Abstract
This article presents a semigroup approach for the mathematical analysis of the inverse
coefficient problems of identifying the unknown coefficient
1 Introduction
Consider the following initial boundary value problem:
where
(C1)
(C2)
Under these conditions, the initial boundary value problem (1) has the unique solution
Consider the inverse problem of determining the unknown coefficient
Here
We denote by
Then the inverse problem [10] with the measured data
We denote by
The aim of this paper is to study a distinguishability of the unknown coefficient
via the above inputoutput mappings. We say that the mapping
The purpose of this paper is to study the distinguishability of the unknown coefficient via the above inputoutput mappings. The results presented here are the first ones, to the knowledge of authors, from the point of view of semigroup approach [11] to inverse problems. This approach sheds more light on the identifiability of the unknown coefficient [12] and shows how much information can be extracted from the measured output data, in particular in the case of constant flux and boundary data [1215].
The paper is organized as follows. In Section 2, the analysis of the semigroup approach
is given for the inverse problem with the measured data
2 Analysis of the inverse problem with measured output data
f
(
t
)
Consider now the inverse problem with one measured output data
Then the initial boundary value problem (1) can be rewritten in the following form:
Here we assume that
which satisfies the following parabolic problem:
Here
Denote by
This problem is called a SturmLiouville problem. We can easily determine that the
eigenvalues are
where
From the definition of the semigroup
The unique solution of the initial value problem (7) in terms of a semigroup
Hence, by using identity (6), the solution
In order to arrange the above solution representation, let us define the following:
Then we can rewrite the solution representation in terms of
Substituting
Taking into account the overmeasured data
which implies that
Differentiating both sides of the above identity with respect to x and using semigroup properties at
Using the boundary condition
Taking limit as
The righthand side of identity (11) defines explicitly the semigroup representation of the inputoutput mapping
Let us differentiate now both sides of identity (8) with respect to t:
Using the semigroup property
Taking
Since
Solving this equation for
Under the determined values
The following lemma implies the relationship between the diffusion coefficients
Lemma 2.1Let
holds, then the outputs
for each
Proof The solutions of the direct problem (5) corresponding to the admissible coefficients
respectively, by using representation (11). From identity (9) it is obvious that
This lemma with identity (14) implies the following.
Corollary 2.1Let conditions of Lemma 2.1 hold. Then
Since the StrumLiouville problem generates a complete orthogonal family of eigenfunctions,
the null space of a semigroup contains only zero function, i.e.,
The combination of the conclusions of Lemma 2.1 and Corollary 2.1 can be given by
the following theorem which states the distinguishability of the inputoutput mapping
Theorem 2.1Let conditions (C1) and (C2) hold. Assume that
3 Analysis of the inverse problem with measured output data
h
(
t
)
Consider now the inverse problem with one measured output data
Then the initial boundary value problem (1) can be rewritten in the following form:
In order to formulate the solution of the above parabolic problem in terms of a semigroup,
let us use the same variable
Here
Denote by
This problem is called a SturmLiouville problem. We can easily determine that the
eigenvalues are
The null space of the semigroup
Since the SturmLiouville problem generates a complete orthogonal family of eigenfunctions,
we can say that the null space of the semigroup
The unique solution of the initial value problem (16) in terms of a semigroup
Hence, by using identity (6), the solution
Defining the following:
The solution representation of the parabolic problem (17) can be rewritten in the following form:
Differentiating both sides of the above identity with respect to x and substituting
Taking into account the overmeasured data
Now we can determine the value
Taking limit as
The righthand side of the above identity defines the semigroup representation of the inputoutput mapping
Differentiating both sides of identity (17) with respect to t, we get
Using semigroup properties, we obtain
Taking
Since
Solving this equation for
Then we can define the admissible set of diffusion coefficients as follows:
The following lemma implies the relation between the coefficients
Lemma 3.1Let
holds, then the outputs
for each
Proof The solutions of the direct problem (15) corresponding to the admissible coefficients
respectively, by using formula (20). From definition (18), it is obvious that
This lemma with identity (23) implies the following conclusion.
Corollary 3.1Let the conditions of Lemma 3.1 hold. Then
hold.
Since the null space of it consists of only zero function, i.e.,
Theorem 3.1Let conditions (C1) and (C2) hold. Assume that
4 The inverse problem with mixed output data
Consider now the inverse problem (1)(2) with two measured output data
On this set, both inputoutput mappings
Corollary 4.1The inputoutput mappings
5 Conclusion
The aim of this study was to analyze distinguishability properties of the inputoutput
mappings
This study shows that boundary conditions and the region on which the problem is defined
have a significant impact on the distinguishability of the inputoutput mappings
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The research was supported in part by the Scientific and Technical Research Council (TUBITAK) and Izmir University of Economics.
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