Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2012, Article ID 921038, 25 pages
doi:10.1155/2012/921038
Research Article
An Inverse Problem for a Class of
Linear Stochastic Evolution Equations
Yuhuan Zhao
College of Science, Civil Aviation University of China, Tianjin 300300, China
Correspondence should be addressed to Yuhuan Zhao, yhzhao@cauc.edu.cn
Received 8 May 2012; Revised 7 August 2012; Accepted 9 August 2012
Academic Editor: Chong Lin
Copyright q 2012 Yuhuan Zhao. 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.
An inverse problem for a linear stochastic evolution equation is researched. The stochastic
evolution equation contains a parameter with values in a Hilbert space. The solution of the
evolution equation depends continuously on the parameter and is Fréchet differentiable with
respect to the parameter. An optimization method is provided to estimate the parameter. A
sufficient condition to ensure the existence of an optimal parameter is presented, and a necessary
condition that the optimal parameter, if it exists, should satisfy is also presented. Finally, two
examples are given to show the applications of the above results.
1. Introduction
The purpose of this paper is to study an inverse problem for the following linear stochastic
evolution equation:
dy A t; p B t; p ydt f t; p dt σ t; p dwt,
yt0 ϕ ξ,
t ∈ t0 , tf ≡ T,
1.1
where tf < ∞, p ∈ Pad ⊂ P is a parameter to be determined, and Pad is a convex domain in
P . The solution of 1.1 corresponding to p can be denoted as y yp yt; p to explicitly
show the dependence of y on p.
The problem of this paper is to determine the unknown parameter p based on the
measurement gt, which is defined by the following:
gt Λ t; p yt,
where V , H, K, W, and P are Hilbert spaces.
t ∈ T,
1.2
2
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There are many papers dealing with parameter estimation problems for stochastic
partial differential equations, for instance, see 1–7
, but only a few papers to estimate directly
parameters involved in stochastic evolution equations in infinite dimensional spaces, for
example, 8, 9
. In particular, Lototsky and Rosovskii 9
consider a problem estimating a
constant parameter and obtain an estimate that is consistent and asymptotically normal.
Denote by LX; Y the linear continuous operator space on X to Y , by ·, ·X the inner
product of X, and by ·, ·X ,X the dual product of X and X, where X is the dual of X.
V , V , and H make up an evolution triple, namely, they should satisfy
V ⊂ H ⊂ V ,
1.3
where each space is dense in the following space and has a continuous injection, H H, and
y, x V ,V y, x H ,
∀x ∈ V, y ∈ H.
1.4
For any t ∈ T and p ∈ Pad , At; p ∈ LV ; V , Bt; p ∈ LV ; H, σt, p ∈ LW; H,
Λt; p ∈ LV ; K, , ft; p ∈ H, and
A t; p z, z V ,V ≥ αz2V ,
z ∈ V,
1.5
where the constant α is independent of p ∈ Pad and t ∈ T . Let Ω, F, μ be a complete
probability space and Ft an increasing family of sub σ-algebras of FF F∞ .
MΩ, μ, Φ denotes the space of random variables with values in a Hilbert space Φ.
L Ω; Φ ≡
r
1/r
x ∈ M Ω, μ, Φ ; x ≡ ExrΦ
L Ω × T ; Φ ≡
r
xt ∈ M Ω, μ, Φ , t ∈ T ; x ≡
1/r
Ω
xωrΦ dμω
Ω
T
xt, ωrΦ dtdμω
<∞ ,
1/r
<∞ .
1.6
wt ∈ MΩ, μ, W is a Wiener process with values in a separable Hilbert space W, which is
adapted to Ft , that is, for all e ∈ W, wt, eW is a real Wiener process and an Ft -martingale,
with the correlation function
E{wt1 , e1 W wt2 , e2 W } mint1 ,t2 Qτe1 , e2 dτ,
1.7
t0
where t0 ≤ t1 , t2 ≤ tf and Qt is a positive self-adjoint nuclear operator almost everywhere
on W. Qt is called the covariance operator. Moreover, assume that Qt satisfies
Q· ∈ L∞ T ; LW,
where ϕ0 ∈ H, ξ ∈ L2 Ω, H is independent of wt.
1.8
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3
Now, determine the parameter p in the system 1.1 and 1.2. It is transformed into an
optimization problem as most researchers expect. That is, seek an optimal parameter p ∈ Pad
such that the cost functional
J p Eg
T
gt − Λ t; p y t; p 2 dt
K
1.9
reaches its minimum over the admissible parameter set Pad at p, that is,
J p min J p ,
p∈Pad
1.10
where Eg is a conditional expectation, that is,
Eg f E f | Gt ,
1.11
and Gt is the sub-σ-algebra induced by the stochastic process gs, 0 ≤ s ≤ t, which is adapted
to Ft .
If there exists a neighbourhood U ⊂ Pad of p0 such that
J p0 inf J p ,
p∈U
1.12
then Po is called a relative optimal parameter.
In Section 2, the base of this paper is given, under certain conditions the function p →
yp is continuous and Fréchet continuously differentiable.
In Section 3, the main results of this paper are proposed. The problem estimating the
parameter is transformed into the optimization problem. The above optimization problem,
such as existence of the optimal parameter and necessary conditions, is studied.
In Section 4, the results in Sections 2 and 3 are applied to parabolic stochastic partial
differential equations to identify certain parameters involved in those equations.
2. Continuity and Differentiability with Respect to a Parameter
In this section the continuity and the differentiability of the solution of the system 1.1 with
respect to the parameter p are studied.
Before studying the properties of the solution to 1.1, it must be shown that the
system 1.1 is well-behaved in some sense on certain conditions. There are many papers
dealing with solvability of Stochastic evolution equation 1.1, for example, see 10–12
. From
Bensoussan 10
the following lemma is useful.
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Lemma 2.1. Besides the assumptions for A, B, f, ϕ, wt, and σ in Section 1, one assumes that, for
any p ∈ Pad ,
A ·; p ∈ L2 T ; L V ; V ∩ L∞ T ; L V ; V ,
B ·; p ∈ L∞ Ω × T ; LV ; H,
f ·, p ∈ L2 Ω × T ; H,
σ ·, p ∈ L2 T, LW; H.
2.1
Then there exists a unique generalized solution, y, in the Ladyzenskaja sense of 1.1 almost every
t ∈ T such that
y ∈ L2 Ω; CT ; H ∩ L2 Ω × T ; V ∩ C T ; L2 Ω; H ≡ S,
2.2
yt is adapted to Ft (as a process with values in H), and yt is Ft measurable with values in V .
In the above lemma the space Ck X, Y with k 0, 1 consists of all continuous functions u : X → Y that have continuous Fréchet derivatives up to order k on X, with the
norm
u k
maxui t.
i0
2.3
t∈X
Remark 2.2. The generalized solution in the Ladyzenskaja sense is the solution of the
following variational equation:
yt, ηt
H
ϕ ξ, ηt0 H t
t0
t
t0
t
yt, dt ηt
t0
V,V A t; p B t; p ytdt, ηt V ,V
σ t; p dwt, ηt H ,
2.4
a.s., ∀η ∈ W 1 T ; V, H,
where the space W 1 T ; V, H is a Hilbert space that is defined by
dη
2
2
∈ L T; V
W T ; V, H η; η ∈ L T ; V ,
,
dt
1
and it is well known that W 1 T ; V, H ⊂ CT ; H.
Now, it is time to give the main results in this section.
2.5
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5
Theorem 2.3. If the assumptions of Lemma 2.1 are satisfied and
A : p −→ L2 T ; L V ; V ,
f : p −→ L2 Ω × T ; H,
B : p −→ L∞ Ω × T ; LV ; H,
2.6
σ : p −→ L2 T, LW; H,
are continuous, the solution of 1.1
y : Pad −→ S
2.7
is continuous, that is, y ∈ CPad ; S or the following equalities are true:
lim y p − y p0 CT ;L2 Ω,H 0,
2.8
lim y p − y p0 2.9
p → p0
p → p0
L2 Ω;CT ;H 0,
lim y p − y p0 L2 Ω×T ;V 0.
p → p0
2.10
Before proving Theorem 2.3, from Bensoussan 10
the following It
o formula in a
Hilbert space is quoted.
Lemma 2.4. Let Φz, t be a functional on H × T , which is twice continuously Fréchet differentiable
in z ∈ H and continuously differentiable in t ∈ T . Assume zt has the stochastic differential:
dzt atdt btdwt,
zt0 z0 ,
2.11
where at is a stochastic process with values in H, which is adapted and satisfies the condition
t
a.s
t0
atH dt < ∞,
∀t ∈ T,
2.12
where bt is an adapted process with values in LW, H such that t, ω → b is measurable and
t1
E
t0
bt2LW,H dt < ∞,
∀t1 < tf ,
2.13
then one has the following It
o formula in the Hilbert space:
t ∂Φ
∂Φ
, as ds , bdws
∂z
∂z
t0
t0
H
H
t
∂2 Φ
∂Φ
ds,
tr b∗ 2 bQ ds ∂z
t0 ∂t
Φzt, t Φzt0 , t0 1
2
t
t0
t 2.14
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Journal of Applied Mathematics
where b∗ is the adjoint of b and the symbol “tr” is the trace operator, of which definition for a nuclear
operator U ∈ LX is as follows:
tr U ∞
Ufn , fn
n1
X
,
2.15
where {fn } is an orthonormal basis of X.
The following two lemmas are obvious, so their proofs are omitted.
Lemma 2.5. Supposes that U ∈ LX is a nuclear operator and B ∈ LX, Y , then
tr B ∗ BU ≤ B2 tr U.
2.16
Lemma 2.6. If ψ ∈ L2 Ω; L2 T ,
ψtdwt 0.
E
2.17
T
Lemma 2.7. If f ∈ L2 Ω; L2 T ; W ≡ L2 Ω × T, W,
2
E
ft, dwt W E
Rtft, ft W dt.
T
2.18
T
Proof. First, suppose that f is a step function, that is,
ft fr ,
tr ≤ t < tr1 , r 0, . . . , N − 1,
2.19
where fr ∈ L2 Ω, W and t0 < t1 < · · · < tN tf .
Let {en } be an orthomomal basis of W. Obviously,
wt ∞
en , wtW en ,
i1
and wn t en , wt is a Wiener process. Furthermore,
2
tr1 ts1 E ft, dwt E
fr , ei dwi
fs , ej dwj
T
i,j
E
tr
r,s
ts
fr , ei wi tr1 − wi tr fs , ej wj ts1 − wj tj
i,j
r,s
2.20
Journal of Applied Mathematics
7
⎛
⎝Rt
E
T
f, ei ei ,
i
⎞
f, ej ej ⎠dt
j
E
T
Rtft, ft W dt,
2.21
So 2.18 is proved.
For the general case, 2.18 can be obtained as most lectures on stochastic integral have
done, which is omitted.
New, the proof of Theorem 2.3 can be given as follows.
Proof of Theorem 2.3. Denote yt yt, p and y0 t yt; p0 , and then
dy A t; p B t; p ydt f t; p dt σ t; p dwt, yt0 ϕ ζ,
dy0 A t; p0 B t; p0 y0 dt f t; p0 dt σ t; p0 dwt, y0 t0 ϕ ζ.
2.22
2.23
Let zt yt − y0 t, and then from the above equalities, it follows
dzt A t; p B t; p zdt f t; p − f t; p0 − A t; p − A t; p0 y0
− B t; p − B t; p0 y0 dt σ t; p − σ t; p0 dwt,
zt0 0.
2.24
Setting Φz 1/2z2 , z ∈ W ! T ; V, H according to Lemma 2.4, it gets
t
1
2
zs, A s; p zs V,V ds
ztH
2
t0
t
−
zs, B s; p zs H ds
t0
−
t
t0
t
zs, A s; p − A s; p0 y0 s V,V ds
zs, f s; p − f s; p0 − B s; p − B s; p0 y0 s H ds
t0
t
t0
1
2
zs, σ s; p − σ s; p0 dws H
t
tr
t0
∗ σ s, p − σ s; p0
σ s, p − σ s; p0 Qs ds,
2.25
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Journal of Applied Mathematics
which is
zt2H
−2
2
t
t0
t
zs, B s; p zs H ds 2
t
t0
−2
t
t
t0
zs, f s; p − f s; p0 H ds
zs; B s; p − B s; p0 y0 s H ds
2.26
zs, σ s; p − σ s; p0 dws H
t0
t
zs, A s; p − A s; p0 y0 s V,V ds
t0
2
zs, A s; p zs V.V ds
t0
−2
t
tr
∗ σ s, p − σ s; p0
σ s, p − σ s; p0 Qs ds.
t0
Taking the expectation from the above and considering Lemma 2.6, it has
Ezt2H 2E
−2E
t
t0
− 2E
t
t0
− 2E
t
t0
t
tr
t
t0
zs, A s; p zs V,V ds
zs, B s; p zs H ds 2E
t
t0
zs, f s; p − f s; p0 H ds
zs, As; p − As; p0 y0 s V,V ds
2.27
zs; B s; p − B s; p0 y0 s H ds
∗ σ s, p − σ s; p0 Qs ds.
σ s, p − σ s; p0
t0
From the assumptions of the spaces H and V , there exists a constant γ such that
xH ≤ γxV ,
∀x ∈ V.
2.28
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9
Furthermore, according to the assumptions of the operator Ap and using 2.28 and 2ab ≤
εa2 b2 /ε, we can obtain
Ezt2H 2αE
≤ αE
t
t0
t
zs2V ds
t0
zs2V ds
4γ
E
α
4γ B ·; p 2 ∞
E
L T ;LV ;H
α
t
t0
zs2H ds
t f s; p − f s; p0 2 A s; p − A s; p0 y0 s2 H
V
2.29
t0
2
B s; p − B s; p0 y0 sH ds.
tr Q
t
σ s, p − σ s; p0 2 ds.
t0
By the assumptions of Theorem 2.3 there is a constant c such that
Ezt2H αE
≤c
t
t0
t
t0
zs2V ds
Ezs2H ds
cE
t t0
c
t
t0
σ s, p − σ s; p0 2
ds
LW;H
2.30
f s; p − f s; p0 2 A s; p − A s; p0 y0 s2 H
V
2
B s; p − B s; p0 y0 sH ds.
Obviously,
Ezt2H
≤c
t
t0
Ezs2H ds
cE
t f s; p − f s; p0 2
t0
H
2
A s; p − A s; p0 y0 sV 2
B s; p − B s; p0 y0 sH ds
c
t
t0
σ s, p − σ s; p0 2
ds.
LW;H
2.31
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Journal of Applied Mathematics
Using the Gronwall inequality see 13
for the definition, and the assumptions of Theorem 2.3, letting p → p0 , it has
maxEzs2H o1,
2.32
t∈T
which is 2.8. Furthermore, from 2.31, it follows
E
T
zs2V o1,
2.33
which is just 2.10.
From 2.26, it has the following estimate:
zt2H
≤
2α
t0
t
α
γ
t0
4γ
α
t
zs2V ds
zs2H ds
t f s; p − f s; p0 2
H
t0
2
A s; p − A s; p0 y0 sV 2
B s; p − B s; p0 y0 sH ds
2
B ·; p L∞ T ;LV ;H
2
t
t0
t
t0
2.34
zs2H ds
zs, σ s; p − σ s; p0 dws H
tr QL∞ T t
σ s, p − σ s; p0 2
t0
LW;H
ds.
Similarly, it also obtains
zt2H
t f s; p − f s; p0 2 A p − A p0 y0 2 ≤c
H
V
t0
2
2 B p − B p0 y0 H σ p − σ p0 LW;H ds
c
t
t0
zs, σ s, p − σ s; p0 dws H ,
t ∈ T.
2.35
Journal of Applied Mathematics
11
On the other hand, the indefinite stochastic integral
Xt t
t0
∗
σ s; p − σ s; p zs, dws H
2.36
is a continuous martingale, so
2 1/2
2
E max Xt ≤ E max |Xt| ≤ E max|Xt|
≤ 2 EX tf t
t∈T
1/2
t
1/2
∗
∗
2.37
2 E
Rt σ s; p − σ s; p0 zs, σ s; p − σ s; p0 zs dt
≤c
T
1/2
σ t; p − σ t; p0 2 dt
,
T
where the following martingale inequality is used:
2
α
α
α
E max|Xt| ≤
EX tf ,
α
−
1
T
2.38
with α 2.
Combining 2.35 with 2.37, it has
E maxzt2H
t∈T
tf f s; p − f s; p0 2 A s; p − A s; p0 y0 s2 ≤ cE
H
V
t0
2
B s; p − B s; p0 y0 sH ds
c
T
σ s, p − σ s; p0 2
ds
LW;H
c
T
2.39
1/2
σ s, p − σ s; p0 2
ds
.
LW;H
Letting p → p0 in P , by the assumptions of Theorem 2.3, it immediately obtains
E maxzt2H o1,
t∈T
2.40
which is 2.9.
Theorem 2.8. Besides the assumptions of Theorem 2.3, suppose that the mappings
A : T × Pad −→ L V ; V ,
f : T × Pad −→ H,
B : T × Pad −→ LV ; H,
σ : T × Pad −→ LW; H
2.41
12
Journal of Applied Mathematics
are continuously Fréchet differentiable, then the solution of 1.1
y : p −→ y p ∈ S
2.42
is continuously Fréchet differentiable and its Fréchet derivative operators at p0 ∈ Pad , y p0 ∈
LP, S, are determined by the following system:
"
! dẏ A t; p0 B t; y0 ẏdt fp t; p0 h − Ap t; p0 hy0 − Bp t; p0 hy0 dt σp t; p0 hdwt,
ẏt0 0,
2.43
where ẏ y p0 h, h ∈ P , y0 yt; p0 is determined by 1.1, and fp t; p0 ∈ LP ; H, Ap p0 ∈
LP ; LV ; V , Bp p0 ∈ LP ; LV ; H, and σp p0 ∈ LP ; LW; H are the Fréchet derivative
operators of p → ft; p, p → Ap, p → Bp, and p → σp, at p p0 , respectively.
Proof. By Lemma 2.1, there exists a unique solution to 2.43, ẏ ∈ S. Taking p0 ∈ Pad , for any
p ∈ Pad , setting h p − p0 and z y − y0 − ẏ, where y yp and y0 yp0 are defined by
2.22 and 2.23, respectively, it has
dz A0 B0 zdt f − f0 − f0 h − A − A0 − A0 h y
− B − B0 − B0 h yA0 h y − y0 − B0 h y − y0 dt σ − σ0 − σ0 h dwt,
zt0 0,
2.44
where A Ap, A0 Ap0 , A0 Ap p0 , . . ., σ σp, σ0 σp0 , and σ0 σp p0 .
Letting p → p0 or h → 0, according to the definition of the Fréchet differentiability,
it has
f − f0 − f h oh,
0
H
A − A0 − A h
oh,
0
LV ;V B − B0 − B h
oh,
0
LV ;H
σ − σ0 − σ h
oh.
0
Lw;H
2.45
Moreover, by Theorem 2.3 it gets y − y0 S o1 and yS y − y0 y0 ≤
cy0 c.
Using the deduction similar to Theorem 2.3 it obtains
zS oh,
which is just y − y0 − y0 oh.
2.46
Journal of Applied Mathematics
13
So, yp is Fréchet differentiable at p0 and its Fréchet derivative operator y0 is determined by 2.43.
The continuity of y p can be proved in a way similar to proof of Theorem 2.3, which
is omitted here.
3. Existence and Necessary Conditions for Optimality
In this section the optimization problem 1.10 is researched. First, we prove that the cost
functional p → Jp is continuous and continuously Fréchet differentiable. Next, we prove
that under certain sufficient conditions there exists an optimal parameter p, at which the cost
functional reaches its minimum over the admissible parameter set Pad , and derive necessary
conditions for optimality, which means that the optimal parameter should satisfy some
inequalities.
Theorem 3.1. Let the assumptions of Theorem 2.3 be satisfied, g ∈ L2 Ω; L2 T ; K L2 Ω × T ; K,
and let
Λ : p → L∞ T ; LV ; K
3.1
J : p → R
3.2
be continuous, then the mapping
is continuous.
Proof. Take p0 , p ∈ Pad and set h p − p0 , y yp, y0 yp0 , Λ Λp, and Λ0 Λp0 , then
it has
J p − J p 0 Λy − g 2 − Λ0 y0 − g 2 dt
Eg
K
K
T
Eg
Λy Λ0 y0 − 2g, Λ y − y0 Λ − Λ0 y0 K dt
T
1/2
1/2 2
Λ y − y0 2 Λ − Λ0 y0 2 dt
≤ 2 Eg
Λy Λ0 y0 − 2g K dt
.
Eg
K
K
T
T
3.3
Letting p → p0 , that is, h → 0, using Theorem 2.3 and the assumptions of Theorem 3.1, it
obtains at once
J p − J p0 o1.
Hence, Jp is continuous.
From Zeidler 14
the following lemma is quoted.
3.4
14
Journal of Applied Mathematics
Lemma 3.2. The minimum problem
minFu α
3.5
u∈M
has a solution, where F : M ⊆ X → −∞, ∞
, M /
∅, if one of the following conditions is fulfilled:
a X is a topological space and F is lower semicompact;
b X is a topological space, M is compact, and F is continuous.
Lemma 3.3. The extreme problem
inf I p ≡ E
p∈Pad
T
Utp − gt2 dt d2 ,
K
U ∈ L2 Ω × T ; LP ; K
3.6
has a solution, p∗ ∈ Pad ⊂ P , if one of the following conditions is fulfilled:
a Pad is a closed subspace of P ;
b Pad is a closed and convex in P .
Proof. Obviously, the condition a is a special case of the condition b. So, it needs only proof
that the conclusion holds in the case b.
Assume Kad UPad . Obviously, Kad is closed and convex.
Suppose {pn } ⊂ Pad is a minimizing sequence, that is, Ipn → d2 .
In order to prove that {Upn } is a Cauchy sequence, we prove the following inequality:
UP − UP 2 dt
E
T
# # 2
2
2
2
2
UP − g dt − d E
UP − g dt − d
≤
E
,
T
3.7
∀p , p ∈ p.
T
Obviously, we have the following inequality:
2 2
2
2
2
2
UP − g K dt−d
UP − g dt−d .
E
E
≤ E
UP − g, UP − g K dt − d
T
T
T
Therefore,
E
UP − UP 2 dt
T
2
2
2
UP − g dt − d − 2 E
E
UP − g, UP − g dt − d
T
2
2
UP − g dt − d
E
T
T
3.8
Journal of Applied Mathematics
2
≤ E UP − g dt − d2
15
T
# # 2
2
2
UP − g dt − d E UP − g dt − d2
2 E
T
T
2
E UP − g dt − d2
T
# 2
# 2
2
2
2
UP − g dt − d E
UP − g dt − d
E
,
T
T
3.9
which is exactly 3.7.
For any m, n ∈ N, from 3.7 it has
Upnm − Upn 2 dt ≤
E
T
# 2
# 2
2
E Upnm − g dt − d2 E Upn − g dt − d2 .
T
T
3.10
Letting n → ∞, it obtains
Upnm − Upn o1.
3.11
Hence, {Upn } ⊂ Kad is a Cauchy sequence. Since Kad is closed, there exists y∗ ∈ Kad such
that
Upn −→ y∗ ,
in L2 Ω; L2 T ; K .
3.12
By the definition of Kad , there exists p∗ ∈ Pad such that y∗ Up∗ . Obviously, it has
I p∗ lim I pn d2 .
n→∞
3.13
Theorem 3.4. Let the assumptions of Theorem 3.1 be true and let one of the following be fulfilled:
a P is a finite-dimensional Euclidean space and Pad is convex, bounded, and closed;
b the function p → yp is linear, Λ is independent of p, and Pad is closed and convex;
c Pad is compact in P ,
then the optimization problem 1.9 has a solution, p∗ ∈ Pad , namely,
J p∗ min J p .
p∈Pad
3.14
16
Journal of Applied Mathematics
Proof. Obviously, the result can be obtained by Lemma 3.2 and Theorem 3.1 when a or c
is fulfilled.
If the condition b holds, there is an operator Y t ∈ LPad ; S, for all t ∈ T , such that
for all p ∈ Pad , Y tp yt; p is the solution of 1.1. Therefore, U ΛY ∈ L2 Ω × T ; LP ; K
satisfies the assumptions of Lemma 3.3. So, by Lemma 3.3 the result immediately can be
obtained.
Furthermore, we also can obtain the smoothness of the mapping Jp.
Theorem 3.5. Let the assumptions of Theorem 2.8 be satisfied, g ∈ L2 Ω × T ; K, and let
Λ : p −→ L∞ T ; LV ; K
3.15
J : p −→ R
3.16
be Fréchet differentiable, then
is continuously Fréchet differentiable and the Fréchet differential of J at p along the direction h ∈ P is
determined by the following:
J p h 2Eg
T
Λ p hy t; p Λ p y t; p h, Λ p y t; p − gt K dt,
3.17
where Λ p ∈ LP ; LS; K and y t; ph ẏt is determined by 2.43.
Proof. Take p, p ∈ Pad , set h p − p, then we have
J p − J p − 2Eg
Eg
T
T
Λ p hy t; p Λ p y t; p h, Λ p y t; p − gt K dt
Λ − Λ − Λp h y Λp h y − y , Λy − g
Λ y − y − yp h , Λy − g
K
K
Λy − g, Λ − Λ − Λp h y Λp h y − y
k
Λy − g, Λ y − y − yp h
k
Λp hy Λyp h, Λ y − y Λ − Λ y
3.18
k
dt,
where y yp and Λ Λp. Letting p → p, that is, h → 0, it has
J p − J p − 2Eg
Λp hy Λy h, Λy − g dt oh.
K
T
3.19
Journal of Applied Mathematics
17
So, the functional Jp is Fréchet differentiable and J ph is determined by 3.17, obviously,
p → J p is continuous.
It is now in the position to state necessary conditions for the optimization problem
1.9.
Theorem 3.6. Let the assumptions of Theorem 3.5 be satisfied. If a point p0 ∈ Pad is a relative optimal
parameter for 1.9, then p0 is characterized by
J p0 p − p0 ≡ 2Eg
Λ p0 p − p0 y t; p0
T
Λ p0 y t; p0 p − p0 , Λ p0 y t; p0 − gt K dt ≥ 0,
p ∈ U ⊂ Pad ,
3.20
where U ∈ Pad is a neighbourhood of p0 .
Proof. Firstly, let p0 be the relative optimal parameter, then, it has
J p0 ≤ J p J p0 J p0 p − p0 o p − p0 ,
∀p ∈ U,
3.21
from which it immediately obtains 3.20.
Alternatively suppose 3.20 is true. Using the Taylor formula with the Peano
remainder
J p J p0 J p0 p − p0 o p − p0 ,
3.22
it at once obtains
J p ≥ J p0 ,
p ∈ U.
3.23
Theorem 3.7. Let the assumptions of Theorem 3.5 be satisfied and let the functional Jp be convex. If
p0 ∈ Pad is an extreme point, then p0 is an optimal parameter and p0 is characterized by the following
inequality
J p0 p − p0 ≥ 0,
∀p ∈ Pad .
In particular, if p → y is linear, 3.24 is true.
(The results of this theorem are obvious, whom proof is omitted here.)
3.24
18
Journal of Applied Mathematics
Theorem 3.8. Let the assumptions of Theorem 3.5 be satisfied and let the observation operator Λ be
independent of p. Then the optimal parameter p0 that minimizes Jp over Pad is characterized by the
following optimization system:
dy0 A p0 B p0 y0 dt f p0 dt σ p0 dwt , y0 ϕ ξ,
∗
−dz A p0 B p0 zdt Λ∗ Λy − g dt, z tf 0,
$
% 1 J p0 p − p0 Eg
fp h − Bp hy0 , z
− Ap hy0 , z dt ≥ 0,
H
v ,v
2
T
3.25
3.26
3.27
p ∈ Pad , where h p − p0 .
Proof. Firstly, using the flow of time reversed changing t to tf − t according to Lemma 2.1, it
is easy to show the problem 3.26 is well posed. By Theorems 3.5 and 3.6
1 0 ≤ J p0 p − p0 Eg
2
T
Eg
T
Λy t; p0 h, Λy t; p0 − gt K dt
y t; p0 h, Λ∗ Λy − g V,V dt,
3.28
p ∈ Pad .
Setting ẏ y t, p0 h and using 3.26 and 2.43, it has
0 ≤ J p0 h Eg
T
Eg
T
ẏ, −dz A B∗ zdt V,V dẏ A Bẏdt, z V ,V
$!
"
%
fp h − Ap hy0 − Bp hy0 dt σp hdw, z
Eg
T
Eg
T
$
%
fp h − Bp hy0 , z
− Ap hy0 , z
H
V ,V
3.29
V V
dt.
4. Applications
In this section the above results are applied to systems governed by stochastic partial
differential equations. The following symbols are used:
D ⊂ Rn : a bounded open set;
∂D: the boundary of D, which is smooth;
dt : the partial differential with respect to t;
∂i ∂/∂xi i 1, ..., n;
W m,2 s: Sobolev space, its definition can be found in 15
.
Journal of Applied Mathematics
19
4.1. The System Governed by a Stochastic Parabolic Partial
Differential Equations
Consider the following stochastic parabolic partial differential equation:
dt ux, t n
∂i aij x∂j u dt − cx, tudt fx, tdt σx, tdwt ,
i,j1
4.1
u|∂D 0,
u |tt0 ϕx ξx,
x, t ∈ D × T,
x ∈ D,
where aij ∈ W m,2 D with m > n/2, c ∈ W m,2 D × T , f ∈ L2 Ω × D × T , σ·, t ∈ LW; L2 D,
&
for all t ∈ T , with T σ·, t2L2 D dt < ∞, ϕ ∈ L2 D, ξ ∈ L2 Ω × D, wt wt is a Wiener
process in W.
The problem we shall deal with is to determine the unknown coefficients aij i, j 1, . . . , n and c based on the measurement
zx u x, tf , x ∈ D.
4.2
Suppose H L2 D and V H01 D, which is the subspace of W 1,2 D consisting of
all elements that vanish on the boundary, then V H −1 D. Note that H H . V , H, and V make up the evolutional tripe.
Take the unknown parameter p a11 , . . . , ann , c and define the parameter space by
P
n
'
W m,2 D × W m,2 D × T ,
4.3
i,j1
with the norm
n 2 p ai,j 2 m,2 c2 m,2
W D×T ,
P
W D
4.4
i,j1
then P is a Hilbert space.
In order to make sense of the problem 4.1, it assumes the admissible parameter set
Pad as follows:
⎧
⎫
n
⎨
⎬
2 2
Pad ≡ p ∈ P ; γ y ≤
ai,j xyi yj ≤ δy , ∀y y1 , . . . , yn ∈ Rn , a.e. x ∈ D , 4.5
⎩
⎭
i,j1
where γ and δ0 < γ ≤ δ are given constants.
Obviously, Pad is a convex and closed set in P . By the Sobolev imbedding theorem see
Chapter 15
, 6, the imbedding
W m,2 D −→ Cλ D ⊂ C D
4.6
is compact, where Cλ D is the Hölder space and λ ∈ 0, m − n/2.
20
Journal of Applied Mathematics
Next, define the operator
A≡−
n
∂i ai,j ∂j · ,
Bt ≡ ct,
4.7
i,j1
where ct ≡ c·, t, and then, obviously, Ap ∈ LV, V . Using the generalized Green
formula, for all y, z ∈ V , it has
.
/
n
∂i ai,j ∂j y , z
Ay, z V ,V −
i,j1
⎛
⎞
n
⎝
ai,j ∂j y, ∂i z⎠
i,j1
H
V ,V
4.8
≤ max maxai,j x∇yL2 ∇zL2
i,j
x∈D
≤ c1 yV zV .
Due to p ∈ Pad , it has
⎛
⎞
n
ai,j ∂j y, ∂i y⎠
Ay, y V V ⎝
i,j1
2
≥ γ ∇yL2 .
4.9
H
By the Poincaré inequality it has
Ay, y
V ,V
2
≥ c2 yV ,
4.10
which, along with 4.8, shows that A satisfies 1.5.
Next, for all z ∈ V
Btz2H ≤
D
|cx, tzx|2 dx ≤ c2L∞ D×T c0 z2H
≤
|zx|2 dx
D
4.11
c1 z2V .
Therefore, Bt ∈ LV, H, for all t ∈ T .
Finally, set ut u·, t, ft f·, t, and σt σ·, t. Thus, 4.1 can be written as
3.25.
Summing up the above reasoning, it has the following theorem.
Theorem 4.1. If f ∈ L2 Ω × D × T , then 4.1 has a unique solution up ux, t; p:
u ∈ L2 Ω; CT ; H ∩ L2 Ω × T ; V ∩ CT ; L2 Ω; H ≡ S,
u·, t is adapted to Ft (as a process with values in H),
u·, t is measurable with values in V .
Journal of Applied Mathematics
21
Theorem 4.2. If f ∈ L2 Ω × D × T , then for any p ∈ Pad , the mapping up is continuously Fréchet
differentiable and its Fréchet differential u ph u̇ h ∈ P is determined by the following system:
dt u̇ −
∂i aij ∂j u̇ dt cu̇dt ∂i Δai,j ∂j u dt − Δcudt,
i,j
i,j
4.12
u̇ |∂Ω 0,
u̇ |tt0 0,
where h Δa11 , . . . , Δann , Δc ∈ P and u is determined by 4.1. Furthermore, the mapping up
is infinitely Fréchet differentiable.
Because u ∈ CT ; L2 Ω; H, u·, tf ∈ L2 Ω; H. So, we can use the following cost
functional:
J p E
u x, tf ; p − zx2 dx,
p ∈ Pad ,
4.13
D
in order to determine p. The operator Λ
Λu u ·, tf ,
x ∈ D,
Λ : C T ; L2 Ω; H −→ L2 Ω; H
4.14
obviously satisfies Λ ∈ LS; L2 Ω; H.
Theorem 4.3. The mapping Jp defined by 4.13 is continuously Fréchet differentiable and its
Fréchet differential is determined by
J p h −2E
⎫
⎬
Δai,j ∂i v∂j u Δcuv dt dx
⎭
T ⎩i,j1
⎧
⎨
n
D
4.15
grad J, h P ,P ,
where h Δa11 , . . . , Δann , Δc ∈ P , the gradient operator is
grad J −2E
∂1 v∂1 u, ∂1 v∂2 u, . . . , ∂n v∂n udt ∈ P ,
4.16
T
u up is the solution of 4.1, and v vp is defined by the following system:
−dt v n
∂j aij ∂i v dt − cvdt,
x, t ∈ D × T,
i,j
v |∂D 0,
v |ttf u|ttf − z.
4.17
22
Journal of Applied Mathematics
4.2. Example 2
Consider the following system:
dt u − Δudt m
ai fi x, tdt σx, tdwt ,
x, t ∈ D × T,
i1
u|∂D 0,
4.18
t ∈ T,
u |tt0 ϕx ζx,
x ∈ D,
where D ⊂ Rn with n ≤ 3.
The problem addressed is to identify the unknown parameter p a1 , . . . , am ∈ Rm ,
which varies in an admissible parameter set Pad ≡ {p ∈ Rm ; 0 ≤ γ ≤ pRm ≤ δ} based on the
approximate point measurement:
a.e. x ∈ Di , ∀t ∈ T, i 1, . . . , L,
zi x, t ux, t,
4.19
where Di is the ε-neighbourhood of xi ∈ D and Di ∩ Dj ∅ i /
j.
In order to make sense of 4.19, it assumes
fi ∈ L2 T ; W 1,2 D ,
i 1, . . . , m,
∂j σt ∈ LW, H,
T
∂j σt2
dt < ∞,
LW,H
j 1, ..., n.
4.20
It is easy to prove that ut ∈ W 2,2 D ∩ H01 D. Using the Sobolev embedding theorem 15
,
it has W 2,2 D ⊂ CD. So 4.19 is sensible.
Introduce the characteristic function of Di :
1,
χDi x 0,
x ∈ Di ,
otherwise,
4.21
and the cost functional
L
min J p ≡
E
p∈Pad
i1
D
T
2
χDi x u x, t; p − zi x, t dtdx.
4.22
Journal of Applied Mathematics
23
In the manner similar to reasoning in Section 4.1 it can be proved that the function up
is continuously Fréchet differentiable and its Fréchet differential, u̇ u ph, h Δa1 ,
. . . , Δam ∈ Rm , is determined by the following:
dt u̇ − Δu̇dt m
Δai fi dt,
x, t ∈ D × T,
i1
4.23
u̇ |∂D 0,
u̇ |tt0 0.
Furthermore, it can also be proved that the functional Jp is continuously Fréchet
differentiable and its Fréchet differential is
L
J p h 2
E
i1
T
χDi x u x, t; p − zi x, t u̇ x, t; p dx dt.
4.24
D
Now, the main results are given as followings.
Theorem 4.4. The functional Jp is continuously Fréchet differentiable and its Fréchet differentiable
is
J p h grad J, h Rm ,
4.25
where grad J is the gradient of Jp determined by the following:
grad J E
vf1 dx dt, . . . , E
vfm dx dt ∈ Rm ,
T
D
T
4.26
D
where v is the solution to the following initial-boundary problem:
−dt v − Δvdt 2
L
χDi x u x, t; p − zx, t dt,
i1
v |∂Ω 0,
and ux, t; p is the solution to the problem 4.18.
v|ttf 0
x, t ∈ D × T,
4.27
24
Journal of Applied Mathematics
Proof. By the formulae 4.24, 4.23, and 4.27, and using the integration by parts, it has
J p h E
−dt v − Δvdtu̇dx
T
D
T
D
E
m
i1
vdt u̇ − Δu̇dtdx E
m
T
Δai E
T
D
vΔai fi dx dt
D i1
4.28
vfi dx dt grad J, h Rm .
5. Conclusion
In this paper we consider solving of the parameter contained in a linear stochastic evolution
equation LSEE by means of smooth optimization methods. We prove that the solution to
the LSEE continuously depends on the parameter and is continuously Fréchet differentiable
with respect to parameter. We also prove that the cost functional is continuously Fréchet
differentiable with respect to parameter. Based on the above results the necessary conditions,
which the optimal parameter should satisfy, are presented. Moreover, the sufficient
conditions, under which there exists an optimal parameter, also are presented. Finally we
apply the results to stochastic partial differential equations with a final measurement and an
approximate point measurement, respectively.
It should be pointed out that we only consider the linear stochastic evolution equation
LSEE without measurement errors. The case with measurement errors, such as filtering of
diffusion processes 16
, is worth investigating in the future.
Acknowledgments
This paper is supported by the Natural Science Foundation of China grant 71171190, in
part by Science Foundation of Civil Aviation University of China 09CAUC-S03 and the
Fundamental Research Funds for the Central Universities ZXH2012K005.
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