Parameter Identification Problem for the Kirchhoff-Type Equation with Viscosity

Abstract

The constant parameter identification problem in the Kirchhoff-type equation with viscosity is studied. The problem is formulated by a minimization of quadratic cost functionals by distributive measurements. The existence of optimal parameters and necessary optimality conditions for the parameters are proved.

1. Introduction

The model of transversal vibration of a string has long history starting from D’ Alembert and Euler. It is widely regarded that the model proposed by D’ Alembert is simple and elementary model describing small transversal vibration of a string in which the effect of elasticity is not considered.

When we take into account the change of length of a string in its small vibration mainly due to the effect of elasticity, the classical model from D’ Alembert is no more correct to cover the more realistic phenomena.

More accurate or appropriate model for the transversal vibration of an elastic string, given by𝜕²𝑦𝜕𝑡²−1𝜌𝑃₀ℎ+𝐸2𝐿^𝐿₀𝜕𝑦𝜕𝑥²𝜕𝑑𝑥²𝑦𝜕𝑥²=0, (1.1) has been proposed by Kirchhoff [1]. Here 𝐿 is the length of the string, ℎ is the area of the cross section, 𝜌 is the mass density, 𝑃₀ is the initial tension, and 𝐸 is the Young’s modulus of a material. For the derivations of (1.1), we can refer to the article of Ferrel and Medeiros [2].

As a general form of (1.1), we consider the following damped equation with appropriate boundary and initial conditions: 𝜕²𝑦𝜕𝑡²−𝑚_Ω∇𝑦²𝑑𝑥Δ𝑦−𝛾𝜕𝑦𝜕𝑡=𝑓, (1.2) where Ω is a smooth domain in 𝐑^𝑛,𝛾>0. Many researches have been devoted to the study of (1.2) for both damped (𝛾>0) or undamped (𝛾=0) cases, see Arosio [3], Spagnolo [4], Pohožaev [5], Lions [6], Nishihara, and Yamada [7] and their long roll of bibliographical references. Those researches are mainly concerned with the well-posedness of solutions in global or local sense under the various data conditions and their decays.

Especially when we take into account the viscosity effect of its vibration due to its inner friction, the damping coefficient 𝛾 in (1.2) is replaced by 𝛾Δ. In this case we can refer to Cavalcanti et al. [8] to show the well-posedness in the Hadamard sense under the data condition (𝑦(0,𝑥),𝜕𝑦(0,𝑥)/𝜕𝑡,𝑓)∈𝐷(Δ)×𝐻¹₀(Ω)×𝐿²(0,𝑇;𝐿²(Ω)). Making use of this result, we are going to study the constant identification problem in the equation of Kirchhoff-type equation with viscosity as follows: 𝜕²𝑦𝜕𝑡²−𝛼+𝛽_Ω∇𝑦²𝑑𝑥Δ𝑦−𝛾Δ𝜕𝑦𝑦()𝜕𝑡=𝑓in𝑄,𝑦=0onΣ,0,𝑥=𝑦₀(𝑥),𝜕𝑦()𝜕𝑡0,𝑥=𝑦₁(𝑥)inΩ, (1.3) where 𝑄=(0,𝑇)×Ω,Σ=(0,𝑇)×𝜕Ω. Here the constants 𝛼 and 𝛽 are physical constants explained above, and 𝛾 stands for the rate of viscosity.

Recently, Hwang and Nakagiri [9] studied optimal control problems for (1.3) under the framework of Lions [10]. And Hwang [11] studied constant parameter identification for the problem of an extensible beam equation. In this paper we will study constant parameter identification problems for (1.3) in the following way.

At first, we assume that the desired state is known, but constant parameters 𝛼,𝛽,𝛾 involved in the above equation are unknown. For more details, we refer to Ha and Nakagiri [12], Hwang and Nakagiri [13]. We show the existence of an optimal parameters in an admissible set and its characterizations, namely, a parameter identification problem in which we use the term optimal parameter to denote the best parameter within any admissible set for which the solution of (1.3) gives a minimum of the given functional. We take this functional by 𝐿²-quadratic norm of observed state minus desired state that is usually regarded as a cost function in optimal control theory.

In this paper we pursue to find necessary conditions for an optimal parameters by using Gâteaux differentiability of the solution mapping and giving variational inequality via an adjoint equation. Proceeding in this way, we can obtain similar results with optimal control problems due to Lions [10]. For more detailed study, we refer to Ahmed [14] for abstract evolution equations.

We explain our identification problem precisely as follows. At first, in order to study parameter identification problem in the framework of optimal control theory due to Lions [10], we need to modify the positive constants, 𝛼,𝛽,𝛾 in (1.3) by 𝛼₀+𝛼,𝛽,𝛾₀+𝛾, respectively, where 𝛼₀ and 𝛾₀ are fixed positive constants, and 𝛼,𝛽,𝛾 are nonnegative constants. Therefore, we take the set 𝒫={(𝑥₁,𝑥₂,𝑥₃)∣𝑥_𝑖≥0, 𝑖=1,2,3} as the set of parameters (𝛼,𝛽,𝛾) in (1.3). By doing this, we can guarantee the well-posedness of (1.3) in verifying the Gâteaux differentiability of the solution mapping from the set of parameters to the corresponding solution space of (1.3).

Let 𝑦(𝑞)=𝑦(𝑞;𝑡,𝑥) be the solution for a given 𝑞=(𝛼,𝛽,𝛾)∈𝒫 and 𝒫_ad⊂𝒫 be an admissible parameter set. We consider the following two quadratic distributive functionals: 𝐽₁𝑞=^𝑇_0Ω𝑦𝑞;𝑡,𝑥−𝑌₁(𝑡,𝑥)²𝑑𝑥𝑑𝑡+_Ω𝑦𝑞;𝑇,𝑥−𝑌^𝑇₁(𝑥)²𝐽𝑑𝑥,₂𝑞=^𝑇_0Ω𝜕𝑦𝑞;𝑡,𝑥𝜕𝑡−𝑌₂(𝑡,𝑥)²𝑑𝑥𝑑𝑡for𝑞∈𝒫_ad, (1.4)(1.5) where 𝑌_𝑖∈𝐿²(𝑄),𝑖=1,2 and 𝑌^𝑇₁∈𝐿²(Ω) are the desired values.

The parameter identification problem for (1.3) with the cost 𝐽=𝐽₁ in (1.4) or 𝐽=𝐽₂ in (1.5) is to find and characterize an optimal parameters 𝑞^∗=(𝛼^∗,𝛽^∗,𝛾^∗)∈𝒫_ad satisfying that 𝐽𝑞^∗=inf𝐽𝑞∶𝑞∈𝒫_ad. (1.6) We prove the existence of an optimal parameter 𝑞^∗ by using the continuity of solutions on parameters and establish the necessary optimality conditions by introducing appropriate adjoint systems for which we prove the strong Gâteaux differentiability of the nonlinear mapping 𝑞→𝑦(𝑞).

Another novelty of this paper is that the first-order Volterra integrodifferential equation is utilized as a proper adjoint system to establish the necessary optimality condition of the velocity’s measurement case (1.5) as in [9, 13].

2. Preliminaries

We consider the following Dirichlet boundary value problem for Kirchhoff-type equation with damping term: 𝜕²𝑦𝜕𝑡²−𝛼+𝛽_Ω∇𝑦²𝑑𝑥Δ𝑦−𝛾Δ𝜕𝑦𝑦()𝜕𝑡=𝑓in𝑄,𝑦=0onΣ,0,𝑥=𝑦₀(𝑥),𝜕𝑦()𝜕𝑡0,𝑥=𝑦₁(𝑥)inΩ, (2.1) where 𝑓 is a forcing function, 𝑦₀ and 𝑦₁ are initial data, and 𝛼,𝛽,𝛾>0 are some physical constants. In this paper we study (2.1) in the class of strong solutions. For the purpose we suppose that 𝑓∈𝐿²(0,𝑇;𝐿²(Ω)), 𝑦₀∈𝐷(Δ)=𝐻²(Ω)∩𝐻¹₀(Ω), and 𝑦₁∈𝐻¹₀(Ω). The solution space 𝑆(0,𝑇) which is the space of strong solutions of (2.1) is defined by 𝑆(0,𝑇)=𝑔∣𝑔∈𝐿²(0,𝑇;𝐷(Δ)),𝑔^′∈𝐿²(0,𝑇;𝐷(Δ)),𝑔^′′∈𝐿²0,𝑇;𝐿²(Ω), (2.2) endowed with the norm 𝑔_𝑆(0,𝑇)=𝑔²_{𝐿2((Δ0,𝑇;𝐷))}+𝑔^′2_{𝐿2((Δ0,𝑇;𝐷))}+𝑔^′′2_{𝐿20,𝑇;𝐿2(Ω)}^1/2. (2.3) Here, 𝑔^′ and 𝑔^′′ denote the first- and second-order distributional derivatives of 𝑔. The scalar products and norms on 𝐿²(Ω) and 𝐻¹₀(Ω) are denoted by (𝜙,𝜓)₂, |𝜙|₂ and (𝜙,𝜓)_𝐻¹_0(Ω), ‖𝜙‖, respectively. The scalar product and norm on [𝐿²(Ω)]^𝑛 are also denoted by (𝜙,𝜓)₂ and |𝜙|₂. Then, the scalar product (𝜙,𝜓)_𝐻¹_0(Ω) and the norm ‖𝜙‖ of 𝐻¹₀(Ω) are given by (∇𝜙,∇𝜓)₂ and ‖𝜙‖=|∇𝜙|₂, respectively. Finally the norm and the scalar product on 𝐷(Δ) are given by (Δ𝜙,Δ𝜓)₂ and ‖𝜙‖_𝐷(Δ)=|Δ𝜙|₂, respectively. The duality pairing between 𝐻¹₀(Ω) and 𝐻⁻¹(Ω) is denoted by ⟨𝜙,𝜓⟩.

Definition 2.1. A function 𝑦 is said to be a strong solution of (2.1) if 𝑦∈𝑆(0,𝑇) and 𝑦 satisfies 𝑦^′′(𝑡)−𝛼+𝛽∇𝑦(𝑡)²₂Δ𝑦(𝑡)−𝛾Δ𝑦^′(𝑡)(𝑡)[],𝑦(0)=𝑓,a.e.𝑡∈0,𝑇=𝑦₀,𝑦^′(0)=𝑦₁. (2.4)

We remark here that 𝑆(0,𝑇) is continuously imbedded in 𝐶([0,𝑇];𝐷(Δ))∩𝐶¹([0,𝑇];𝐻¹₀(Ω)) (cf. Dautray and Lions [15, page 555]).

The following variational formulation is used to define the weak solution of (2.1).

Definition 2.2. A function 𝑦 is said to be a weak solution of (2.1) if 𝑦∈𝑊(0,𝑇)≡{𝑔∣𝑔∈𝐿²(0,𝑇;𝐻¹₀(Ω)),𝑔′∈𝐿²(0,𝑇;𝐻¹₀(Ω)),𝑔^′′∈𝐿²(0,𝑇;𝐻⁻¹(Ω))}, and 𝑦 satisfies 𝑦^′′(⋅),𝜙+𝛼+𝛽∇𝑦(⋅)²₂∇𝑦(⋅),∇𝜙+𝛾∇𝑦^′(⋅)=𝑓(⋅),∇𝜙,𝜙∀𝜙∈𝐻¹₀(Ω)inthesenseof𝐷^′(),𝑦(0)0,𝑇=𝑦₀,𝑦^′(0)=𝑦₁. (2.5)

In order to verify the well-posedness of (2.1), we refer to the results in [8, 9]. The well-posedness in the sense of Hadamard can be given as follows.

Theorem 2.3. Assume that 𝑓∈𝐿²(0,𝑇;𝐿²(Ω)) and 𝑦₀∈𝐷(Δ),𝑦₁∈𝐻¹₀(Ω). Then the problem (2.1) has a unique strong solution 𝑦 in 𝑆(0,𝑇). And the solution mapping 𝑝=(𝑦₀,𝑦₁,𝑓)→𝑦(𝑝) of 𝑃≡𝐷(Δ)×𝐻¹₀(Ω)×𝐿²(0,𝑇;𝐿²(Ω)) into 𝑆(0,𝑇) is strongly continuous. Further, for each 𝑝₁=(𝑦¹₀,𝑦¹₁,𝑓₁)∈𝑃 and 𝑝₂=(𝑦²₀,𝑦²₁,𝑓₂)∈𝑃, we have the following inequality: ∇𝑦^′𝑝₁;𝑡−𝑦^′𝑝₂;𝑡²₂+Δ𝑦𝑝₁;𝑡−𝑦𝑝₂;𝑡²₂+^𝑡₀Δ𝑦^′𝑝₁;𝑠−𝑦^′𝑝₂;𝑠²₂Δ𝑦𝑑𝑠≤𝐶¹₀−𝑦²₀²₂+∇𝑦¹₁−𝑦²₁²₂+𝑓₁−𝑓₂²_{𝐿20,𝑇;𝐿2(Ω)}, (2.6) where 𝐶 is a constant and 𝑡∈[0,𝑇].

Proof (see Hwang and Nakagiri [9]). We will omit writing the integral variables in the definite integral without any confusion. For example, in (2.6), we will write ∫^𝑡₀|∇𝑦′(𝑝₁)|²𝑑𝑠 instead of ∫^𝑡₀|∇𝑦′(𝑝₁;𝑠)|²𝑑𝑠.

3. Identification Problems

In this section we study the identification problem for the unknown parameters 𝑞=(𝛼,𝛽,𝛾)∈𝒫 in the problem 𝜕²𝑦𝜕𝑡²−𝛼₀+𝛼+𝛽∇𝑦²₂Δ𝑦−𝛾₀+𝛾Δ𝜕𝑦𝑦()𝜕𝑡=𝑓in𝑄,𝑦=0onΣ,0,𝑥=𝑦₀(𝑥),𝜕𝑦()𝜕𝑡0,𝑥=𝑦₁(𝑥)inΩ, (3.1) where 𝛼₀,𝛾₀>0, 𝑦₀∈𝐷(Δ), 𝑦₁∈𝐻¹₀(Ω), and 𝑓∈𝐿²(0,𝑇;𝐿²(Ω)) are fixed. The physical constants 𝑞=(𝛼,𝛽,𝛾) in (3.1) are an unknown parameter that should be identified. In this setting we take 𝒫={(𝑥₁,𝑥₂,𝑥₃)∣𝑥_𝑖≥0, 𝑖=1,2,3} to be the space of parameters 𝑞=(𝛼,𝛽,𝛾) with the Euclidian norm. By Theorem 2.3 we have that for each 𝑞∈𝒫 there exists a unique solution 𝑦=𝑦(𝑞)∈𝑆(0,𝑇) of (3.1).

At first we show the continuous dependence of solutions on parameters 𝑞=(𝛼,𝛽,𝛾).

Theorem 3.1. The solution map 𝑞→𝑦(𝑞) from 𝒫={(𝑥₁,𝑥₂,𝑥₃)∣𝑥_𝑖≥0, 𝑖=1,2,3} into 𝑆(0,𝑇) is continuous.

Proof. Let 𝑞=(𝛼,𝛽,𝛾) be arbitrarily fixed. Suppose that 𝑞_𝑚=(𝛼_𝑚,𝛽_𝑚,𝛾_𝑚)→𝑞=(𝛼,𝛽,𝛾) in 𝒫. Let 𝑦_𝑚=𝑦(𝑞_𝑚) and 𝑦=𝑦(𝑞) be the solutions of (3.1) for 𝑞=𝑞_𝑚 and for 𝑞, respectively. Since {𝑞_𝑚} is bounded in 𝒫, by Theorem 2.3, we see that Δ𝑦^′_𝑚(𝑡)²₂+Δ𝑦_𝑚(𝑡)²₂+^𝑡₀Δ𝑦^′_𝑚²₂𝑑𝑠≤𝐶₀<∞,∀𝑡∈[0,𝑇], (3.2) where 𝐶₀>0 is a constant depending only on 𝛼₀,𝛽₀,𝛾₀,𝑦₀,𝑦₁, and 𝑓. Applying (3.2) to (3.1), we can deduce by choosing appropriate subsequence of {𝑦_𝑚} denoted again by {𝑦_𝑚} that 𝑦_𝑚⟶𝑦weaklyin𝑆(0,𝑇)as𝑚⟶∞. (3.3) Since 𝐷(Δ)↪𝐻¹₀(Ω) is compact, we can deduce from [16, pages 273–278] that the space 𝑆(0,𝑇) is compactly imbedded in 𝐿²(0,𝑇;𝐻¹₀(Ω)). Therefore, we can take a subsequence {𝑦_𝑚𝑘} of {𝑦_𝑚}, if necessary, such that 𝑦_𝑚𝑘⟶𝑦stronglyin𝐿²0,𝑇;𝐻¹₀(Ω)as𝑘⟶∞. (3.4) Equation (3.4) implies that ∇𝑦_𝑚𝑘(⋅)²₂⟶∇𝑦(⋅)²₂a.e.in(0,𝑇)as𝑘⟶∞. (3.5) Taking into account (3.3) and (3.5) and coming back to (3.1), we deduce that 𝑦 is the solution of (3.1) corresponding to the parameter 𝑞.
In order to obtain strong convergency, we set 𝜓_𝑚=𝑦_𝑚−𝑦. Then, in weak sense, 𝜓_𝑚 satisfies 𝜕²𝜓_𝑚𝜕𝑡²−𝛼₀+𝛼_𝑚+𝛽_𝑚∇𝑦_𝑚²₂Δ𝜓_𝑚−𝛾₀+𝛾_𝑚Δ𝜕𝜓_𝑚𝜕𝑡=ℱ_𝑚𝜓in𝑄,_𝑚𝜓=0onΣ,_𝑚(0,𝑥)=0,𝜕𝜓_𝑚()𝜕𝑡0,𝑥=0inΩ, (3.6) where ℱ_𝑚=𝛼_𝑚−𝛼+𝛽_𝑚∇𝑦_𝑚²₂−𝛽∇𝑦²₂Δ𝑦−𝛾−𝛾_𝑚Δ𝜕𝑦.𝜕𝑡 (3.7) Using (3.5) and the Lebesgue-dominated convergence theorem, we can verify that ℱ_𝑚⟶0in𝐿²0,𝑇;𝐿²(Ω)as𝑚→∞. (3.8) Multiplying (3.6) by Δ𝜓^′_𝑚, and integrating it over Ω×[0,𝑡], we have ∇𝜓^′_𝑚(𝑡)²₂+𝛼₀+𝛼_𝑚+𝛽_𝑚∇𝑦_𝑚(𝑡)²₂Δ𝜓_𝑚(𝑡)²₂+2𝛾₀+𝛾_𝑚^𝑡₀Δ𝜓^′_𝑚²₂𝑑𝑠=−2^𝑡₀ℱ_𝑚,Δ𝜓^′_𝑚2𝑑𝑠+^𝑡₀2𝛽_𝑚∇𝑦_𝑚,∇𝑦^′_𝑚2Δ𝜓_𝑚²₂𝑑𝑠. (3.9) By the Cauchy-Schwarz inequality and the fact that 𝑦_𝑚∈𝑆(0,𝑇)↪𝐶([0,𝑇];𝐷(Δ))∩𝐶¹([0,𝑇];𝐻¹₀(Ω)), we have following inequalities: −2^𝑡₀ℱ_𝑚,∇𝜓^′_𝑚2𝑑𝑠≤2^𝑡₀ℱ_𝑚2Δ𝜓^′_𝑚2≤1𝑑𝑠𝛾₀ℱ_𝑚²_{𝐿20,𝑇;𝐿2(Ω)}+𝛾₀^𝑡₀Δ𝜓^′_𝑚²₂𝑑𝑠,_𝑡02𝛽_𝑚∇𝑦_𝑚,∇𝑦^′_𝑚2Δ𝜓_𝑚²₂𝑑𝑠≤2𝛽_𝑚^𝑡₀∇𝑦_𝑚2∇𝑦^′_𝑚2Δ𝜓_𝑚²₂𝑑𝑠≤2𝛽_𝑚𝑦_{𝑚𝐶([0,𝑇];𝐻10(Ω))}𝑦^′_{𝑚𝐶([0,𝑇];𝐻10(Ω))}^𝑡₀Δ𝜓_𝑚²₂𝑑𝑠. (3.10) Then by (3.9) and (3.10), we can obtain ∇𝜓^′_𝑚(𝑡)²₂+Δ𝜓_𝑚(𝑡)²₂+^𝑡₀Δ𝜓^′_𝑚²₂𝑑𝑠≤𝐶ℱ_𝑚²_{𝐿20,𝑇;𝐿2(Ω)}+𝐶^𝑡₀Δ𝜓_𝑚²₂𝑑𝑠, (3.11) where 𝐶>0. Hence by applying Gronwall’s inequality to (3.11), we have ∇𝜓^′_𝑚(𝑡)²₂+Δ𝜓_𝑚(𝑡)²₂+^𝑡₀Δ𝜓^′_𝑚²₂𝑑𝑠≤𝐶exp(𝐶𝑇)ℱ_𝑚²_{𝐿2(0,𝑇;𝐿2(Ω))}. (3.12) Combining (3.8) and (3.12), we have 𝜓_𝑚](Δ,𝜓⟶0in𝐶([0,𝑇;𝐷))_′𝑚⟶0in𝐶[0,𝑇];𝐻¹₀(Ω)∩𝐿²(0,𝑇;𝐷(Δ)), (3.13) so that 𝑦_𝑚(⋅)⟶𝑦(⋅)stronglyin𝑆(0,𝑇). (3.14) This proves Theorem 3.1.

As explained before, we choose the 𝐿² objective costs to be minimized for the identification of 𝑞=(𝛼,𝛽,𝛾) which are given by 𝐽₁𝑞=𝑦(𝑞)−𝑌₁²_{𝐿20,𝑇;𝐿2(Ω)}+𝑦𝑞;𝑇−𝑌^𝑇₁²₂,𝐽₂𝑞=𝑦′(𝑞)−𝑌₂²_{𝐿20,𝑇;𝐿2(Ω)}for𝑞∈𝒫_ad, (3.15)(3.16) where 𝑌_𝑖∈𝐿²(0,𝑇;𝐿²(Ω)),𝑖=1,2, and 𝑌^𝑇₁∈𝐿²(Ω).

If 𝒫_ad is compact, then for the minimizing sequence {𝑞_𝑚} such as 𝐽(𝑞_𝑚)→𝐽^∗=inf{𝐽(𝑞)∶𝑞∈𝒫_ad} we can choose a subsequence {𝑞_𝑚𝑗} of {𝑞_𝑚} such that 𝑞_𝑚𝑗→𝑞^∗∈𝒫_ad and 𝑦(𝑞_𝑚𝑗)→𝑦(𝑞^∗) strongly in 𝑆(0,𝑇) by Theorem 3.1. Due to the continuous imbedding 𝑆(0,𝑇)↪𝐶([0,𝑇];𝐷(Δ))∩𝐶¹([0,𝑇];𝐻¹₀(Ω)) we have 𝐽^∗=𝐽(𝑞^∗) for the costs (3.15) and (3.16). Thus we have the following corollary.

Corollary 3.2. If 𝒫_ad is compact, then there exists at least one optimal parameter 𝑞^∗∈𝒫_ad for the cost 𝐽₁ in (3.15) or 𝐽₂ in (3.16).

Let the admissible set 𝒫_ad be compact and convex in 𝒫, and let 𝑞^∗=(𝛼^∗,𝛽^∗,𝛾^∗) be an optimal parameter on 𝒫_ad for the cost 𝐽(𝑞). As is well known the necessary optimality condition of an optimal parameter 𝑞^∗=(𝛼^∗,𝛽^∗,𝛾^∗) for the cost 𝐽 is given by 𝐷𝐽𝑞^∗𝑞−𝑞^∗≥0∀𝑞∈𝒫_ad, (3.17) where 𝐷𝐽(𝑞^∗) denotes the Gâteaux derivative of 𝐽(𝑞) at 𝑞=𝑞^∗.

The Gâteaux differentiability of the above quadratic costs 𝐽_𝑖(𝑞),𝑖=1,2 follows from that of the nonlinear solution mapping 𝑞→𝑦(𝑞) of 𝒫_ad into 𝑆(0,𝑇). The following theorem proves the Gâteaux differentiability of the nonlinear solution mapping 𝑞→𝑦(𝑞) and gives its characterization.

Theorem 3.3. The map 𝑞→𝑦(𝑞) of 𝒫_ad into 𝑆(0,𝑇) is Gâteaux differentiable at 𝑞=𝑞^∗ and such the Gâteaux derivative of 𝑦(𝑞) at 𝑞=𝑞^∗ in the direction 𝑞−𝑞^∗∈𝒫, say 𝑧=𝐷𝑦(𝑞^∗)(𝑞−𝑞^∗), is a unique solution of the following linear problem: 𝜕²𝑧𝜕𝑡²−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑧−2𝛽^∗∇𝑧,∇𝑦^∗₂Δ𝑦^∗−𝛾₀+𝛾^∗Δ𝜕𝑧𝜕𝑡=𝒢𝑞−𝑞^∗;𝑦^∗𝑧()in𝑄,𝑧=0𝑜𝑛Σ,0,𝑥=0,𝜕𝑧()𝜕𝑡0,𝑥=0inΩ, (3.18) where 𝑦^∗=𝑦(𝑞^∗) and 𝒢𝑞−𝑞^∗;𝑦^∗=𝛼−𝛼^∗Δ𝑦^∗+𝛽−𝛽^∗∇𝑦^∗2₂Δ𝑦^∗+𝛾−𝛾^∗Δ𝜕𝑦^∗.𝜕𝑡 (3.19)

Proof. Let 𝜆∈[0,1], and let 𝑦_𝜆 and 𝑦^∗ be the solutions of (3.1) corresponding to 𝑞^∗+𝜆(𝑞−𝑞^∗) and 𝑞^∗, respectively. We set 𝑧_𝜆=𝜆⁻¹(𝑦_𝜆−𝑦^∗),𝜆≠0. Then 𝑧_𝜆 satisfies the following problem in the weak sense: 𝜕²𝑧_𝜆𝜕𝑡²−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑧_𝜆−𝛽^∗∇𝑧_𝜆,∇𝑦_𝜆+∇𝑦^∗₂Δ𝑦_𝜆−𝛾₀+𝛾^∗Δ𝜕𝑧_𝜆𝜕𝑡=𝒢𝑞−𝑞^∗;𝑦_𝜆𝑧in𝑄,_𝜆𝑧=0onΣ,_𝜆(0,𝑥)=0,𝜕𝑧_𝜆()𝜕𝑡0,𝑥=0inΩ, (3.20) where 𝒢𝑞−𝑞^∗;𝑦_𝜆=𝛼−𝛼^∗Δ𝑦_𝜆+𝛽−𝛽^∗∇𝑦_𝜆²₂Δ𝑦_𝜆+𝛾−𝛾^∗Δ𝜕𝑦_𝜆.𝜕𝑡 (3.21) Since 𝑦_𝜆∈𝑆(0,𝑇) we can easily know that 𝒢(𝑞−𝑞^∗;𝑦_𝜆)∈𝐿²(0,𝑇;𝐿²(Ω)) and 𝒢𝑞−𝑞^∗;𝑦_{𝜆𝐿2(0,𝑇;𝐿2(Ω))}≤𝐶₀𝑞−𝑞^∗𝑦_{𝜆𝑆(0,𝑇)}<𝐶₁, (3.22) where 𝐶_𝑖,𝑖=0,1 are positive constants.
By similar arguments in the proof of Theorem 3.1, multiplying the both sides of (3.20) by −Δ𝑧^′_𝜆 and integrating it over Ω×[0,𝑡], we can obtain the following inequality: Δ𝑧_𝜆(𝑡)²₂+∇𝑧^′_𝜆(𝑡)²₂+^𝑡₀Δ𝑧^′_𝜆²₂𝑑𝑠≤𝐾𝒢𝑞−𝑞^∗;𝑦_𝜆²_{𝐿20,𝑇;𝐿2(Ω)} (3.23) for some 𝐾>0. Therefore, combining (3.20) and (3.23), we can deduce that there exists a 𝑧∈𝑆(0,𝑇) and a sequence {𝜆_𝑘}⊂[0,1] tending to 0 such that 𝑧_𝜆𝑘⟶𝑧weaklyin𝑆(0,𝑇). (3.24) By Theorem 3.1, ∇𝑦_𝜆𝑘⟶∇𝑦^∗stronglyin𝐶[0,𝑇];𝐿²(Ω)^𝑛as𝑘⟶∞ (3.25) so that by (3.24) and by the compact imbedding theorem given in [16, pages 273–278], we can know that 𝑧_𝜆𝑘⟶𝑧stronglyin𝐿²0,𝑇;𝐻¹₀(Ω)as𝑘⟶∞. (3.26) Combining (3.25) and (3.26), we can have ∇𝑧_𝜆𝑘(𝑡),∇𝑦_𝜆𝑘(𝑡)₂⟶∇𝑧(𝑡),∇𝑦^∗(𝑡)₂a.e.in(0,𝑇) (3.27) as 𝑘→∞. Therefore by Theorem 3.1, (3.27), and the Lebesgue-dominated convergence theorem we can verify that ∇𝑧_𝜆𝑘,∇𝑦_𝜆𝑘+∇𝑦^∗₂Δ𝑦_𝜆𝑘⟶2∇𝑧,∇𝑦^∗₂Δ𝑦^∗stronglyin𝐿²0,𝑇;𝐿²(Ω) (3.28) as 𝑘→∞. At the same time, we can also verify that 𝒢𝑞−𝑞^∗;𝑦_𝜆𝑘⟶𝒢𝑞−𝑞^∗;𝑦^∗≡𝛼−𝛼^∗Δ𝑦^∗+𝛽−𝛽^∗∇𝑦^∗2₂Δ𝑦^∗+𝛾−𝛾^∗Δ𝜕𝑦^∗𝜕𝑡stronglyin𝐿²0,𝑇;𝐿²(Ω) (3.29) as 𝜆_𝑘→0. Hence we can see from (3.24) to (3.29) that 𝑧_𝜆𝑘→𝑧=𝐷𝑦(𝑢)𝑤 weakly in 𝑆(0,𝑇) as 𝜆_𝑘→0 in which 𝑧 is a strong solution of (3.18).
This convergency can be improved by showing the strong convergence of {𝑧_𝜆} in the strong topology of 𝑆(0,𝑇). Subtracting (3.18) from (3.20) and denoting 𝑧_𝜆−𝑧 by 𝜙_𝜆, we see that 𝜕²𝜙_𝜆𝜕𝑡²−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝜙_𝜆−𝛾₀+𝛾^∗Δ𝜕𝜙_𝜆𝜕𝑡=𝒢𝑞−𝑞^∗;𝑦_𝜆−𝒢𝑞−𝑞^∗;𝑦^∗+𝛿𝑦_𝜆,𝑧_𝜆𝜙in𝑄,_𝜆𝜙=0onΣ,_𝜆(0,𝑥)=0,𝜕𝜙_𝜆()𝜕𝑡0,𝑥=0inΩ, (3.30) where 𝛿𝑦_𝜆,𝑧_𝜆=𝛽^∗∇𝑧_𝜆,∇𝑦_𝜆+∇𝑦^∗₂Δ𝑦_𝜆−2𝛽^∗∇𝑧,∇𝑦^∗₂Δ𝑦^∗. (3.31) Estimating 𝜙_𝜆 as in (3.23), we can easily deduce that ∇𝜙^′_𝜆(𝑡)²₂+Δ𝜙_𝜆(𝑡)²₂+^𝑡₀Δ𝜙^′_𝜆²₂𝑑𝑠≤𝐶₂𝒢𝑞−𝑞^∗;𝑦_𝜆−𝒢𝑞−𝑞^∗;𝑦^∗+𝛿𝑦_𝜆,𝑧_𝜆²_{𝐿20,𝑇;𝐿2(Ω)}≤2𝐶₂𝒢𝑞−𝑞^∗;𝑦_𝜆−𝒢𝑞−𝑞^∗;𝑦^∗2_{𝐿20,𝑇;𝐿2(Ω)}+𝛿𝑦_𝜆,𝑧_𝜆²_{𝐿20,𝑇;𝐿2(Ω)}, (3.32) where 𝐶₂ is a positive constant. By virtue of (3.28), (3.29), and (3.30), we can deduce that 𝜙_𝜆](Δ𝜙⟶0in𝐶([0,𝑇;𝐷))as𝜆⟶0,_′𝜆⟶0in𝐶[0,𝑇];𝐻¹₀(Ω)∩𝐿²(0,𝑇;𝐷(Δ))as𝜆⟶0. (3.33) Finally, by means of (3.30) and (3.33) it is followed that 𝑧_𝜆(⋅)⟶𝑧(⋅)stronglyin(0,𝑇)as𝜆⟶0. (3.34) This completes the proof.

3.1. Case of Distributive and Terminal Value Observations

The cost functional 𝐽₁ in (3.15) is represented by 𝐽₁𝑞=^𝑇₀𝑦𝑞;𝑡−𝑌₁(𝑡)²₂𝑑𝑡+𝑦𝑞;𝑇−𝑌^𝑇₁²₂,𝑞∈𝒫. (3.35) Then it is easily verified that the optimality condition (3.17) is written as ^𝑇₀𝑦𝑞^∗;𝑡−𝑌₁(𝑡),𝐷𝑦𝑞^∗𝑞−𝑞^∗(𝑡)₂𝑑𝑡+𝑦𝑞;𝑇−𝑌^𝑇₁,𝐷𝑦𝑞^∗𝑞−𝑞^∗(𝑇)₂≥0,∀𝑞∈𝒫_ad, (3.36) where 𝑞^∗=(𝛼^∗,𝛽^∗,𝛾^∗) is the optimal parameter for (3.35), and 𝑧=𝐷𝑦(𝑞^∗)(𝑞−𝑞^∗) is a solution of (3.18). The necessary condition for the optimal parameter 𝑞^∗=(𝛼^∗,𝛽^∗,𝛾^∗) is given in the following theorem.

Theorem 3.4. The optimal parameter 𝑞^∗=(𝛼^∗,𝛽^∗,𝛾^∗) for (3.35) is characterized by the following system of equations and inequality: 𝜕²𝑦^∗𝜕𝑡²−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑦^∗−𝛾₀+𝛾^∗Δ𝜕𝑦^∗𝑦𝜕𝑡=𝑓𝑖𝑛𝑄,_∗𝑦=0𝑜𝑛Σ,_∗(0,𝑥)=𝑦₀(𝑥),𝜕𝑦^∗()𝜕𝑡0,𝑥=𝑦₁(𝑥)𝜕𝑖𝑛Ω,₂𝑝𝜕𝑡²−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑝−2𝛽^∗∇𝑝,∇𝑦^∗₂Δ𝑦^∗+𝛾₀+𝛾^∗Δ𝜕𝑝𝜕𝑡=𝑦^∗−𝑌₁𝑝()𝑖𝑛𝑄,𝑝=0𝑜𝑛Σ,𝑇,𝑥=0,𝜕𝑝()𝜕𝑡𝑇,𝑥=−𝑦^∗(𝑇)+𝑌^𝑇₁𝑖𝑛Ω,_𝑄𝑝𝒢𝑞−𝑞^∗;𝑦^∗𝑑𝑥𝑑𝑡≥0,∀𝑞=𝛼,𝛽,𝛾∈𝒫_ad. (3.37)(3.38)

Proof. Since 𝑦^∗−𝑌₁∈𝐿²(0,𝑇;𝐿²(Ω)) and 𝑦^∗(𝑇)−𝑌^𝑇₁∈𝐿²(Ω), it is verified by the time reversion 𝑡→𝑇−𝑡, and there is a unique weak solution 𝑝∈𝑊(0,𝑇) of (3.37) (cf. [15, pages 558–574]). Multiplying both sides of the weak form of (3.37) by 𝑧=𝐷𝑦(𝑞^∗)(𝑞−𝑞^∗) and integrating it by parts on [0,𝑇], we have that ^𝑇₀𝑦^∗−𝑌₁,𝑧₂=𝑑𝑡_𝑇0𝑝^′′−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑝−2𝛽^∗∇𝑝,∇𝑦^∗₂Δ𝑦^∗+𝛾₀+𝛾^∗Δ𝑝^′,𝑧₂=𝑑𝑡_𝑇0𝑝,𝑧^′′−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑧−2𝛽^∗∇𝑧,∇𝑦^∗₂Δ𝑦^∗−𝛾₀+𝛾^∗Δ𝑧^′₂−𝑦𝑑𝑡𝑞;𝑇−𝑌^𝑇₁,𝐷𝑦𝑞^∗𝑞−𝑞^∗(𝑇)₂=^𝑇₀𝑝,𝒢𝑞−𝑞^∗;𝑦^∗₂𝑑𝑡−𝑦𝑞;𝑇−𝑌^𝑇₁,𝐷𝑦𝑞^∗𝑞−𝑞^∗(𝑇)₂. (3.39) Therefore, (3.39) and (3.36) imply that the required optimality condition (3.36) is equivalent to the condition (3.38). This proves Theorem 3.4.

3.2. Case of Velocity Observations

The cost functional 𝐽₂ in (3.16) is represented by 𝐽₂𝑞=^𝑇₀𝑦^′𝑞;𝑡−𝑌₂(𝑡)²₂𝑑𝑡,𝑞∈𝒫. (3.40) The optimality condition (3.17) for (3.40) is given by ^𝑇₀𝑦^′𝑞^∗;𝑡−𝑌₂(𝑡),𝐷𝑦𝑞^∗𝑞−𝑞^∗′(𝑡)₂𝑑𝑡≥0,∀𝑞∈𝒫_ad, (3.41) where 𝑧=𝐷𝑦(𝑞^∗)(𝑞−𝑞^∗) is a solution of (3.18).

Remark 3.5. As indicated in [13], if we derive a formal second-order adjoint system of this quasilinear system related to the velocity observation with the cost (3.40), then it is hard to explain whether it is well-posed or not. In order to overcome this difficulty, we follow the idea given in Hwang and Nakagiri [17] in which it is adopted that the first-order integrodifferential system as an appropriate adjoint-system of a quasilinear system instead of the formal second-order adjoint system.
For this reason, we introduce an adjoint-system represented by the following first-order integrodifferential equation: 𝜕𝑝+𝜕𝑡^𝑇_𝑡𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑝+2𝛽^∗∇𝑝,∇𝑦^∗₂Δ𝑦^∗𝑑𝑠+𝛾₀+𝛾^∗Δ𝑝=𝜕𝑦^∗𝜕𝑡−𝑌₂𝑝()in𝑄,𝑝=0onΣ,𝑇,𝑥=0inΩ. (3.42) Since (𝜕𝑦^∗/𝜕𝑡)−𝑌₂∈𝐿²(𝑄)=𝐿²(0,𝑇;𝐿²(Ω)), by reversing the direction of time 𝑡→𝑇−𝑡 and applying the result of [15, pages 656–662] to the problem (3.42), we can assert that (3.42) admits a unique weak solution 𝑝 satisfying 𝑝∈𝑊𝐻¹₀(Ω),𝐿²(Ω)∩𝐶[0,𝑇];𝐻¹₀(Ω), (3.43) where the solution space 𝑊(𝐻¹₀(Ω),𝐿²(Ω)) is defined by 𝑊𝐻¹₀(Ω),𝐿²(Ω)=𝑔∣𝑔∈𝐿²0,𝑇;𝐻¹₀(Ω),𝑔^′∈𝐿²0,𝑇;𝐿²(Ω). (3.44)

Theorem 3.6. The optimal parameter 𝑞^∗=(𝛼^∗,𝛽^∗,𝛾^∗) for (3.40) is characterized by the following system of equations and inequality: 𝜕²𝑦^∗𝜕𝑡²−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑦^∗−𝛾₀+𝛾^∗Δ𝜕𝑦^∗𝑦𝜕𝑡=𝑓𝑖𝑛𝑄,_∗𝑦=0𝑜𝑛Σ,_∗(0,𝑥)=𝑦₀(𝑥),𝜕𝑦^∗()𝜕𝑡0,𝑥=𝑦₁(𝑥)𝑖𝑛Ω,𝜕𝑝+𝜕𝑡^𝑇_𝑡𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑝+2𝛽^∗∇𝑝,∇𝑦^∗₂Δ𝑦^∗𝑑𝑠+𝛾₀+𝛾^∗Δ𝑝=𝜕𝑦^∗𝜕𝑡−𝑌₂𝑝()𝑖𝑛𝑄,𝑝=0𝑜𝑛Σ,𝑇,𝑥=0𝑖𝑛Ω,_𝑄𝑝𝒢𝑞−𝑞^∗;𝑦^∗𝑑𝑥𝑑𝑡≤0,∀𝑞=𝛼,𝛽,𝛾∈𝒫_ad. (3.45)(3.46)

Proof. Multiplying both sides of the weak form of (3.45) by 𝑧′=𝐷𝑦(𝑞^∗)(𝑞−𝑞^∗)′, taking dual pairing between 𝐻¹₀(Ω) and 𝐻⁻¹(Ω) and integrating it by parts on [0,𝑇], we have that ^𝑇₀𝑦^∗′−𝑌₂,𝑧^′₂=𝑑𝑡_𝑇0𝑝^′+^𝑇_𝑡𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑝+2𝛽^∗∇𝑝,∇𝑦^∗₂Δ𝑦^∗𝑑𝑠+𝛾₀+𝛾^∗Δ𝑝,𝑧^′𝑑𝑡=−^𝑇₀𝑝,𝑧^′′−𝛼₀+𝛼^∗+𝛽^∗∇𝑦^∗2₂Δ𝑧−2𝛽^∗∇𝑧,∇𝑦^∗₂Δ𝑦^∗−𝛾₀+𝛾^∗Δ𝑧′₂𝑑𝑡=−^𝑇₀𝑝,𝒢𝑞−𝑞^∗;𝑦^∗₂𝑑𝑡. (3.47) Thus, (3.47) and (3.41) imply that the required optimality condition is given by (3.46).

Acknowledgment

This research was supported by the Daegu University Research Grant, 2010.