,q
· m,q,Ω , and seminorm | · |m,q,Ω . We set W0 Ω {w ∈ W m,q Ω : w|∂Ω 0}. We denote
W m,2 ΩW0m,2 Ω by H m ΩH0m Ω, with norm · m,Ω , and seminorm | · |m,Ω . In addition
c or C denotes a general positive constant independent of h.
We denote by Ls 0, T ; W m,q Ω the Banach space of all Ls integrable functions from
T
1/s
0, T into W m,q Ω with norm vLs 0,T ;W m,q Ω 0 vsW m,q Ω dt for s ∈ 1, ∞ and the
standard modification for s ∞. Similarly, one can define the spaces H 1 0, T ; W m,q Ω and
Ck 0, T ; W m,q Ω. The details can be found in 23. To fix idea, we will take the state space
W L2 0, T ; V with V H01 Ω and the control space X L2 0, T ; U. Let the observation
space be Y L2 0, T ; H with H L2 Ω.
Let
f1 , f2 Ω
f1 f2 ,
∀ f1 , f2 ∈ H × H,
u, vU az, ω αz, ω,
ΩU
uv,
2.1
∀u, v ∈ U × U,
ct, τ; z, ω γt, τz, ω ,
∀z, w ∈ V × V.
Assume that there are constants c and C, such that for all t, τ ∈ 0, T :
a az, z cz21,Ω ,
∀z ∈ V,
b |az, w| Cz1,Ω w1,Ω ,
∀z, w ∈ V,
c |ct, τ; z, w| Cz1,Ω w1,Ω ,
2.2
∀z, w ∈ V.
We will assume the following convexity conditions:
h u − h v, u − v cu − v20,ΩU ,
that is to say h· is uniformly convex.
∀u, v ∈ L2 ΩU ,
2.3
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Discrete Dynamics in Nature and Society
Noting that g· is convex, it is easy to see that
g u − g v, u − v 0,
∀u, v ∈ H 1 Ω.
2.4
∀v ∈ L2 ΩU , u ∈ H 1 Ω
2.5
Also, we have that
|Bv, w| cv0,ΩU w0,Ω ,
because B is a bounded linear operator.
Therefore the above-mentioned control problem 1.1-1.2 can be restated as follows
OCP:
min J u, yu T
u∈Uad
∂y
,w
∂t
a y, w t
g y hu dt,
0
c t, τ; yτ, w dτ f Bu, w ,
0
∀w ∈ V, t ∈ 0, T ,
yt0 y0 ,
2.6
where B is a linear bounded operator from L2 ΩU to L2 Ω and independent with t.
From Yanik and Fairweather 19, we know that the above state equation has at least
one solution y ∈ W0, T {w ∈ L2 0, T ; H 1 Ω, wt ∈ L2 0, T ; H −1 Ω}. For the existence
and uniqueness of the solution of the system 2.6, we have the following lemmas.
Lemma 2.1. For the optimal control problem 2.6, there exists the unique solution u, y, such that
u ∈ X and y ∈ L∞ 0, T ; L2 Ω ∩ L2 0, T ; H01 Ω and ∂y/∂t ∈ L2 0, T ; H −1 Ω.
Proof. Assume that {un , yn }∞
n1 is a minimization sequence for the problem 2.6. It follows
2
2
n ∞
that {un }∞
n1 are bounded in L 0, T ; L ΩU . Therefore there is a subsequence of {u }n1 still
n ∞
n
∗
2
2
denoted by {u }n1 such that u converges to u weakly in L 0, T ; L ΩU . It is clear that
for the subsequence un
∂yn
,w
∂t
a yn , w t
c t, τ; yn τ, wt dτ f Bun , w ,
∀w ∈ V, t ∈ 0, T .
0
2.7
By taking w yn and integrating time from 0 to t in 2.7, and applying Gronwall’s inequality,
we have
T
0
2
n 2
y dτ C y0 1,Ω
2
max yn t0,Ω
0tT
0,Ω
2
C y0 0,Ω
0
T
0
2
2
n
f eCT C∗ < ∞,
u 0,ΩU
−1,Ω
T
2
n 2
f u
0,Ω
−1,Ω
U
T t
0
0
n 2
y τ dτ dt
1,Ω
C.
2.8
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This infers that un ∈ L2 0, T ; L2 ΩU and yn ∈ L∞ 0, T ; L2 Ω ∩ L2 0, T; H 1 Ω, and
weakly in L2 0, T ; L2 ΩU ,
un −→ u
yn −→ y
L2 0, T ; H 1 Ω ,
weakly in L∞ 0, T ; L2 Ω
2.9
yn T −→ yT weakly in L2 Ω.
Now let us integrate time from 0 to T in 2.7 and take limits as n → ∞. Clearly we have
yT , wT − y0 , w0 −
T
T
0
f Bu, w dt,
y, wt
dt T
a y, w dt T t
0
0
c t, τ; yτ, wt dτ dt
0
∀w ∈ W0, T .
0
2.10
Therefore
T
0
T
T t
∂y
, w dt a y, w dt c t, τ; yτ, wt dτ dt
∂t
0
0 0
T
f Bu, w dt, ∀w ∈ W0, T .
2.11
0
Furthermore, we have
T ∂y ∂t L2 0,T ;H −1 Ω
∂y/∂t, w dt
0
sup
w∈L2 0,T ;H01 Ω
2
C y0 0,Ω
wL2 0,T ;H01 Ω
2
2
f u0,ΩU dt eCT .
−1,Ω
T
0
2.12
It follows that ∂y/∂t ∈ L2 0, T ; H −1 Ω.
Since g· is a convex function on space L2 0, T ; L2 Ω and h· is a strictly convex
function on Uad , we have
T
0
g y hu dt lim
n→∞
T
n
g y hun dt.
2.13
0
It follows that u, y is one solution of 2.6. Because Ju, yu is a strictly convex function
on Uad , we have that the solution for the minimization problem 2.6 is unique. The proof of
Lemma 2.1 is completed.
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Discrete Dynamics in Nature and Society
Remark 2.2. Here we suppose that the operator α is independent of time variable t. The above
results can also be applied to the case α αx, t provided that suitable conditions for the
operator α are to be imposed.
3. Finite Element Approximation of Control
In this section, we firstly state the optimality conditions and set up the finite element
approximation for optimal control problems governed by parabolic integrodifferential
equation.
It follows from 24 that we can similarly deduce the following optimality conditions
of the problem 2.6.
Theorem 3.1. A pair y, u ∈ L2 0, T ; H01 Ω ∩ L∞ 0, T ; L2 Ω × L2 0, T ; L2 ΩU is the
solution of the optimal control problem 2.6, if and only if there exists a costate p ∈ L2 0, T ; H01 Ω∩
L∞ 0, T ; L2 Ω such that the triple y, p, u satisfies the following optimality conditions:
∂y
,w
∂t
a y, w t
c t, τ; yτ, wt dτ f Bu, w ,
∀w ∈ V, t ∈ 0, T ,
0
3.1
y|t0 y0 ,
∂p
− q,
∂t
a q, p T
c τ, t; qt, pτ dτ y − zd , q ,
∀q ∈ V, t ∈ 0, T ,
t
3.2
p|tT 0,
T
h u B∗ p, v − u
0
U
dt 0,
3.3
∀v ∈ Uad ,
where B∗ is the adjoint operator of B.
In the following, we construct the semidiscrete finite element approximation of the
control problem 2.6 by approximating the optimality conditions.
Let Ωh be a polygonal approximation to Ω with boundary ∂Ωh . For simplicity, we
assume that Ω is a convex polygon so that Ω Ωh . Let T h be a partitioning of Ωh into disjoint
h
regular n-simplices τ, so that Ω τ∈T h τ. Each element has at most one face on ∂Ωh , and τ
and τ have either only one common vertex or a whole edge or face if τ and τ ∈ T h . As usual,
h denotes the diameter of the triangulation T h .
h
Associated with T h is a finite-dimensional subspace Sh of CΩ , such that χ|τ are
polynomials of m m 1 order for all χ ∈ Sh and τ ∈ T h . Let V h Sh ∩ H01 Ω. Note that we
do not impose a continuity requirement. It is easy to see that V h ⊂ V , W h ⊂ W.
h
Let TUh be a partitioning of ΩhU into disjoint regular n-simplices τU , so that ΩU h
τU ∈T h τ U . For simplicity, we again assume that ΩU is a convex polygon so that ΩU ΩU .
U
τ U and τ U have either only one common vertex or a whole edge or face if τ U and τ U ∈ TUh .
Let hτ hτU denote the maximum diameter of the element ττU in T h TUh .
Associated with TUh is another finite-dimensional subspace Uh of L2 ΩhU , such that
χ|τU are polynomials of order m m 0 for all χ ∈ Uh and τU ∈ TUh . Here there is no
Discrete Dynamics in Nature and Society
7
requirement for the continuity. Let X h L2 0, T ; Uh . It is easy to see that X h ⊂ X. Let hτ hτU denote the maximum diameter of the element ττU in T h TUh .
In this paper we only consider the piecewise constant finite element space for the
approximation of the control for the reason of the limited regularity of the optimal control
h
⊂ Uad ∩ X h .
u in general. For ease of exposition, in this paper we assume that Uad
In order to derive a posteriori error estimates of residual type, we need the following
important lemmas.
Lemma 3.2 see 12. Let πh be the standard Lagrange interpolation operator. For m 0 or 1,
q > n/2 and v ∈ W 2,q Ω,
|v − πh v|m,q,Ω Ch2−m |v|2,q,Ω .
3.4
Lemma 3.3 see 25. Let πh be the average interpolation operator defined in [25]. For m 0 or 1,
1 q ∞ and for all v ∈ W 1,q Ωh |v − πh v|m,q,τ τ
Ch1−m
|v|1,q,τ .
τ
τ/
∅
3.5
Lemma 3.4 see 26. For all v ∈ W 1,q Ω, 1 q < ∞
−1/q
1−1/q
|v|1,q,τ .
v0,q,∂τ C hτ v0,q,τ hτ
3.6
Then a possible semidiscrete finite element approximation of OCP is thus defined by
OCP :
h
min J uh , yh h
uh ∈Uad
T
g yh huh dt
3.7
0
subject to
∂yh
, wh
∂t
a yh , wh t
0
c t, τ; yh τ, wh t dτ f Buh , wh ,
yh t0 yh0 ,
∀wh ∈ V h , t ∈ 0, T ,
3.8
where yh ∈ W h , yh0 ∈ V h is the approximation of y0 .
Since this is a finite dimensional linear control problem and the reduced objective
function is convex, we can easily prove that the above problem 3.7-3.8 has a unique
h
.
solution yh , uh ∈ W h × Uad
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h
is a solution of
By applying 24 again we can show that a pair yh , uh ∈ W h × Uad
h
3.7-3.8, if and only if there exists a costate ph ∈ W such that the triple yh , ph , uh satisfies
the following optimality conditions, which we will label OCP − OPTh :
∂yh
, wh
∂t
a yh , wh t
0
c t, τ; yh τ, wh t dτ f Buh , wh ,
∀wh ∈ V h ,
yh t0 yh0 ,
T
∂ph
a qh , ph c τ, t; qh , ph τ dτ yh − zd , qh , ∀qh ∈ V h ,
− qh ,
∂t
t
ph tT 0,
T
h
.
h uh B∗ ph , vh − uh U dt 0, ∀vh ∈ Uad
3.9
3.10
3.11
0
The optimality conditions in 3.9–3.11 are the semidiscrete approximation to the problem
3.1–3.3.
4. A Posteriori Error Estimates for First-Order Derivatives
Adaptive finite element approximation has been found very useful in computing optimal
control, as mentioned in Introduction. It uses an a posteriori error indicator to guide the mesh
refinement procedure. Furthermore it has been recently found that for constrained control
problems, different adaptive meshes are often needed for the control and the states; see 27.
Using different adaptive meshes for the control and the state allows very coarse meshes to
be used in solving the state and costate equations. Thus much computational work can be
saved since one of the major computational loads is to solve the state and costate equations
repeatedly. In this section, we derive the upper a posteriori error estimates for the optimal
control problem allowing different meshes to be used for the states and the control.
For simplicity, we will only consider the case of quadratic objective functionals as
follows:
J u, y T
g y hu dt 0
T
T
2
β
1
2
y − zd u0,ΩU .
0,Ω
2 0
2 0
4.1
Here
1
g y 2
T
β
hu 2
0
y − zd 2 ,
0,Ω
T
0
4.2
u20,ΩU ,
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9
where γ is a positive regularity constant. By differential theory, we have
h uv βu, v .
4.3
Then the inequality 3.3 and 3.11 in optimality condition can be restated as follows:
T
T
βu B∗ p, v − u
0
U
βuh B∗ ph , vh − uh
0
dt 0,
U
4.4
∀v ∈ Uad ,
dt 0,
4.5
h
∀vh ∈ Uad
.
In this paper, we consider the integration obstacle type control constraint:
Uad v ∈ X;
ΩU
4.6
v 0, t ∈ 0, T .
By computation and from 28, we know that the solutions of inequality 4.4 and 4.5 yield
∗
ΩU
βu −B p max 0, B∗ p
ΩU
1
∗
βuh Ph −B ph max 0,
,
ΩU
B ∗ ph
ΩU
1
,
4.7
where Ph is the L2 -projection from L2 Ω to Uh .
And we will consider the special case of the differential operator of α and γ:
αy − div A∇y ,
γt, τy − div Ct, τ∇y ,
n×n
where A Ax ai,j ·n×n ∈ C∞ Ω
4.8
, such that there are constants c > 0 satisfying
X t AX cX2Rn ,
∀X ∈ Rn
4.9
n×n
and C Cx, t, τ ci,j x, t, τn×n ∈ C∞ 0, T ; L2 Ω .
In this case, we have the following bilinear form
az, w αz, w − divA∇z, w A∇z, ∇w,
ct, τ; z, w γt, τz, w − divCt, τ∇z, w Ct, τ∇z, ∇w.
4.10
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4.1. Main Results
We first state the main results of this section.
Theorem 4.1. Let y, u and yh , uh be the solutions of OCP and OCP h , respectively. Let p
and ph be the solution of the costate equations 3.2 and 3.10. Then there hold the a posteriori error
estimates
2
2
2
u − uh 2L2 0,T ;L2 Ω y − yh L∞ 0,T ;L2 Ω y − yh L2 0,T ;H 1 Ω p − ph L∞ 0,T ;L2 Ω
U
6
2
p − ph L2 0,T ;H 1 Ω C ηi2 ,
i1
4.11
where η12 , . . . , η62 is defined as follows:
η12
0
η22 η32
τU
⎧
T ⎨
0
⎩
∗
∗
B ph − Ph B ph
2
dt,
τU
h2τ
τ
τ
∂
ph div A∗ ∇ph ∂t
T
⎫
⎬
2
div C∗ τ, t∇ph τ dτ yh − zd
dτ
t
dt,
⎭
2
T T
∗
∗
hl
C τ, t∇ph τ · ndτ dl dt,
A ∇ph · n 0
η42 η52
T τ
⎧
T ⎨
0⎩
t
∂τ
h2τ
τ
τ
∂
yh − div A∇yh −
∂t
t
div Ct, τ∇yh τ dτ − f − Buh
0
2 ⎫
⎬
dt,
⎭
2
T t
hl
Ct, τ∇yh τ · ndτ dl dt,
A∇yh · n 0
τ
0
∂τ
2
η62 y0h − y0 2
L Ω
,
4.12
where l is a face of an element τ, hl is the maximum diameter of l, ∇ph · n and ∇yh · n are the
normal derivative jumps over the interior face l, defined by
∇ph · n l ∇ph hτ 1 − ∇ph τ 2 · n,
l
l
∇yh · n l ∇yh τ 1 − ∇yh τ 2 · n,
l
4.13
l
where n is the unit normal vector on l τl1 ∩ τl2 outwards τl1 . For later convenience, one defines
∇ph · nl 0 and ∇yh · nl 0 when l ⊂ ∂Ω.
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11
In the following subsections, we will prove Theorem 4.1. To this end, we first give some
lemmas in the following subsection. The proofs of Theorem 4.1 are put in the last subsection.
4.2. Some Lemmas
We have the following.
Lemma 4.2. Let y, u and yh , uh be the solutions of OCP and OCP h , respectively. Let p and
ph be the solution of the costate equations 3.2 and 3.10. Then
2
u − uh 2L2 0,T ;L2 Ω Cη12 Cph − puh L2 0,T ;L2 Ω ,
U
4.14
where puh is defined by the following system:
t
∂
yuh , ω a yuh , ω c t, τ; yuh τ, ωt dτ f Buh , ω ,
∂t
0
∀ω ∈ V,
4.15
yuh 0 y0h x, x ∈ Ω,
T
∂
c τ, t; qt, puh τ dτ yuh − zd , q ,
− q, puh a q, puh ∂t
t
4.16
∀q ∈ V.
4.17
Proof. It follows from 4.4 that we have
βu, u − uh
U
− B∗ p, u − uh U .
4.18
Then by 4.5 and 4.18
βu −
uh 2L2 0,T ;L2 Ω U T
0
βu, u − uh U − βuh , u − uh U dt
T
0
−
∗
− B p, u − uh U − βuh , u − uh U dt T
B∗ ph βuh , u − vh
0
T
0
∗
∗
U
dt −
B ph − B puh , u − uh
T
0
U
dt T
0
− B∗ p βuh , u − uh U dt
B∗ ph βuh , vh − uh
T
0
U
dt
B∗ puh − B∗ p, u − uh
U
dt
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Discrete Dynamics in Nature and Society
T
inf
h
vh ∈Uad
T
B∗ ph βuh , vh − u
U
0
∗
B ph − puh , u − uh
U
0
dt
dt T
0
B∗ puh − p , u − uh U dt
I1 I 2 I 3 .
4.19
Since Ph is the L2 -projection, then, for any v ∈ Uad , we have
ΩU
Ph v − vφ 0,
∀φ ∈ X h , t ∈ 0, T .
4.20
h
Note that ΩU v 0 and ΩU Ph v − v 0. Then we have ΩU Ph v 0, thus Ph v ∈ Uad
. So that
we can take vh Ph u in I1 and by 4.7, we have
I1 T
B∗ ph βuh , Ph u − u
0
T 0
τU
∗
Ph −B ph max 0,
ΩU
B ∗ ph
∗
∗
B ph Ph u − u dt
1
∗
−Ph B ph B ph Ph u − u dt
τU
T 4.21
−Ph B∗ ph B∗ ph Ph u − uh − u − uh dt
τU
τU
Cδ
ΩU
T 0
dt
τU
τU
0
U
T 0
τU
∗
∗
−Ph B ph B ph
2
τU
Cη12 δu − uh 2L2
0,T ;L2 ΩU dt δu − uh 2L2
0,T ;L2 ΩU ,
2
I2 Cph − puh L2 0,T ;L2 Ω δu − uh 2L2
0,T ;L2 ΩU .
4.22
From 4.15 and 3.1, we have for t ∈ 0, T t
∂
y − yuh , ω a y − yuh , w c t, τ; y − yuh τ, ωt dτ
∂t
0
Bu − uh , ω,
∀ω ∈ V
4.23
Discrete Dynamics in Nature and Society
13
and from 4.17 and 3.2
T
∂
c τ, t; qt, p − puh τ dτ
− q,
p − puh a q, p − puh ∂t
t
y − yuh , q , ∀q ∈ V.
4.24
Then from 4.23, 4.24 and integrating by part
I3 T
puh − p, Bu − uh 0
U
dt
T 0
∂
y − yuh , puh − p a y − yuh , puh − p
∂t
t
c t, τ; y − yuh τ, puh − p t dτ dt
0
T
∂
p − puh a y − yuh , puh − p
−y − yuh ,
∂t
0
T
c τ, t; y − yuh t, puh − p τ dτ dt
4.25
t
T
− y − yuh , y − yuh dt 0.
0
Following from 4.21–4.25, let δ be small enough:
2
u − uh 2L2 0,T ;L2 Ω Cη12 Cph − puh L2 0,T ;L2 Ω .
U
4.26
The proof of Lemma 4.2 is completed.
Lemma 4.3. Let y, u and yh , uh be the solutions of OCP and OCP h , respectively. Let p and
ph be the solution of the costate equations 3.2 and 3.10. Then there hold the a posteriori error
estimates
2
2
yh − yuh 2 ∞
yh − yuh L2 0,T ;H 1 Ω ph − puh L∞ 0,T ;L2 Ω
L 0,T ;L2 Ω
ph − puh 2L2
0,T ;H 1 Ω
C
6
4.27
ηi2 .
i2
Proof. Let
T
∂
c τ, t; vt, ph − puh τ dτ
ph − puh a v, ph − puh Euh , v − v,
∂t
t
4.28
14
Discrete Dynamics in Nature and Society
and πh be the average interpolation operator defined as in 27 and e ph − puh . Then from
3.10 and 4.17
T
∂
− qh ,
c τ, t; qh t, ph − puh τ dτ
ph − puh a qh , ph − puh ∂t
t
h
yh − yuh , qh , ∀qh ∈ V .
4.29
So we have
T
∂
c τ, t; vt, ph − puh τ dτ
− v,
ph − puh a v, ph − puh ∂t
t
T
∂
c τ, t; v − πh vt, ph τ dτ
− v − πh v, ph a v − πh v, ph ∂t
t
∂
v − πh v, puh − a v − πh v, puh ∂t
T
−
c τ, t; v − πh vt, puh τ dτ yh − yuh , πh v
t
⎧
⎨ ⎩
τ
τ
h2τ
hl
∂
− ph − div A∗ ∇ph −
∂t
A∗ ∇ph · n ∗
2 ⎫1/2
⎬
2
div C τ, t∇ph τ dτ − yh zd
t
C∗ τ, t∇ph τ · ndτ
t
∂τ
τ
T
T
⎭
v1,Ω yh − yuh , v .
4.30
By letting v ph − puh in 4.30 and from 2.2, we have
−
1 d
ph − puh 2 cph − puh 2
0,Ω
1,Ω
2 dt
⎧
2
T
⎨ ∗
∗
∂
2
h − ph − div A ∇ph −
div C τ, t∇ph τ dτ − yh zd
⎩ τ τ τ
∂t
t
τ
hl
A∗ ∇ph · n T
∂τ
yh − yuh , ph − puh −
2 ⎫1/2
⎬ ph − puh C∗ τ, t∇ph τ · ndτ
1,Ω
⎭
t
T
t
c τ, t; ph − puh t, ph − puh τ dτ.
4.31
Discrete Dynamics in Nature and Society
15
Integrating time from t to T in 4.31 and by Schwartz inequality, Lemmas 3.3 and 3.4, we
have
T
1
ph − puh 2 dτ
ph − puh 2 c
0,Ω
1,Ω
2
t
2
T T
∗
∗
∂
2
ph div A ∇ph hτ
div C s, τ∇ph s ds yh − zd dτ
∂t
t τ
τ
τ
2
T
T T
∗
∗
ph − puh 2 dτ
hl
C s, τ∇ph s · nds dτ δ
A ∇ph · n 1,Ω
t
C
T
t
τ
∂τ
τ
yh − yuh 2 dτ C
0,Ω
t
T T
t
τ
ph − puh s2 ds dτ.
1,Ω
4.32
Letting δ be small enough, we obtain
T
t
ph − puh 2 dτ
1,Ω
2
T T
∗
∗
∂
2
ph div A ∇ph hτ
div C s, τ∇ph s ds yh − zd dτ
C
∂t
t τ
τ
τ
2
T T
∗
∗
hl
C
C s, τ∇ph s · nds dτ
A ∇ph · n t
C
T
t
τ
τ
∂τ
yh − yuh 2 dτ C
0,Ω
4.33
T T
t
τ
ph − puh s2 ds dτ.
1,Ω
By Gronwall inequality and 4.30–4.33
2
ph − puh 2 2
Cη22 Cη32 Cyh − yuh L2 0,T ;L2 Ω .
L 0,T ;H 1 Ω
4.34
Similarly, we have
2
ph − puh 2 ∞
C η22 η32 yh − yuh L2 0,T ;L2 Ω
L 0,T ;L2 Ω
C
T T
0
t
ph − puh τ2 dτ dt
1,Ω
2
Cη22 Cη32 Cyh − yuh L2 0,T ;L2 Ω .
4.35
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Discrete Dynamics in Nature and Society
Then from 4.30, 4.34, and 4.35
T ∂
p − puh , v
0 ∂/∂t h
sup
∂t ph − puh 2
vL2 0,T ;H 1 Ω
L 0,T ;H −1 Ω
v∈L2 0,T ;H 1 Ω
0
T T −
pu
c
τ,
t;
vt,
p
−
pu
v
a
v,
p
,
τ dτ dt
−Eu
h
h
h
h
h
0
t
sup
vL2 0,T ;H 1 Ω
v∈L2 0,T ;H01 Ω
≤ Cη2 Cη3 Cyh − yuh L2 0,T ;L2 Ω Cph − puh L2 0,T ;H 1 Ω
≤ Cη2 Cη3 Cyh − yuh L2 0,T ;L2 Ω .
4.36
Similarly by analysis of yh − yuh L2 0,T ;H 1 Ω , we let
Quh , v ∂
yh − yuh , v
∂t
a yh − yuh , v t
c t, τ; yh − yuh τ, vt dτ.
0
4.37
By 3.9 and 4.15
ωh ,
t
∂
yh − yuh a yh − yuh , ωh c t, τ; yh − yuh τ, ωh t dτ
∂t
0
4.38
∀ωh ∈ V .
0,
h
So
t
∂
yh − yuh , v a yh − yuh , v c t, τ; yh − yuh τ, vt dτ
∂t
0
t
∂
yh , v − πh v a yh , v − πh v c t, τ; yh τ, v − πh vt dτ − f Buh , v − πh v
∂t
0
⎧
2
t
⎨ ∂
2
y − div A∇yh − div Ct, τ∇yh dτ − f − Buh
h
⎩ τ τ τ ∂t h
0
τ
hl
∂τ
A∇yh · n t
0
Ct, τ∇yh · ndτ
2 ⎫1/2
⎬
⎭
v1,Ω .
4.39
Discrete Dynamics in Nature and Society
17
By letting v yh − yuh and applying Swartz inequality, we have
1 d
yh − yuh 2 cyh − yuh 2
0,Ω
1,Ω
2 dt
2
t
∂
2
yh − div A∇yh − div Ct, τ∇yh dτ − f − Buh
hτ
∂t
0
τ
τ
τ
A∇yh · n hl
t
2
4.40
Ct, τ∇yh · ndτ
0
∂τ
2
δyh − yuh 1,Ω −
t
c t, τ; yh − yuh τ, yh − yuh t dτ.
0
By integrating time from 0 to t in 4.40
yh − yuh 2 c
0,Ω
C
t
h2τ
0 τ
t
0
τ
yh − yuh 2 dτ
1,Ω
∂
yh − div A∇yh −
∂t
τ
div Cτ, s∇yh ds − f − Buh
2
dτ
0
t t
τ
2
2
A∇yh · n hl
Cτ, s∇yh · nds dt δ yh − yuh 1,Ω dτ
0 τ
C
t τ
0
0
∂τ
0
0
2
yh − yuh 2 ds dτ C
y0 − y0h .
1,Ω
0,Ω
4.41
Since δ is small, then from 4.41 and Gronwall inequality we have
t
0
yh − yuh 2 dt
1,Ω
C
t τ
2
∂
yh − div A∇yh −
h2τ
div Cτ, s∇yh ds − f − Buh dτ
∂t
0 τ
0
τ
C
4.42
t τ
2
2
hl
A∇yh · n Cτ, s∇yh · nds dt Cy0 − y0h 0 τ
∂τ
0
0,Ω
and we obtain
2
2
2
yh − yuh 2 2
,
η
η
C
η
1
5
6
4
L 0,T ;H Ω
yh − yuh 2 ∞
C η42 η52 η62 .
L 0,T ;L2 Ω
4.43
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Discrete Dynamics in Nature and Society
In the same way of getting 4.36, we also have
∂
2
C η42 η52 η62 .
−
yu
y
h ∂t h
L2 0,T ;H −1 Ω
4.44
Then the desired results 4.27 follow from 4.34–4.36 and 4.43–4.44.
4.3. Proof of Theorem 4.1
By Lemmas 4.2 and 4.3, we prove Theorem 4.1 in the following.
Proof. By using the triangle inequality, Lemmas 4.2 and 4.3, and from 4.29 and 4.38, 17–
19, using the following stability results
2
y − yuh 2 ∞
y − yuh L2 0,T ;H 1 Ω Cu − uh 2L2
L 0,T ;L2 Ω
0,T ;L2 ΩU ,
2
2
p − puh 2 ∞
p − puh L2 0,T ;H 1 Ω Cy − yuh L2 0,T ;L2 Ω
L 0,T ;L2 Ω
Cu − uh 2L2
0,T ;L2 ΩU 4.45
,
we can easily obtain 4.11. The proof of Theorem 4.1 is completed.
5. A Posteriori Error Estimates in Integral
Often we need sharper a posteriori error estimates. Then we need deriving the a posteriori
error estimates in L2 0, T ; L2 Ω-norm. To this end we first state the main results in this
paper.
5.1. Main Results
We have the following results.
Theorem 5.1. Let y, u and yh , uh be the solutions of OCP and OCP h , respectively. Let p
and ph be the solution of the costate equations 3.2 and 3.10. Then there hold the a posteriori error
estimates
u −
uh 2L2 0,T ;L2 Ω U 5
2
2
2
2
2
y − yh L2 0,T ;L2 Ω p − ph L2 0,T ;L2 Ω C η1 ξi η6 ,
i2
5.1
Discrete Dynamics in Nature and Society
19
where η12 , η62 is defined in Theorem 4.1 and
⎧
2 ⎫
T ⎨ T
⎬
∂
dt,
ph div A∗ ∇ph h4τ
div C∗ τ, t∇ph τ dτ yh − zd
ξ22 ⎭
∂t
0⎩ τ
t
τ
ξ32
2
T
T ∗
∗
3
C τ, t∇ph τ · ndτ dl dt,
A ∇ph · n hl
0
τ
t
∂τ
⎧
2 ⎫
t
T ⎨ ⎬
∂
ξ42 yh − div A∇yh − div Ct, τ∇yh τ dτ − f − Buh
h4τ
dt,
⎭
∂t
0⎩ τ
0
τ
ξ52
5.2
2
T t
3
hl
Ct, τ∇yh τ · ndτ dl dt.
A∇yh · n 0
τ
0
∂τ
In order to prove Theorem 5.1, we need the following dual equations and lemmas.
5.2. Dual Equations and Some Lemmas
For given F ∈ L2 0, T ; L2 Ω and the following equation:
∂
φ − div A∇φ −
∂t
t
div Ct, τ∇φτ dτ F,
x, t ∈ Ω × 0, T ,
0
φ∂Ω 0,
5.3
t ∈ 0, T ,
φx, 0 0,
x ∈ Ω,
we have its dual equation:
∂
− ψ − div A∗ ∇ψ −
∂t
T
div C∗ τ, t∇ψτ dτ F,
t
ψ∂Ω 0,
t ∈ 0, T ,
ψx, T 0,
x, t ∈ Ω × 0, T ,
5.4
x ∈ Ω.
In order to derive a posteriori error estimates in L2 -norm, it is necessary to have some stability
results of the dual equations. From 17–19, 29, we have the following stability results.
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Discrete Dynamics in Nature and Society
Lemma 5.2. Assume that φ and ψ are the solution of 5.3 and 5.4, respectively. Then
φ CFL2 0,T ;L2 Ω ,
L∞ 0,T ;L2 Ω
∇φ 2
CFL2 0,T ;L2 Ω ,
L 0,T ;L2 Ω
2 CFL2 0,T ;L2 Ω ,
D φ 2
2
L 0,T ;L Ω
∂ φ
CFL2 0,T ;L2 Ω ,
∂t 2
L 0,T ;L2 Ω
ψ ∞
CFL2 0,T ;L2 Ω ,
L 0,T ;L2 Ω
∇ψ 2
CFL2 0,T ;L2 Ω ,
L 0,T ;L2 Ω
2 CFL2 0,T ;L2 Ω ,
D ψ 2
2
5.5
L 0,T ;L Ω
∂ ψ
CFL2 0,T ;L2 Ω ,
∂t 2
L 0,T ;L2 Ω
where D2 φ ∂2 φ/∂xi ∂xj , 1 i, j n, and D2 ψ is defined similarly.
It follows from Lemmas 4.2 and 5.2 that we have the following upper bounds.
Lemma 5.3. Let y, u and yh , uh be the solutions of OCP and OCP h , respectively. Let p and
ph be the solution of the costate equations 3.2 and 3.10. Then there hold the a posteriori error
estimates
5
2
2
2
2
yh − yuh 2
ξi η6 .
ph − puh L2 0,T ;L2 Ω C
L 0,T ;L2 Ω
5.6
i2
Proof. It follows from Lemma 4.2 that we only need to estimate ph − puh 2L2 0,T ;L2 Ω .
Let φ be the solution of 5.3 with F ph − puh and φI πh φ be the interpolation of
φ in Lemma 3.2.
It follows from 5.3, 4.29 and by integrating by parts
ph − puh 2 2
L 0,T ;L2 Ω
T
0
Ft, ph − puh t dt
T t
∂
−
ph − puh , φ a φ, ph − puh c t, τ; φτ, ph − puh t dτ dt
∂t
0
0
Discrete Dynamics in Nature and Society
T
T ∂
−
ph , φ − φI a φ − φI , ph c τ, t; φ − φI t, ph τ dτ
∂t
0
t
∂
puh , φ − φI
∂t
− a φ − φI , puh −
T
21
c τ, t; φ − φI t, puh τ dτ
t
T
∂
c τ, t; φI t, ph − puh τ dτ dt
−
ph − puh , φI a φI , ph − puh ∂t
t
T T
∗
∗
∂
ph div A ∇ph div C τ, t∇ph τ dτ yh − zd φ − φI dt
∂t
0
t
τ
τ
T T
T
∗
∗
C τ, t∇ph τ · ndτ φ − φI dl dt
yh − yuh , φ dt
A ∇ph · n
0
t
∂τ
τ
0
J1 J2 J3 .
5.7
By Lemmas 3.2, 3.4, and 5.2, we obtain
J1 Cδ
⎧
T ⎨
0
δ
T
0
⎩
h4τ
τ
τ
∂
ph div A∗ ∇ph ∂t
T
div C∗ τ, t∇ph τ dτ yh − zd
2 ⎫
⎬
t
dt
⎭
2
φ dt Cδξ2 δph − puh 2 2
,
2
2,Ω
L 0,T ;L2 Ω
2
T T
T
2
∗
∗
3
φ dt
hl
J2 Cδ
C τ, t∇ph τ · ndτ dl dt δ
A ∇ph · n 0,Ω
0
τ
∂τ
t
0
2
Cδξ32 δph − puh L2 0,T ;L2 Ω .
5.8
By Schwartz inequality
2
2
J3 Cδyh − yuh L2 0,T ;L2 Ω δph − puh L2 0,T ;L2 Ω .
5.9
Letting δ be small enough, it follows from 5.7–5.9
3
2
ph − puh 2 2
ξi2 Cyh − yuh L2 0,T ;L2 Ω .
C
2
L 0,T ;L Ω
5.10
i2
Similarly, let ψ be the solution of 5.4 with F yh − yuh and ψI πh φ be the interpolation
of φ in Lemma 3.2.
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Discrete Dynamics in Nature and Society
From 4.38 and by integrating by parts, we have that
yh − yuh 2 2
L 0,T ;L2 Ω
T
Ft, yh − yuh t dt
0
T T
∂
c τ, t; yh − yuh t, ψτ dτ dt
yh − yuh , ψ a yh − yuh , ψ ∂t
0
t
T ∂
t h
yh , ψ −ψI a yh , ψ − ψI c t, τ; yh τ, ψ −ψI t dτ dt
y0 −y0 , ψ0 ∂t
0
0
−
T 0
∂
yuh , ψ − ψI a yuh , ψ − ψI
∂t
t
c t, τ; yuh τ, ψ − ψI t dτ dt y0h − y0 , ψ0
0
T t
∂
yh − div A∇yh − div Cτ, t∇yh τ dτ − f − Buh ψ − ψI dt
∂t
0
0
τ
τ
T t
Ct, τ∇yh τ · n dτ ψ − ψI dl dt
A∇yh · n −
0
0
∂τ
τ
y0h − y0 , ψ0 D1 D2 D3 .
5.11
Similarly, it follows from Lemmas 3.2, 3.4, and 5.2 that
⎧
2 ⎫
T ⎨ t
⎬
∂
dt
D1 Cδ
yh − div A∇yh − div Cτ, t∇yh τ dτ − f − Buh
h4τ
⎭
∂t
0⎩ τ
0
τ
δ
T
0
D2 Cδ
2
ψ dt Cξ2 δyh − yuh 2 2
,
4
2,Ω
L 0,T ;L2 Ω
T 0
τ
h3l
A∇yh · n −
∂τ
t
0
2
Ct, τ∇yh τ · n dl dt δ
T
0
2
ψ dt
2,Ω
2
Cξ52 δyh − yuh L2 0,T ;L2 Ω ,
2
D3 Cη62 δyh − yuh L2 0,T ;L2 Ω .
5.12
Discrete Dynamics in Nature and Society
23
Letting δ be small enough, then from 5.11-5.12 we have
yh − yuh 2 2
L
2
2
2
ξ
η
C
ξ
5
6 .
4
0,T ;L2 Ω
5.13
Then 5.6 follows from 5.10–5.13. The proof of Lemma 5.3 is completed.
5.3. Proof of Theorem 5.1
From Lemmas 4.2 and 5.3, we can easily prove Theorem 5.1.
Proof. By triangle inequality, 4.45, Lemmas 4.2 and 5.3, we can easily prove 5.1 in the same
way of getting 4.11. The proof of Theorem 5.1 is completed.
6. Conclusion
In this paper, we first briefly introduce optimal control problem governed by parabolic
integrodifferential equations and give the weak formulation for this optimal control problem.
For this formulation, we prove the existence and uniqueness of the solution. Then by the
theory of optimal control problem, we present the optimality conditions and semidiscrete
finite element approximation scheme. The upper a posteriori error estimates for firstorder derivative and L2 0, T ; L2 Ω-norm are derived for both the state and the control
approximation for the case of an integral obstacle constraint. We will research on the upper
and lower a posteriori error estimates for full discrete finite element approximations of this
control problem.
Acknowledgments
This research was supported by Science and Technology Development Planning Project
of Shandong Province no. 2011GGH20118 and Shandong Province Natural Science
Foundation nos. ZR2012AQ024 and ZR2012GM018.
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