An Inverse Problem for a Class of Linear Stochastic Evolution Equations

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.

1. Introduction

The purpose of this paper is to study an inverse problem for the following linear stochastic evolution equation: 𝑑𝑦+𝐴𝑡;𝑝+𝐵𝑡;𝑝𝑦𝑑𝑡=𝑓𝑡;𝑝𝑑𝑡+𝜎𝑡;𝑝𝑑𝑤(𝑡),𝑡∈𝑡₀,𝑡_𝑓𝑦𝑡≡𝑇,₀=𝜑+𝜉, (1.1) where 𝑡_𝑓<∞, 𝑝∈𝑃_𝑎𝑑⊂𝑃 is a parameter to be determined, and 𝑃_𝑎𝑑 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 parameter 𝑝 based on the measurement 𝑔(𝑡), which is defined by the following: 𝑔(𝑡)=Λ𝑡;𝑝𝑦(𝑡),𝑡∈𝑇, (1.2) 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 by ℒ(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 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∈𝑃_𝑎𝑑, 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 𝐴𝑡;𝑝𝑧,𝑧_𝑉′,𝑉≥𝛼‖𝑧‖²_𝑉,𝑧∈𝑉, (1.5) where the constant 𝛼 is independent of 𝑝∈𝑃_𝑎𝑑 and 𝑡∈𝑇. Let (Ω,ℱ,𝜇) be a complete probability space and ℱ^𝑡 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. 𝐿^𝑟(Ω;Φ)≡𝑥∈𝑀Ω,𝜇,Φ;‖𝑥‖≡𝐸‖𝑥‖^𝑟_Φ^1/𝑟=_Ω‖𝑥(𝜔)‖^𝑟_Φ𝑑𝜇(𝜔)^1/𝑟,𝐿<∞_𝑟(Ω×𝑇;Φ)≡𝑥(𝑡)∈𝑀Ω,𝜇,Φ,𝑡∈𝑇;‖𝑥‖≡_Ω𝑇‖𝑥(𝑡,𝜔)‖^𝑟_Φ𝑑𝑡𝑑𝜇(𝜔)^1/𝑟<∞. (1.6)𝑤(𝑡)∈𝑀(Ω,𝜇,𝑊) is a Wiener process with values in a separable Hilbert space 𝑊, which is adapted to ℱ^𝑡, that is, for all𝑒∈𝑊, (𝑤(𝑡),𝑒)_𝑊 is a real Wiener process and an ℱ^𝑡-martingale, with the correlation function 𝐸𝑤𝑡₁,𝑒_1𝑊𝑤𝑡₂,𝑒_2𝑊=^{min(𝑡1,𝑡2)}_𝑡0𝑄(𝜏)𝑒₁,𝑒₂𝑑𝜏, (1.7) where 𝑡₀≤𝑡₁, 𝑡₂≤𝑡_𝑓 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 𝑄(⋅)∈𝐿^∞(𝑇;ℒ(𝑊)), (1.8) where 𝜑₀∈𝐻, 𝜉∈𝐿²(Ω,𝐻) is independent of 𝑤(𝑡).

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∈𝑃_𝑎𝑑 such that the cost functional 𝐽𝑝=𝐸_𝑔𝑇𝑔(𝑡)−Λ𝑡;𝑝𝑦𝑡;𝑝²_𝐾𝑑𝑡 (1.9) reaches its minimum over the admissible parameter set 𝑃_𝑎𝑑 at 𝑝, that is, 𝐽𝑝=min_{𝑝∈𝑃𝑎𝑑}𝐽𝑝, (1.10) where 𝐸_𝑔 is a conditional expectation, that is, 𝐸_𝑔𝑓=𝐸𝑓∣𝐺^𝑡, (1.11) and 𝐺^𝑡 is the sub-𝜎-algebra induced by the stochastic process 𝑔(𝑠), 0≤𝑠≤𝑡, which is adapted to ℱ^𝑡.

If there exists a neighbourhood 𝑈⊂𝑃_𝑎𝑑 of 𝑝₀ such that 𝐽𝑝₀=inf_𝑝∈𝑈𝐽𝑝, (1.12) then 𝑃_𝑜 is called a relative optimal parameter.

In Section 2, the base of this paper is given, under certain conditions the function 𝑝→𝑦(𝑝) 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.

Lemma 2.1. Besides the assumptions for 𝐴, 𝐵, 𝑓, 𝜑, 𝑤(𝑡), and 𝜎 in Section 1, one assumes that, for any 𝑝∈𝑃_𝑎𝑑, 𝐴⋅;𝑝∈𝐿²𝑇;ℒ𝑉;𝑉^′∩𝐿^∞𝑇;ℒ𝑉;𝑉^′,𝐵⋅;𝑝∈𝐿^∞((,𝑓Ω×𝑇;ℒ𝑉;𝐻))⋅,𝑝∈𝐿²(),𝜎Ω×𝑇;𝐻⋅,𝑝∈𝐿²(𝑇,ℒ(𝑊;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;𝐶(𝑇;𝐻))∩𝐿²(Ω×𝑇;𝑉)∩𝐶𝑇;𝐿²(Ω;𝐻)≡𝔖, (2.2)𝑦(𝑡) is adapted to ℱ^𝑡 (as a process with values in 𝐻), and 𝑦(𝑡) is ℱ^𝑡 measurable with values in 𝑉.

In the above lemma the space 𝐶^𝑘(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_𝑖=0max_𝑡∈𝑋𝑢^(𝑖)(𝑡). (2.3)

Remark 2.2. The generalized solution in the Ladyzenskaja sense is the solution of the following variational equation: 𝑦(𝑡),𝜂(𝑡)_𝐻=𝜑+𝜉,𝜂𝑡_0𝐻+^𝑡_𝑡0𝑦(𝑡),𝑑_𝑡𝜂(𝑡)_𝑉,𝑉′+^𝑡_𝑡0𝐴𝑡;𝑝+𝐵𝑡;𝑝𝑦(𝑡)𝑑𝑡,𝜂(𝑡)_𝑉′,𝑉+^𝑡_𝑡0𝜎𝑡;𝑝𝑑𝑤(𝑡),𝜂(𝑡)_𝐻,a.s.,∀𝜂∈𝑊¹(𝑇;𝑉,𝐻), (2.4) where the space 𝑊¹(𝑇;𝑉,𝐻) is a Hilbert space that is defined by 𝑊¹(𝑇;𝑉,𝐻)=𝜂;𝜂∈𝐿²(𝑇;𝑉),𝑑𝜂𝑑𝑡∈𝐿²𝑇;𝑉^′, (2.5) 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 Lemma 2.1 are satisfied and 𝐴∶𝑝→𝐿²𝑇;ℒ𝑉;𝑉^′,𝐵∶𝑝→𝐿^∞(Ω×𝑇;ℒ(𝑉;𝐻)),𝑓∶𝑝→𝐿²(Ω×𝑇;𝐻),𝜎∶𝑝→𝐿²(𝑇,ℒ(𝑊;𝐻)), (2.6) are continuous, the solution of (1.1) 𝑦∶𝑃_𝑎𝑑→𝔖 (2.7) is continuous, that is, 𝑦∈𝐶(𝑃_𝑎𝑑;𝔖) or the following equalities are true: lim_𝑝→𝑝0𝑦𝑝−𝑦𝑝_{0𝐶𝑇;𝐿2()Ω,𝐻}=0,lim_𝑝→𝑝0𝑦𝑝−𝑦𝑝_{0𝐿2Ω;𝐶𝑇;𝐻}=0,lim_𝑝→𝑝0𝑦𝑝−𝑦𝑝_{0𝐿2()Ω×𝑇;𝑉}=0. (2.8)(2.9)(2.10)

Before proving Theorem 2.3, from Bensoussan [10] the following 𝐼𝑡𝑜̂ formula in a Hilbert space is quoted.

Lemma 2.4. Let Φ(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: 𝑑𝑧(𝑡)=𝑎(𝑡)𝑑𝑡+𝑏(𝑡)𝑑𝑤(𝑡),𝑧𝑡₀=𝑧₀, (2.11) where 𝑎(𝑡) is a stochastic process with values in 𝐻, which is adapted and satisfies the condition a.s^𝑡_𝑡0‖𝑎(𝑡)‖_𝐻𝑑𝑡<+∞,∀𝑡∈𝑇, (2.12) where 𝑏(𝑡) is an adapted process with values in ℒ(𝑊,𝐻) such that 𝑡, 𝜔→𝑏 is measurable and 𝐸^𝑡1_𝑡0‖𝑏(𝑡)‖²_{ℒ()𝑊,𝐻}𝑑𝑡<∞,∀𝑡₁<𝑡_𝑓, (2.13) then one has the following 𝐼𝑡𝑜̂ formula in the Hilbert space: Φ(𝑧(𝑡),𝑡)=Φ𝑧𝑡₀,𝑡₀+^𝑡_𝑡0𝜕Φ(𝑠)𝜕𝑧,𝑎_𝐻𝑑𝑠+^𝑡_𝑡0𝜕Φ(𝑠)𝜕𝑧,𝑏𝑑𝑤_𝐻+12^𝑡_𝑡0tr𝑏^∗𝜕²Φ𝜕𝑧²𝑏𝑄𝑑𝑠+^𝑡_𝑡0𝜕Φ𝜕𝑡𝑑𝑠, (2.14) 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: tr𝑈=^∞_𝑛=1𝑈𝑓_𝑛,𝑓_𝑛𝑋, (2.15) where {𝑓_𝑛} is an orthonormal basis of 𝑋.

The following two lemmas are obvious, so their proofs are omitted.

Lemma 2.5. Supposes that 𝑈∈ℒ(𝑋) is a nuclear operator and 𝐵∈ℒ(𝑋,𝑌), then tr𝐵^∗𝐵𝑈≤‖𝐵‖²tr𝑈. (2.16)

Lemma 2.6. If 𝜓∈𝐿²(Ω;𝐿²(𝑇)), 𝐸_𝑇𝜓(𝑡)𝑑𝑤(𝑡)=0. (2.17)

Lemma 2.7. If 𝑓∈𝐿²(Ω;𝐿²(𝑇;𝑊))≡𝐿²(Ω×𝑇,𝑊), 𝐸_𝑇𝑓(𝑡),𝑑𝑤(𝑡)_𝑊²=𝐸_𝑇𝑅(𝑡)𝑓(𝑡),𝑓(𝑡)_𝑊𝑑𝑡. (2.18)

Proof. First, suppose that 𝑓 is a step function, that is, 𝑓(𝑡)=𝑓_𝑟,𝑡_𝑟≤𝑡<𝑡_𝑟+1,𝑟=0,…,𝑁−1, (2.19) where 𝑓_𝑟∈𝐿²(Ω,𝑊) and 𝑡₀<𝑡₁<⋯<𝑡_𝑁=𝑡_𝑓.
Let {𝑒_𝑛} be an orthomomal basis of 𝑊. Obviously, 𝑤(𝑡)=^∞_𝑖=1𝑒_𝑛,𝑤(𝑡)_𝑊𝑒_𝑛, (2.20) and 𝑤_𝑛(𝑡)=(𝑒_𝑛,𝑤(𝑡)) is a Wiener process. Furthermore, 𝐸_𝑇(𝑓(𝑡),𝑑𝑤(𝑡))²=𝐸_{𝑖,𝑗𝑟,𝑠}^𝑡𝑟+1_𝑡𝑟𝑓_𝑟,𝑒_𝑖𝑑𝑤_𝑖^𝑡𝑠+1_𝑡𝑠𝑓_𝑠,𝑒_𝑗𝑑𝑤_𝑗=𝐸_{𝑖,𝑗𝑟,𝑠}𝑓_𝑟,𝑒_𝑖𝑤_𝑖𝑡_𝑟+1−𝑤_𝑖𝑡_𝑟𝑓_𝑠,𝑒_𝑗𝑤_𝑗𝑡_𝑠+1−𝑤_𝑗𝑡_𝑗=𝐸_𝑇𝑅(𝑡)_𝑖𝑓,𝑒_𝑖𝑒_𝑖,_𝑗𝑓,𝑒_𝑗𝑒_𝑗𝑑𝑡=𝐸_𝑇𝑅(𝑡)𝑓(𝑡),𝑓(𝑡)_𝑊𝑑𝑡, (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 𝑦(𝑡)=𝑦(𝑡,𝑝) and 𝑦₀(𝑡)=𝑦(𝑡;𝑝₀), and then 𝑑𝑦+𝐴𝑡;𝑝+𝐵𝑡;𝑝𝑦𝑑𝑡=𝑓𝑡;𝑝𝑑𝑡+𝜎𝑡;𝑝𝑑𝑤(𝑡),𝑦𝑡₀=𝜑+𝜁,𝑑𝑦₀+𝐴𝑡;𝑝₀+𝐵𝑡;𝑝₀𝑦₀𝑑𝑡=𝑓𝑡;𝑝₀𝑑𝑡+𝜎𝑡;𝑝₀𝑑𝑤(𝑡),𝑦₀𝑡₀=𝜑+𝜁. (2.22)(2.23) Let 𝑧(𝑡)=𝑦(𝑡)−𝑦₀(𝑡), and then from the above equalities, it follows 𝑑𝑧(𝑡)+𝐴𝑡;𝑝+𝐵𝑡;𝑝𝑧𝑑𝑡=𝑓𝑡;𝑝−𝑓𝑡;𝑝₀−𝐴𝑡;𝑝−𝐴𝑡;𝑝₀𝑦₀−𝐵𝑡;𝑝−𝐵𝑡;𝑝₀𝑦₀𝑑𝑡+𝜎𝑡;𝑝−𝜎𝑡;𝑝₀(𝑡),𝑧𝑡𝑑𝑤₀=0. (2.24)
Setting Φ(𝑧)=(1/2)‖𝑧‖²,𝑧∈𝑊^!(𝑇;𝑉,𝐻) according to Lemma 2.4, it gets 12‖𝑧(𝑡)‖²_𝐻+^𝑡_𝑡0𝑧(𝑠),𝐴𝑠;𝑝𝑧(𝑠)_𝑉,𝑉′𝑑𝑠=−^𝑡_𝑡0𝑧(𝑠),𝐵𝑠;𝑝𝑧(𝑠)_𝐻−𝑑𝑠_𝑡𝑡0𝑧(𝑠),𝐴𝑠;𝑝−𝐴𝑠;𝑝₀𝑦₀(𝑠)_𝑉,𝑉′+𝑑𝑠_𝑡𝑡0𝑧(𝑠),𝑓𝑠;𝑝−𝑓𝑠;𝑝₀−𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)_𝐻+𝑑𝑠_𝑡𝑡0𝑧(𝑠),𝜎𝑠;𝑝−𝜎𝑠;𝑝₀𝑑𝑤(𝑠)_𝐻+12^𝑡_𝑡0tr𝜎𝑠,𝑝−𝜎𝑠;𝑝₀^∗𝜎𝑠,𝑝−𝜎𝑠;𝑝₀𝑄(𝑠)𝑑𝑠, (2.25) which is ‖𝑧(𝑡)‖²_𝐻+2^𝑡_𝑡0𝑧(𝑠),𝐴𝑠;𝑝𝑧(𝑠)_𝑉.𝑉′𝑑𝑠=−2^𝑡_𝑡0𝑧(𝑠),𝐵𝑠;𝑝𝑧(𝑠)_𝐻𝑑𝑠+2^𝑡_𝑡0𝑧(𝑠),𝑓𝑠;𝑝−𝑓𝑠;𝑝_0𝐻𝑑𝑠−2^𝑡_𝑡0𝑧(𝑠),𝐴𝑠;𝑝−𝐴𝑠;𝑝₀𝑦₀(𝑠)_𝑉,𝑉′𝑑𝑠−2^𝑡_𝑡0𝑧(𝑠);𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)_𝐻𝑑𝑠+2^𝑡_𝑡0𝑧(𝑠),𝜎𝑠;𝑝−𝜎𝑠;𝑝₀𝑑𝑤(𝑠)_𝐻+^𝑡_𝑡0tr𝜎𝑠,𝑝−𝜎𝑠;𝑝₀^∗𝜎𝑠,𝑝−𝜎𝑠;𝑝₀𝑄(𝑠)𝑑𝑠. (2.26)
Taking the expectation from the above and considering Lemma 2.6, it has 𝐸‖𝑧(𝑡)‖²_𝐻+2𝐸^𝑡_𝑡0𝑧(𝑠),𝐴𝑠;𝑝𝑧(𝑠)_𝑉,𝑉′𝑑𝑠=−2𝐸^𝑡_𝑡0𝑧(𝑠),𝐵𝑠;𝑝𝑧(𝑠)_𝐻𝑑𝑠+2𝐸^𝑡_𝑡0𝑧(𝑠),𝑓𝑠;𝑝−𝑓𝑠;𝑝_0𝐻𝑑𝑠−2𝐸^𝑡_𝑡0𝑧(𝑠),[𝐴(𝑠;𝑝)−𝐴(𝑠;𝑝₀)]𝑦₀(𝑠)_𝑉,𝑉′𝑑𝑠−2𝐸^𝑡_𝑡0𝑧(𝑠);𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)_𝐻+𝑑𝑠_𝑡𝑡0tr𝜎𝑠,𝑝−𝜎𝑠;𝑝₀^∗𝜎𝑠,𝑝−𝜎𝑠;𝑝₀𝑄(𝑠)𝑑𝑠. (2.27)
From the assumptions of the spaces 𝐻 and 𝑉, there exists a constant 𝛾 such that ‖𝑥‖_𝐻≤𝛾‖𝑥‖_𝑉,∀𝑥∈𝑉. (2.28) Furthermore, according to the assumptions of the operator 𝐴(𝑝) and using (2.28) and 2𝑎𝑏≤𝜀𝑎²+𝑏²/𝜀, we can obtain 𝐸‖𝑧(𝑡)‖²_𝐻+2𝛼𝐸^𝑡_𝑡0‖𝑧(𝑠)‖²_𝑉𝑑𝑠≤𝛼𝐸^𝑡_𝑡0‖𝑧(𝑠)‖²_𝑉𝑑𝑠+4𝛾𝛼𝐵⋅;𝑝²_{𝐿∞((𝑇;ℒ𝑉;𝐻))}𝐸^𝑡_𝑡0‖𝑧(𝑠)‖²_𝐻+𝑑𝑠4𝛾𝛼𝐸^𝑡_𝑡0𝑓𝑠;𝑝−𝑓𝑠;𝑝₀²_𝐻+𝐴𝑠;𝑝−𝐴𝑠;𝑝₀𝑦₀(𝑠)²_𝑉′+𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)²_𝐻+‖‖𝑑𝑠.tr𝑄^𝑡_𝑡0𝜎𝑠,𝑝−𝜎𝑠;𝑝₀²𝑑𝑠. (2.29) By the assumptions of Theorem 2.3 there is a constant 𝑐 such that 𝐸‖𝑧(𝑡)‖²_𝐻+𝛼𝐸^𝑡_𝑡0‖𝑧(𝑠)‖²_𝑉𝑑𝑠≤𝑐^𝑡_𝑡0𝐸‖𝑧(𝑠)‖²_𝐻𝑑𝑠+𝑐^𝑡_𝑡0𝜎𝑠,𝑝−𝜎𝑠;𝑝₀²_{ℒ()𝑊;𝐻}𝑑𝑠+𝑐𝐸^𝑡_𝑡0𝑓𝑠;𝑝−𝑓𝑠;𝑝₀²_𝐻+𝐴𝑠;𝑝−𝐴𝑠;𝑝₀𝑦₀(𝑠)²_𝑉′+𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)²_𝐻𝑑𝑠. (2.30) Obviously, 𝐸‖𝑧(𝑡)‖²_𝐻≤𝑐^𝑡_𝑡0𝐸‖𝑧(𝑠)‖²_𝐻𝑑𝑠+𝑐𝐸^𝑡_𝑡0𝑓𝑠;𝑝−𝑓𝑠;𝑝₀²_𝐻+𝐴𝑠;𝑝−𝐴𝑠;𝑝₀𝑦₀(𝑠)²_𝑉′+𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)²_𝐻𝑑𝑠+𝑐^𝑡_𝑡0𝜎𝑠,𝑝−𝜎𝑠;𝑝₀²_{ℒ()𝑊;𝐻}𝑑𝑠. (2.31) Using the Gronwall inequality (see [13] for the definition), and the assumptions of Theorem 2.3, letting 𝑝→𝑝₀, it has max_𝑡∈𝑇𝐸‖𝑧(𝑠)‖²_𝐻=𝑜(1), (2.32) which is (2.8). Furthermore, from (2.31), it follows 𝐸_𝑇‖𝑧(𝑠)‖²_𝑉=𝑜(1), (2.33) which is just (2.10).
From (2.26), it has the following estimate: ‖𝑧(𝑡)‖²_𝐻+2𝛼^𝑡_𝑡0‖𝑧(𝑠)‖²_𝑉≤𝛼𝑑𝑠𝛾^𝑡_𝑡0‖𝑧(𝑠)‖²_𝐻+𝑑𝑠4𝛾𝛼^𝑡_𝑡0𝑓𝑠;𝑝−𝑓𝑠;𝑝₀²_𝐻+𝐴𝑠;𝑝−𝐴𝑠;𝑝₀𝑦₀(𝑠)²_𝑉′+𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)²_𝐻+𝐵𝑑𝑠⋅;𝑝²_{𝐿∞(𝑇;ℒ(𝑉;𝐻))}^𝑡_𝑡0‖𝑧(𝑠)‖²_𝐻𝑑𝑠+2^𝑡_𝑡0𝑧(𝑠),𝜎𝑠;𝑝−𝜎𝑠;𝑝₀𝑑𝑤(𝑠)_𝐻+‖tr𝑄‖_𝐿^∞_(𝑇)^𝑡_𝑡0𝜎𝑠,𝑝−𝜎𝑠;𝑝₀²_{ℒ()𝑊;𝐻}𝑑𝑠. (2.34) Similarly, it also obtains ‖𝑧(𝑡)‖²_𝐻≤𝑐^𝑡_𝑡0𝑓𝑠;𝑝−𝑓𝑠;𝑝₀²_𝐻+𝐴𝑝−𝐴𝑝₀𝑦₀²_𝑉′+𝐵𝑝−𝐵𝑝₀𝑦₀²_𝐻+𝜎𝑝−𝜎𝑝₀²_{ℒ()𝑊;𝐻}𝑑𝑠+𝑐^𝑡_𝑡0𝑧(𝑠),𝜎𝑠,𝑝−𝜎𝑠;𝑝₀𝑑𝑤(𝑠)_𝐻,𝑡∈𝑇. (2.35) On the other hand, the indefinite stochastic integral 𝑋(𝑡)=^𝑡_𝑡0𝜎𝑠;𝑝−𝜎𝑠;𝑝^∗𝑧(𝑠),𝑑𝑤(𝑠)_𝐻 (2.36) is a continuous martingale, so 𝐸max_𝑡∈𝑇𝑋(𝑡)≤𝐸max_𝑡|𝑋(𝑡)|≤𝐸max_𝑡|𝑋(𝑡)|^21/2≤2𝐸𝑋𝑡_𝑓^21/2=2𝐸_𝑇𝑅(𝑡)𝜎𝑠;𝑝−𝜎𝑠;𝑝₀^∗𝑧(𝑠),𝜎𝑠;𝑝−𝜎𝑠;𝑝₀^∗𝑧(𝑠)𝑑𝑡^1/2≤𝑐_𝑇𝜎𝑡;𝑝−𝜎𝑡;𝑝₀²𝑑𝑡^1/2, (2.37) where the following martingale inequality is used: 𝐸max_𝑇|𝑋(𝑡)|^𝛼≤𝛼𝛼−1²𝐸𝑋𝑡_𝑓^𝛼, (2.38) with 𝛼=2.
Combining (2.35) with (2.37), it has 𝐸max_𝑡∈𝑇‖𝑧(𝑡)‖²_𝐻≤𝑐𝐸^𝑡𝑓_𝑡0𝑓𝑠;𝑝−𝑓𝑠;𝑝₀²_𝐻+𝐴𝑠;𝑝−𝐴𝑠;𝑝₀𝑦₀(𝑠)²_𝑉′+𝐵𝑠;𝑝−𝐵𝑠;𝑝₀𝑦₀(𝑠)²_𝐻𝑑𝑠+𝑐_𝑇𝜎𝑠,𝑝−𝜎𝑠;𝑝₀²_{ℒ()𝑊;𝐻}𝑑𝑠+𝑐_𝑇𝜎𝑠,𝑝−𝜎𝑠;𝑝₀²_{ℒ()𝑊;𝐻}𝑑𝑠^1/2. (2.39) Letting 𝑝→𝑝₀ in 𝑃, by the assumptions of Theorem 2.3, it immediately obtains 𝐸max_𝑡∈𝑇‖𝑧(𝑡)‖²_𝐻=𝑜(1), (2.40) which is (2.9).

Theorem 2.8. Besides the assumptions of Theorem 2.3, suppose that the mappings 𝐴∶𝑇×𝑃_𝑎𝑑→ℒ𝑉;𝑉^′,𝐵∶𝑇×𝑃_𝑎𝑑→ℒ(𝑉;𝐻),𝑓∶𝑇×𝑃_𝑎𝑑→𝐻,𝜎∶𝑇×𝑃_𝑎𝑑→ℒ(𝑊;𝐻) (2.41) are continuously Fréchet differentiable, then the solution of (1.1) 𝑦∶𝑝→𝑦𝑝∈𝔖 (2.42) is continuously Fréchet differentiable and its Fréchet derivative operators at 𝑝₀∈𝑃_𝑎𝑑, 𝑦′(𝑝₀)∈ℒ(𝑃,𝔖), are determined by the following system: 𝑑𝑦̇+𝐴𝑡;𝑝₀+𝐵𝑡;𝑦₀𝑦𝑑̇𝑡=𝑓^′_𝑝𝑡;𝑝₀ℎ−𝐴^′_𝑝𝑡;𝑝₀ℎ𝑦₀−𝐵^′_𝑝𝑡;𝑝₀ℎ𝑦₀𝑑𝑡+𝜎^′_𝑝𝑡;𝑝₀(𝑡),𝑦̇𝑡ℎ𝑑𝑤₀=0, (2.43) where 𝑦̇=𝑦^′(𝑝₀)ℎ, ℎ∈𝑃, 𝑦₀=𝑦(𝑡;𝑝₀) is determined by (1.1), and 𝑓^′_𝑝(𝑡;𝑝₀)∈ℒ(𝑃;𝐻), 𝐴^′_𝑝(𝑝₀)∈ℒ(𝑃;ℒ(𝑉;𝑉^′)), 𝐵^′_𝑝(𝑝₀)∈ℒ(𝑃;ℒ(𝑉;𝐻)), and 𝜎^′_𝑝(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 Lemma 2.1, there exists a unique solution to (2.43), 𝑦̇∈𝔖. Taking 𝑝₀∈𝑃_𝑎𝑑, for any 𝑝∈𝑃_𝑎𝑑, setting ℎ=𝑝−𝑝₀ and 𝑧=𝑦−𝑦₀−𝑦̇, where 𝑦=𝑦(𝑝) and 𝑦₀=𝑦(𝑝₀) are defined by (2.22) and (2.23), respectively, it has 𝑑𝑧+𝐴₀+𝐵₀𝑧𝑑𝑡=𝑓−𝑓₀−𝑓^′₀ℎ−𝐴−𝐴₀−𝐴^′₀ℎ𝑦−𝐵−𝐵₀−𝐵^′₀ℎ𝑦𝐴^′₀ℎ𝑦−𝑦₀−𝐵^′₀ℎ𝑦−𝑦₀𝑑𝑡+𝜎−𝜎₀−𝜎^′₀ℎ(𝑡),𝑧𝑡𝑑𝑤₀=0, (2.44) where 𝐴=𝐴(𝑝), 𝐴₀=𝐴(𝑝₀), 𝐴^′₀=𝐴^′_𝑝(𝑝₀),…, 𝜎=𝜎(𝑝), 𝜎₀=𝜎(𝑝₀), and 𝜎^′₀=𝜎^′_𝑝(𝑝₀).
Letting 𝑝→𝑝₀ or ‖ℎ‖→0, according to the definition of the Fréchet differentiability, it has 𝑓−𝑓₀−𝑓^′₀ℎ_𝐻=𝑜(‖ℎ‖),𝐴−𝐴₀−𝐴^′₀ℎ_{ℒ𝑉;𝑉′}=𝑜(‖ℎ‖),𝐵−𝐵₀−𝐵^′₀ℎ_{ℒ()𝑉;𝐻}=𝑜(‖ℎ‖),𝜎−𝜎₀−𝜎^′₀ℎ_{ℒ()𝑤;𝐻}=𝑜(‖ℎ‖). (2.45)
Moreover, by Theorem 2.3 it gets ‖𝑦−𝑦₀‖_𝔖=𝑜(1) and ‖𝑦‖_𝔖=‖𝑦−𝑦₀‖+‖𝑦₀‖≤𝑐‖𝑦₀‖+𝑐.
Using the deduction similar to Theorem 2.3 it obtains ‖𝑧‖_𝔖=𝑜‖ℎ‖, (2.46) which is just ‖𝑦−𝑦₀−𝑦^′₀‖=𝑜(‖ℎ‖).
So, 𝑦(𝑝) is Fréchet differentiable at 𝑝₀ and its Fréchet derivative operator 𝑦^′₀ 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→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 𝑃_𝑎𝑑, 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;ℒ(𝑉;𝐾)) (3.1) be continuous, then the mapping 𝐽∶𝑝→𝑅⁺ (3.2) is continuous.

Proof. Take 𝑝₀, 𝑝∈𝑃_𝑎𝑑 and set ℎ=𝑝−𝑝₀, 𝑦=𝑦(𝑝), 𝑦₀=𝑦(𝑝₀), Λ=Λ(𝑝), and Λ₀=Λ(𝑝₀), then it has 𝐽𝑝−𝐽𝑝₀=𝐸_𝑔𝑇Λ𝑦−𝑔²_𝐾−Λ₀𝑦₀−𝑔²_𝐾=𝐸𝑑𝑡_𝑔𝑇Λ𝑦+Λ₀𝑦₀−2𝑔,Λ𝑦−𝑦₀+Λ−Λ₀𝑦_0𝐾𝐸𝑑𝑡≤2_𝑔𝑇Λ𝑦+Λ₀𝑦₀−2𝑔²_𝐾𝑑𝑡^1/2𝐸_𝑔𝑇Λ𝑦−𝑦₀²_𝐾+Λ−Λ₀𝑦₀²_𝐾𝑑𝑡^1/2. (3.3) Letting 𝑝→𝑝₀, that is, ‖ℎ‖→0, using Theorem 2.3 and the assumptions of Theorem 3.1, it obtains at once 𝐽𝑝−𝐽𝑝₀=𝑜(1). (3.4) Hence, 𝐽(𝑝) is continuous.

From Zeidler [14] the following lemma is quoted.

Lemma 3.2. The minimum problem min_𝑢∈𝑀𝐹(𝑢)=𝛼 (3.5) 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 inf_{𝑝∈𝑃𝑎𝑑}𝐼𝑝≡𝐸_𝑇𝑈(𝑡)𝑝−𝑔(𝑡)²_𝐾𝑑𝑡=𝑑²,𝑈∈𝐿²(Ω×𝑇;ℒ(𝑃;𝐾)) (3.6) has a solution, 𝑝^∗∈𝑃_𝑎𝑑⊂𝑃, if one of the following conditions is fulfilled:(a)𝑃_𝑎𝑑 is a closed subspace of 𝑃;(b)𝑃_𝑎𝑑 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 𝐾_𝑎𝑑=𝑈𝑃_𝑎𝑑. Obviously, 𝐾_𝑎𝑑 is closed and convex.
Suppose {𝑝_𝑛}⊂𝑃_𝑎𝑑 is a minimizing sequence, that is, 𝐼(𝑝_𝑛)→𝑑².
In order to prove that {𝑈𝑝_𝑛} is a Cauchy sequence, we prove the following inequality: 𝐸_𝑇𝑈𝑃^′−𝑈𝑃^′′2≤𝑑𝑡𝐸_𝑇𝑈𝑃^′−𝑔²𝑑𝑡−𝑑²+𝐸_𝑇𝑈𝑃^′′−𝑔²𝑑𝑡−𝑑²²,∀𝑝^′,𝑝^′′∈𝑝. (3.7) Obviously, we have the following inequality: 𝐸_𝑇𝑈𝑃^′−𝑔,𝑈𝑃^′′−𝑔_𝐾𝑑𝑡−𝑑²²≤𝐸_𝑇𝑈𝑃^′−𝑔²_𝐾𝑑𝑡−𝑑²𝐸_𝑇𝑈𝑃^′′−𝑔²𝑑𝑡−𝑑². (3.8) Therefore, 𝐸_𝑇𝑈𝑃^′−𝑈𝑃^′′2=𝐸𝑑𝑡_𝑇𝑈𝑃^′−𝑔²𝑑𝑡−𝑑²−2𝐸_𝑇𝑈𝑃^′−𝑔,𝑈𝑃^′′−𝑔𝑑𝑡−𝑑²+𝐸_𝑇𝑈𝑃^′′−𝑔²𝑑𝑡−𝑑²≤𝐸_𝑇𝑈𝑃^′−𝑔²𝑑𝑡−𝑑²+2𝐸_𝑇𝑈𝑃^′−𝑔²𝑑𝑡−𝑑²𝐸_𝑇𝑈𝑃^′′−𝑔²𝑑𝑡−𝑑²+𝐸_𝑇𝑈𝑃^′′−𝑔²𝑑𝑡−𝑑²=𝐸_𝑇𝑈𝑃^′−𝑔²𝑑𝑡−𝑑²+𝐸_𝑇𝑈𝑃^′′−𝑔²𝑑𝑡−𝑑²², (3.9) which is exactly (3.7).
For any 𝑚, 𝑛∈ℕ, from (3.7) it has 𝐸_𝑇𝑈𝑝_𝑛+𝑚−𝑈𝑝_𝑛²𝑑𝑡≤𝐸_𝑇𝑈𝑝_𝑛+𝑚−𝑔²𝑑𝑡−𝑑²+𝐸_𝑇𝑈𝑝_𝑛−𝑔²𝑑𝑡−𝑑²². (3.10) Letting 𝑛→+∞, it obtains 𝑈𝑝_𝑛+𝑚−𝑈𝑝_𝑛=𝑜(1). (3.11) Hence, {𝑈𝑝_𝑛}⊂𝐾_𝑎𝑑 is a Cauchy sequence. Since 𝐾_𝑎𝑑 is closed, there exists 𝑦^∗∈𝐾_𝑎𝑑 such that 𝑈𝑝_𝑛→𝑦^∗,in𝐿²Ω;𝐿²(𝑇;𝐾). (3.12) By the definition of 𝐾_𝑎𝑑, there exists 𝑝^∗∈𝑃_𝑎𝑑 such that 𝑦^∗=𝑈𝑝^∗. Obviously, it has 𝐼𝑝^∗=lim_𝑛→∞𝐼𝑝_𝑛=𝑑². (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 𝑃_𝑎𝑑 is convex, bounded, and closed;(b)the function 𝑝→𝑦(𝑝) is linear, Λ is independent of 𝑝, and 𝑃_𝑎𝑑 is closed and convex;(c)𝑃_𝑎𝑑 is compact in 𝑃, then the optimization problem (1.9) has a solution, 𝑝^∗∈𝑃_𝑎𝑑, namely, 𝐽𝑝^∗=min_{𝑝∈𝑃𝑎𝑑}𝐽𝑝. (3.14)

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)∈ℒ(𝑃_𝑎𝑑;𝔖), forall𝑡∈𝑇, such that forall𝑝∈𝑃_𝑎𝑑, 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 mapping 𝐽(𝑝).

Theorem 3.5. Let the assumptions of Theorem 2.8 be satisfied, 𝑔∈𝐿²(Ω×𝑇;𝐾), and let Λ∶𝑝→𝐿^∞(𝑇;ℒ(𝑉;𝐾)) (3.15) be Fréchet differentiable, then 𝐽∶𝑝→ℝ⁺ (3.16) is continuously Fréchet differentiable and the Fréchet differential of 𝐽 at 𝑝 along the direction ℎ∈𝑃 is determined by the following: 𝐽^′𝑝ℎ=2𝐸_𝑔𝑇Λ^′𝑝ℎ𝑦𝑡;𝑝+Λ𝑝𝑦^′𝑡;𝑝ℎ,Λ𝑝𝑦𝑡;𝑝−𝑔(𝑡)_𝐾𝑑𝑡, (3.17) where Λ^′(𝑝)∈ℒ(𝑃;ℒ(𝔖;𝐾)) and 𝑦′(𝑡;𝑝)ℎ=𝑦(̇𝑡) is determined by (2.43).

Proof. Take 𝑝, 𝑝∈𝑃_𝑎𝑑, set ℎ=𝑝−𝑝, then we have 𝐽𝑝−𝐽𝑝−2𝐸_𝑔𝑇Λ^′𝑝ℎ𝑦𝑡;𝑝+Λ𝑝𝑦^′𝑡;𝑝ℎ,Λ𝑝𝑦𝑡;𝑝−𝑔(𝑡)_𝐾𝑑𝑡=𝐸_𝑔𝑇Λ−Λ−Λ^′_𝑝ℎ𝑦+Λ^′_𝑝ℎ𝑦−𝑦,Λ𝑦−𝑔_𝐾+Λ𝑦−𝑦−𝑦^′_𝑝ℎ,Λ𝑦−𝑔_𝐾+Λ𝑦−𝑔,Λ−Λ−Λ^′_𝑝ℎ𝑦+Λ^′_𝑝ℎ𝑦−𝑦_𝑘+Λ𝑦−𝑔,Λ𝑦−𝑦−𝑦^′_𝑝ℎ_𝑘+Λ^′_𝑝ℎ𝑦+Λ𝑦^′_𝑝ℎ,Λ𝑦−𝑦+Λ−Λ𝑦_𝑘𝑑𝑡, (3.18) where 𝑦=𝑦(𝑝) and Λ=Λ(𝑝). Letting 𝑝→𝑝, that is, ‖ℎ‖→0, it has 𝐽𝑝−𝐽𝑝−2𝐸_𝑔𝑇Λ^′_𝑝ℎ𝑦+Λ𝑦^′ℎ,Λ𝑦−𝑔_𝐾𝑑𝑡=𝑜‖ℎ‖. (3.19) 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 Theorem 3.5 be satisfied. If a point 𝑝₀∈𝑃_𝑎𝑑 is a relative optimal parameter for (1.9), then 𝑝₀ is characterized by 𝐽^′𝑝₀𝑝−𝑝₀≡2𝐸_𝑔𝑇Λ^′𝑝₀𝑝−𝑝₀𝑦𝑡;𝑝₀+Λ𝑝₀𝑦^′𝑡;𝑝₀𝑝−𝑝₀,Λ𝑝₀𝑦𝑡;𝑝₀−𝑔(𝑡)_𝐾𝑑𝑡≥0,𝑝∈𝑈⊂𝑃_𝑎𝑑, (3.20) where 𝑈∈𝑃_𝑎𝑑 is a neighbourhood of 𝑝₀.

Proof. Firstly, let 𝑝₀ be the relative optimal parameter, then, it has 𝐽𝑝₀≤𝐽𝑝=𝐽𝑝₀+𝐽^′𝑝₀𝑝−𝑝₀+𝑜𝑝−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 𝐽𝑝=𝐽𝑝₀+𝐽^′𝑝₀𝑝−𝑝₀+𝑜𝑝−𝑝₀, (3.22) it at once obtains 𝐽𝑝≥𝐽𝑝₀,𝑝∈𝑈. (3.23)

Theorem 3.7. Let the assumptions of Theorem 3.5 be satisfied and let the functional 𝐽(𝑝) be convex. If 𝑝₀∈𝑃_𝑎𝑑 is an extreme point, then 𝑝₀ is an optimal parameter and 𝑝₀ is characterized by the following inequality 𝐽^′𝑝₀𝑝−𝑝₀≥0,∀𝑝∈𝑃_𝑎𝑑. (3.24) 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 Theorem 3.5 be satisfied and let the observation operator Λ be independent of 𝑝. Then the optimal parameter 𝑝₀ that minimizes 𝐽(𝑝) over 𝑃_𝑎𝑑 is characterized by the following optimization system: 𝑑𝑦₀+𝐴𝑝₀+𝐵𝑝₀𝑦₀𝑑𝑡=𝑓𝑝₀𝑑𝑡+𝜎𝑝₀𝑑𝑤_𝑡,𝑦₀𝐴𝑝=𝜑+𝜉,−𝑑𝑧+₀+𝐵𝑝₀^∗𝑧𝑑𝑡=Λ^∗Λ𝑦−𝑔𝑑𝑡,𝑧𝑡_𝑓1=0,2𝐽^′𝑝₀𝑝−𝑝₀=𝐸_𝑔𝑇𝑓^′_𝑝ℎ−𝐵^′_𝑝ℎ𝑦₀,𝑧_𝐻−𝐴^′_𝑝ℎ𝑦₀,𝑧_𝑣′,𝑣𝑑𝑡≥0, (3.25)(3.26)(3.27)𝑝∈𝑃_𝑎𝑑, where ℎ=𝑝−𝑝₀.

Proof. Firstly, using the flow of time reversed (changing 𝑡 to 𝑡_𝑓−𝑡) according to Lemma 2.1, it is easy to show the problem (3.26) is well posed. By Theorems 3.5 and 3.610≤2𝐽^′𝑝₀𝑝−𝑝₀=𝐸_𝑔𝑇Λ𝑦^′𝑡;𝑝₀ℎ,Λ𝑦𝑡;𝑝₀−𝑔(𝑡)_𝐾𝑑𝑡=𝐸_𝑔𝑇𝑦^′𝑡;𝑝₀ℎ,Λ^∗Λ𝑦−𝑔_𝑉,𝑉′𝑑𝑡,𝑝∈𝑃_𝑎𝑑. (3.28) Setting 𝑦̇=𝑦^′(𝑡,𝑝₀)ℎ and using (3.26) and (2.43), it has 0≤𝐽^′𝑝₀ℎ=𝐸_𝑔𝑇𝑦,̇−𝑑𝑧+(𝐴+𝐵)^∗𝑧𝑑𝑡_𝑉,𝑉′=𝐸_𝑔𝑇𝑑𝑦̇+(𝐴+𝐵)𝑦𝑑̇𝑡,𝑧_𝑉′,𝑉=𝐸_𝑔𝑇𝑓^′_𝑝ℎ−𝐴^′_𝑝ℎ𝑦₀−𝐵^′_𝑝ℎ𝑦₀𝑑𝑡+𝜎^′_𝑝ℎ𝑑𝑤,𝑧_𝑉′𝑉=𝐸_𝑔𝑇𝑓^′_𝑝ℎ−𝐵^′_𝑝ℎ𝑦₀,𝑧_𝐻−𝐴^′_𝑝ℎ𝑦₀,𝑧_𝑉′,𝑉𝑑𝑡. (3.29)

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: 𝑑_𝑡𝑢(𝑥,𝑡)=^𝑛_𝑖,𝑗=1𝜕_𝑖𝑎_𝑖𝑗(𝑥)𝜕_𝑗𝑢𝑑𝑡−𝑐(𝑥,𝑡)𝑢𝑑𝑡+𝑓(𝑥,𝑡)𝑑𝑡+𝜎(𝑥,𝑡)𝑑𝑤_𝑡,(𝑥,𝑡)∈𝐷×𝑇,𝑢∣_𝜕𝐷=0,𝑢∣_𝑡=𝑡0=𝜑(𝑥)+𝜉(𝑥),𝑥∈𝐷, (4.1) where 𝑎_𝑖𝑗∈𝑊^𝑚,2(𝐷) with 𝑚>𝑛/2, 𝑐∈𝑊^𝑚,2(𝐷×𝑇), 𝑓∈𝐿²(Ω×𝐷×𝑇), 𝜎(⋅,𝑡)∈ℒ(𝑊;𝐿²(𝐷)), for all 𝑡∈𝑇, with ∫_𝑇‖𝜎(⋅,𝑡)‖²_𝐿2(𝐷)𝑑𝑡<∞, 𝜑∈𝐿²(𝐷), 𝜉∈𝐿²(Ω×𝐷), 𝑤_𝑡=𝑤(𝑡) is a Wiener process in 𝑊.

The problem we shall deal with is to determine the unknown coefficients 𝑎_𝑖𝑗(𝑖,𝑗=1,…,𝑛) and 𝑐 based on the measurement 𝑧(𝑥)=𝑢𝑥,𝑡_𝑓,𝑥∈𝐷. (4.2)

Suppose 𝐻=𝐿²(𝐷) and 𝑉=𝐻¹₀(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,…,𝑎_𝑛𝑛,𝑐) and define the parameter space by 𝑃=^𝑛_𝑖,𝑗=1𝑊^𝑚,2(𝐷)×𝑊^𝑚,2(𝐷×𝑇), (4.3) with the norm 𝑝²_𝑃=^𝑛_𝑖,𝑗=1𝑎_𝑖,𝑗²_{𝑊𝑚,2(𝐷)}+‖𝑐‖²_{𝑊𝑚,2()𝐷×𝑇}, (4.4) then 𝑃 is a Hilbert space.

In order to make sense of the problem (4.1), it assumes the admissible parameter set 𝑃_𝑎𝑑 as follows: 𝑃_𝑎𝑑≡𝑝∈𝑃;𝛾𝑦²≤^𝑛_𝑖,𝑗=1𝑎_𝑖,𝑗(𝑥)𝑦_𝑖𝑦_𝑗≤𝛿𝑦²,∀𝑦=𝑦₁,…,𝑦_𝑛∈ℝ^𝑛,a.e.𝑥∈𝐷, (4.5) where 𝛾 and 𝛿(0<𝛾≤𝛿) are given constants.

Obviously, 𝑃_𝑎𝑑 is a convex and closed set in 𝑃. By the Sobolev imbedding theorem (see Chapter [15], 6), the imbedding 𝑊^𝑚,2(𝐷)→𝐶^𝜆𝐷⊂𝐶𝐷 (4.6) is compact, where 𝐶^𝜆(𝐷) is the Hölder space and 𝜆∈(0,𝑚−𝑛/2).

Next, define the operator 𝐴≡−^𝑛_𝑖,𝑗=1𝜕_𝑖𝑎_𝑖,𝑗𝜕_𝑗⋅,𝐵(𝑡)≡𝑐(𝑡), (4.7) 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 𝐴𝑦,𝑧_𝑉′,𝑉=−^𝑛_𝑖,𝑗=1𝜕_𝑖𝑎_𝑖,𝑗𝜕_𝑗𝑦,𝑧_𝑉′,𝑉=^𝑛_𝑖,𝑗=1𝑎_𝑖,𝑗𝜕_𝑗𝑦,𝜕_𝑖𝑧_𝐻≤max_𝑖,𝑗max_𝑥∈𝐷𝑎_𝑖,𝑗(𝑥)∇𝑦_𝐿2‖∇𝑧‖_𝐿2≤𝑐₁𝑦_𝑉‖𝑧‖_𝑉. (4.8) Due to 𝑝∈𝑃_𝑎𝑑, it has 𝐴𝑦,𝑦_𝑉′𝑉=^𝑛_𝑖,𝑗=1𝑎_𝑖,𝑗𝜕_𝑗𝑦,𝜕_𝑖𝑦_𝐻≥𝛾∇𝑦²_𝐿2. (4.9) By the Poincaré inequality it has 𝐴𝑦,𝑦_𝑉′,𝑉≥𝑐₂𝑦²_𝑉, (4.10) which, along with (4.8), shows that 𝐴 satisfies (1.5).

Next, for all 𝑧∈𝑉‖𝐵(𝑡)𝑧‖²_𝐻=_𝐷|𝑐(𝑥,𝑡)𝑧(𝑥)|²𝑑𝑥≤‖𝑐‖²_𝐿^∞_{()𝐷×𝑇𝐷}|𝑧(𝑥)|²𝑑𝑥≤𝑐₀‖𝑧‖²_𝐻≤𝑐₁‖𝑧‖²_𝑉. (4.11) Therefore, 𝐵(𝑡)∈ℒ(𝑉,𝐻), for all 𝑡∈𝑇.

Finally, set 𝑢(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. If 𝑓∈𝐿²(Ω×𝐷×𝑇), then (4.1) has a unique solution 𝑢(𝑝)=𝑢(𝑥,𝑡;𝑝):
𝑢∈𝐿²(Ω;𝐶(𝑇;𝐻)∩𝐿²(Ω×𝑇;𝑉)∩𝐶(𝑇;𝐿²(Ω;𝐻))≡𝔖,
𝑢(⋅,𝑡) is adapted to ℱ^𝑡 (as a process with values in 𝐻),
𝑢(⋅,𝑡) is measurable with values in 𝑉.

Theorem 4.2. If 𝑓∈𝐿²(Ω×𝐷×𝑇), then for any 𝑝∈𝑃_𝑎𝑑, the mapping 𝑢(𝑝) is continuously Fréchet differentiable and its Fréchet differential 𝑢^′(𝑝)ℎ=𝑢̇(ℎ∈𝑃) is determined by the following system: 𝑑_𝑡𝑢̇−_𝑖,𝑗𝜕_𝑖𝑎_𝑖𝑗𝜕_𝑗𝑢̇𝑑𝑡+𝑐𝑢̇𝑑𝑡=_𝑖,𝑗𝜕_𝑖Δ𝑎_𝑖,𝑗𝜕_𝑗𝑢̇𝑑𝑡−Δ𝑐𝑢𝑑𝑡,𝑢∣_𝜕Ω̇=0,𝑢∣_𝑡=𝑡0=0, (4.12) where ℎ=(Δ𝑎₁₁,…,Δ𝑎_𝑛𝑛,Δ𝑐)∈𝑃 and 𝑢 is determined by (4.1). Furthermore, the mapping 𝑢(𝑝) is infinitely Fréchet differentiable.

Because 𝑢∈𝐶(𝑇;𝐿²(Ω;𝐻)), 𝑢(⋅,𝑡_𝑓)∈𝐿²(Ω;𝐻). So, we can use the following cost functional: 𝐽𝑝=𝐸_𝐷𝑢𝑥,𝑡_𝑓;𝑝−𝑧(𝑥)²𝑑𝑥,𝑝∈𝑃_𝑎𝑑, (4.13) in order to determine 𝑝. The operator ΛΛ𝑢=𝑢⋅,𝑡_𝑓,𝑥∈𝐷,Λ∶𝐶𝑇;𝐿²(Ω;𝐻)→𝐿²(Ω;𝐻) (4.14) obviously satisfies Λ∈ℒ(𝔖;𝐿²(Ω;𝐻)).

Theorem 4.3. The mapping 𝐽(𝑝) defined by (4.13) is continuously Fréchet differentiable and its Fréchet differential is determined by 𝐽^′𝑝ℎ=−2𝐸_𝐷𝑇^𝑛_𝑖,𝑗=1Δ𝑎_𝑖,𝑗𝜕_𝑖𝑣𝜕_𝑗=𝑢+Δ𝑐𝑢𝑣𝑑𝑡𝑑𝑥grad𝐽,ℎ_𝑃′,𝑃, (4.15) where ℎ=(Δ𝑎₁₁,…,Δ𝑎_𝑛𝑛,Δ𝑐)∈𝑃, the gradient operator is grad𝐽=−2𝐸_𝑇𝜕₁𝑣𝜕₁𝑢,𝜕₁𝑣𝜕₂𝑢,…,𝜕_𝑛𝑣𝜕_𝑛𝑢𝑑𝑡∈𝑃^′, (4.16)𝑢=𝑢(𝑝) is the solution of (4.1), and 𝑣=𝑣(𝑝) is defined by the following system: −𝑑_𝑡𝑣=^𝑛_𝑖,𝑗𝜕_𝑗𝑎_𝑖𝑗𝜕_𝑖𝑣𝑑𝑡−𝑐𝑣𝑑𝑡,(𝑥,𝑡)∈𝐷×𝑇,𝑣∣_𝜕𝐷=0,𝑣∣_{𝑡=𝑡𝑓}=𝑢∣_{𝑡=𝑡𝑓}−𝑧. (4.17)

4.2. Example 2

Consider the following system: 𝑑_𝑡𝑢−Δ𝑢𝑑𝑡=^𝑚_𝑖=1𝑎_𝑖𝑓_𝑖(𝑥,𝑡)𝑑𝑡+𝜎(𝑥,𝑡)𝑑𝑤_𝑡,(𝑥,𝑡)∈𝐷×𝑇,𝑢∣_𝜕𝐷=0,𝑡∈𝑇,𝑢∣_𝑡=𝑡0=𝜑(𝑥)+𝜁(𝑥),𝑥∈𝐷, (4.18) where 𝐷⊂ℝ^𝑛 with 𝑛≤3.

The problem addressed is to identify the unknown parameter 𝑝=(𝑎₁,…,𝑎_𝑚)∈ℝ^𝑚, which varies in an admissible parameter set 𝑃_𝑎𝑑≡{𝑝∈ℝ^𝑚;0≤𝛾≤‖𝑝‖_𝑅𝑚≤𝛿} based on the approximate point measurement: 𝑧_𝑖(𝑥,𝑡)=𝑢(𝑥,𝑡),a.e.𝑥∈𝐷_𝑖,∀𝑡∈𝑇,𝑖=1,…,𝐿, (4.19) where 𝐷_𝑖 is the 𝜀-neighbourhood of 𝑥_𝑖∈𝐷 and 𝐷_𝑖∩𝐷_𝑗=∅(𝑖≠𝑗).

In order to make sense of (4.19), it assumes 𝑓_𝑖∈𝐿²𝑇;𝑊^1,2(𝐷𝜕),𝑖=1,…,𝑚,_𝑗𝜎(𝑡)∈ℒ(𝑊,𝐻),_𝑇𝜕_𝑗𝜎(𝑡)²_{ℒ()𝑊,𝐻}𝑑𝑡<∞,𝑗=1,...,𝑛. (4.20) It is easy to prove that 𝑢(𝑡)∈𝑊^2,2(𝐷)∩𝐻¹₀(𝐷). Using the Sobolev embedding theorem [15], it has 𝑊^2,2(𝐷)⊂𝐶(𝐷). So (4.19) is sensible.

Introduce the characteristic function of 𝐷_𝑖: 𝜒_𝐷𝑖(𝑥)=1,𝑥∈𝐷_𝑖,0,otherwise, (4.21) and the cost functional min_{𝑝∈𝑃𝑎𝑑}𝐽𝑝≡^𝐿_𝑖=1𝐸_𝐷𝑇𝜒_𝐷𝑖(𝑥)𝑢𝑥,𝑡;𝑝−𝑧_𝑖(𝑥,𝑡)²𝑑𝑡𝑑𝑥. (4.22) 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, 𝑢̇=𝑢^′(𝑝)ℎ,ℎ=(Δ𝑎₁,…,Δ𝑎_𝑚)∈ℝ^𝑚, is determined by the following: 𝑑_𝑡𝑢̇−Δ𝑢̇𝑑𝑡=^𝑚_𝑖=1Δ𝑎_𝑖𝑓_𝑖()̇𝑑𝑡,𝑥,𝑡∈𝐷×𝑇,𝑢∣_𝜕𝐷̇=0,𝑢∣_𝑡=𝑡0=0. (4.23) Furthermore, it can also be proved that the functional 𝐽(𝑝) is continuously Fréchet differentiable and its Fréchet differential is 𝐽^′𝑝ℎ=2^𝐿_𝑖=1𝐸_𝑇𝐷𝜒_𝐷𝑖(𝑥)𝑢𝑥,𝑡;𝑝−𝑧_𝑖(𝑥,𝑡)𝑢̇𝑥,𝑡;𝑝𝑑𝑥𝑑𝑡. (4.24) Now, the main results are given as followings.

Theorem 4.4. The functional 𝐽(𝑝) is continuously Fréchet differentiable and its Fréchet differentiable is 𝐽^′𝑝ℎ=grad𝐽,ℎ_ℝ𝑚, (4.25) where grad𝐽 is the gradient of 𝐽(𝑝) determined by the following: grad𝐽=𝐸_𝑇𝐷𝑣𝑓₁𝑑𝑥𝑑𝑡,…,𝐸_𝑇𝐷𝑣𝑓_𝑚𝑑𝑥𝑑𝑡∈ℝ^𝑚, (4.26) where 𝑣 is the solution to the following initial-boundary problem: −𝑑_𝑡𝑣−Δ𝑣𝑑𝑡=2^𝐿_𝑖=1𝜒_𝐷𝑖(𝑥)𝑢𝑥,𝑡;𝑝−𝑧(𝑥,𝑡)𝑑𝑡,(𝑥,𝑡)∈𝐷×𝑇,𝑣∣_𝜕Ω=0,𝑣∣_{𝑡=𝑡𝑓}=0 (4.27) 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 𝐽^′𝑝ℎ=𝐸_𝑇𝐷−𝑑_𝑡𝑣−Δ𝑣𝑑𝑡𝑢̇𝑑𝑥=𝐸_𝑇𝐷𝑣𝑑_𝑡𝑢̇−Δ𝑢̇𝑑𝑡𝑑𝑥=𝐸_𝑇𝐷^𝑚_𝑖=1𝑣Δ𝑎_𝑖𝑓_𝑖=𝑑𝑥𝑑𝑡_𝑚𝑖=1Δ𝑎_𝑖𝐸_𝑇𝐷𝑣𝑓_𝑖𝑑𝑥𝑑𝑡=grad𝐽,ℎ_ℝ𝑚. (4.28)

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).