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

- Ebru Ozbilge
^{1}Email author and - Ali Demir
^{2}

**2013**:43

https://doi.org/10.1186/1687-2770-2013-43

© Ozbilge and Demir; licensee Springer. 2013

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

## 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 $k(0){u}_{x}(0,t)={\psi}_{0}$, $u(1,t)={\psi}_{1}$. The aim of this paper is to investigate the distinguishability of the input-output mappings $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$, $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ 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 $\mathrm{\Phi}[\cdot ]$ and $\mathrm{\Psi}[\cdot ]$ 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 ${k}^{\prime}(0)$ and ${k}^{\prime}(1)$ 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 $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$, $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ are given explicitly in terms of the semigroup.

## Keywords

- Inverse Problem
- Null Space
- Distinguishability Property
- Initial Boundary
- Parabolic Problem

## 1 Introduction

where ${\mathrm{\Omega}}_{T}=\{(x,t)\in {R}^{2}:0<x<1,0<t\le T\}$. The left flux ${\psi}_{0}$ and the right boundary condition ${\psi}_{1}$ are assumed to be constants. The functions ${c}_{1}>k(x)\ge {c}_{0}>0$ and $g(x)$ satisfy the following conditions:

(C1) $k(x)\in {H}^{1,2}[0,1]$;

(C2) $g(x)\in {H}^{2,2}[0,1]$, ${g}^{\prime}(0)={\psi}_{0}$, $g(1)={\psi}_{1}$.

Under these conditions, the initial boundary value problem (1) has the unique solution $u(x,t)\in {H}^{2,2}[0,1]\cap {H}^{1,2}[0,1]$ [1–4].

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

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 $f(0)=g(0)$, ${h}^{\prime}(0)=k(1){g}^{\prime}(1)$.

We denote by $\mathcal{K}:=\{k(x)\in {H}^{1,2}[0,1]:{c}_{1}>k(x)\ge {c}_{0}>0,x\in [0,1]\}\subset {H}^{1,2}[0,1]$, the set of admissible coefficients $k=k(x)$. The monotonicity, continuity and hence invertibility of the input-output mappings $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ and $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ 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 $\mathrm{\Phi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$ (or $\mathrm{\Psi}[\cdot ]:\mathcal{K}\to {H}^{1,2}[0,T]$) has the distinguishability property if $\mathrm{\Phi}[{k}_{1}]\ne \mathrm{\Phi}[{k}_{2}]$ ($\mathrm{\Psi}[{k}_{1}]\ne \mathrm{\Psi}[{k}_{2}]$) implies ${k}_{1}(x)\ne {k}_{2}(x)$. This, in particular, means injectivity of the inverse mappings ${\mathrm{\Phi}}^{-1}$ and ${\mathrm{\Psi}}^{-1}$.

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 $f(t)$

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

*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:

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 $k(x)$.

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

*x*and using semigroup properties at $x=0$ yield

*the semigroup representation of the input-output mapping*$\mathrm{\Phi}[k]$ on the set of admissible unknown diffusion coefficients $\mathcal{K}$:

*t*:

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

**Lemma 2.1**

*Let*${u}_{1}(x,t)=u(x,t;{k}_{1})$

*and*${u}_{2}(x,t)=u(x,t;{k}_{2})$

*be solutions of the direct problem*(5)

*corresponding to the admissible coefficients*${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$.

*Suppose that*${f}_{j}(t)=u(0,t;{k}_{j})$, $j=1,2$,

*are the corresponding outputs and denote by*$\mathrm{\Delta}f(t)={f}_{1}(t)-{f}_{2}(t)$, $\mathrm{\Delta}\xi (x,t)={\xi}^{1}(x,t)-{\xi}^{2}(x,t)$.

*If the condition*

*holds*,

*then the outputs*${f}_{j}(t)$, $j=1,2$,

*satisfy the following integral identity*:

*for each* $\tau \in (0,T]$.

*Proof*The solutions of the direct problem (5) corresponding to the admissible coefficients ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$ can be written at $x=0$ as follows:

respectively, by using representation (11). From identity (9) it is obvious that ${\zeta}^{1}(0,\tau )={\zeta}^{2}(0,\tau )$ for each $\tau \in (0,T]$. 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*${f}_{1}(t)={f}_{2}(t)$, $\mathrm{\forall}t\in [0,T]$,

*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.*, $N(T)=\{0\}$. Thus Corollary 2.1 states that ${f}_{1}\equiv {f}_{2}$ if and only if ${\xi}^{1}(x,t)-{\xi}^{2}(x,t)=0$ for all $(x,t)\in {\mathrm{\Omega}}_{T}$. The definition of $\xi (x,t)$ implies that ${k}_{1}(x)={k}_{2}(x)$ for all $x\in [0,1]$.

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 $\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{0}\to {H}^{1,2}[0,T]$.

**Theorem 2.1**

*Let conditions*(C1)

*and*(C2)

*hold*.

*Assume that*$\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{0}\to {H}^{1,2}[0,T]$

*is the input*-

*output mapping defined by*(3)

*and corresponding to the measured output*$f(t):=u(0,t)$.

*Then the mapping*$\mathrm{\Phi}[k]$

*has the distinguishability property in the class of admissible coefficients*${\mathcal{K}}_{0}$,

*i*.

*e*.,

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

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

*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:

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.*, $N(S)=\mathrm{\varnothing}$. This result is very important for the uniqueness of the unknown coefficient $k(x)$.

*x*and substituting $x=1$ yield

*the semigroup representation of the input-output mapping*$\mathrm{\Psi}[k]$ on the set of admissible unknown diffusion coefficient $\mathcal{K}$:

*t*, we get

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

**Lemma 3.1**

*Let*${u}_{1}(x,t)=u(x,t;{k}_{1})$

*and*${u}_{2}(x,t)=u(x,t;{k}_{2})$

*be solutions of the direct problem*(16)

*corresponding to the admissible coefficients*${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$.

*Suppose that*${h}_{j}(t)=u(1,t;{k}_{j})$, $j=1,2$,

*are the corresponding outputs and denote by*$\mathrm{\Delta}h(t)={h}_{1}(t)-{h}_{2}(t)$, $\mathrm{\Delta}{w}_{1}(x,t,s)={w}_{1}^{1}(x,t,s)-{w}_{1}^{2}(x,t,s)$.

*If the condition*

*holds*,

*then the outputs*${h}_{j}(t)$, $j=1,2$,

*satisfy the following integral identity*:

*for each* $\tau \in (0,T]$.

*Proof*The solutions of the direct problem (15) corresponding to the admissible coefficients ${k}_{1}(x),{k}_{2}(x)\in \mathcal{K}$ can be written at $x=1$ as follows:

respectively, by using formula (20). From definition (18), it is obvious that ${z}_{1}^{1}(1,\tau )={z}_{1}^{2}(1,\tau )$ for each $\tau \in (0,T]$. Hence the difference of these formulas implies the desired result. □

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

**Corollary 3.1**

*Let the conditions of Lemma*3.1

*hold*.

*Then*${h}_{1}(t)={h}_{2}(t)$, $\mathrm{\forall}t\in [0,T]$,

*if and only if*

*hold*.

Since the null space of it consists of only zero function, *i.e.*, $N(S)=\{0\}$, Corollary 3.1 states that ${h}_{1}\equiv {h}_{2}$ if and only if ${\chi}^{1}(x,t)-{\chi}^{2}(x,t)=0$ for all $(x,t)\in {\mathrm{\Omega}}_{T}$. The definition of $\chi (x,t)$ implies that ${k}_{1}(x)={k}_{2}(x)$ for all $x\in (0,1]$.

**Theorem 3.1**

*Let conditions*(C1)

*and*(C2)

*hold*.

*Assume that*$\mathrm{\Psi}[\cdot ]:{\mathcal{K}}_{1}\to {C}^{1}[0,T]$

*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*$\mathrm{\Psi}[k]$

*has the distinguishability property in the class of admissible coefficients*${\mathcal{K}}_{1}$,

*i*.

*e*.,

## 4 The inverse problem with mixed output data

On this set, both input-output mappings $\mathrm{\Phi}[k]$ and $\mathrm{\Psi}[k]$ have distinguishability property.

**Corollary 4.1**

*The input*-

*output mappings*$\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$

*and*$\mathrm{\Psi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$

*distinguish any two functions*${k}_{1}(x)\ne {k}_{2}(x)$

*from the set*${\mathcal{K}}_{2}$,

*i*.

*e*.,

## 5 Conclusion

The aim of this study was to analyze distinguishability properties of the input-output mappings $\mathrm{\Phi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$ and $\mathrm{\Psi}[\cdot ]:{\mathcal{K}}_{2}\to {H}^{1,2}[0,T]$ 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 $\mathrm{\Phi}[\cdot ]$ and $\mathrm{\Psi}[\cdot ]$ 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 $\mathrm{\Phi}[\cdot ]$ and $\mathrm{\Psi}[\cdot ]$ since these key elements determine the structure of the semigroups $T(t)$ and $S(t)$ of linear operators and their null spaces.

## Declarations

### 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.

## Authors’ Affiliations

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