# Parameter Identification Problem for the Kirchhoff-Type Equation with Viscosity

Authors: Jinsoo Hwang
Journal: Abstract and Applied Analysis
 (2012)

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## 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.


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## Body

## 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(1.1)∂2y∂t2-1ρ(P0h+E2L∫0L(∂y∂x)2dx)∂2y∂x2=0,
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:
(1.2)∂2y∂t2-m(∫Ω‍|∇y|2dx)Δy-γ∂y∂t=f,
where Ω is a smooth domain in Rn,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 coefficientγ 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)×H01(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:
(1.3)∂2y∂t2-(α+β∫Ω‍|∇y|2dx)Δy-γΔ∂y∂t=finQ,y=0on⁡Σ,y(0,x)=y0(x),∂y∂t(0,x)=y1(x)inΩ,
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 parametersα,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)∣xi≥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).Lety(q)=y(q;t,x) be the solution for a given q=(α,β,γ)∈𝒫 and 𝒫ad⊂𝒫 be an admissible parameter set. We consider the following two quadratic distributive functionals:
(1.4)J1(q)=∫0T‍∫Ω‍|y(q;t,x)-Y1(t,x)|2dxdt+∫Ω‍|y(q;T,x)-Y1T(x)|2dx,(1.5)J2(q)=∫0T‍∫Ω‍|∂y(q;t,x)∂t-Y2(t,x)|2dxdtforq∈Pad,
where Yi∈L2(Q),i=1,2 and Y1T∈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*)∈𝒫ad satisfying that
(1.6)J(q*)=inf⁡{J(q):q∈Pad}.
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:(2.1)∂2y∂t2-(α+β∫Ω|∇y|2dx)Δy-γΔ∂y∂t=finQ,y=0on⁡Σ,y(0,x)=y0(x),∂y∂t(0,x)=y1(x)inΩ,
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)∩H01(Ω), and y1∈H01(Ω). The solution space S(0,T) which is the space of strong solutions of (2.1) is defined by
(2.2)S(0,T)={g∣g∈L2(0,T;D(Δ)),g′∈L2(0,T;D(Δ)),g′′∈L2(0,T;L2(Ω))},
endowed with the norm
(2.3)‖g‖S(0,T)=(‖g‖L2(0,T;D(Δ))2+‖g′‖L2(0,T;D(Δ))2+‖g′′‖L2(0,T;L2(Ω))2)1/2.
Here, g′ and g′′ denote the first- and second-order distributional derivatives of g. The scalar products and norms on L2(Ω) and H01(Ω) are denoted by (ϕ,ψ)2, |ϕ|2 and (ϕ,ψ)H01(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)H01(Ω) and the norm ∥ϕ∥ of H01(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 H01(Ω) and H-1(Ω) is denoted by 〈ϕ,ψ〉.Definition 2.1.
A functiony is said to be a strong solution of (2.1) if y∈S(0,T) and y satisfies
(2.4)y′′(t)-(α+β|∇y(t)|22)Δy(t)-γΔy′(t)=f(t),a.e.t∈[0,T],y(0)=y0,y′(0)=y1.We remark here thatS(0,T) is continuously imbedded in C([0,T];D(Δ))∩C1([0,T];H01(Ω)) (cf. Dautray and Lions [15, page 555]).The following variational formulation is used to define theweak solution of (2.1).Definition 2.2.
A functiony is said to be a weak solution of (2.1) if y∈W(0,T)≡{g∣g∈L2(0,T;H01(Ω)),g′∈L2(0,T;H01(Ω)),g′′∈L2(0,T;H-1(Ω))}, and y satisfies
(2.5)〈y′′(⋅),ϕ〉+(α+β|∇y(⋅)|22)(∇y(⋅),∇ϕ)+γ(∇y′(⋅),∇ϕ)=(f(⋅),ϕ)∀ϕ∈H01(Ω)inthesenseofD′(0,T),y(0)=y0,y′(0)=y1.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 thatf∈L2(0,T;L2(Ω)) and y0∈D(Δ),y1∈H01(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)×H01(Ω)×L2(0,T;L2(Ω)) into S(0,T) is strongly continuous. Further, for each p1=(y01,y11,f1)∈P and p2=(y02,y12,f2)∈P, we have the following inequality:
(2.6)|∇(y′(p1;t)-y′(p2;t))|22+|Δ(y(p1;t)-y(p2;t))|22+∫0t‍|Δ(y′(p1;s)-y′(p2;s))|22ds≤C(|Δ(y01-y02)|22+|∇(y11-y12)|22+‖f1-f2‖L2(0,T;L2(Ω))2),
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 ∫0t‍|∇y′(p1)|2ds instead of ∫0t‍|∇y′(p1;s)|2ds.


## 3. Identification Problems

In this section we study the identification problem for the unknown parametersq=(α,β,γ)∈𝒫 in the problem
(3.1)∂2y∂t2-(α0+α+β|∇y|22)Δy-(γ0+γ)Δ∂y∂t=finQ,y=0on⁡Σ,y(0,x)=y0(x),∂y∂t(0,x)=y1(x)inΩ,
where α0,γ0>0, y0∈D(Δ), y1∈H01(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)∣xi≥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 parametersq=(α,β,γ).Theorem 3.1.
The solution mapq→y(q) from 𝒫={(x1,x2,x3)∣xi≥0, i=1,2,3} into S(0,T) is continuous.Proof.
Letq=(α,β,γ) be arbitrarily fixed. Suppose that qm=(αm,βm,γm)→q=(α,β,γ) in 𝒫. Let ym=y(qm) and y=y(q) be the solutions of (3.1) for q=qm and for q, respectively. Since {qm} is bounded in 𝒫, by Theorem 2.3, we see that
(3.2)|Δym′(t)|22+|Δym(t)|22+∫0t‍|Δym′|22ds≤C0<∞,∀t∈[0,T],
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 {ym} denoted again by {ym} that
(3.3)ym⟶yweaklyinS(0,T)asm⟶∞.
Since D(Δ)↪H01(Ω) is compact, we can deduce from [16, pages 273–278] that the space S(0,T) is compactly imbedded in L2(0,T;H01(Ω)). Therefore, we can take a subsequence {ymk} of {ym}, if necessary, such that
(3.4)ymk⟶ystronglyinL2(0,T;H01(Ω))ask⟶∞.
Equation (3.4) implies that
(3.5)|∇ymk(⋅)|22⟶|∇y(⋅)|22a.e.in(0,T)ask⟶∞.
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 setψm=ym-y. Then, in weak sense, ψm satisfies
(3.6)∂2ψm∂t2-(α0+αm+βm|∇ym|22)Δψm-(γ0+γm)Δ∂ψm∂t=FminQ,ψm=0on⁡Σ,ψm(0,x)=0,∂ψm∂t(0,x)=0inΩ,
where
(3.7)Fm=(αm-α+βm|∇ym|22-β|∇y|22)Δy-(γ-γm)Δ∂y∂t.
Using (3.5) and the Lebesgue-dominated convergence theorem, we can verify that
(3.8)Fm⟶0inL2(0,T;L2(Ω))asm→∞.
Multiplying (3.6) by Δψm′, and integrating it over Ω×[0,t], we have
(3.9)|∇ψm′(t)|22+(α0+αm+βm|∇ym(t)|22)|Δψm(t)|22+2(γ0+γm)∫0t‍|Δψm′|22ds=-2∫0t‍(Fm,Δψm′)2ds+∫0t‍2βm(∇ym,∇ym′)2|Δψm|22ds.
By the Cauchy-Schwarz inequality and the fact that ym∈S(0,T)↪C([0,T];D(Δ))∩C1([0,T];H01(Ω)), we have following inequalities:
(3.10)|-2∫0t‍(Fm,∇ψm′)2ds|≤2∫0t‍|Fm|2|Δψm′|2ds≤1γ0‖Fm‖L2(0,T;L2(Ω))2+γ0∫0t‍|Δψm′|22ds,|∫0t‍2βm(∇ym,∇ym′)2|Δψm|22ds|≤2βm∫0t‍|∇ym|2|∇ym′|2|Δψm|22ds≤2βm‖ym‖C([0,T];H01(Ω))‖ym′‖C([0,T];H01(Ω))∫0t‍|Δψm|22ds.
Then by (3.9) and (3.10), we can obtain
(3.11)|∇ψm′(t)|22+|Δψm(t)|22+∫0t‍|Δψm′|22ds≤C‖Fm‖L2(0,T;L2(Ω))2+C∫0t‍|Δψm|22ds,
where C>0. Hence by applying Gronwall’s inequality to (3.11), we have
(3.12)|∇ψm′(t)|22+|Δψm(t)|22+∫0t‍|Δψm′|22ds≤Cexp⁡(CT)‖Fm‖L2(0,T;L2(Ω))2.
Combining (3.8) and (3.12), we have
(3.13)ψm⟶0inC([0,T];D(Δ)),ψm′⟶0inC([0,T];H01(Ω))∩L2(0,T;D(Δ)),
so that
(3.14)ym(⋅)⟶y(⋅)stronglyinS(0,T).
This proves Theorem 3.1.As explained before, we choose theL2 objective costs to be minimized for the identification of q=(α,β,γ) which are given by
(3.15)J1(q)=‖y(q)-Y1‖L2(0,T;L2(Ω))2+|y(q;T)-Y1T|22,(3.16)J2(q)=‖y′(q)-Y2‖L2(0,T;L2(Ω))2forq∈Pad,
where Yi∈L2(0,T;L2(Ω)),i=1,2, and Y1T∈L2(Ω).If𝒫ad is compact, then for the minimizing sequence {qm} such as J(qm)→J*=inf{J(q):q∈𝒫ad} we can choose a subsequence {qmj} of {qm} such that qmj→q*∈𝒫ad and y(qmj)→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];H01(Ω)) we have J*=J(q*) for the costs (3.15) and (3.16). Thus we have the following corollary.Corollary 3.2.
If𝒫ad is compact, then there exists at least one optimal parameter q*∈𝒫ad for the cost J1 in (3.15) or J2 in (3.16).Let the admissible set𝒫ad be compact and convex in 𝒫, and let q*=(α*,β*,γ*) be an optimal parameter on 𝒫ad 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
(3.17)DJ(q*)(q-q*)≥0∀q∈Pad,
where DJ(q*) denotes the Gâteaux derivative of J(q) at q=q*.The Gâteaux differentiability of the above quadratic costsJi(q),i=1,2 follows from that of the nonlinear solution mapping q→y(q) of 𝒫ad 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 mapq→y(q) of 𝒫ad 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:
(3.18)∂2z∂t2-(α0+α*+β*|∇y*|22)Δz-2β*(∇z,∇y*)2Δy*-(γ0+γ*)Δ∂z∂t=G(q-q*;y*)inQ,z=0onΣ,z(0,x)=0,∂z∂t(0,x)=0inΩ,
where y*=y(q*) and
(3.19)G(q-q*;y*)=(α-α*)Δy*+(β-β*)|∇y*|22Δy*+(γ-γ*)Δ∂y*∂t.Proof.
Letλ∈[0,1], and let yλ and y* be the solutions of (3.1) corresponding to q*+λ(q-q*) and q*, respectively. We set zλ=λ-1(yλ-y*),λ≠0. Then zλ satisfies the following problem in the weak sense:
(3.20)∂2zλ∂t2-(α0+α*+β*|∇y*|22)Δzλ-β*(∇zλ,∇yλ+∇y*)2Δyλ-(γ0+γ*)Δ∂zλ∂t=G(q-q*;yλ)inQ,zλ=0on⁡Σ,zλ(0,x)=0,∂zλ∂t(0,x)=0inΩ,
where
(3.21)G(q-q*;yλ)=(α-α*)Δyλ+(β-β*)|∇yλ|22Δyλ+(γ-γ*)Δ∂yλ∂t.
Since yλ∈S(0,T) we can easily know that 𝒢(q-q*;yλ)∈L2(0,T;L2(Ω)) and
(3.22)‖G(q-q*;yλ)‖L2(0,T;L2(Ω))≤C0|q-q*|‖yλ‖S(0,T)<C1,
where Ci,i=0,1 are positive constants.
By similar arguments in the proof of Theorem3.1, multiplying the both sides of (3.20) by -Δzλ′ and integrating it over Ω×[0,t], we can obtain the following inequality:
(3.23)|Δzλ(t)|22+|∇zλ′(t)|22+∫0t‍|Δzλ′|22ds≤K‖G(q-q*;yλ)‖L2(0,T;L2(Ω))2
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 {λk}⊂[0,1] tending to 0 such that
(3.24)zλk⟶zweaklyinS(0,T).
By Theorem 3.1,
(3.25)∇yλk⟶∇y*stronglyin[C([0,T];L2(Ω))]nask⟶∞
so that by (3.24) and by the compact imbedding theorem given in [16, pages 273–278], we can know that
(3.26)zλk⟶zstronglyinL2(0,T;H01(Ω))ask⟶∞.
Combining (3.25) and (3.26), we can have
(3.27)(∇zλk(t),∇yλk(t))2⟶(∇z(t),∇y*(t))2a.e.in(0,T)
as k→∞. Therefore by Theorem 3.1, (3.27), and the Lebesgue-dominated convergence theorem we can verify that
(3.28)(∇zλk,∇yλk+∇y*)2Δyλk⟶2(∇z,∇y*)2Δy*stronglyinL2(0,T;L2(Ω))
as k→∞. At the same time, we can also verify that
(3.29)G(q-q*;yλk)⟶G(q-q*;y*)≡(α-α*)Δy*+(β-β*)|∇y*|22Δy*+(γ-γ*)Δ∂y*∂tstronglyinL2(0,T;L2(Ω))
as λk→0. Hence we can see from (3.24) to (3.29) that zλk→z=Dy(u)w weakly in S(0,T) as λk→0 in which z is a strong solution of (3.18).
This convergency can be improved by showing the strong convergence of{zλ} in the strong topology of S(0,T). Subtracting (3.18) from (3.20) and denoting zλ-z by ϕλ, we see that (3.30)∂2ϕλ∂t2-(α0+α*+β*|∇y*|22)Δϕλ-(γ0+γ*)Δ∂ϕλ∂t=G(q-q*;yλ)-G(q-q*;y*)+δ(yλ,zλ)inQ,ϕλ=0on⁡Σ,ϕλ(0,x)=0,∂ϕλ∂t(0,x)=0inΩ,
where
(3.31)δ(yλ,zλ)=β*(∇zλ,∇yλ+∇y*)2Δyλ-2β*(∇z,∇y*)2Δy*.
Estimating ϕλ as in (3.23), we can easily deduce that
(3.32)|∇ϕλ′(t)|22+|Δϕλ(t)|22+∫0t‍|Δϕλ′|22ds≤C2‖G(q-q*;yλ)-G(q-q*;y*)+δ(yλ,zλ)‖L2(0,T;L2(Ω))2≤2C2(‖G(q-q*;yλ)-G(q-q*;y*)‖L2(0,T;L2(Ω))2+‖δ(yλ,zλ)‖L2(0,T;L2(Ω))2),
where C2 is a positive constant. By virtue of (3.28), (3.29), and (3.30), we can deduce that
(3.33)ϕλ⟶0inC([0,T];D(Δ))asλ⟶0,ϕλ′⟶0inC([0,T];H01(Ω))∩L2(0,T;D(Δ))asλ⟶0.
Finally, by means of (3.30) and (3.33) it is followed that
(3.34)zλ(⋅)⟶z(⋅)stronglyin(0,T)asλ⟶0.
This completes the proof.


### 3.1. Case of Distributive and Terminal Value Observations

The cost functionalJ1 in (3.15) is represented by
(3.35)J1(q)=∫0T‍|y(q;t)-Y1(t)|22dt+|y(q;T)-Y1T|22,q∈P.
Then it is easily verified that the optimality condition (3.17) is written as
(3.36)∫0T‍(y(q*;t)-Y1(t),Dy(q*)(q-q*)(t))2dt+(y(q;T)-Y1T,Dy(q*)(q-q*)(T))2≥0,∀q∈Pad,
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 parameterq*=(α*,β*,γ*) for (3.35) is characterized by the following system of equations and inequality:
(3.37)∂2y*∂t2-(α0+α*+β*|∇y*|22)Δy*-(γ0+γ*)Δ∂y*∂t=finQ,y*=0onΣ,y*(0,x)=y0(x),∂y*∂t(0,x)=y1(x)inΩ,∂2p∂t2-(α0+α*+β*|∇y*|22)Δp-2β*(∇p,∇y*)2Δy*+(γ0+γ*)Δ∂p∂t=y*-Y1inQ,p=0onΣ,p(T,x)=0,∂p∂t(T,x)=-y*(T)+Y1TinΩ,(3.38)∫Q‍pG(q-q*;y*)dxdt≥0,∀q=(α,β,γ)∈Pad.Proof.
Sincey*-Y1∈L2(0,T;L2(Ω)) and y*(T)-Y1T∈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
(3.39)∫0T‍(y*-Y1,z)2dt=∫0T‍(p′′-(α0+α*+β*|∇y*|22)Δp-2β*(∇p,∇y*)2Δy*+(γ0+γ*)Δp′,z)2dt=∫0T‍(p,z′′-(α0+α*+β*|∇y*|22)Δz-2β*(∇z,∇y*)2Δy*-(γ0+γ*)Δz′)2dt-(y(q;T)-Y1T,Dy(q*)(q-q*)(T))2=∫0T‍(p,G(q-q*;y*))2dt-(y(q;T)-Y1T,Dy(q*)(q-q*)(T))2.
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 functionalJ2 in (3.16) is represented by
(3.40)J2(q)=∫0T‍|y′(q;t)-Y2(t)|22dt,q∈P.
The optimality condition (3.17) for (3.40) is given by
(3.41)∫0T‍(y′(q*;t)-Y2(t),Dy(q*)(q-q*)′(t))2dt≥0,∀q∈Pad,
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:(3.42)∂p∂t+∫tT‍{(α0+α*+β*|∇y*|22)Δp+2β*(∇p,∇y*)2Δy*}ds+(γ0+γ*)Δp=∂y*∂t-Y2inQ,p=0on⁡Σ,p(T,x)=0inΩ.
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
(3.43)p∈W(H01(Ω),L2(Ω))∩C([0,T];H01(Ω)),
where the solution space W(H01(Ω),L2(Ω)) is defined by
(3.44)W(H01(Ω),L2(Ω))={g∣g∈L2(0,T;H01(Ω)),g′∈L2(0,T;L2(Ω))}.Theorem 3.6.
The optimal parameterq*=(α*,β*,γ*) for (3.40) is characterized by the following system of equations and inequality:
(3.45)∂2y*∂t2-(α0+α*+β*|∇y*|22)Δy*-(γ0+γ*)Δ∂y*∂t=finQ,y*=0onΣ,y*(0,x)=y0(x),∂y*∂t(0,x)=y1(x)inΩ,∂p∂t+∫tT{(α0+α*+β*|∇y*|22)Δp+2β*(∇p,∇y*)2Δy*}ds+(γ0+γ*)Δp=∂y*∂t-Y2inQ,p=0onΣ,p(T,x)=0inΩ,(3.46)∫QpG(q-q*;y*)dxdt≤0,∀q=(α,β,γ)∈Pad.Proof.
Multiplying both sides of the weak form of (3.45) by z′=Dy(q*)(q-q*)′, taking dual pairing between H01(Ω) and H-1(Ω) and integrating it by parts on [0,T], we have that
(3.47)∫0T‍(y*′-Y2,z′)2dt=∫0T‍〈p′+∫tT‍{(α0+α*+β*|∇y*|22)Δp+2β*(∇p,∇y*)2Δy*}ds+(γ0+γ*)Δp,z′〉dt=-∫0T‍(p,z′′-(α0+α*+β*|∇y*|22)Δz-2β*(∇z,∇y*)2Δy*-(γ0+γ*)Δz′)2dt=-∫0T‍(p,G(q-q*;y*))2dt.
Thus, (3.47) and (3.41) imply that the required optimality condition is given by (3.46).


## 3.1. Case of Distributive and Terminal Value Observations

The cost functionalJ1 in (3.15) is represented by
(3.35)J1(q)=∫0T‍|y(q;t)-Y1(t)|22dt+|y(q;T)-Y1T|22,q∈P.
Then it is easily verified that the optimality condition (3.17) is written as
(3.36)∫0T‍(y(q*;t)-Y1(t),Dy(q*)(q-q*)(t))2dt+(y(q;T)-Y1T,Dy(q*)(q-q*)(T))2≥0,∀q∈Pad,
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 parameterq*=(α*,β*,γ*) for (3.35) is characterized by the following system of equations and inequality:
(3.37)∂2y*∂t2-(α0+α*+β*|∇y*|22)Δy*-(γ0+γ*)Δ∂y*∂t=finQ,y*=0onΣ,y*(0,x)=y0(x),∂y*∂t(0,x)=y1(x)inΩ,∂2p∂t2-(α0+α*+β*|∇y*|22)Δp-2β*(∇p,∇y*)2Δy*+(γ0+γ*)Δ∂p∂t=y*-Y1inQ,p=0onΣ,p(T,x)=0,∂p∂t(T,x)=-y*(T)+Y1TinΩ,(3.38)∫Q‍pG(q-q*;y*)dxdt≥0,∀q=(α,β,γ)∈Pad.Proof.
Sincey*-Y1∈L2(0,T;L2(Ω)) and y*(T)-Y1T∈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
(3.39)∫0T‍(y*-Y1,z)2dt=∫0T‍(p′′-(α0+α*+β*|∇y*|22)Δp-2β*(∇p,∇y*)2Δy*+(γ0+γ*)Δp′,z)2dt=∫0T‍(p,z′′-(α0+α*+β*|∇y*|22)Δz-2β*(∇z,∇y*)2Δy*-(γ0+γ*)Δz′)2dt-(y(q;T)-Y1T,Dy(q*)(q-q*)(T))2=∫0T‍(p,G(q-q*;y*))2dt-(y(q;T)-Y1T,Dy(q*)(q-q*)(T))2.
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 functionalJ2 in (3.16) is represented by
(3.40)J2(q)=∫0T‍|y′(q;t)-Y2(t)|22dt,q∈P.
The optimality condition (3.17) for (3.40) is given by
(3.41)∫0T‍(y′(q*;t)-Y2(t),Dy(q*)(q-q*)′(t))2dt≥0,∀q∈Pad,
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:(3.42)∂p∂t+∫tT‍{(α0+α*+β*|∇y*|22)Δp+2β*(∇p,∇y*)2Δy*}ds+(γ0+γ*)Δp=∂y*∂t-Y2inQ,p=0on⁡Σ,p(T,x)=0inΩ.
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
(3.43)p∈W(H01(Ω),L2(Ω))∩C([0,T];H01(Ω)),
where the solution space W(H01(Ω),L2(Ω)) is defined by
(3.44)W(H01(Ω),L2(Ω))={g∣g∈L2(0,T;H01(Ω)),g′∈L2(0,T;L2(Ω))}.Theorem 3.6.
The optimal parameterq*=(α*,β*,γ*) for (3.40) is characterized by the following system of equations and inequality:
(3.45)∂2y*∂t2-(α0+α*+β*|∇y*|22)Δy*-(γ0+γ*)Δ∂y*∂t=finQ,y*=0onΣ,y*(0,x)=y0(x),∂y*∂t(0,x)=y1(x)inΩ,∂p∂t+∫tT{(α0+α*+β*|∇y*|22)Δp+2β*(∇p,∇y*)2Δy*}ds+(γ0+γ*)Δp=∂y*∂t-Y2inQ,p=0onΣ,p(T,x)=0inΩ,(3.46)∫QpG(q-q*;y*)dxdt≤0,∀q=(α,β,γ)∈Pad.Proof.
Multiplying both sides of the weak form of (3.45) by z′=Dy(q*)(q-q*)′, taking dual pairing between H01(Ω) and H-1(Ω) and integrating it by parts on [0,T], we have that
(3.47)∫0T‍(y*′-Y2,z′)2dt=∫0T‍〈p′+∫tT‍{(α0+α*+β*|∇y*|22)Δp+2β*(∇p,∇y*)2Δy*}ds+(γ0+γ*)Δp,z′〉dt=-∫0T‍(p,z′′-(α0+α*+β*|∇y*|22)Δz-2β*(∇z,∇y*)2Δy*-(γ0+γ*)Δz′)2dt=-∫0T‍(p,G(q-q*;y*))2dt.
Thus, (3.47) and (3.41) imply that the required optimality condition is given by (3.46).



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*Source: 971796-2012-07-01.xml*