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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

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2013/1/43


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

© 2013 Ozbilge and Demir; licensee Springer

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a> in the linear parabolic equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M2">View MathML</a> with mixed boundary conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M4">View MathML</a>. The aim of this paper is to investigate the distinguishability of the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M6">View MathML</a> via semigroup theory. In this paper, we show that if the null space of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> consists of only zero function, then the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9">View MathML</a> 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) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M10">View MathML</a> or/and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M11">View MathML</a>, the values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13">View MathML</a> of the unknown diffusion coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a>, respectively, can be determined explicitly. In addition to these, the values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M18">View MathML</a> of the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a>, respectively, are also determined via the input data. Furthermore, it is shown that measured output data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> can be determined analytically by an integral representation. Hence the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M25">View MathML</a> are given explicitly in terms of the semigroup.

1 Introduction

Consider the following initial boundary value problem:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M26">View MathML</a>

(1)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M27">View MathML</a>. The left flux <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M28">View MathML</a> and the right boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M29">View MathML</a> are assumed to be constants. The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M30">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31">View MathML</a> satisfy the following conditions:

(C1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M32">View MathML</a>;

(C2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M33">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M34">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M35">View MathML</a>.

Under these conditions, the initial boundary value problem (1) has the unique solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M36">View MathML</a>[1-4].

Consider the inverse problem of determining the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M37">View MathML</a>[5-9] from the following observations at the boundaries <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M40">View MathML</a>

(2)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M41">View MathML</a> is the solution of the parabolic problem (1). The functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a>. It is assumed that the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> belong to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M48">View MathML</a> and satisfy the consistency conditions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M49">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M50">View MathML</a>.

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M51">View MathML</a>, the set of admissible coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M37">View MathML</a> and introduce the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M53">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M25">View MathML</a>, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M55">View MathML</a>

(3)

Then the inverse problem [10] with the measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> can be formulated as the following operator equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M58">View MathML</a>

(4)

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M59">View MathML</a>, the set of admissible coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M37">View MathML</a>. The monotonicity, continuity and hence invertibility of the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M62">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M5">View MathML</a> (or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M64">View MathML</a>) has the distinguishability property if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M65">View MathML</a> (<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M66">View MathML</a>) implies <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M67">View MathML</a>. This, in particular, means injectivity of the inverse mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M68">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M69">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a>. A similar analysis is applied to the inverse problem with the single measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> given at the point <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a> in Section 3. The inverse problem with two Neumann measured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> is discussed in Section 4. Finally, some concluding remarks are given in Section 5.

2 Analysis of the inverse problem with measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a>

Consider now the inverse problem with one measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a>. 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M78">View MathML</a>

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

(5)

Here we assume that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12">View MathML</a> was known. Later we will determine the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12">View MathML</a>. In order to formulate the solution of the parabolic problem (5) in terms of a semigroup, we need to define the following function:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M82">View MathML</a>

(6)

which satisfies the following parabolic problem:

(7)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M84">View MathML</a> is a second-order differential operator, its domain is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M85">View MathML</a>. Since the initial value function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M87">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M88">View MathML</a>.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M92">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M93">View MathML</a> and the corresponding eigenfunctions are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M94">View MathML</a>. In this case, the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> can be represented in the following way:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M96">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M97">View MathML</a>. The null space of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> of the linear operators can be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M99">View MathML</a>

From the definition of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a>, we can say that the null space of it is an empty set, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M101">View MathML</a>. This result is very important for the uniqueness of the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a>.

The unique solution of the initial value problem (7) in terms of a semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> can be represented in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M104">View MathML</a>

Hence, by using identity (6), the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M105">View MathML</a> of the parabolic problem (5) in terms of a semigroup can be written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M106">View MathML</a>

(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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M109','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M109">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M110','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M110">View MathML</a> in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M111">View MathML</a>

Substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> into this solution representation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M113">View MathML</a>

Taking into account the overmeasured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M114">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M115">View MathML</a>

(11)

which implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> can be determined analytically.

Differentiating both sides of the above identity with respect to x and using semigroup properties at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M118">View MathML</a>

Using the boundary condition <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M3">View MathML</a>, we can write <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M120">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M121','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M121">View MathML</a> which can be rewritten in terms of a semigroup in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M122">View MathML</a>

Taking limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M123">View MathML</a> in the above identity, we obtain the following explicit formula for the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12">View MathML</a> of the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M126">View MathML</a>

The right-hand side of identity (11) defines explicitly the semigroup representation of the input-output mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M127">View MathML</a> on the set of admissible unknown diffusion coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M128">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M129">View MathML</a>

(12)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M130">View MathML</a>

Using the semigroup property <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M131','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M131">View MathML</a>, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M132">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> in the above identity, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M134">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M135">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M136">View MathML</a>. Taking into account this and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M137">View MathML</a> yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M138">View MathML</a>

Solving this equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17">View MathML</a> and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M140">View MathML</a>, we obtain the following explicit formula for the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17">View MathML</a> of the first derivative <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M142">View MathML</a> of the unknown coefficient at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M144','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M144">View MathML</a>

(13)

Under the determined values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M17">View MathML</a>, the set of admissible coefficients can be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M147">View MathML</a>

The following lemma implies the relationship between the diffusion coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M148">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> and the corresponding outputs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M150">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151">View MathML</a>.

Lemma 2.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M152">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M153">View MathML</a>be solutions of the direct problem (5) corresponding to the admissible coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M154">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M155">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151">View MathML</a>, are the corresponding outputs and denote by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M157">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M158">View MathML</a>. If the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M159">View MathML</a>

holds, then the outputs<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M160">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151">View MathML</a>, satisfy the following integral identity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M162','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M162">View MathML</a>

(14)

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M163">View MathML</a>.

Proof The solutions of the direct problem (5) corresponding to the admissible coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M164">View MathML</a> can be written at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M15">View MathML</a> as follows:

respectively, by using representation (11). From identity (9) it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M167">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M168">View MathML</a>. 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M169','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M169">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M170">View MathML</a>, if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M171','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M171">View MathML</a>

Since the Strum-Liouville problem generates a complete orthogonal family of eigenfunctions, the null space of a semigroup contains only zero function, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M172','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M172">View MathML</a>. Thus Corollary 2.1 states that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M173','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M173">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M174','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M174">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M175">View MathML</a>. The definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M176','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M176">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M177">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M178','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M178">View MathML</a>.

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M179">View MathML</a>.

Theorem 2.1Let conditions (C1) and (C2) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M179','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M179">View MathML</a>is the input-output mapping defined by (3) and corresponding to the measured output<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M10">View MathML</a>. Then the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M127">View MathML</a>has the distinguishability property in the class of admissible coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M183','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M183">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M184','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M184">View MathML</a>

3 Analysis of the inverse problem with measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a>

Consider now the inverse problem with one measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a>. As in the previous section, let us arrange the parabolic equation as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M188','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M188">View MathML</a>

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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M190','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M190">View MathML</a> in identity (6), which satisfies the following parabolic problem:

(16)

Here <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M192','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M192">View MathML</a> is a second-order differential operator, its domain is <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M193','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M193">View MathML</a>. Since the initial value function <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M31">View MathML</a> belongs to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M195','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M195">View MathML</a>, it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M196','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M196">View MathML</a>.

Denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> 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:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M199','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M199">View MathML</a>

This problem is called a Sturm-Liouville problem. We can easily determine that the eigenvalues are <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M200','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M200">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M201','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M201">View MathML</a> and the corresponding eigenfunctions become <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M202','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M202">View MathML</a>. Hence the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> can be represented in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M204','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M204">View MathML</a>

The null space of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> of the linear operators can be defined as follows:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M206','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M206">View MathML</a>

Since the Sturm-Liouville problem generates a complete orthogonal family of eigenfunctions, we can say that the null space of the semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> is an empty set, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M208','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M208">View MathML</a>. This result is very important for the uniqueness of the unknown coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M1">View MathML</a>.

The unique solution of the initial value problem (16) in terms of a semigroup <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> can be represented in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M211','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M211">View MathML</a>

Hence, by using identity (6), the solution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M105">View MathML</a> of the parabolic problem (15) in terms of a semigroup can be written in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M213','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M213">View MathML</a>

(17)

Defining the following:

(18)

(19)

(20)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M217','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M217">View MathML</a>

Differentiating both sides of the above identity with respect to x and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a> yield

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M219','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M219">View MathML</a>

Taking into account the overmeasured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M220">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M221','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M221">View MathML</a>

(21)

Now we can determine the value <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13">View MathML</a>. From the overmeasured data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M220','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M220">View MathML</a>, the identity <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M224','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M224">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M225','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M225">View MathML</a> can be rewritten in terms of a semigroup in the following form:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M226','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M226">View MathML</a>

Taking limit as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M123">View MathML</a> in the above identity yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M228','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M228">View MathML</a>

The right-hand side of the above identity defines the semigroup representation of the input-output mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M229">View MathML</a> on the set of admissible unknown diffusion coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M128">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M231','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M231">View MathML</a>

(22)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M232','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M232">View MathML</a>

Using semigroup properties, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M233','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M233">View MathML</a>

Taking <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a> in the above identity, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M235','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M235">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M236','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M236">View MathML</a>, we have <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M237','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M237">View MathML</a>. Taking into account this and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M137">View MathML</a>, we get

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M239','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M239">View MathML</a>

Solving this equation for <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M18">View MathML</a> and substituting <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M241','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M241">View MathML</a>, we reach the following result:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M242','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M242">View MathML</a>

(23)

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

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M243','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M243">View MathML</a>

The following lemma implies the relation between the coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M164">View MathML</a> at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a> and the corresponding outputs <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M246','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M246">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151">View MathML</a>.

Lemma 3.1Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M152">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M153">View MathML</a>be solutions of the direct problem (16) corresponding to the admissible coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M154">View MathML</a>. Suppose that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M251','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M251">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151">View MathML</a>, are the corresponding outputs and denote by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M253','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M253">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M254','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M254">View MathML</a>. If the condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M255','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M255">View MathML</a>

holds, then the outputs<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M256','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M256">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M151">View MathML</a>, satisfy the following integral identity:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M258','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M258">View MathML</a>

(24)

for each<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M163">View MathML</a>.

Proof The solutions of the direct problem (15) corresponding to the admissible coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M164">View MathML</a> can be written at <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M16">View MathML</a> as follows:

respectively, by using formula (20). From definition (18), it is obvious that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M263','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M263">View MathML</a> for each <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M163">View MathML</a>. 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<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M265','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M265">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M170','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M170">View MathML</a>, if and only if

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M267','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M267">View MathML</a>

hold.

Since the null space of it consists of only zero function, i.e., <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M268','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M268">View MathML</a>, Corollary 3.1 states that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M269','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M269">View MathML</a> if and only if <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M270','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M270">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M175','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M175">View MathML</a>. The definition of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M272','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M272">View MathML</a> implies that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M177','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M177">View MathML</a> for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M274','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M274">View MathML</a>.

Theorem 3.1Let conditions (C1) and (C2) hold. Assume that<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M275','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M275">View MathML</a>is the input-output mapping defined by (3) and corresponding to the measured output<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M11">View MathML</a>. Then the mapping<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M229">View MathML</a>has the distinguishability property in the class of admissible coefficients<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M278','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M278">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M279','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M279">View MathML</a>

4 The inverse problem with mixed output data

Consider now the inverse problem (1)-(2) with two measured output data <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M22">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M23">View MathML</a>. As shown before, having these two data, the values <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M12">View MathML</a> as well as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M13">View MathML</a> can be defined by the above explicit formulas. Based on this result, let us define now the set of admissible coefficients <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M284">View MathML</a> as an intersection:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M285','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M285">View MathML</a>

On this set, both input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M127">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M229','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M229">View MathML</a> have distinguishability property.

Corollary 4.1The input-output mappings<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M288">View MathML</a>and<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M289">View MathML</a>distinguish any two functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M290','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M290">View MathML</a>from the set<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M284','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M284">View MathML</a>, i.e.,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M292','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M292">View MathML</a>

5 Conclusion

The aim of this study was to analyze distinguishability properties of the input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M288','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M288">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M289','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M289">View MathML</a> which are naturally determined by the measured output data. In this paper we show that if the null spaces of the semigroups <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> include only zero function then the corresponding input-output mappings <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9">View MathML</a> 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 <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M8">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M9">View MathML</a> since these key elements determine the structure of the semigroups <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M7">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2013/1/43/mathml/M197">View MathML</a> 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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