Parameter Identification Problem for the Kirchhoff-Type Equation with Viscosity

and Applied Analysis 3 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 . Let y 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:


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 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, P 0 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: 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 ×H 1 0 Ω ×L 2 0, T; L 2 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: 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 L 2 -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 α 0 α, β, γ 0 γ, respectively, where α 0 and γ 0 are fixed positive constants, and α, β, γ are nonnegative constants.Therefore, we take the set P { x 1 , x 2 , x 3 | 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: where are the desired values.The parameter identification problem for 1.3 with the cost J J 1 in 1.4 or J J 2 in 1.5 is to find and characterize an optimal parameters q * α * , β * , γ * ∈ P ad satisfying that J q * inf J q : 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 .

Preliminaries
We consider the following Dirichlet boundary value problem for Kirchhoff-type equation with damping term: where f is a forcing function, y 0 and y 1 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 The solution space S 0, T which is the space of strong solutions of 2.1 is defined by endowed with the norm Here, g and g denote the first-and second-order distributional derivatives of g.The scalar products and norms on L 2 Ω and H 1 0 Ω are denoted by φ, ψ 2 , |φ| 2 and φ, ψ H 1 0 Ω , φ , respectively.The scalar product and norm on L 2 Ω n are also denoted by φ, ψ 2 and |φ| 2 .Then, the scalar product φ, ψ H 1 0 Ω and the norm φ of H 1 0 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 H 1 0 Ω and H −1 Ω is denoted by φ, ψ .

2.4
We remark here that S 0, T is continuously imbedded in C 0, T ; D Δ ∩ C 1 0, T ; H 1 0 Ω cf.Dautray and Lions 15, page 555 .The following variational formulation is used to define the weak solution of 2.1 .

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 ∈ L 2 0, T; L 2 Ω and y 0 ∈ D Δ , y 1 ∈ H 1 0 Ω .Then the problem 2.1 has a unique strong solution y in S 0, T .And the solution mapping p y 0 , y 1 , f → y p 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

Identification Problems
In this section we study the identification problem for the unknown parameters q α, β, γ ∈ P in the problem

3.1
where α 0 , γ 0 > 0, y 0 ∈ D Δ , y 1 ∈ H 1 0 Ω , and f ∈ L 2 0, T; L 2 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 { x 1 , x 2 , x 3 | 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 { 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 where C 0 > 0 is a constant depending only on α 0 , β 0 , γ 0 , y 0 , y 1 , 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 −→ y weakly in S 0, T as m −→ ∞.

3.3
Since D Δ → H 1 0 Ω is compact, we can deduce from 16, pages 273-278 that the space S 0, T is compactly imbedded in L 2 0, T; H 1 0 Ω .Therefore, we can take a subsequence {y m k } of {y m }, if necessary, such that a.e. in 0, T as k −→ ∞.

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

3.6
where Using 3.5 and the Lebesgue-dominated convergence theorem, we can verify that

3.9
Abstract and Applied Analysis 7 By the Cauchy-Schwarz inequality and the fact that y m ∈ S 0, T → C 0, T ; D Δ ∩ C 1 0, T ; H 1 0 Ω , we have following inequalities:

3.10
Then by 3.9 and 3.10 , we can obtain Δψ m t where C > 0. Hence by applying Gronwall's inequality to 3.11 , we have Δψ m t Combining 3.8 and 3.12 , we have
As explained before, we choose the L 2 objective costs to be minimized for the identification of q α, β, γ which are given by for q ∈ P ad , 3.16 where 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 ∩ C 1 0, T ; H 1 0 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 J 1 in 3.15 or J 2 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 DJ q * 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:

3.18
where y * y q * and

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:

3.21
Since y λ ∈ S 0, T we can easily know that G q − q * ; y λ ∈ L 2 0, T; L 2 Ω and 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:

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 λ k −→ z weakly in S 0, T .

3.24
By Theorem 3.1, so that by 3.24 and by the compact imbedding theorem given in 16, pages 273-278 , we can know that as λ k → 0. Hence we can see from 3.24 to 3.29 that z λ k → 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 Δφ λ t where C 2 is a positive constant.By virtue of 3.28 , 3.29 , and 3.30 , we can deduce that

3.34
This completes the proof.

Case of Distributive and Terminal Value Observations
The cost functional J 1 in 3.15 is represented by , q ∈ P.

3.35
Then it is easily verified that the optimality condition 3.17 is written as T 0 y q * ; t − Y 1 t , Dy q * q − q * t 2 dt y q; T − Y T 1 , Dy q * 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: , 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
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.

Case of Velocity Observations
The cost functional J 2 in 3.16 is represented by dt, q ∈ P.

3.40
The optimality condition 3.17 for 3.40 is given by T 0 y q * ; t − Y 2 t , Dy q * q − q * t 2 dt ≥ 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.

3.46
Proof.Multiplying both sides of the weak form of 3.45 by z Dy q * q − q * , taking dual pairing between H 1 0 Ω and H −1 Ω and integrating it by parts on 0, T , we have that p, G q − q * ; y * 2 dt.

3.47
Thus, 3.47 and 3.41 imply that the required optimality condition is given by 3.46 .

Definition 2 . 1 . 2 2
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 Δy t − γΔy t f t , a.e.t ∈ 0,