Source degenerate identification problems with smoothing overdetermination
 Fadi Awawdeh^{1, 2}Email author,
 AK Alomari^{3} and
 Ibraheem AbuFalahah^{1}
https://doi.org/10.1186/s136620171403z
© The Author(s) 2017
Received: 13 July 2017
Accepted: 13 October 2017
Published: 26 October 2017
Abstract
We consider degenerate identification problems with smoothing overdetermination in abstract spaces. We establish an identifiability result using a projection method and suitable hypotheses on the operators involved and develop an identification method by reformulating the problem into a nondegenerate problem. Then we use perturbation results for linear operators to solve the regular problem. The introduced identification method permits one to solve the problems under the minimum restrictions on the input data. Finally, we provide applications to degenerate differential equations that appear in mathematical physics to support the theoretical results.
1 Introduction
In certain systems in physics and engineering, there arise differential equations with degeneracy. The system with a noninvertible operator at the derivative is considered as a good example of such systems. Most research in the literature was devoted to nondegenerate systems which do not cover fully the diversity of problems arising in theory and applications. In the case of degenerate problems, the analysis becomes more complicated, and somewhat different techniques are required. On the other hand, most of the important problems modeled by nonlinear degenerate partial differential equations are truly challenging and require further new approaches and techniques, and need our full attention and further efforts.
 (\(\mathcal{IP}1\)):

Given \(v_{0}\in X\) and \(g\in \mathcal{C}^{1}([0, \tau ];\mathbb{R} )\), find \(f\in \mathcal{C}([0,\tau ];\mathbb{R} )\) and a strict solution \(v\in \mathcal{C}^{1}([0,\tau ];X)\) to the degenerate Cauchy problemsatisfying the additional condition$$ \textstyle\begin{cases} \frac{dMv}{dt}=Lv(t)+f(t)z,&0\leq t\leq \tau, \\ Mv(0)=Mv_{0}, \end{cases} $$(1)More precisely, we are concerned with the determination of the conditions under which we can identify \(f\in \mathcal{C}([0,\tau ]; \mathbb{R} )\) such that v is a strict solution to the above problem, i.e.,$$ \phi \bigl[ Mv(t)\bigr]=g(t),\quad 0\leq t\leq \tau . $$(2)$$ Mv\in \mathcal{C}^{1}\bigl([0,\tau ];X\bigr), \qquad Lv\in \mathcal{C} \bigl([0, \tau ];X\bigr). $$
It should be emphasized that our identification problem (\(\mathcal{IP}1\)) is related to applications in control theory. There are some ways to consider the (\(\mathcal{IP}1\)) as being naturally connected with an optimal control problem [1].
1.1 Background and related work
In many applications concerning identification problems for PDEs, the source term may be unknown. The best known applications of these types of problems occur when finding a pollution source intensity by measurements of the pollutant concentrations and for identifying the laser beam intensity and trajectory in the heat equation corresponding to a giveninadvance temperature distribution [2].
One difficulty in inverse source problems is the absence of identifiability (uniqueness) of an arbitrary source as was shown in [3]. Thus, a wellposed source identification problem can be obtained if some a priori information is known. This additional information may be described by certain conditions on the admissible sources induced by the physical problem. For example, measuring the temperature by a perfect sensor of finite size in the heat source identification problem. A spacedependent source identification problem is considered by Cannon in [4]. Farcas and Lesnic studied a source dependent only on time [5]. Separable sources are analyzed in detail by Engl et al. [6]. A moving source whose spatial support is contained within a ball with a given radius is treated by Kusiak and Weatherwax [7]. The sum of m stationary sources with timevarying intensities are considered in [3].
It is worth noting that the identification problem, with linear source, related to nondegenerate systems is widely studied in the literature concerning inverse problems for PDEs. For more details on this subject, we refer to the monograph of Prilepko et al. [8, Chapter 7], the two papers by Orlovsky [9, 10] and the references given there.
The direct problem of (\(\mathcal{IP}1\)), i.e., \(M=I\), was first considered by Lorenzi [11]. In this paper an explicit solution formula was established under the condition that the operator L is bounded. As a matter of fact, more results were obtained by Prilepko et al. [8], where the operator L was supposed to be a generator of a \(c_{0}\)semigroup. In the work of Lorenzi [11] it was supposed with regard to problem (\(\mathcal{IP}1\)) that \(f(t)\) is known, z is unknown and the semigroup generated by L is analytic, that is, the case of parabolic equation occurred. A direct nonlinear version of (\(\mathcal{IP}1\)) with L being the infinitesimal generator of a compact \(c_{0}\)semigroup of contractions was discussed by Lorenzi [1]. General results for secondorder identification differential problems were obtained by Awawdeh [12].
It is important to mention that there are few studies about abstract degenerate fractional differential equations since they are essentially complicated and their theory is still in its early stages. The first paper on this subject was provided in Fedorov and Ivanova [21] for studying an identification problem for timefractional order partial differential equations. After that, some works have been published. In [22], Kostić considered abstract degenerate fractional differential inclusions in Banach spaces and gave uniqueness results. We refer the reader to [23–25] for further information about abstract degenerate differential equations with integer order derivatives.
1.2 Contributions
This paper builds on the works of Al Horani [13–16] and Favini and Marinoschi [26]. The objective of these works was to obtain theoretical identifiability and local stability results for degenerate differential equations. That is, solution schemes that work well even in the case of systems with a noninvertible operator at the derivative. It is worth noting here that the classical theory of \(c_{0}\)semigroups is not obviously applicable for problem (\(\mathcal{IP}1\)) because \(M^{1}\) is not continuous in general. Another option is to use an operational method. These methods have been developed mainly for linear systems arising from elliptic and parabolic PDEs.
We propose an identification method based on reformulating the inverse problem (\(\mathcal{IP}1\)) into an equivalent nondegenerate problem. As a first step, a projection method is used to reduce the problem to a regular abstract inverse problem. The problem is then handled with the help of some perturbation results for linear operators. Our smoother result is based on constructing an exact representation of the solution avoiding the calculations of some resolvent estimates of the involved operators and the study of some properties of the multivalued linear operator \(LM^{1}\).
We apply the abstract results for problems of mathematical physics. The method performs well with differential equations of the Sobolev type. Thus, despite the restrictive assumptions we impose in order to obtain the identifiability results, we are able to construct a scheme that is efficient for a broader set of degenerate identification problems.
1.3 Organization of the paper
The rest of the paper is organized as follows. In Section 2 the existence and uniqueness of the solution to the identification problems (\(\mathcal{IP}1 \)) and (\(\mathcal{IP}2\)) are proved under suitable assumptions on the data. Furthermore, an implicit representation of the solution (v, f) is obtained. An identification problem for a firstorder differential equation of the Sobolev type, subjected to an overdetermination expressed by means of a Lebesgue integral, is presented in Section 3. The results so found are then applied to such a problem. Section 4 offers one possible way of applying the abstract results to problems of mathematical physics.
2 Main results
 (\(\mathcal{IP}2\)):

Given \(w_{0}\in X\), \(\phi :[0,\tau ]\rightarrow \mathbb{R} _{+}\), \(\tau >0\), a functional on X, and \(g\in \mathcal{C}^{1}([0,\tau ];\mathbb{R} )\) determine the conditions under which we can identify \(f\in \mathcal{C}([0,\tau ];\mathbb{R} )\) such that w is a strict solution to the Cauchy problem$$\begin{aligned}& \frac{d}{dt}Bww =f(t)z,\quad 0< t\leq \tau, \end{aligned}$$(3)satisfying the additional condition$$\begin{aligned}& Bw(0) =w_{0}, \end{aligned}$$(4)Here B is a closed linear operator in X which has \(\lambda =0\) as a pole of \((\lambda B)^{1}\) of order \(k+1\).$$ \phi \bigl[ Bw(t)\bigr]=g(t), \quad 0< t\leq \tau . $$(5)
As the next stage, we are going to use a method coupled with perturbation theory for linear operators for the solvability of the inverse problem (10)(12).
Theorem 1
([29])
Let X be a Banach space, and let A be the infinitesimal generator of a c _{0}semigroup \(T(t)\) on X. If \(B:X_{A}\longrightarrow X _{A}\) is a continuous linear operator, then \(A+B\) is the infinitesimal generator of a c _{0}semigroup on X.
In accordance with what has been said, we arrive at the following assertions.
Theorem 2
Theorem 3
We are now ready to pass to the statement of the main result.
Theorem 4
Proof
Consider the identification problem (\(\mathcal{IP}1\)) where M and L are two closed linear operators in X with \(\mathcal{D}(L)\subset \mathcal{D}(M) \), L being invertible, with the property that \(\lambda =0\) is a pole of order \(k+1\), \(k=0,1,2,\ldots \) , of the bounded operator \(L(\lambda LM)^{1}\), \(z,v_{0}\in X\), \(g\in \mathcal{C}^{1}([0, \tau ];\mathbb{R} )\) and \(\phi \in X^{\ast }\).
3 Differential equations of Sobolev type
Sobolevtype equations appear in a variety of physical problems such as flow of fluid through fissured rocks, thermodynamics, heat conduction involving two temperatures and soil mechanics. There is an extensive literature in which Sobolevtype equations are investigated in the abstract framework; see, for instance, [30–33]. As far as we know, the present paper is the first one to consider identification problems for differential equations of Sobolev type.
4 Applications
We finally consider the abstract results as a source of existence and uniqueness theorems for the identification problems related to some degenerate systems.
Example 5
Example 6
Let B be a closed linear operator with a compact resolvent. If \(0\in \sigma (B)\), then it is an isolated eigenvalue with finite multiplicity [28, p.181], and therefore Theorem 4 applies again.
Example 7
Example 8
If X is a Hilbert space and \(B_{0}\) is a compact linear operator from X into itself such that \(\lambda_{0}\neq 0\) is an element of \(\sigma (B_{0})\), then Theorem 4 applies with \(B=\lambda_{0}B _{0}\).
Declarations
Authors’ contributions
All authors contributed equally and significantly in this paper. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
 Lorenzi, A, Vrabie, I: Identification for a semilinear evolution equation in a Banach space. Inverse Probl. 26, 116 (2010) View ArticleMATHMathSciNetGoogle Scholar
 Johansson, BT, Lesnic, D: A variational method for identifying a spacewisedependent heat source. IMA J. Appl. Math. 72, 748760 (2007) View ArticleMATHMathSciNetGoogle Scholar
 El Badia, A, HaDuong, T: On an inverse source problem for the heat equation. Application to a pollution detection problem. J. Inverse IllPosed Probl. 10, 585599 (2002) MATHMathSciNetGoogle Scholar
 Cannon, JR: Determination of an unknown heat source from overspecified boundary data. SIAM J. Numer. Anal. 5, 275286 (1968) View ArticleMATHMathSciNetGoogle Scholar
 Farcas, A, Lesnic, D: The boundaryelement method for the determination of a heat source dependent on one variable. J. Eng. Math. 54, 375388 (2005) View ArticleMATHMathSciNetGoogle Scholar
 Engl, HW, Scherzer, O, Yamamoto, M: Uniqueness and stable determination of forcing terms in linear partial differential equations with overspecified boundary data. Inverse Probl. 10, 12531276 (1994) View ArticleMATHMathSciNetGoogle Scholar
 Kusiak, S, Weatherwax, J: Identification and characterization of a mobile source in a general parabolic differential equation with constant coefficients. SIAM J. Appl. Math. 68, 784805 (2008) View ArticleMATHMathSciNetGoogle Scholar
 Prilepko, I, Orlovsky, G, Vasin, A: Methods for Solving Inverse Problems in Mathematical Physics. Dekker, New York (2000) MATHGoogle Scholar
 Orlovsky, D: An inverse problem for a second order differential equation in a Banach space. Differ. Equ. 25, 10001009 (1989) MathSciNetGoogle Scholar
 Orlovsky, D: Weak and strong solutions of inverse problems for differential equations in a Banach space. Differ. Equ. 27, 867874 (1991) MathSciNetGoogle Scholar
 Lorenzi, A: An Introduction to Identification Problems, Via Functional Analysis. VSP, Utrecht (2001) View ArticleGoogle Scholar
 Awawdeh, F: Perturbation method for abstract secondorder inverse problems. Nonlinear Anal. 72, 13791386 (2010) View ArticleMATHMathSciNetGoogle Scholar
 Al Horani, M: An identification problem for some degenerate differential equation. Matematiche 57(1), 217227 (2002) MATHMathSciNetGoogle Scholar
 Al Horani, M, Favini, A: An identification problem for firstorder degenerate differential equations. J. Optim. Theory Appl. 130(1), 4160 (2006) View ArticleMATHMathSciNetGoogle Scholar
 Al Horani, M: Projection method for solving degenerate firstorder identification problem. J. Math. Anal. Appl. 364(1), 204208 (2010) View ArticleMATHMathSciNetGoogle Scholar
 Al Horani, M, Favini, M, Lorenzi, A: Secondorder degenerate identification differential problems. J. Optim. Theory Appl. 141(1), 1336 (2009) View ArticleMATHMathSciNetGoogle Scholar
 Favini, A, Yagi, A: Degenerate Differential Equations in Banach Spaces. Dekker, New York (1999) MATHGoogle Scholar
 Awawdeh, F, Obiedat, HM: Source identification problem for degenerate differential equations. UPB Sci. Bull., Ser. A 73(1), 6172 (2011) MATHMathSciNetGoogle Scholar
 Fedorov, VE, Ivanova, ND: Identification problem for a degenerate evolution equation with overdetermination on the solution semigroup kernel. Discrete Contin. Dyn. Syst., Ser. S 9(3), 687696 (2016) View ArticleMATHMathSciNetGoogle Scholar
 Favini, A, Lorenzi, A: Identification problems for singular integrodifferential equations of parabolic type II. Nonlinear Anal. 56, 879904 (2004) View ArticleMATHMathSciNetGoogle Scholar
 Fedorov, VE, Ivanova, ND: Identification problem for degenerate evolution equations of fractional order. Fract. Calc. Appl. Anal. 20, 706721 (2017) View ArticleMATHMathSciNetGoogle Scholar
 Kostić, M: Abstract degenerate fractional differential inclusions in Banach spaces. Appl. Anal. Discrete Math. 11, 3961 (2017) View ArticleMathSciNetGoogle Scholar
 Carroll, RW, Showalter, RE: Singular and Degenerate Cauchy Problems. Academic Press, New York (1976) Google Scholar
 Kostić, M: Abstract Degenerate Volterra IntegroDifferential Equations. Book manuscript (2016) Google Scholar
 Sviridyuk, GA, Fedorov, VE: Linear Sobolev Type Equations and Degenerate SemiGroups of Operators. VSP, Utrecht (2003) View ArticleMATHGoogle Scholar
 Favini, A, Marinoschi, G: Identification for degenerate problems of hyperbolic type. Appl. Anal. 91(8), 15111527 (2012) View ArticleMATHMathSciNetGoogle Scholar
 Yoshida, K: Functional Analysis, 6th edn. Springer, Berlin (1980) Google Scholar
 Kato, T: Perturbation Theory of Linear Operators. Springer, Berlin (1966) View ArticleMATHGoogle Scholar
 Desch, W, Schappacher, W: On relatively bounded perturbations of linear c_{0}semigroups. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 11(2), 327341 (1984) MATHGoogle Scholar
 Krishnan, B, Subbarayan, K: Regularity of solutions of Sobolevtype semilinear integrodifferential equations in Banach spaces. Electron. J. Differ. Equ. 2003, 114 (2003) MATHMathSciNetGoogle Scholar
 Lightbourne, JH, Rankin, SM: A partial functional differential equation of Sobolev type. J. Math. Anal. Appl. 93, 328337 (1983) View ArticleMATHMathSciNetGoogle Scholar
 Showalter, RE: A nonlinear parabolic Sobolev equation. J. Math. Anal. Appl. 50, 183190 (1975) View ArticleMATHMathSciNetGoogle Scholar
 Sobolev, S: Some new problems in mathematical physics. Izv. Akad. Nauk SSSR, Ser. Mat. 18, 350 (1954) MATHMathSciNetGoogle Scholar
 Schechter, M: Principles of Functional Analysis. Academic Press, New York (1971) MATHGoogle Scholar
 Schechter, M: Basic theory of Fredholm operators. Ann. Sc. Norm. Super. Pisa 21, 361380 (1967) MATHMathSciNetGoogle Scholar