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# Semigroup approach for identification of the unknown diffusion coefficient in a linear parabolic equation with mixed output data

Ebru Ozbilge1* and Ali Demir2

### Author affiliations

1 Department of Mathematics, Faculty of Science and Literature, Izmir University of Economics, Sakarya Caddesi, No. 156, Balcova, Izmir, 35330, Turkey

2 Department of Mathematics, Kocaeli University, Umuttepe, Izmit, Kocaeli, 41380, Turkey

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Boundary Value Problems 2013, 2013:43  doi:10.1186/1687-2770-2013-43

 Received: 3 January 2013 Accepted: 14 February 2013 Published: 1 March 2013

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

This article presents a semigroup approach for the mathematical analysis of the inverse coefficient problems of identifying the unknown coefficient k ( x ) in the linear parabolic equation u t ( x , t ) = ( k ( x ) u x ( x , t ) ) x with mixed boundary conditions k ( 0 ) u x ( 0 , t ) = ψ 0 , u ( 1 , t ) = ψ 1 . The aim of this paper is to investigate the distinguishability of the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] , Ψ [ ] : K H 1 , 2 [ 0 , T ] via semigroup theory. In this paper, we show that if the null space of the semigroup T ( t ) consists of only zero function, then the input-output mappings Φ [ ] and Ψ [ ] have the distinguishability property. It is shown that the types of the boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability property of these mappings. Moreover, in the light of measured output data (boundary observations) f ( t ) : = u ( 0 , t ) or/and h ( t ) : = k ( 1 ) u x ( 1 , t ) , the values k ( 0 ) and k ( 1 ) of the unknown diffusion coefficient k ( x ) at x = 0 and x = 1 , respectively, can be determined explicitly. In addition to these, the values k ( 0 ) and k ( 1 ) of the unknown coefficient k ( x ) at x = 0 and x = 1 , respectively, are also determined via the input data. Furthermore, it is shown that measured output data f ( t ) and h ( t ) can be determined analytically by an integral representation. Hence the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] , Ψ [ ] : K H 1 , 2 [ 0 , T ] are given explicitly in terms of the semigroup.

### 1 Introduction

Consider the following initial boundary value problem:

{ u t ( x , t ) = ( k ( x ) u x ( x , t ) ) x , ( x , t ) Ω T , u ( x , 0 ) = g ( x ) , 0 < x < 1 , k ( 0 ) u x ( 0 , t ) = ψ 0 , u ( 1 , t ) = ψ 1 , 0 < t < T , (1)

where Ω T = { ( x , t ) R 2 : 0 < x < 1 , 0 < t T } . The left flux ψ 0 and the right boundary condition ψ 1 are assumed to be constants. The functions c 1 > k ( x ) c 0 > 0 and g ( x ) satisfy the following conditions:

(C1) k ( x ) H 1 , 2 [ 0 , 1 ] ;

(C2) g ( x ) H 2 , 2 [ 0 , 1 ] , g ( 0 ) = ψ 0 , g ( 1 ) = ψ 1 .

Under these conditions, the initial boundary value problem (1) has the unique solution u ( x , t ) H 2 , 2 [ 0 , 1 ] H 1 , 2 [ 0 , 1 ] [1-4].

Consider the inverse problem of determining the unknown coefficient k = k ( x ) [5-9] from the following observations at the boundaries x = 0 and x = 1 :

u ( 0 , t ) = f ( t ) , k ( 1 ) u x ( 1 , t ) = h ( t ) , t ( 0 , T ] . (2)

Here u = u ( x , t ) is the solution of the parabolic problem (1). The functions f ( t ) , h ( t ) are assumed to be noisy free measured output data. In this context, the parabolic problem (1) will be referred to as a direct (forward) problem with the inputs g ( x ) and k ( x ) . It is assumed that the functions f ( t ) and h ( t ) belong to H 1 , 2 [ 0 , T ] and satisfy the consistency conditions f ( 0 ) = g ( 0 ) , h ( 0 ) = k ( 1 ) g ( 1 ) .

We denote by K : = { k ( x ) H 1 , 2 [ 0 , 1 ] : c 1 > k ( x ) c 0 > 0 , x [ 0 , 1 ] } H 1 , 2 [ 0 , 1 ] , the set of admissible coefficients k = k ( x ) and introduce the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] , Ψ [ ] : K H 1 , 2 [ 0 , T ] , where

Φ [ k ] = u ( x , t ; k ) | x = 0 , Ψ [ k ] = k ( x ) u x ( x , t ; k ) | x = 1 , k K , f ( t ) , h ( t ) H 1 , 2 [ 0 , T ] . (3)

Then the inverse problem [10] with the measured data f ( t ) and h ( t ) can be formulated as the following operator equations:

Φ [ k ] = f , Ψ [ k ] = h , k K , f , h H 1 , 2 [ 0 , T ] . (4)

We denote by K : = { k ( x ) H 1 , 2 [ 0 , 1 ] : c 1 > k ( x ) c 0 > 0 , x [ 0 , 1 ] } H 1 , 2 [ 0 , 1 ] , the set of admissible coefficients k = k ( x ) . The monotonicity, continuity and hence invertibility of the input-output mappings Φ [ ] : K H 1 , 2 [ 0 , T ] and Ψ [ ] : K H 1 , 2 [ 0 , T ] are given in [3,4].

The aim of this paper is to study a distinguishability of the unknown coefficient via the above input-output mappings. We say that the mapping Φ [ ] : K H 1 , 2 [ 0 , T ] (or Ψ [ ] : K H 1 , 2 [ 0 , T ] ) has the distinguishability property if Φ [ k 1 ] Φ [ k 2 ] ( Ψ [ k 1 ] Ψ [ k 2 ] ) implies k 1 ( x ) k 2 ( x ) . This, in particular, means injectivity of the inverse mappings Φ 1 and Ψ 1 .

The purpose of this paper is to study the distinguishability of the unknown coefficient via the above input-output 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 [12-15].

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 f ( t ) . A similar analysis is applied to the inverse problem with the single measured output data h ( t ) given at the point x = 1 in Section 3. The inverse problem with two Neumann measured data f ( t ) and h ( t ) is discussed in Section 4. Finally, some concluding remarks are given in Section 5.

### 2 Analysis of the inverse problem with measured output data f ( t )

Consider now the inverse problem with one measured output data f ( t ) at x = 0 . 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:

u t ( x , t ) ( k ( 0 ) u x ( x , t ) ) x = ( ( k ( x ) k ( 0 ) ) u x ( x , t ) ) x , ( x , t ) Ω T .

Then the initial boundary value problem (1) can be rewritten in the following form:

(5)

Here we assume that k ( 0 ) was known. Later we will determine the value k ( 0 ) . In order to formulate the solution of the parabolic problem (5) in terms of a semigroup, we need to define the following function:

v ( x , t ) = u ( x , t ) ψ 0 k ( 0 ) x + ψ 0 ψ 1 , x [ 0 , 1 ] (6)

which satisfies the following parabolic problem:

(7)

Here A [ v ( x , t ) ] : = k ( 0 ) d 2 v ( x , t ) / d x 2 is a second-order differential operator, its domain is D A = { u H 2 , 2 ( 0 , 1 ) H 1 , 2 [ 0 , 1 ] : u x ( 0 ) = u ( 1 ) = 0 } . Since the initial value function g ( x ) belongs to C 2 [ 0 , 1 ] , it is obvious that g ( x ) D A .

Denote by T ( t ) 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 T ( t ) 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 λ n = k ( 0 ) ( 2 n 1 ) 2 π 2 / 4 for all n = 1 , and the corresponding eigenfunctions are ϕ n ( x ) = 2 cos ( ( 2 n 1 ) x π / 2 ) . In this case, the semigroup T ( t ) can be represented in the following way:

T ( t ) U ( x , s ) = n = 0 ϕ n ( x ) , U ( x , s ) e λ n t ϕ n ( x ) ,

where ϕ n ( x ) , U ( x , s ) = 0 1 ϕ n ( x ) U ( x , s ) d x . The null space of the semigroup T ( t ) of the linear operators can be defined as follows:

N ( T ) = { U ( x , s ) : ϕ n ( x ) , U ( x , s ) = 0 ,  for all  n = 1 , 2 , 3 , } .

From the definition of the semigroup T ( t ) , we can say that the null space of it is an empty set, i.e., N ( T ) = { 0 } . This result is very important for the uniqueness of the unknown coefficient k ( x ) .

The unique solution of the initial value problem (7) in terms of a semigroup T ( t ) can be represented in the following form:

v ( x , t ) = T ( t ) v ( x , 0 ) + 0 t T ( t s ) ( ( k ( x ) k ( 0 ) ) ( v x ( x , t ) + ψ 0 k ( 0 ) ) ) x d s .

Hence, by using identity (6), the solution u ( x , t ) of the parabolic problem (5) in terms of a semigroup can be written in the following form:

u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 + T ( t ) ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 ψ 1 ) + 0 t T ( t s ) ( ( k ( x ) k ( 0 ) ) u x ( x , s ) ) x d s . (8)

In order to arrange the above solution representation, let us define the following:

(9)

(10)

Then we can rewrite the solution representation in terms of ζ ( x ) and ξ ( x , s ) in the following form:

u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 + T ( t ) ζ ( x ) + 0 t T ( t s ) ξ ( x , s ) d s .

Substituting x = 0 into this solution representation yields

u ( 0 , t ) = ψ 1 ψ 0 + T ( t ) ζ ( 0 ) + 0 t T ( t s ) ξ ( 0 , s ) d s .

Taking into account the overmeasured data u ( 0 , t ) = f ( t ) , we get

f ( t ) = ( ψ 1 ψ 0 + T ( t ) ζ ( 0 ) + 0 t T ( t s ) ξ ( 0 , s ) d s ) , (11)

which implies that f ( t ) can be determined analytically.

Differentiating both sides of the above identity with respect to x and using semigroup properties at x = 0 yield

u x ( 0 , t ) = ψ 0 k ( 0 ) + z ( 0 , t ) + 0 t w ( 0 , t s , s ) d s .

Using the boundary condition k ( 0 ) u x ( 0 , t ) = ψ 0 , we can write k ( 0 ) = ψ 0 / u x ( 0 , t ) for all t 0 which can be rewritten in terms of a semigroup in the following form:

k ( 0 ) = ψ 0 / ( ψ 0 + ψ 1 + z ( 0 , t ) + 0 t w ( 0 , t s , s ) d s ) .

Taking limit as t 0 in the above identity, we obtain the following explicit formula for the value k ( 0 ) of the unknown coefficient k ( x ) :

k ( 0 ) = ψ 0 / ( ψ 0 + ψ 1 + z ( 0 , 0 ) ) .

The right-hand side of identity (11) defines explicitly the semigroup representation of the input-output mapping Φ [ k ] on the set of admissible unknown diffusion coefficients K :

Φ [ k ] ( x ) : = ψ 1 ψ 0 + T ( t ) ζ ( 0 ) + 0 t T ( t s ) ξ ( 0 , s ) d s , t [ 0 , T ] . (12)

Let us differentiate now both sides of identity (8) with respect to t:

u t ( x , t ) = T ( t ) A ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 ψ 1 ) + ( ( k ( x ) k ( 0 ) ) u x ( x , t ) ) x + 0 t A T ( t s ) ( ( k ( x ) k ( 0 ) ) u x ( x , s ) ) x d s .

Using the semigroup property 0 t A T ( s ) u ( x , s ) d s = T ( t ) u ( x , t ) T ( 0 ) u ( x , t ) , we obtain

u t ( x , t ) = k ( 0 ) T ( t ) g ( x ) 2 T ( 0 ) ( ( k ( x ) k ( 0 ) ) u x ( x , t ) ) x + T ( t ) ( ( k ( x ) k ( 0 ) ) u x ( x , 0 ) ) x .

Taking x = 0 in the above identity, we get

u t ( 0 , t ) = k ( 0 ) T ( t ) g ( 0 ) T ( 0 ) k ( 0 ) u x ( 0 , 0 ) + T ( t ) ( k ( 0 ) u x ( 0 , 0 ) ) T ( 0 ) ( k ( 0 ) u x ( 0 , t ) ) .

Since u ( 0 , t ) = f ( t ) , we have u t ( 0 , t ) = f ( t ) . Taking into account this and substituting t = 0 yield

f ( 0 ) = k ( 0 ) g ( 0 ) k ( 0 ) g ( 0 ) k ( 0 ) .

Solving this equation for k ( 0 ) and substituting u x ( 0 , 0 ) = g ( 0 ) / k ( 0 ) , we obtain the following explicit formula for the value k ( 0 ) of the first derivative k ( x ) of the unknown coefficient at x = 0 :

k ( 0 ) = k 2 ( 0 ) g ( 0 ) k ( 0 ) f ( 0 ) g ( 0 ) . (13)

Under the determined values k ( 0 ) and k ( 0 ) , the set of admissible coefficients can be defined as follows:

K 0 : = { k K : k ( 0 ) = ψ 0 ψ 0 + ψ 1 + z ( 0 , 0 ) , k ( 0 ) = k 2 ( 0 ) g ( 0 ) k ( 0 ) f ( 0 ) g ( 0 ) } .

The following lemma implies the relationship between the diffusion coefficients k 1 ( x ) , k 2 ( x ) K at x = 0 and the corresponding outputs f j ( t ) : = u ( 0 , t ; k j ) , j = 1 , 2 .

Lemma 2.1Let u 1 ( x , t ) = u ( x , t ; k 1 ) and u 2 ( x , t ) = u ( x , t ; k 2 ) be solutions of the direct problem (5) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K . Suppose that f j ( t ) = u ( 0 , t ; k j ) , j = 1 , 2 , are the corresponding outputs and denote by Δ f ( t ) = f 1 ( t ) f 2 ( t ) , Δ ξ ( x , t ) = ξ 1 ( x , t ) ξ 2 ( x , t ) . If the condition

k 1 ( 0 ) = k 2 ( 0 ) : = k ( 0 )

holds, then the outputs f j ( t ) , j = 1 , 2 , satisfy the following integral identity:

Δ f ( τ ) = 0 τ T ( τ s ) Δ ξ ( 0 , s ) d s d s (14)

for each τ ( 0 , T ] .

Proof The solutions of the direct problem (5) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K can be written at x = 0 as follows:

respectively, by using representation (11). From identity (9) it is obvious that ζ 1 ( 0 , τ ) = ζ 2 ( 0 , τ ) for each τ ( 0 , T ] . Hence the difference of these formulas implies the desired result. □

This lemma with identity (14) implies the following.

Corollary 2.1Let conditions of Lemma 2.1 hold. Then f 1 ( t ) = f 2 ( t ) , t [ 0 , T ] , if and only if

ϕ n ( x ) , Δ ξ ( 0 , s ) = ϕ n ( x ) , ξ 1 ( x , t ) ξ 2 ( x , t ) = 0 , t ( 0 , T ] , n = 0 , 1 ,

Since the Strum-Liouville problem generates a complete orthogonal family of eigenfunctions, the null space of a semigroup contains only zero function, i.e., N ( T ) = { 0 } . Thus Corollary 2.1 states that f 1 f 2 if and only if ξ 1 ( x , t ) ξ 2 ( x , t ) = 0 for all ( x , t ) Ω T . The definition of ξ ( x , t ) implies that k 1 ( x ) = k 2 ( x ) for all x [ 0 , 1 ] .

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 Φ [ ] : K 0 H 1 , 2 [ 0 , T ] .

Theorem 2.1Let conditions (C1) and (C2) hold. Assume that Φ [ ] : K 0 H 1 , 2 [ 0 , T ] is the input-output mapping defined by (3) and corresponding to the measured output f ( t ) : = u ( 0 , t ) . Then the mapping Φ [ k ] has the distinguishability property in the class of admissible coefficients K 0 , i.e.,

Φ [ k 1 ] Φ [ k 2 ] k 1 , k 2 K 0 , k 1 ( x ) k 2 ( x ) .

### 3 Analysis of the inverse problem with measured output data h ( t )

Consider now the inverse problem with one measured output data h ( t ) at x = 1 . As in the previous section, let us arrange the parabolic equation as follows:

u t ( x , t ) ( k ( 1 ) u x ( x , t ) ) x = ( ( k ( x ) k ( 1 ) ) u x ( x , t ) ) x , ( x , t ) Ω T .

Then the initial boundary value problem (1) can be rewritten in the following form:

(15)

In order to formulate the solution of the above parabolic problem in terms of a semigroup, let us use the same variable v ( x , t ) in identity (6), which satisfies the following parabolic problem:

(16)

Here B [ v ( x , t ) ] : = k ( 1 ) d 2 v ( x , t ) / d x 2 is a second-order differential operator, its domain is D B = { u H 2 , 2 ( 0 , 1 ) H 1 , 2 [ 0 , 1 ] : u x ( 0 ) = u ( 1 ) = 0 } . Since the initial value function g ( x ) belongs to H 2 , 2 [ 0 , 1 ] , it is obvious that g ( x ) D B .

Denote by S ( t ) the semigroup of linear operators generated by the operator −A[5,6]. As in the previous section, we can easily find the eigenvalues and eigenfunctions of the differential operator B. Furthermore, the semigroup S ( t ) can be easily constructed by using the eigenvalues and eigenfunctions of the differential operator B. For this reason, we first consider the following eigenvalue problem:

B ϕ ( x ) = λ ϕ ( x ) , ϕ x ( 0 ) = 0 ; ϕ ( 1 ) = 0 .

This problem is called a Sturm-Liouville problem. We can easily determine that the eigenvalues are λ n = k ( 1 ) ( 2 n 1 ) 2 π 2 / 4 for all n = 0 , 1 , and the corresponding eigenfunctions become ϕ n ( x ) = 2 cos ( ( 2 n 1 ) π / 2 x ) . Hence the semigroup S ( t ) can be represented in the following form:

S ( t ) U ( x , s ) = n = 0 ϕ n ( x ) , U ( x , s ) e λ n t ϕ n ( x ) .

The null space of the semigroup S ( t ) of the linear operators can be defined as follows:

N ( S ) = { U ( x , s ) : ϕ n ( x ) , U ( x , s ) = 0 ,  for all  n = 1 , 2 , 3 , } .

Since the Sturm-Liouville problem generates a complete orthogonal family of eigenfunctions, we can say that the null space of the semigroup S ( t ) is an empty set, i.e., N ( S ) = . This result is very important for the uniqueness of the unknown coefficient k ( x ) .

The unique solution of the initial value problem (16) in terms of a semigroup S ( t ) can be represented in the following form:

v ( x , t ) = S ( t ) v ( x , 0 ) + 0 t S ( t s ) ( ( k ( x ) k ( 1 ) ) ( v x ( x , s ) + ψ 0 k ( 0 ) ) ) x d s .

Hence, by using identity (6), the solution u ( x , t ) of the parabolic problem (15) in terms of a semigroup can be written in the following form:

u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 k ( 0 ) + S ( t ) ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 k ( 0 ) ψ 1 ) + 0 t S ( t s ) ( ( k ( x ) k ( 1 ) ) u x ( x , s ) ) x d s . (17)

Defining the following:

(18)

(19)

(20)

The solution representation of the parabolic problem (17) can be rewritten in the following form:

u ( x , t ) = ψ 0 k ( 0 ) x + ψ 1 ψ 0 k ( 0 ) + S ( t ) ζ ( x ) + 0 t S ( t s ) χ ( x , s ) d s .

Differentiating both sides of the above identity with respect to x and substituting x = 1 yield

u x ( 1 , t ) = ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s .

Taking into account the overmeasured data k ( 1 ) u x ( 1 , t ) = h ( t ) , we get

h ( t ) = k ( 1 ) ( ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s ) . (21)

Now we can determine the value k ( 1 ) . From the overmeasured data k ( 1 ) u x ( 1 , t ) = h ( t ) , the identity k ( 1 ) = h ( t ) / u x ( 1 , t ) for all t > 0 can be rewritten in terms of a semigroup in the following form:

k ( 1 ) = h ( t ) ( ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s ) .

Taking limit as t 0 in the above identity yields

k ( 1 ) = h ( 0 ) / ( ψ 0 k ( 0 ) + z 1 ( 1 , 0 ) ) .

The right-hand side of the above identity defines the semigroup representation of the input-output mapping Ψ [ k ] on the set of admissible unknown diffusion coefficient K :

Ψ [ k ] ( x ) : = k ( 1 ) ( ψ 0 k ( 0 ) + z 1 ( 1 , t ) + 0 t w 1 ( 1 , t s , s ) d s ) , t [ 0 , T ] . (22)

Differentiating both sides of identity (17) with respect to t, we get

u t ( x , t ) = S ( t ) B ( g ( x ) ψ 0 k ( 0 ) x + ψ 0 k ( 0 ) ψ 1 ) S ( 0 ) ( ( k ( x ) k ( 1 ) ) u x ( x , t ) ) x + 0 t B S ( t s ) ( ( k ( x ) k ( 1 ) ) u x ( x , s ) ) x d s .

Using semigroup properties, we obtain

u t ( x , t ) = S ( t ) k ( 1 ) g ( x ) S ( 0 ) ( k ( x ) u x ( x , t ) + ( k ( x ) k ( 1 ) ) u x x ( x , t ) ) + S ( t ) ( ( k ( x ) k ( 1 ) ) u x ( x , 0 ) ) x S ( 0 ) ( ( k ( x ) k ( 1 ) ) u x ( x , t ) ) x .

Taking x = 1 in the above identity, we get

u t ( 1 , t ) = S ( t ) k ( 1 ) g ( 1 ) 2 S ( 0 ) k ( 1 ) u x ( 1 , t ) + S ( t ) k ( 1 ) u x ( 1 , 0 ) .

Since u ( 1 , t ) = ψ 1 , we have u t ( 1 , t ) = 0 . Taking into account this and substituting t = 0 , we get

0 = k ( 1 ) g ( 1 ) k ( 1 ) u x ( 1 , 0 ) .

Solving this equation for k ( 1 ) and substituting u x ( 1 , 0 ) = h ( 0 ) / k ( 1 ) , we reach the following result:

k ( 1 ) = k 2 ( 1 ) g ( 1 ) h ( 0 ) . (23)

Then we can define the admissible set of diffusion coefficients as follows:

K 1 : = { k K : k ( 1 ) = h ( 0 ) ( ψ 0 k ( 0 ) + z 1 ( 1 , 0 ) ) , k ( 1 ) = k 2 ( 1 ) g ( 1 ) h ( 0 ) } .

The following lemma implies the relation between the coefficients k 1 ( x ) , k 2 ( x ) K at x = 1 and the corresponding outputs h j ( t ) : = k j ( 1 ) u x ( 1 , t ; k j ) , j = 1 , 2 .

Lemma 3.1Let u 1 ( x , t ) = u ( x , t ; k 1 ) and u 2 ( x , t ) = u ( x , t ; k 2 ) be solutions of the direct problem (16) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K . Suppose that h j ( t ) = u ( 1 , t ; k j ) , j = 1 , 2 , are the corresponding outputs and denote by Δ h ( t ) = h 1 ( t ) h 2 ( t ) , Δ w 1 ( x , t , s ) = w 1 1 ( x , t , s ) w 1 2 ( x , t , s ) . If the condition

k 1 ( 1 ) = k 2 ( 1 ) : = k ( 1 )

holds, then the outputs h j ( t ) , j = 1 , 2 , satisfy the following integral identity:

Δ h ( τ ) = k ( 1 ) 0 τ Δ w 1 ( 1 , τ s , s ) d s d s (24)

for each τ ( 0 , T ] .

Proof The solutions of the direct problem (15) corresponding to the admissible coefficients k 1 ( x ) , k 2 ( x ) K can be written at x = 1 as follows:

respectively, by using formula (20). From definition (18), it is obvious that z 1 1 ( 1 , τ ) = z 1 2 ( 1 , τ ) for each τ ( 0 , T ] . Hence the difference of these formulas implies the desired result. □

This lemma with identity (23) implies the following conclusion.

Corollary 3.1Let the conditions of Lemma 3.1 hold. Then h 1 ( t ) = h 2 ( t ) , t [ 0 , T ] , if and only if

ϕ n ( x ) , χ 1 ( x , t ) χ 2 ( x , t ) = 0 , t ( 0 , T ] , n = 0 , 1 ,

hold.

Since the null space of it consists of only zero function, i.e., N ( S ) = { 0 } , Corollary 3.1 states that h 1 h 2 if and only if χ 1 ( x , t ) χ 2 ( x , t ) = 0 for all ( x , t ) Ω T . The definition of χ ( x , t ) implies that k 1 ( x ) = k 2 ( x ) for all x ( 0 , 1 ] .

Theorem 3.1Let conditions (C1) and (C2) hold. Assume that Ψ [ ] : K 1 C 1 [ 0 , T ] is the input-output mapping defined by (3) and corresponding to the measured output h ( t ) : = k ( 1 ) u x ( 1 , t ) . Then the mapping Ψ [ k ] has the distinguishability property in the class of admissible coefficients K 1 , i.e.,

Ψ [ k 1 ] Ψ [ k 2 ] k 1 , k 2 K 1 , k 1 ( x ) k 2 ( x ) .

### 4 The inverse problem with mixed output data

Consider now the inverse problem (1)-(2) with two measured output data f ( t ) and h ( t ) . As shown before, having these two data, the values k ( 0 ) as well as k ( 1 ) can be defined by the above explicit formulas. Based on this result, let us define now the set of admissible coefficients K 2 as an intersection:

K 2 : = K 0 K 1 = { k K : k ( 0 ) = ψ 0 ψ 0 + ψ 1 + z ( 0 , 0 ) , k ( 1 ) = h ( 0 ) ψ 0 / k ( 0 ) + z 1 ( 1 , 0 ) , k ( 0 ) = k 2 ( 0 ) g ( 0 ) k ( 0 ) f ( 0 ) g ( 0 ) , k ( 1 ) = k 2 ( 1 ) g ( 1 ) h ( 0 ) } .

On this set, both input-output mappings Φ [ k ] and Ψ [ k ] have distinguishability property.

Corollary 4.1The input-output mappings Φ [ ] : K 2 H 1 , 2 [ 0 , T ] and Ψ [ ] : K 2 H 1 , 2 [ 0 , T ] distinguish any two functions k 1 ( x ) k 2 ( x ) from the set K 2 , i.e.,

k 1 ( x ) , k 2 ( x ) K 2 , k 1 ( x ) k 2 ( x ) , Φ [ k 1 ] Φ [ k 2 ] , Ψ [ k 1 ] Ψ [ k 2 ] .

### 5 Conclusion

The aim of this study was to analyze distinguishability properties of the input-output mappings Φ [ ] : K 2 H 1 , 2 [ 0 , T ] and Ψ [ ] : K 2 H 1 , 2 [ 0 , T ] which are naturally determined by the measured output data. In this paper we show that if the null spaces of the semigroups T ( t ) and S ( t ) include only zero function then the corresponding input-output mappings Φ [ ] and Ψ [ ] have distinguishability property.

This study shows that boundary conditions and the region on which the problem is defined have a significant impact on the distinguishability of the input-output mappings Φ [ ] and Ψ [ ] since these key elements determine the structure of the semigroups T ( t ) and S ( t ) of linear operators and their null spaces.

### 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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