Consider now the inverse problem with one measured output data at . In order to formulate the solution of the parabolic problem (1) in terms of a semigroup, let us first arrange the parabolic equation as follows:
Then the initial boundary value problem (1) can be rewritten in the following form:
Here we assume that was known. Later we will determine the value . In order to formulate the solution of the parabolic problem (5) in terms of a semigroup, we need to define the following function:
which satisfies the following parabolic problem:
Here is a second-order differential operator, its domain is . Since the initial value function belongs to , it is obvious that .
Denote by the semigroup of linear operators generated by the operator −A [5, 6]. Note that we can easily find the eigenvalues and eigenfunctions of the differential operator A. Furthermore, the semigroup can be easily constructed by using the eigenvalues and eigenfunctions of a differential operator A. For this reason, we first consider the following eigenvalue problem:
This problem is called a Sturm-Liouville problem. We can easily determine that the eigenvalues are for all and the corresponding eigenfunctions are . In this case, the semigroup can be represented in the following way:
where . The null space of the semigroup of the linear operators can be defined as follows:
From the definition of the semigroup , we can say that the null space of it is an empty set, i.e., . This result is very important for the uniqueness of the unknown coefficient .
The unique solution of the initial value problem (7) in terms of a semigroup can be represented in the following form:
Hence, by using identity (6), the solution of the parabolic problem (5) in terms of a semigroup can be written in the following form:
In order to arrange the above solution representation, let us define the following:
Then we can rewrite the solution representation in terms of and in the following form:
Substituting into this solution representation yields
Taking into account the overmeasured data , we get
which implies that can be determined analytically.
Differentiating both sides of the above identity with respect to x and using semigroup properties at yield
Using the boundary condition , we can write for all which can be rewritten in terms of a semigroup in the following form:
Taking limit as in the above identity, we obtain the following explicit formula for the value of the unknown coefficient :
The right-hand side of identity (11) defines explicitly the semigroup representation of the input-output mapping on the set of admissible unknown diffusion coefficients :
Let us differentiate now both sides of identity (8) with respect to t:
Using the semigroup property , we obtain
Taking in the above identity, we get
Since , we have . Taking into account this and substituting yield
Solving this equation for and substituting , we obtain the following explicit formula for the value of the first derivative of the unknown coefficient at :
Under the determined values and , the set of admissible coefficients can be defined as follows:
The following lemma implies the relationship between the diffusion coefficients at and the corresponding outputs , .
Lemma 2.1 Let and be solutions of the direct problem (5) corresponding to the admissible coefficients . Suppose that , , are the corresponding outputs and denote by , . If the condition
holds, then the outputs , , satisfy the following integral identity:
for each .
Proof The solutions of the direct problem (5) corresponding to the admissible coefficients can be written at as follows:
respectively, by using representation (11). From identity (9) it is obvious that for each . Hence the difference of these formulas implies the desired result. □
This lemma with identity (14) implies the following.
Corollary 2.1 Let conditions of Lemma 2.1 hold. Then , , if and only if
Since the Strum-Liouville problem generates a complete orthogonal family of eigenfunctions, the null space of a semigroup contains only zero function, i.e., . Thus Corollary 2.1 states that if and only if for all . The definition of implies that for all .
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 input-output mapping .
Theorem 2.1 Let conditions (C1) and (C2) hold. Assume that is the input-output mapping defined by (3) and corresponding to the measured output . Then the mapping has the distinguishability property in the class of admissible coefficients , i.e.,