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Abstract and Applied Analysis
Volume 2012, Article ID 971796, 15 pages
http://dx.doi.org/10.1155/2012/971796
Research Article

Parameter Identification Problem for the Kirchhoff-Type Equation with Viscosity

Department of Mathematics Education, Daegu University, Jillyang, Gyeongsan, Gyeongbuk 712-714, Republic of Korea

Received 14 February 2012; Accepted 8 May 2012

Academic Editor: Xiaodong Yan

Copyright © 2012 Jinsoo Hwang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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 has been proposed by Kirchhoff [1]. Here is the length of the string, is the area of the cross section, is the mass density, is the initial tension, and 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 is a smooth domain in . Many researches have been devoted to the study of (1.2) for both damped or undamped 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 in (1.2) is replaced by . In this case we can refer to Cavalcanti et al. [8] to show the well-posedness in the Hadamard sense under the data condition . 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: where . Here the constants and are physical constants explained above, and 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 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 -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, in (1.3) by , respectively, where and are fixed positive constants, and are nonnegative constants. Therefore, we take the set , as the set of parameters 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 be the solution for a given and be an admissible parameter set. We consider the following two quadratic distributive functionals: where and are the desired values.

The parameter identification problem for (1.3) with the cost in (1.4) or in (1.5) is to find and characterize an optimal parameters satisfying that We prove the existence of an optimal parameter 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 .

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: where is a forcing function, and are initial data, and are some physical constants. In this paper we study (2.1) in the class of strong solutions. For the purpose we suppose that , , and . The solution space which is the space of strong solutions of (2.1) is defined by endowed with the norm Here, and denote the first- and second-order distributional derivatives of . The scalar products and norms on and are denoted by , and , , respectively. The scalar product and norm on are also denoted by and . Then, the scalar product and the norm of are given by and , respectively. Finally the norm and the scalar product on are given by and , respectively. The duality pairing between and is denoted by .

Definition 2.1. A function is said to be a strong solution of (2.1) if and satisfies

We remark here that is continuously imbedded in (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 is said to be a weak solution of (2.1) if ,, and satisfies

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 and . Then the problem (2.1) has a unique strong solution in . And the solution mapping of into is strongly continuous. Further, for each and , we have the following inequality: where is a constant and .

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 instead of .

3. Identification Problems

In this section we study the identification problem for the unknown parameters in the problem where , , , and are fixed. The physical constants in (3.1) are an unknown parameter that should be identified. In this setting we take , to be the space of parameters with the Euclidian norm. By Theorem 2.3 we have that for each there exists a unique solution of (3.1).

At first we show the continuous dependence of solutions on parameters .

Theorem 3.1. The solution map from , into is continuous.

Proof. Let be arbitrarily fixed. Suppose that in . Let and be the solutions of (3.1) for and for , respectively. Since is bounded in , by Theorem 2.3, we see that where is a constant depending only on , and . Applying (3.2) to (3.1), we can deduce by choosing appropriate subsequence of denoted again by that Since is compact, we can deduce from [16, pages 273–278] that the space is compactly imbedded in . Therefore, we can take a subsequence of , if necessary, such that Equation (3.4) implies that Taking into account (3.3) and (3.5) and coming back to (3.1), we deduce that is the solution of (3.1) corresponding to the parameter .
In order to obtain strong convergency, we set . Then, in weak sense, satisfies where Using (3.5) and the Lebesgue-dominated convergence theorem, we can verify that Multiplying (3.6) by , and integrating it over , we have By the Cauchy-Schwarz inequality and the fact that , we have following inequalities: Then by (3.9) and (3.10), we can obtain where . Hence by applying Gronwall’s inequality to (3.11), we have Combining (3.8) and (3.12), we have so that This proves Theorem 3.1.

As explained before, we choose the objective costs to be minimized for the identification of which are given by where , and .

If is compact, then for the minimizing sequence such as we can choose a subsequence of such that and strongly in by Theorem 3.1. Due to the continuous imbedding we have for the costs (3.15) and (3.16). Thus we have the following corollary.

Corollary 3.2. If is compact, then there exists at least one optimal parameter for the cost in (3.15) or in (3.16).

Let the admissible set be compact and convex in , and let be an optimal parameter on for the cost . As is well known the necessary optimality condition of an optimal parameter for the cost is given by where denotes the Gâteaux derivative of at .

The Gâteaux differentiability of the above quadratic costs follows from that of the nonlinear solution mapping of into . The following theorem proves the Gâteaux differentiability of the nonlinear solution mapping and gives its characterization.

Theorem 3.3. The map of into is Gâteaux differentiable at and such the Gâteaux derivative of at in the direction , say , is a unique solution of the following linear problem: where and

Proof. Let , and let and be the solutions of (3.1) corresponding to and , respectively. We set . Then satisfies the following problem in the weak sense: where Since we can easily know that and where are positive constants.
By similar arguments in the proof of Theorem 3.1, multiplying the both sides of (3.20) by and integrating it over , we can obtain the following inequality: for some . Therefore, combining (3.20) and (3.23), we can deduce that there exists a and a sequence tending to such that 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 Combining (3.25) and (3.26), we can have as . Therefore by Theorem 3.1, (3.27), and the Lebesgue-dominated convergence theorem we can verify that as . At the same time, we can also verify that as . Hence we can see from (3.24) to (3.29) that weakly in as in which is a strong solution of (3.18).
This convergency can be improved by showing the strong convergence of in the strong topology of . Subtracting (3.18) from (3.20) and denoting by , we see that where Estimating as in (3.23), we can easily deduce that where is a positive constant. By virtue of (3.28), (3.29), and (3.30), we can deduce that Finally, by means of (3.30) and (3.33) it is followed that This completes the proof.

3.1. Case of Distributive and Terminal Value Observations

The cost functional in (3.15) is represented by Then it is easily verified that the optimality condition (3.17) is written as where is the optimal parameter for (3.35), and is a solution of (3.18). The necessary condition for the optimal parameter is given in the following theorem.

Theorem 3.4. The optimal parameter for (3.35) is characterized by the following system of equations and inequality:

Proof. Since and , it is verified by the time reversion , and there is a unique weak solution of (3.37) (cf. [15, pages 558–574]). Multiplying both sides of the weak form of (3.37) by and integrating it by parts on , we have that 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 in (3.16) is represented by The optimality condition (3.17) for (3.40) is given by where 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: Since , by reversing the direction of time 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 satisfying where the solution space is defined by

Theorem 3.6. The optimal parameter for (3.40) is characterized by the following system of equations and inequality:

Proof. Multiplying both sides of the weak form of (3.45) by , taking dual pairing between and and integrating it by parts on , we have that 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.

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