Identification problems of retarded differential systems in Hilbert spaces
type: Item
instance of: scholarly article
title: Identification problems of retarded differential systems in Hilbert spaces
zbMATH Open document ID: 1359.35226
author: Jin-Mun Jeong
author: Seong Ho Cho
published in: Evolution Equations and Control Theory
publication date: 2017-01-09T00:00:00Z
review text: Denote by \(\Omega\) a bounded domain in \(\mathbb R^n\) with smooth boundary \(\partial\Omega\), the operator \(A_0=-A(x,D_x)\) where \(A(x,D_x)\) is the elliptic operator in \(L^1(\Omega)\), defined as follows \[ A(x,D_x)=-\sum\limits_{i,j=1}^n\frac{\partial}{\partial x_j}(a_{i,j}(x)\frac{\partial}{\partial x_i})+\sum^n_{i=1}b_i(x)\frac{\partial}{\partial x_i}+c(x). \] The authors consider the inverse problem for the following retarded functional differential equation \[ u'(t)=A_0u(t)+\gamma A_0u(t-h)+\int^0_{-h}a(s)A_0u(t+s)ds \] with initial conditions \[ u(0)=g^0,\,u(s)=g^1(s),\,s\in[-h,0). \] The main task of the paper is to identify the unknown quantities \(A_0\), \(\gamma\) and \(a(\cdot)\). This identifiability can be regarded as a kind of inverse problem in many applications. Sufficient conditions for the identification problem are charaterized in terms of the initial values and eigenvectors of the adjoint operator. The authors use an approach which is a combination of the solution semigroup, the structural operators, the representations of spectral projections and the completeness of generalized eigenspaces.
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 93B30
Mathematics Subject Classification ID: 35J15
zbMATH DE Number: 6672806
zbMATH Keywords: identifiability
zbMATH Keywords: retarded differential system
zbMATH Keywords: inverse problem
zbMATH Keywords: completeness
zbMATH Keywords: \(\zeta\)-convex space
reviewed by: Xiangdong Yang
MaRDI profile type: Publication
full work available at URL: eect.2017005
OpenAlex ID: W2560376816
cites work: On the eigenfunctions and on the eigenvalues of general elliptic boundary value problems
cites work: Some remarks on Banach spaces in which martingale difference sequences are unconditional
cites work: Q4200361
cites work: \(L^ 2\)-regularity for parabolic partial integrodifferential equations with delay in the highest-order derivatives
cites work: Identification problems for integro-differential delay equations
cites work: On the closedness of the sum of two closed operators
cites work: Q4277763
cites work: Q4859687
cites work: Spectral properties of the operator associated with a retarded functional differential equation in Hilbert space
cites work: Identifiability of Spatially-Varying and Constant Parameters in Distributed Systems of Parabolic Type
cites work: Identifiability of Spatially-Varying Conductivity from Point Observation as an Inverse Sturm–Liouville Problem
cites work: Stability of functional partial differential equations
cites work: Completeness and F-completeness of eigenfunctions associated with retarded functional differential equations
cites work: Q3041011
cites work: Q3810201
cites work: Q4280513
cites work: Structural properties of functional differential equations in Banach spaces
cites work: Structural operators and eigenmanifold decomposition for functional differential equations in Hilbert spaces
cites work: Unique Identification of Eigenvalues and Coefficients in a Parabolic Problem
cites work: Norms and Domains of the Complex Powers A B z
cites work: Interpolation in $L^{p}$ with boundary conditions
cites work: Uniqueness and nonuniqueness in an inverse problem for the parabolic equation
cites work: Q5687543
cites work: Q4169288
cites work: Identifiability of operators for evolution equations in Banach spaces with an application to transport equations
cites work: Q5332803
DOI: 10.3934/EECT.2017005
description: scientific article; zbMATH DE number 6672806