# An Inverse Problem for a Class of Linear Stochastic Evolution Equations

Authors: Yuhuan Zhao
Journal: Journal of Applied Mathematics
 (2012)

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## Abstract

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.


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## Body

## 1. Introduction

The purpose of this paper is to study an inverse problem for the following linear stochastic evolution equation:(1.1)dy+[A(t;p)+B(t;p)]ydt=f(t;p)dt+σ(t;p)dw(t),t∈(t0,tf)≡T,y(t0)=φ+ξ,
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=y(p)=y(t;p) to explicitly show the dependence of y on p.The problem of this paper is to determine the unknown parameterp based on the measurement g(t), which is defined by the following:
(1.2)g(t)=Λ(t;p)y(t),t∈T,
where V, H, K, W, and P are Hilbert spaces.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 byℒ(X;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
(1.3)V⊂H⊂V′,
where each space is dense in the following space and has a continuous injection, H′=H, and
(1.4)〈y,x〉V′,V=(y,x)H,∀x∈V,y∈H.For anyt∈T and p∈Pad, A(t;p)∈ℒ(V;V'), B(t;p)∈ℒ(V;H), σ(t,p)∈ℒ(W;H), Λ(t;p)∈ℒ(V;K,),  f(t;p)∈H, and
(1.5)〈A(t;p)z,z〉V′,V≥α∥z∥V2,z∈V,
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 ℱ(ℱ=ℱ∞).M
(
Ω
,
μ
,
Φ
) denotes the space of random variables with values in a Hilbert space Φ.
(1.6)Lr(Ω;Φ)≡{x∈M(Ω,μ,Φ);∥x∥≡[E∥x∥Φr]1/r=[∫Ω‍∥x(ω)∥Φrdμ(ω)]1/r<∞},Lr(Ω×T;Φ)≡{x(t)∈M(Ω,μ,Φ),t∈T;∥x∥≡[∫Ω‍∫T‍∥x(t,ω)∥Φrdtdμ(ω)]1/r<∞}.w(t)∈M(Ω,μ,W) is a Wiener process with values in a separable Hilbert space W, which is adapted to ℱt, that is, for alle∈W, (w(t),e)W is a real Wiener process and an ℱt-martingale, with the correlation function
(1.7)E{(w(t1),e1)W(w(t2),e2)W}=∫t0min(t1,t2)‍(Q(τ)e1,e2)dτ,
where t0≤t1, t2≤tf and Q(t) is a positive self-adjoint nuclear operator almost everywhere on W. Q(t) is called the covariance operator. Moreover, assume that Q(t) satisfies
(1.8)Q(·)∈L∞(T;ℒ(W)),
where φ0∈H, ξ∈L2(Ω,H) is independent of w(t).Now, determine the parameterp 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
(1.9)J(p)=Eg∫T‍∥g(t)-Λ(t;p)y(t;p)∥K2dt
reaches its minimum over the admissible parameter set Pad at p^, that is,
(1.10)J(p^)=minp∈PadJ(p),
where Eg is a conditional expectation, that is,
(1.11)Egf=E{f∣Gt},
and Gt is the sub-σ-algebra induced by the stochastic process g(s), 0≤s≤t, which is adapted to ℱt.If there exists a neighbourhoodU⊂Pad of p0 such that
(1.12)J(p0)=infp∈UJ(p),
then Po is called a relative optimal parameter.In Section2, the base of this paper is given, under certain conditions the function p→y(p) is continuous and Fréchet continuously differentiable.In Section3, 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 Section4, 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.Lemma 2.1.
Besides the assumptions forA, B, f, φ, w(t), and σ in Section 1, one assumes that, for any p∈Pad,
(2.1)A(·;p)∈L2(T;ℒ(V;V′))∩L∞(T;ℒ(V;V′)),B(·;p)∈L∞(Ω×T;ℒ(V;H)),f(·,p)∈L2(Ω×T;H),σ(·,p)∈L2(T,ℒ(W;H)).
Then there exists a unique generalized solution, y, in the Ladyzenskaja sense of (1.1) almost every t∈T such that
(2.2)y∈L2(Ω;C(T;H))∩L2(Ω×T;V)∩C(T-;L2(Ω;H))≡𝔖,y(t) is adapted to ℱt (as a process with values in H), and y(t) is ℱt measurable with values in V.In the above lemma the spaceCk(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
(2.3)∥u∥=∑i=0k‍maxt∈X∥u(i)(t)∥.Remark 2.2.
The generalized solution in the Ladyzenskaja sense is the solution of the following variational equation:(2.4)(y(t),η(t))H=(φ+ξ,η(t0))H+∫t0t‍〈y(t),dtη(t)〉V,V′+∫t0t‍〈[A(t;p)+B(t;p)]y(t)dt,η(t)〉V′,V+∫t0t‍(σ(t;p)dw(t),η(t))H,a.s.,∀η∈W1(T;V,H),
where the space W1(T;V,H) is a Hilbert space that is defined by
(2.5)W1(T;V,H)={η;η∈L2(T;V),dηdt∈L2(T;V′)},
and it is well known that W1(T;V,H)⊂C(T-;H).
Now, it is time to give the main results in this section.Theorem 2.3.
If the assumptions of Lemma2.1 are satisfied and
(2.6)A:p→L2(T;ℒ(V;V′)),B:p→L∞(Ω×T;ℒ(V;H)),f:p→L2(Ω×T;H),σ:p→L2(T,ℒ(W;H)),
are continuous, the solution of (1.1)
(2.7)y:Pad→𝔖
is continuous, that is, y∈C(Pad;𝔖) or the following equalities are true:
(2.8)limp→p0∥y(p)-y(p0)∥C(T-;L2(Ω,H))=0,(2.9)limp→p0∥y(p)-y(p0)∥L2(Ω;C(T-;H))=0,(2.10)limp→p0∥y(p)-y(p0)∥L2(Ω×T;V)=0.Before proving Theorem2.3, from Bensoussan [10] the following Ito^ formula in a Hilbert space is quoted.Lemma 2.4.
LetΦ(z,t) be a functional on H×T-, which is twice continuously Fréchet differentiable in z∈H and continuously differentiable in t∈T. Assume z(t) has the stochastic differential:
(2.11)dz(t)=a(t)dt+b(t)dw(t),z(t0)=z0,
where a(t) is a stochastic process with values in H, which is adapted and satisfies the condition
(2.12)a.s∫t0t‍∥a(t)∥Hdt<+∞,∀t∈T,
where b(t) is an adapted process with values in ℒ(W,H) such that t, ω→b is measurable and
(2.13)E∫t0t1‍∥b(t)∥ℒ(W,H)2dt<∞,∀t1<tf,
then one has the following Ito^ formula in the Hilbert space:
(2.14)Φ(z(t),t)=Φ(z(t0),t0))+∫t0t‍(∂Φ∂z,a(s))Hds+∫t0t‍(∂Φ∂z,bdw(s))H+12∫t0t‍tr[b*∂2Φ∂z2bQ]ds+∫t0t‍∂Φ∂tds,
where b* is the adjoint of b and the symbol “tr” is the trace operator,  of which definition for a nuclear operator U∈ℒ(X) is as follows:
(2.15)trU=∑n=1∞‍(Ufn,fn)X,
where {fn} is an orthonormal basis of X.The following two lemmas are obvious, so their proofs are omitted.Lemma 2.5.
Supposes thatU∈ℒ(X) is a nuclear operator and B∈ℒ(X,Y), then
(2.16)tr(B*BU)≤∥B∥2trU.Lemma 2.6.
Ifψ∈L2(Ω;L2(T)),
(2.17)E∫T‍ψ(t)dw(t)=0.Lemma 2.7.
Iff∈L2(Ω;L2(T;W))≡L2(Ω×T,W),
(2.18)E|∫T‍(f(t),dw(t))W|2=E∫T‍(R(t)f(t),f(t))Wdt.Proof.
First, suppose thatf is a step function, that is,
(2.19)f(t)=fr,tr≤t<tr+1,r=0,…,N-1,
where fr∈L2(Ω,W) and t0<t1<⋯<tN=tf.
Let{en} be an orthomomal basis of W. Obviously,
(2.20)w(t)=∑i=1∞‍(en,w(t))Wen,
and wn(t)=(en,w(t)) is a Wiener process. Furthermore,
(2.21)E|∫T‍(f(t),dw(t))|2=E∑i,j‍∑r,s‍∫trtr+1‍(fr,eidwi)∫tsts+1‍(fs,ejdwj)=E∑i,j‍∑r,s‍(fr,ei)[wi(tr+1)-wi(tr)](fs,ej)[wj(ts+1)-wj(tj)]=E∫T‍(R(t)∑i‍(f,ei)ei,∑j‍(f,ej)ej)dt=E∫T(R(t)f(t),f(t))Wdt,
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 Theorem2.3 can be given as follows.Proof of Theorem2.3.
Denotey(t)=y(t,p) and y0(t)=y(t;p0), and then
(2.22)dy+[A(t;p)+B(t;p)]ydt=f(t;p)dt+σ(t;p)dw(t),y(t0)=φ+ζ,(2.23)dy0+[A(t;p0)+B(t;p0)]y0dt=f(t;p0)dt+σ(t;p0)dw(t),y0(t0)=φ+ζ.
Let z(t)=y(t)-y0(t), and then from the above equalities, it follows
(2.24)dz(t)+[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)]dw(t),z(t0)=0.
SettingΦ(z)=(1/2)∥z∥2,z∈W!(T;V,H) according to Lemma 2.4, it gets
(2.25)12∥z(t)∥H2+∫t0t‍〈z(s),A(s;p)z(s)〉V,V′ds=-∫t0t‍(z(s),B(s;p)z(s))Hds-∫t0t‍〈z(s),[A(s;p)-A(s;p0)]y0(s)〉V,V′ds+∫t0t‍(z(s),{f(s;p)-f(s;p0)-[B(s;p)-B(s;p0)]y0(s)})Hds+∫t0t‍(z(s),[σ(s;p)-σ(s;p0)]dw(s))H+12∫t0t‍tr{[σ(s,p)-σ(s;p0)]*[σ(s,p)-σ(s;p0)]Q(s)}ds,
which is
(2.26)∥z(t)∥H2+2∫t0t‍〈z(s),A(s;p)z(s)〉V.V′ds=-2∫t0t‍(z(s),B(s;p)z(s))Hds+2∫t0t‍(z(s),f(s;p)-f(s;p0))Hds-2∫t0t‍〈z(s),[A(s;p)-A(s;p0)]y0(s)〉V,V′ds-2∫t0t‍(z(s);[B(s;p)-B(s;p0)]y0(s))Hds+2∫t0t‍(z(s),[σ(s;p)-σ(s;p0)]dw(s))H+∫t0t‍tr{[σ(s,p)-σ(s;p0)]*[σ(s,p)-σ(s;p0)]Q(s)}ds.
Taking the expectation from the above and considering Lemma2.6, it has
(2.27)E∥z(t)∥H2+2E∫t0t‍〈z(s),A(s;p)z(s)〉V,V′ds=-2E∫t0t‍(z(s),B(s;p)z(s))Hds+2E∫t0t‍(z(s),f(s;p)-f(s;p0))Hds-2E∫t0t‍〈z(s),[A(s;p)-A(s;p0)]y0(s)〉V,V′ds-2E∫t0t‍(z(s);[B(s;p)-B(s;p0)]y0(s))Hds+∫t0t‍tr{[σ(s,p)-σ(s;p0)]*[σ(s,p)-σ(s;p0)]Q(s)}ds.
From the assumptions of the spacesH and V,  there exists a constant γ such that
(2.28)∥x∥H≤γ∥x∥V,∀x∈V.
Furthermore, according to the assumptions of the operator A(p) and using (2.28) and 2ab≤εa2+b2/ε, we can obtain
(2.29)E∥z(t)∥H2+2αE∫t0t‍∥z(s)∥V2ds≤αE∫t0t‍∥z(s)∥V2ds+4γα∥B(·;p)∥L∞(T;ℒ(V;H))2E∫t0t‍∥z(s)∥H2ds+4γαE∫t0t‍{∥f(s;p)-f(s;p0)∥H2+∥[A(s;p)-A(s;p0)]y0(s)∥V′2+∥[B(s;p)-B(s;p0)]y0(s)∥H2}ds.+∥trQ∥∫t0t‍∥σ(s,p)-σ(s;p0)∥2ds.
By the assumptions of Theorem 2.3 there is a constant c such that
(2.30)E∥z(t)∥H2+αE∫t0t‍∥z(s)∥V2ds≤c∫t0t‍E∥z(s)∥H2ds+c∫t0t‍∥σ(s,p)-σ(s;p0)∥ℒ(W;H)2ds+cE∫t0t‍{∥f(s;p)-f(s;p0)∥H2+∥[A(s;p)-A(s;p0)]y0(s)∥V′2+∥[B(s;p)-B(s;p0)]y0(s)∥H2}ds.
Obviously,
(2.31)E∥z(t)∥H2≤c∫t0t‍E∥z(s)∥H2ds+cE∫t0t{∥f(s;p)-f(s;p0)∥H2+∥[A(s;p)-A(s;p0)]y0(s)∥V′2+∥[B(s;p)-B(s;p0)]y0(s)∥H2}ds+c∫t0t‍∥σ(s,p)-σ(s;p0)∥ℒ(W;H)2ds.
Using the Gronwall inequality (see [13] for the definition),  and the assumptions of Theorem 2.3, letting p→p0, it has
(2.32)maxt∈T-E∥z(s)∥H2=o(1),
which is (2.8). Furthermore, from (2.31), it follows
(2.33)E∫T‍∥z(s)∥V2=o(1),
which is just (2.10).
From (2.26), it has the following estimate:
(2.34)∥z(t)∥H2+2α∫t0t‍∥z(s)∥V2ds≤αγ∫t0t‍∥z(s)∥H2ds+4γα∫t0t{∥f(s;p)-f(s;p0)∥H2‍+∥[A(s;p)-A(s;p0)]y0(s)∥V′2+∥[B(s;p)-B(s;p0)]y0(s)∥H2}ds+∥B(·;p)∥L∞(T;ℒ(V;H))2∫t0t‍∥z(s)∥H2ds+2∫t0t‍(z(s),[σ(s;p)-σ(s;p0)]dw(s))H+∥trQ∥L∞(T)∫t0t‍∥σ(s,p)-σ(s;p0)∥ℒ(W;H)2ds.
Similarly, it also obtains
(2.35)∥z(t)∥H2≤c∫t0t‍{∥f(s;p)-f(s;p0)∥H2+∥[A(p)-A(p0)]y0∥V′2+∥[B(p)-B(p0)]y0∥H2+∥σ(p)-σ(p0)∥ℒ(W;H)2}ds+c∫t0t‍(z(s),[σ(s,p)-σ(s;p0)]dw(s))H,t∈T-.
On the other hand, the indefinite stochastic integral
(2.36)X(t)=∫t0t‍([σ(s;p)-σ(s;p)]*z(s),dw(s))H
is a continuous martingale, so
(2.37)Emaxt∈T-X(t)≤Emaxt|X(t)|≤[E(maxt|X(t)|)2]1/2≤2{E|X(tf)|2}1/2=2{E∫T‍(R(t)[σ(s;p)-σ(s;p0)]*z(s),[σ(s;p)-σ(s;p0)]*z(s))dt}1/2≤c{∫T‍∥σ(t;p)-σ(t;p0)∥2dt}1/2,
where the following martingale inequality is used:
(2.38)E{maxT-|X(t)|α}≤(αα-1)2E|X(tf)|α,
with α=2.
Combining (2.35) with (2.37), it has
(2.39)Emaxt∈T-∥z(t)∥H2≤cE∫t0tf‍{∥f(s;p)-f(s;p0)∥H2+∥[A(s;p)-A(s;p0)]y0(s)∥V′2+∥[B(s;p)-B(s;p0)]y0(s)∥H2}ds+c∫T‍∥σ(s,p)-σ(s;p0)∥ℒ(W;H)2ds+c{∫T‍∥σ(s,p)-σ(s;p0)∥ℒ(W;H)2ds}1/2.
Letting p→p0 in P, by the assumptions of Theorem 2.3, it immediately obtains
(2.40)Emaxt∈T-∥z(t)∥H2=o(1),
which is (2.9).Theorem 2.8.
Besides the assumptions of Theorem2.3, suppose that the mappings
(2.41)A:T×Pad→ℒ(V;V′),B:T×Pad→ℒ(V;H),f:T×Pad→H,σ:T×Pad→ℒ(W;H)
are continuously Fréchet differentiable, then the solution of (1.1)
(2.42)y:p→y(p)∈𝔖
is continuously Fréchet differentiable and its Fréchet derivative operators at p0∈Pad, y'(p0)∈ℒ(P,𝔖),  are determined by the following system:
(2.43)dy˙+[A(t;p0)+B(t;y0)]y˙dt=[fp′(t;p0)h-Ap′(t;p0)hy0-Bp′(t;p0)hy0]dt+σp′(t;p0)hdw(t),y˙(t0)=0,
where y˙=y′(p0)h, h∈P, y0=y(t;p0) is determined by (1.1), and fp′(t;p0)∈ℒ(P;H), Ap′(p0)∈ℒ(P;ℒ(V;V′)), Bp′(p0)∈ℒ(P;ℒ(V;H)), and σp′(p0)∈ℒ(P;ℒ(W;H)) are the Fréchet derivative operators of p→f(t;p), p→A(p), p→B(p), and p→σ(p), at p=p0, respectively.Proof.
By Lemma2.1, there exists a unique solution to (2.43), y˙∈𝔖. Taking p0∈Pad, for any p∈Pad, setting h=p-p0 and z=y-y0-y˙, where y=y(p) and y0=y(p0) are defined by (2.22) and (2.23), respectively, it has
(2.44)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)dw(t),z(t0)=0,
where A=A(p), A0=A(p0), A0′=Ap′(p0),…, σ=σ(p), σ0=σ(p0), and σ0′=σp′(p0).
Lettingp→p0 or ∥h∥→0, according to the definition of the Fréchet differentiability, it has
(2.45)∥f-f0-f0′h∥H=o(∥h∥),∥A-A0-A0′h∥ℒ(V;V′)=o(∥h∥),∥B-B0-B0′h∥ℒ(V;H)=o(∥h∥),∥σ-σ0-σ0′h∥ℒ(w;H)=o(∥h∥).
Moreover, by Theorem2.3 it gets ∥y-y0∥𝔖=o(1) and ∥y∥𝔖=∥y-y0∥+∥y0∥≤c∥y0∥+c.
Using the deduction similar to Theorem2.3 it obtains
(2.46)∥z∥𝔖=o(∥h∥),
which is just ∥y-y0-y0′∥=o(∥h∥).
So,y(p) is Fréchet differentiable at p0 and its Fréchet derivative operator y0′ is determined by (2.43).
The continuity ofy′(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→J(p) 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 Theorem2.3 be satisfied, g∈L2(Ω;L2(T;K))=L2(Ω×T;K), and let
(3.1)Λ:p→L∞(T;ℒ(V;K))
be continuous, then the mapping
(3.2)J:p→R+
is continuous.Proof.
Takep0, p∈Pad and set h=p-p0, y=y(p), y0=y(p0), Λ=Λ(p), and Λ0=Λ(p0), then it has
(3.3)|J(p)-J(p0)|=|Eg∫T‍{∥Λy-g∥K2-∥Λ0y0-g∥K2}dt|=|Eg∫T‍(Λy+Λ0y0-2g,Λ(y-y0)+(Λ-Λ0)y0)Kdt|≤2(Eg∫T‍∥Λy+Λ0y0-2g∥K2dt)1/2(Eg∫T‍{∥Λ(y-y0)∥K2+∥(Λ-Λ0)y0∥K2}dt)1/2.
Letting p→p0, that is, ∥h∥→0, using Theorem 2.3 and the assumptions of Theorem 3.1, it obtains at once
(3.4)|J(p)-J(p0)|=o(1).
Hence, J(p) is continuous.From Zeidler [14] the following lemma is quoted.Lemma 3.2.
The minimum problem(3.5)minu∈MF(u)=α
has a solution, where F:M⊆X→(-∞,+∞], M≠∅, if one of the following conditions is fulfilled:(a)
Xis 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(3.6)infp∈PadI(p)≡E∫T‍∥U(t)p-g(t)∥K2dt=d2,U∈L2(Ω×T;ℒ(P;K))
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).
AssumeKad=UPad. Obviously, Kad is closed and convex.
Suppose{pn}⊂Pad is a minimizing sequence, that is, I(pn)→d2.
In order to prove that{Upn} is a Cauchy sequence, we prove the following inequality:
(3.7)E∫T∥UP′-UP′′∥2dt≤{E∫T∥UP′-g∥2dt-d2+E∫T∥UP′′-g∥2dt-d2}2,∀p′,p′′∈p.
Obviously, we have the following inequality:
(3.8){E∫T(UP′-g,UP′′-g)Kdt-d2}2≤{E∫T∥UP′-g∥K2dt-d2}{E∫T∥UP′′-g∥2dt-d2}.
Therefore,
(3.9)E∫T∥UP′-UP′′∥2dt={E∫T∥UP′-g∥2dt-d2}-2{E∫T(UP′-g,UP′′-g)dt-d2}+{E∫T∥UP′′-g∥2dt-d2}≤{E∫T∥UP′-g∥2dt-d2}+2E∫T∥UP′-g∥2dt-d2E∫T∥UP′′-g∥2dt-d2+{E∫T∥UP′′-g∥2dt-d2}={E∫T∥UP′-g∥2dt-d2+E∫T∥UP′′-g∥2dt-d2}2,
which is exactly (3.7).
For anym, n∈ℕ, from (3.7) it has
(3.10)E∫T‍∥Upn+m-Upn∥2dt≤{E∫T‍∥Upn+m-g∥2dt-d2+E∫T‍∥Upn-g∥2dt-d2}2.
Letting n→+∞, it obtains
(3.11)Upn+m-Upn=o(1).
Hence, {Upn}⊂Kad is a Cauchy sequence. Since Kad is closed, there exists y*∈Kad such that
(3.12)Upn→y*,inL2(Ω;L2(T;K)).
By the definition of Kad, there exists p*∈Pad such that y*=Up*. Obviously, it has
(3.13)I(p*)=limn→∞I(pn)=d2.Theorem 3.4.
Let the assumptions of Theorem3.1 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 functionp→y(p) 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,
(3.14)J(p*)=minp∈PadJ(p).Proof.
Obviously, the result can be obtained by Lemma3.2 and Theorem 3.1 when (a) or (c) is fulfilled.
If the condition (b) holds, there is an operatorY(t)∈ℒ(Pad;𝔖), forallt∈T, such that forallp∈Pad, Y(t)p=y(t;p) is the solution of (1.1). Therefore, U=ΛY∈L2(Ω×T;ℒ(P;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 mappingJ(p).Theorem 3.5.
Let the assumptions of Theorem2.8 be satisfied, g∈L2(Ω×T;K), and let
(3.15)Λ:p→L∞(T;ℒ(V;K))
be Fréchet differentiable, then
(3.16)J:p→ℝ+
is continuously Fréchet differentiable and the Fréchet differential of J at p along the direction h∈P is determined by the following:
(3.17)J′(p)h=2Eg∫T‍(Λ′(p)hy(t;p)+Λ(p)y′(t;p)h,Λ(p)y(t;p)-g(t))Kdt,
where Λ′(p)∈ℒ(P;ℒ(𝔖;K)) and y'(t;p)h=y˙(t) is determined by (2.43).Proof.
Takep, p-∈Pad, set h=p--p, then  we have
(3.18)J(p-)-J(p)-2Eg∫T‍(Λ′(p)hy(t;p)+Λ(p)y′(t;p)h,Λ(p)y(t;p)-g(t))Kdt=Eg∫T‍{((Λ--Λ-Λp′h)y-+Λp′h(y--y),Λ-y--g)K+(Λ(y--y-yp′h),Λ-y--g)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)k}dt,
where y-=y(p-) and Λ-=Λ(p-). Letting p-→p, that is,  ∥h∥→0, it has
(3.19)|J(p-)-J(p)-2Eg∫T‍(Λp′hy+Λy′h,Λy-g)Kdt|=o(∥h∥).
So, the functional J(p) is Fréchet differentiable and J′(p)h 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 Theorem3.5 be satisfied. If a point p0∈Pad is a relative optimal parameter for (1.9), then p0 is characterized by
(3.20)J′(p0)(p-p0)≡2Eg∫T‍(Λ′(p0)(p-p0)y(t;p0)+Λ(p0)y′(t;p0)(p-p0),Λ(p0)y(t;p0)-g(t))Kdt≥0,p∈U⊂Pad,
where U∈Pad is a neighbourhood of p0.Proof.
Firstly, letp0 be the relative optimal parameter, then, it has
(3.21)J(p0)≤J(p)=J(p0)+J′(p0)(p-p0)+o(∥p-p0∥),∀p∈U,
from which it immediately obtains (3.20).
Alternatively suppose (3.20) is true. Using the Taylor formula with the Peano remainder
(3.22)J(p)=J(p0)+J′(p0)(p-p0)+o(∥p-p0∥),
it at once obtains
(3.23)J(p)≥J(p0),p∈U.Theorem 3.7.
Let the assumptions of Theorem3.5 be satisfied and let the functional J(p) be convex. If p0∈Pad is an extreme point, then p0 is an optimal parameter and p0 is characterized by the following inequality
(3.24)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.)Theorem 3.8.
Let the assumptions of Theorem3.5 be satisfied and let the observation operator Λ be independent of p. Then the optimal parameter p0 that minimizes J(p) over Pad is characterized by the following optimization system:
(3.25)dy0+[A(p0)+B(p0)]y0dt=f(p0)dt+σ(p0)dwt,y0=φ+ξ,(3.26)-dz+[A(p0)+B(p0)]*zdt=Λ*(Λy-g)dt,z(tf)=0,(3.27)12J′(p0)(p-p0)=Eg∫T‍{(fp′h-Bp′hy0,z)H-〈Ap′hy0,z〉v′,v}dt≥0,p∈Pad, where h=p-p0.Proof.
Firstly, using the flow of time reversed (changingt 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(3.28)0≤12J′(p0)(p-p0)=Eg∫T‍(Λy′(t;p0)h,Λy(t;p0)-g(t))Kdt=Eg∫T‍〈y′(t;p0)h,Λ*[Λy-g]〉V,V′dt,p∈Pad.
Setting y˙=y′(t,p0)h and using (3.26) and (2.43), it has
(3.29)0≤J′(p0)h=Eg∫T‍〈y˙,-dz+(A+B)*zdt〉V,V′=Eg∫T‍〈dy˙+(A+B)y˙dt,z〉V′,V=Eg∫T‍〈[fp′h-Ap′hy0-Bp′hy0]dt+σp′hdw,z〉V′V=Eg∫T‍{(fp′h-Bp′hy0,z)H-〈Ap′hy0,z〉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
⊂
ℝ
n: a bounded open set;∂
D: the boundary of D, which is smooth;d
t: the partial differential with respect to t;∂
i
=
(
∂
/
∂
x
i
)
(
i
=
1
,
.
.
.
,
n
);W
m
,
2
(
s
): Sobolev space, its definition can be found in [15].


### 4.1. The System Governed by a Stochastic Parabolic Partial Differential Equations

Consider the following stochastic parabolic partial differential equation:(4.1)dtu(x,t)=∑i,j=1n‍∂i(aij(x)∂ju)dt-c(x,t)udt+f(x,t)dt+σ(x,t)dwt,(x,t)∈D×T,u∣∂D=0,u∣t=t0=φ(x)+ξ(x),x∈D,
where aij∈Wm,2(D) with m>n/2, c∈Wm,2(D×T), f∈L2(Ω×D×T), σ(·,t)∈ℒ(W;L2(D)), for all t∈T, with ∫T‍∥σ(·,t)∥L2(D)2dt<∞, φ∈L2(D), ξ∈L2(Ω×D), wt=w(t) is a Wiener process in W.The problem we shall deal with is to determine the unknown coefficientsaij(i,j=1,…,n) and c based on the measurement
(4.2)z(x)=u(x,tf),x∈D.SupposeH=L2(D) and V=H01(D), which is the subspace of W1,2(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 parameterp=(a11,…,ann,c) and define the parameter space by
(4.3)P=∏i,j=1nWm,2(D)×Wm,2(D×T),
with the norm
(4.4)∥p∥P2=∑i,j=1n‍∥ai,j∥Wm,2(D)2+∥c∥Wm,2(D×T)2,
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:
(4.5)Pad≡{p∈P;γ|y|2≤∑i,j=1n‍ai,j(x)yiyj≤δ|y|2,∀y=(y1,…,yn)∈ℝn,a.e.x∈D},
where γ and δ(0<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
(4.6)Wm,2(D)→Cλ(D-)⊂C(D-)
is compact, where Cλ(D-) is the Hölder space and λ∈(0,m-n/2).Next, define the operator(4.7)A≡-∑i,j=1n‍∂i(ai,j∂j·),B(t)≡c(t),
where c(t)≡c(·,t), and then, obviously, A(p)∈ℒ(V,V′). Using the generalized Green formula, for all y, z∈V, it has
(4.8)|〈Ay,z〉V′,V|=|〈-∑i,j=1n‍∂i(ai,j∂jy),z〉V′,V|=|(∑i,j=1n‍ai,j∂jy,∂iz)H|≤maxi,jmaxx∈D-|ai,j(x)|∥∇y∥L2∥∇z∥L2≤c1∥y∥V∥z∥V.
Due to p∈Pad, it has
(4.9)〈Ay,y〉V′V=(∑i,j=1n‍ai,j∂jy,∂iy)H≥γ∥∇y∥L22.
By the Poincaré inequality it has
(4.10)〈Ay,y〉V',V≥c2∥y∥V2,
which, along with (4.8), shows that A satisfies (1.5).Next, for allz∈V(4.11)∥B(t)z∥H2=∫D‍|c(x,t)z(x)|2dx≤∥c∥L∞(D×T)2∫D‍|z(x)|2dx≤c0∥z∥H2≤c1∥z∥V2.
Therefore, B(t)∈ℒ(V,H), for all t∈T.Finally, setu(t)=u(·,t), f(t)=f(·,t), and σ(t)=σ(·,t). Thus, (4.1) can be written as (3.25).Summing up the above reasoning, it has the following theorem.Theorem 4.1.
Iff∈L2(Ω×D×T), then (4.1) has a unique solution u(p)=u(x,t;p):
u
∈
L
2
(
Ω
;
C
(
T
;
H
)
∩
L
2
(
Ω
×
T
;
V
)
∩
C
(
T
-
;
L
2
(
Ω
;
H
)
)
≡
𝔖,
u
(
·
,
t
) is adapted to ℱt (as a process with values in  H),
u
(
·
,
t
) is measurable with values in V.Theorem 4.2.
Iff∈L2(Ω×D×T), then for any p∈Pad, the mapping u(p) is continuously Fréchet differentiable and its Fréchet differential u′(p)h=u˙(h∈P) is determined by the following system:
(4.12)dtu˙-∑i,j‍∂i(aij∂ju˙)dt+cu˙dt=∑i,j‍∂i(Δai,j∂ju)dt-Δcudt,u˙∣∂Ω=0,u˙∣t=t0=0,
where h=(Δa11,…,Δann,Δc)∈P and u is determined by (4.1). Furthermore, the mapping u(p) is infinitely Fréchet differentiable.Becauseu∈C(T-;L2(Ω;H)), u(·,tf)∈L2(Ω;H). So, we can use the following cost functional:
(4.13)J(p)=E∫D‍|u(x,tf;p)-z(x)|2dx,p∈Pad,
in order to determine p. The operator Λ(4.14)Λu=u(·,tf),x∈D,Λ:C(T-;L2(Ω;H))→L2(Ω;H)
obviously satisfies Λ∈ℒ(𝔖;L2(Ω;H)).Theorem 4.3.
The mappingJ(p) defined by (4.13) is continuously Fréchet differentiable and its Fréchet differential is determined by
(4.15)J′(p)h=-2E∫D‍∫T‍{∑i,j=1n‍Δai,j∂iv∂ju+Δcuv}dtdx=〈gradJ,h〉P′,P,
where h=(Δa11,…,Δann,Δc)∈P, the gradient operator is
(4.16)gradJ=-2E∫T‍(∂1v∂1u,∂1v∂2u,…,∂nv∂nu)dt∈P′,u=u(p) is the solution of (4.1), and v=v(p) is defined by the following system:
(4.17)-dtv=∑i,jn‍∂j(aij∂iv)dt-cvdt,(x,t)∈D×T,v∣∂D=0,v∣t=tf=u∣t=tf-z.


### 4.2. Example 2

Consider the following system:(4.18)dtu-Δudt=∑i=1m‍aifi(x,t)dt+σ(x,t)dwt,(x,t)∈D×T,u∣∂D=0,t∈T,u∣t=t0=φ(x)+ζ(x),x∈D,
where D⊂ℝn with n≤3.The problem addressed is to identify the unknown parameterp=(a1,…,am)∈ℝm, which varies in an admissible parameter set Pad≡{p∈ℝm;0≤γ≤∥p∥Rm≤δ} based on the approximate point measurement:
(4.19)zi(x,t)=u(x,t),a.e.x∈Di,∀t∈T,i=1,…,L,
where Di is the ε-neighbourhood of xi∈D and Di∩Dj=∅(i≠j).In order to make sense of (4.19), it assumes
(4.20)fi∈L2(T;W1,2(D)),i=1,…,m,∂jσ(t)∈ℒ(W,H),∫T‍∥∂jσ(t)∥ℒ(W,H)2dt<∞,j=1,...,n.
It is easy to prove that u(t)∈W2,2(D)∩H01(D). Using the Sobolev embedding theorem [15], it has W2,2(D)⊂C(D-). So (4.19) is sensible.Introduce the characteristic function ofDi:
(4.21)χDi(x)={1,x∈Di,0,otherwise,
and the cost functional
(4.22)minp∈PadJ(p)≡∑i=1L‍E∫D‍∫T‍[χDi(x)[u(x,t;p)-zi(x,t)]]2dtdx.
In the manner similar to reasoning in Section 4.1 it can be proved that the function u(p) is continuously Fréchet differentiable and its Fréchet differential, u˙=u′(p)h,h=(Δa1,…,Δam)∈ℝm, is determined by the following:
(4.23)dtu˙-Δu˙dt=∑i=1m‍Δaifidt,(x,t)∈D×T,u˙∣∂D=0,u˙∣t=t0=0.
Furthermore, it can also be proved that the functional J(p) is continuously Fréchet differentiable and its Fréchet differential is
(4.24)J′(p)h=2∑i=1L‍E∫T‍∫D‍χDi(x)[u(x,t;p)-zi(x,t)]u˙(x,t;p)dxdt.
Now, the main results are given as followings.Theorem 4.4.
The functionalJ(p) is continuously Fréchet differentiable and its Fréchet differentiable is
(4.25)J′(p)h=(gradJ,h)ℝm,
where gradJ is the gradient of J(p) determined by the following:
(4.26)gradJ=(E∫T‍∫D‍vf1dxdt,…,E∫T‍∫D‍vfmdxdt)∈ℝm,
where v is the solution to the following initial-boundary problem:
(4.27)-dtv-Δvdt=2∑i=1L‍χDi(x)[u(x,t;p)-z(x,t)]dt,(x,t)∈D×T,v∣∂Ω=0,v∣t=tf=0
and u(x,t;p) is the solution to the problem (4.18).Proof.
By the formulae (4.24), (4.23), and (4.27), and using the integration by parts, it has
(4.28)J′(p)h=E∫T‍∫D‍(-dtv-Δvdt)u˙dx=E∫T‍∫D‍v(dtu˙-Δu˙dt)dx=E∫T‍∫D‍∑i=1m‍vΔaifidxdt=∑i=1m‍ΔaiE∫T‍∫D‍vfidxdt=(gradJ,h)ℝm.


## 4.1. The System Governed by a Stochastic Parabolic Partial Differential Equations

Consider the following stochastic parabolic partial differential equation:(4.1)dtu(x,t)=∑i,j=1n‍∂i(aij(x)∂ju)dt-c(x,t)udt+f(x,t)dt+σ(x,t)dwt,(x,t)∈D×T,u∣∂D=0,u∣t=t0=φ(x)+ξ(x),x∈D,
where aij∈Wm,2(D) with m>n/2, c∈Wm,2(D×T), f∈L2(Ω×D×T), σ(·,t)∈ℒ(W;L2(D)), for all t∈T, with ∫T‍∥σ(·,t)∥L2(D)2dt<∞, φ∈L2(D), ξ∈L2(Ω×D), wt=w(t) is a Wiener process in W.The problem we shall deal with is to determine the unknown coefficientsaij(i,j=1,…,n) and c based on the measurement
(4.2)z(x)=u(x,tf),x∈D.SupposeH=L2(D) and V=H01(D), which is the subspace of W1,2(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 parameterp=(a11,…,ann,c) and define the parameter space by
(4.3)P=∏i,j=1nWm,2(D)×Wm,2(D×T),
with the norm
(4.4)∥p∥P2=∑i,j=1n‍∥ai,j∥Wm,2(D)2+∥c∥Wm,2(D×T)2,
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:
(4.5)Pad≡{p∈P;γ|y|2≤∑i,j=1n‍ai,j(x)yiyj≤δ|y|2,∀y=(y1,…,yn)∈ℝn,a.e.x∈D},
where γ and δ(0<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
(4.6)Wm,2(D)→Cλ(D-)⊂C(D-)
is compact, where Cλ(D-) is the Hölder space and λ∈(0,m-n/2).Next, define the operator(4.7)A≡-∑i,j=1n‍∂i(ai,j∂j·),B(t)≡c(t),
where c(t)≡c(·,t), and then, obviously, A(p)∈ℒ(V,V′). Using the generalized Green formula, for all y, z∈V, it has
(4.8)|〈Ay,z〉V′,V|=|〈-∑i,j=1n‍∂i(ai,j∂jy),z〉V′,V|=|(∑i,j=1n‍ai,j∂jy,∂iz)H|≤maxi,jmaxx∈D-|ai,j(x)|∥∇y∥L2∥∇z∥L2≤c1∥y∥V∥z∥V.
Due to p∈Pad, it has
(4.9)〈Ay,y〉V′V=(∑i,j=1n‍ai,j∂jy,∂iy)H≥γ∥∇y∥L22.
By the Poincaré inequality it has
(4.10)〈Ay,y〉V',V≥c2∥y∥V2,
which, along with (4.8), shows that A satisfies (1.5).Next, for allz∈V(4.11)∥B(t)z∥H2=∫D‍|c(x,t)z(x)|2dx≤∥c∥L∞(D×T)2∫D‍|z(x)|2dx≤c0∥z∥H2≤c1∥z∥V2.
Therefore, B(t)∈ℒ(V,H), for all t∈T.Finally, setu(t)=u(·,t), f(t)=f(·,t), and σ(t)=σ(·,t). Thus, (4.1) can be written as (3.25).Summing up the above reasoning, it has the following theorem.Theorem 4.1.
Iff∈L2(Ω×D×T), then (4.1) has a unique solution u(p)=u(x,t;p):
u
∈
L
2
(
Ω
;
C
(
T
;
H
)
∩
L
2
(
Ω
×
T
;
V
)
∩
C
(
T
-
;
L
2
(
Ω
;
H
)
)
≡
𝔖,
u
(
·
,
t
) is adapted to ℱt (as a process with values in  H),
u
(
·
,
t
) is measurable with values in V.Theorem 4.2.
Iff∈L2(Ω×D×T), then for any p∈Pad, the mapping u(p) is continuously Fréchet differentiable and its Fréchet differential u′(p)h=u˙(h∈P) is determined by the following system:
(4.12)dtu˙-∑i,j‍∂i(aij∂ju˙)dt+cu˙dt=∑i,j‍∂i(Δai,j∂ju)dt-Δcudt,u˙∣∂Ω=0,u˙∣t=t0=0,
where h=(Δa11,…,Δann,Δc)∈P and u is determined by (4.1). Furthermore, the mapping u(p) is infinitely Fréchet differentiable.Becauseu∈C(T-;L2(Ω;H)), u(·,tf)∈L2(Ω;H). So, we can use the following cost functional:
(4.13)J(p)=E∫D‍|u(x,tf;p)-z(x)|2dx,p∈Pad,
in order to determine p. The operator Λ(4.14)Λu=u(·,tf),x∈D,Λ:C(T-;L2(Ω;H))→L2(Ω;H)
obviously satisfies Λ∈ℒ(𝔖;L2(Ω;H)).Theorem 4.3.
The mappingJ(p) defined by (4.13) is continuously Fréchet differentiable and its Fréchet differential is determined by
(4.15)J′(p)h=-2E∫D‍∫T‍{∑i,j=1n‍Δai,j∂iv∂ju+Δcuv}dtdx=〈gradJ,h〉P′,P,
where h=(Δa11,…,Δann,Δc)∈P, the gradient operator is
(4.16)gradJ=-2E∫T‍(∂1v∂1u,∂1v∂2u,…,∂nv∂nu)dt∈P′,u=u(p) is the solution of (4.1), and v=v(p) is defined by the following system:
(4.17)-dtv=∑i,jn‍∂j(aij∂iv)dt-cvdt,(x,t)∈D×T,v∣∂D=0,v∣t=tf=u∣t=tf-z.


## 4.2. Example 2

Consider the following system:(4.18)dtu-Δudt=∑i=1m‍aifi(x,t)dt+σ(x,t)dwt,(x,t)∈D×T,u∣∂D=0,t∈T,u∣t=t0=φ(x)+ζ(x),x∈D,
where D⊂ℝn with n≤3.The problem addressed is to identify the unknown parameterp=(a1,…,am)∈ℝm, which varies in an admissible parameter set Pad≡{p∈ℝm;0≤γ≤∥p∥Rm≤δ} based on the approximate point measurement:
(4.19)zi(x,t)=u(x,t),a.e.x∈Di,∀t∈T,i=1,…,L,
where Di is the ε-neighbourhood of xi∈D and Di∩Dj=∅(i≠j).In order to make sense of (4.19), it assumes
(4.20)fi∈L2(T;W1,2(D)),i=1,…,m,∂jσ(t)∈ℒ(W,H),∫T‍∥∂jσ(t)∥ℒ(W,H)2dt<∞,j=1,...,n.
It is easy to prove that u(t)∈W2,2(D)∩H01(D). Using the Sobolev embedding theorem [15], it has W2,2(D)⊂C(D-). So (4.19) is sensible.Introduce the characteristic function ofDi:
(4.21)χDi(x)={1,x∈Di,0,otherwise,
and the cost functional
(4.22)minp∈PadJ(p)≡∑i=1L‍E∫D‍∫T‍[χDi(x)[u(x,t;p)-zi(x,t)]]2dtdx.
In the manner similar to reasoning in Section 4.1 it can be proved that the function u(p) is continuously Fréchet differentiable and its Fréchet differential, u˙=u′(p)h,h=(Δa1,…,Δam)∈ℝm, is determined by the following:
(4.23)dtu˙-Δu˙dt=∑i=1m‍Δaifidt,(x,t)∈D×T,u˙∣∂D=0,u˙∣t=t0=0.
Furthermore, it can also be proved that the functional J(p) is continuously Fréchet differentiable and its Fréchet differential is
(4.24)J′(p)h=2∑i=1L‍E∫T‍∫D‍χDi(x)[u(x,t;p)-zi(x,t)]u˙(x,t;p)dxdt.
Now, the main results are given as followings.Theorem 4.4.
The functionalJ(p) is continuously Fréchet differentiable and its Fréchet differentiable is
(4.25)J′(p)h=(gradJ,h)ℝm,
where gradJ is the gradient of J(p) determined by the following:
(4.26)gradJ=(E∫T‍∫D‍vf1dxdt,…,E∫T‍∫D‍vfmdxdt)∈ℝm,
where v is the solution to the following initial-boundary problem:
(4.27)-dtv-Δvdt=2∑i=1L‍χDi(x)[u(x,t;p)-z(x,t)]dt,(x,t)∈D×T,v∣∂Ω=0,v∣t=tf=0
and u(x,t;p) is the solution to the problem (4.18).Proof.
By the formulae (4.24), (4.23), and (4.27), and using the integration by parts, it has
(4.28)J′(p)h=E∫T‍∫D‍(-dtv-Δvdt)u˙dx=E∫T‍∫D‍v(dtu˙-Δu˙dt)dx=E∫T‍∫D‍∑i=1m‍vΔaifidxdt=∑i=1m‍ΔaiE∫T‍∫D‍vfidxdt=(gradJ,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.



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*Source: 921038-2012-09-27.xml*