Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts

Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts

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F R O N T I E R S IN APPLIED MATHEMATICS The SIAM series on Frontiers in Applied Mathematics publishes monographs dealing with creative work in a substantive field involving applied mathematics or scientific computation. All works focus on emerging or rapidly developing research areas that report on new techniques to solve mainstream problems in science or engineering. The goal of the series is to promote, through short, inexpensive, expertly written monographs, cutting edge research poised to have a substantial impact on the solutions of problems that advance science and technology. The volumes encompass a broad spectrum of topics important to the applied mathematical areas of education, government, and industry.

EDITORIAL BOARD H.T. Banks, Editor-in-Chief, North Carolina State University Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB Carlos Castillo Chavez, Cornell University Doina Cioranescu, Universite Pierre et Marie Curie (Paris VI) Pat Hagan, NumeriX, New York Matthias Heinkenschloss, Rice University Belinda King, Virginia Polytechnic Institute and State University Jeffrey Sachs, Merck Research Laboratories, Merck and Co., Inc. Ralph Smith, North Carolina State University Anna Tsao, Institute for Defense Analyses, Center for Computing Sciences

BOOKS PUBLISHED IN FRONTIERS IN A P P L I E D MATHEMATICS

Banks, H.T., Buksas, M.W., and Lin, T., Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts Oostveen, Job, Strongly Stabilizable Distributed Parameter Systems Griewank, Andreas, Evaluating Derivatives: Principles andTechniques of Algorithmic Differentiation Kelley, C. T., Iterative Methods for Optimization Greenbaum.Anne, Iterative Methods for Solving Linear Systems Kelley, C. T., Iterative Methods for Linear and Nonlinear Equations Bank, Randolph E., PLTMG:A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 7.0 More, Jorge J. and Wright, Stephen J., Optimization Software Guide Rude, Ulrich, Mathematical and Computational Techniques for Multilevel Adaptive Methods Cook, L. Pamela, Transonic Aerodynamics: Problems in Asymptotic Theory Banks, H.T., Control and Estimation in Distributed Parameter Systems Van Loan, Charles, Computational Frameworks for the Fast Fourier Transform Van Huffel, Sabine and Vandewalle, Joos, The Total Least Squares Problem: Computational Aspects and Analysis Castillo, Jose E., Mathematical Aspects of Numerical Grid Generation Bank, R. E., PLTMG: A Software Package for Solving Elliptic Partial Differential Equations. Users'Guide 6.0 McCormick, Stephen F., Multilevel Adaptive Methods for Partial Differential Equations Grossman, Robert, Symbolic Computation: Applications to Scientific Computing Coleman,Thomas F. and Van Loan, Charles, Handbook for Matrix Computations McCormick, Stephen F., Multigrid Methods Buckmaster.John D., The Mathematics of Combustion Ewing, Richard E., The Mathematics of Reservoir Simulation

Electromagnetic Material Interrogation Using Conductive Interfaces and Acoustic Wavefronts

H.T. Banks North Carolina State University Raleigh, North Carolina

M.W. Buksas Los Alamos National Laboratory Los Alamos, New Mexico

T.Lin Virginia Polytechnic Institute and State University Blacksburg, Virginia

Siam Society for Industrial and Applied Mathematics Philadelphia

Copyright © 2000 by the Society for Industrial and Applied Mathematics. 1098765432 I All rights reserved. Printed in the United States of America. No part of this book may be reproduced, stored, or transmitted in any manner without the written permission of the publisher. For information, write the Society for Industrial and Applied Mathematics, 3600 University City Science Center, Philadelphia, PA 19104-2688.

Library of Congress Cataloging-in-Publication Data Banks, H.T. Electromagnetic material interrogation using conductive interfaces and acoustic wavefronts / H.T. Banks, M.W. Buksas.T. Lin. p. cm.– (Frontiers in applied mathematics ;21) Includes bibliographical references and index. ISBN 0-89871-459-1 1.Acoustic emission testing. 2. Electromagnetic waves-Scattering. 3. Surfaces (Physics) I. Buksas, M.W. II. Lin.T. Ill.Title. IV. Series TA4I8.84.B362000 620.1'127-dc2l 00-032944

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trademark.

Contents Foreword

ix

Preface

xi

1 Introduction

1

2 Problem Formulation and Physical Modeling 2.1 Motivation 2.2 Estimation Methodology 2.3 Reduction to Specific Problems 2.3.1 Acoustic Reflectors

7 7 15 18 23

3 Well-Posedness 3.1 A Variational Formulation 3.2 A Semigroup Formulation 3.3 Enhanced Regularity of Solutions 3.4 Convergence of Finite Element Approximations 3.5 Inverse Problem Methodology

27 27 35 42 47 48

4 Computational Methods for Dielectrics with Supraconductive Backing 4.1 The Forward Problem for the Debye Polarization Model 4.2 The Inverse Problem for a Debye Medium 4.2.1 Sample Results 4.2.2 Reconstruction with Inferior Accuracy 4.2.3 Identification of Material Depth 4.2.4 Results from the Two-Step Algorithm 4.2.5 Estimating Depth and Physical Parameters 4.2.6 Results for the Simultaneous Estimation of Depth and Dielectric Parameters 4.3 The Forward Problem for the Lorentz Model 4.4 The Inverse Problem for a Lorentz Medium vii

57 57 63 65 71 72 74 76 77 79 85

viii

Contents

5 Computational Methods for General Polarization Models 89 5.1 The Forward Problem 89 5.1.1 Galerkin Methods for the History Approximation 91 5.1.2 Approximating the History ofe ( t ) 92 5.1.3 Implementing the Hysteresis Term 95 5.1.4 Specific Implementation: Constant Material Parameters 96 5.2 Results of Simulations with the General Model 97 5.2.1 Dielectric Response Function Representation of the Debye Model 97 5.2.2 DRF Representation of the Lorentz Model 99 5.3 The Inverse Problem with the General Polarization Model . . . . 101 5.3.1 Practical Conclusions 105 6 Computational Methods for Acoustically Backed Dielectrics 6.1 The Forward Problem with an Acoustically Backed Layer 6.1.1 Numerical Methods 6.1.2 Simulation Results 6.2 Identification of Material Depth 7 Concluding Summary and Remarks on Potential Applications 7.1 Summary of Mathematical and Computational Results 7.2 Limitations and Unanswered Questions 7.3 Future Technological Possibilities

107 107 108 109 119 129 129 131 133

Bibliography

137

Index

145

Foreword

The human extension of visual ability has been very important to technology, society, and culture. This extension of vision has been to the very small, the distant, and the internal. Microscopy—viewing the small—continues to open new doors into the structure of materials, including biological materials in their relation to human disease. Radar systems and other airborne sensor modalities enable environmental monitoring important to the health of our planet and essential to flying safety. Internal imaging is central to medical diagnosis, on one hand, and to the nondestructive assessment of the integrity of nonliving structures, on the other. This volume in the SIAM Frontiers in Applied Mathematics series is a milestone in the human endeavor to extend vision. It is concerned with imaging the interior of materials or objects and shows, in a systematic way, how internal reflecting surfaces can be exploited to aid the process of "seeing within." Convincing evidence is presented that highly reflective items, such as metal surfaces, can substantially aid characterization of materials interrogated by short electromagnetic pulses. Also, and perhaps most intriguing, the work herein suggests that slowly moving acoustic wave structures can also serve as useful reflecting surfaces and can be tracked by fast moving electromagnetic pulses to enhance internal visualization. In short, I believe that this volume can foster a significant increment in imaging technology using low energy electromagnetic radiation. The use of electromagnetic pulses interacting with specially placed reflective surfaces, whether solid or acoustic, is a new dimension that will substantively impact medical imaging, subsoil investigation, and structure evaluation. The text is accessible to the advanced undergraduate or early graduate engineering,bioengineering, geology, and mathematics or physics student. Its strengths include a clear discussion of materials properties from the electromagnetic point of view, a careful formulation of the imaging problems addressed, solid treatment of mathematical issues, and useful illustration of computational methods and results. While confined to internal vision in onedimensional settings, this volume will stimulate further developments in internal vision to include two- and three-dimensional interior assessments. It is an excellent and robust source of applied mathematics and engineering research challenges for the future.

ix

In the next quarter-century one can hope for the early diagnosis of cancer in such inaccessible organs as the ovary and the pancreas to enable curative action, and for improved, more convenient intestinal, heart, and breast surveillance. Imaging of structures in our cities, including bridges, buildings, road beds, and tunnel supports, must progress if structural aging processes are to be detected and arrested. Soil, lake, and river imaging is essential for pollution control and remedy. "Internal seeing" will be increasingly important in law enforcement in the search for concealed weapons, explosives, and biological or chemical threats. This volume materially addresses these current and emerging needs.

Richard Albanese U.S. Air Force Research Laboratory Brooks Air Force Base San Antonio, Texas

x

Preface This monograph contributes to the general area of electromagnetic imaging. Our treatment has several distinguishing features. First we explore with some care the possibility of using interfaces, either with supraconductive boundary materials or with acoustic wavefronts, as reflectors to determine geometry as well as dielectric material characteristics of a "target." Moreover, we use windowed microwave pulses from an antenna-like source to provide nonharmonic time dependent interrogating signals. These are inputs to a general time dependent form of Maxwell's equations containing Ohmic conductivity along with quite general material polarization constitutive laws to represent dispersiveness in dielectric media. Mathematically, we offer a rigorous treatment of the resulting integro-partial differential equation system with delta function inputs in a variational framework. We provide both theoretical and computational analysis in the context of inverse or parameter estimation problems. Our efforts are at the level of fundamental research on the basic question: Can supraconductive interfaces or acoustic waves be used as "electromagnetic mirrors" in imaging scenarios? While we believe we provide an affirmative answer to this question in the abstract, we also believe our specific examples provide computational validation that such an approach holds direct promise for several classes of interrogation applications (these are discussed in Chapter 7). The literature addressing electromagnetic imaging is extensive and the mass of relevant information can be viewed as even larger by including work on acoustic imaging. However, we are unaware of texts or research articles that systematically exploit interface phenomena, the electrodynamics of material responses, and time dependent interrogating signals in an integrated manner. We perceive that it is the variational framework of our effort that permits this integrated approach. We hope that this integrated approach will evolve into a standard for the field with particular extension to more complex targets. Electromagnetic theory offers fascination and challenge from both a physical and mathematical perspective. For the nonspecialist in electromagnetic theory (with which the authors of this monograph readily identify themselves) an effort in electromagnetic imaging can appear daunting. However, we have tried xi

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Preface

to make our presentation accessible to a wide audience (the mature upperclassman, and especially graduate students or postdoctoral students in mathematics, physics, and engineering). We do not derive the free space Maxwell's equations from first principles (i.e., Coulomb's law plus relativity theory), but we do summarize the needed background in Chapter 2, where we formulate the basic mathematics and physics associated with material responses and imaging. Our brief literature survey in Chapter 1 can be skipped on a first reading, but a reading of Chapter 2 is an essential prerequisite for the remainder of the monograph. For the theoretical foundations of Chapter 3 (the details of which can be skipped or read only cursorily), we assume a background of analysis usually taught in a first graduate course in functional analysis. The computational material which is the heart of Chapters 4, 5, and 6 should be accessible to anyone with a standard course in numerical analysis which includes some numerical optimization. Finally, an elementary understanding of electromagnetics and acoustics would be helpful but we feel it is not necessary if leaders are willing to either accept our summaries of the physics in Chapters 2 and 6 or to pursue extra reading (references are given throughout the presentation) on their own. The first author has (successfully he thinks!) covered the material in Chapter 2 with graduate students who had no special electromagnetic training. We believe that electromagnetic imaging provides exciting challenges to which modern mathematicians, engineers, and physicists with computational skills can contribute at the frontiers. The tremendous advances in mathematical and computational capabilities over the past several decades permit consideration of time domian problems that were completely out of reach for our predecessors. This monograph represents our attempt to share our enthusiasm for exciting scientific opportunities in electromagnetic imaging, and, hopefully, to stimulate others to join the pursuit. This monograph grew out of the efforts of the second author in his Ph.D. thesis of August 1998 at North Carolina State University and several related research projects of the first author. The major stimulation in all cases came from Dr. Richard Albanese, U.S. Air Force Research Laboratory, Brooks AFB, San Antonio, Texas. The authors are deeply indebted to Dr. Albanese for his continual encouragement and numerous specific technical discussions throughout the course of the research discussed in this monograph. His many suggestions and questions provided significant impetus for our research efforts. Moreover, he read with extreme care several earlier versions of the manuscript. His thoughtful challenges and comments always improved our presentations and in some cases provoked us to obtain new results. Finally, he made a major contribution to the speculative comments in Chapter 7. (Dick, it is indeed exciting and great fun to dream with you!) We would also like to thank Dr. Mac Hyman of Los Alamos National Laboratory for his support of the second author during the completion of this manuscript. His encouragement and guidance have been most helpful.

Preface

xiii

Our research has been strongly supported by the U.S. Air Force Office of Scientific Research under grants AFOSR F49620-95-1-0236, AFOSR F49620-981-0180, AFOSR F49620-95-1-0375; in part by the U.S. Department of Education under a GAANN Fellowship (MWB) through grant P200A40730; and the Department of Energy (MWB) under contract W-7405-ENG-36. The AFOSR grants were under the auspices of Dr. Marc Q. Jacobs, who as a Program Manager has unfailingly encouraged our (and many others') serious involvement with AFRL scientists and engineers through the years. For this we are most grateful. A number of postdocs and graduate students (particular thanks to Julie Raye) in the Center for Research in Scientific Computation made suggestions and corrected typos in earlier versions. Finally, we wish to acknowledge a debt to Dr. Yun Wang. It was through her enthusiasm for electromagnetic problems while working with Dr. Albanese at Brooks AFB that our collaboration with Dr. Albanese and his group began.

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Chapter 1 Introduction A survey of the mathematical literature reveals considerable interest in the identification of material parameters describing electromagnetic phenomenon. For our purposes, we categorize the materials and the models employed to describe them as either dispersive or nondispersive, where dispersive materials are denned as those in which planar electromagnetic waves propagate with phase velocities that depend on the frequency of the waves. Thus an incident transient pulse in a dispersive medium will spread and change shape, even in a homogeneous medium. When modeled in the frequency domain, this is manifested as parameters which depend explicitly on frequency. In time domain models, the same phenomenon can be captured with constitutive laws in which the electric and/or magnetic polarizations are expressed in terms of the convolution of the history of the electric and magnetic fields. The equivalence of the two in the case of electric polarization dispersion is shown by Jackson [Jac75, p. 306]. Alternatively, one can introduce dispersiveness through Ohmic conductivity, or more generally, by a conductivity denned via a convolution with the electric field [APM89]. Simple Ohmic conductivity results in a dissipation term in the electromagnetic equations. In identification of electromagnetic material parameters, the emphasis has been on one-dimensional scattering problems where planar electromagnetic waves impinge on dielectric slabs. In a series of four papers [KK86a, KK86b, KK87, KK89] Kristenson and Krueger examine this problem with a wave splitting technique and the derivation of scattering operators which satisfy imbedding equations. A nice summary of this general technique involving invariant imbedding coupled with wave splitting (discussed also in the earlier papers [CK83, CDK83]) can be found in [AMP94]. The reconstruction of the functions representing the physical parameters is carried out via the deconvolution of the imbedding equations which relate incident and reflected waves through a convolution kernel. The physical model in these papers includes dissipativeness via Ohmic conduc1

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Banks, Buksas, and Lin

tivity but no polarization-induced dispersiveness; it also covers stratified media, meaning that the material is inhomogeneous only in the direction of the propagation of the waves. Scattering techniques using wave splitting were applied to a general hyperbolic model by Weston [Wes72], who considered a dissipative wave equation equivalent to the problem of planar waves in stratified media. These results were extended by Krueger [Kru76, Kru78, Kru81] to cover media in multiple slabs, thus containing multiple discontunities in the material parameters. Corones and Sun [CS93] used the same method of wave splitting and invariant imbedding to reconstruct coefficients in a one dimensional wave equation with a smooth source term. In another paper, He and Strom [HS91] also employed wave splitting in a scattering problem for stratified materials illuminated with waves generated by a magnetic dipole. Some progress has also been made with more general geometries, largely due to the increasing sophistication of wave splitting techniques in higher dimensions. In a paper by Weston [Wes88], a decomposition of solutions of the dissipative wave equation in R3 is given and integro-differential equations are derived for the reflection operator. The reconstruction of the velocity and dissipation coefficients from the kernel of this operator are demonstrated. Inverse problems involving polarization-based dispersive materials follow a similar pattern of development in moving from one dimensional scattering with planar waves to more general settings. Beezley and Krueger [BK85] began investigation into these problems in one dimension by employing a method similar to that described above for conductivity-based dissipative wave equations. Imbedding equations are derived for the reflection operator relating the incident and scattered parts of the split wave solution. The dielectric response kernel is reconstructed from the imbedding equations derived for homogeneous semiinfinite and finite slabs. The reconstruction is also carried out numerically in the presence of noise applied to the reflection kernel. Well-posedness results for these problems involving wave splitting and imbedding equations are developed by Bui [Bui95] in which a time domain model involving both conductivity and electric polarization is considered. Under stringent smoothness (regularity) assumptions on the electric field and on the displacement and conductivity susceptibility kernels, theorems governing the existence, uniqueness, and continuous dependence of solutions (to the imbedding equations) in the context of both the forward and inverse problems are given. In [Sun92] Sun combined wave splitting/invariant imbedding techniques with Green's function methods to identify the source current embedded inside a dispersive material using a time-domain approach in systems similar to those of Beezley and Krueger. In a paper by Lerche [Ler86], a different integral equation is derived relating the dielectric response function to an operator representing the frequency domain absorption characteristics of a material. Wolfers-

Introduction

3

dorf [Wol91] extends these results to finite and semi-infinite slabs and derives exact solutions to the integral equations for the dielectric response. A geometry different from the usual slab was considered by Kreider [Kre89], who posed the problem of reconstruction of the displacement susceptibility kernel from reflected data in a stratified cylinder. The problem is, however, rendered one-dimensional through angular symmetry of the cylinder and the fields. The dispersive material is permitted inhomogeneities in space, although the dielectric parameter is restricted to functions which are separable between the variables of space and frequency: e(x,w;) = e\(x)E 2 ((w]. Another treatment which allows for spatially inhomogeneous dispersive materials is considered in [HFL96]. In this case an optimization approach is applied to match simulations to experimental as well as synthetic data. The constitutive equations permit convolution terms (i.e., displacement and conductivity susceptibility kernels) in both the electric polarization and the conductivity. The limitations of simultaneous reconstruction of the spatial and time-varying parts of the kernel functions is considered when both transmission and reflection data are available. Another inhomogeneous two-dimensional problem is considered by Colton and Monk [CM94a, CM95] who use a frequency domain approach in modeling the interaction of electromagnetic waves with human tissue for the detection of leukemia in bone marrow. The geometry of the medium is presumed known and the two dimensional domain is further partitioned into subdomains of known geometry. The only unknowns are the constitutive parameters describing the bone marrow, and these are considered functions of space as well. The reconstruction of the bone marrow parameters is demonstrated and is shown to be robust in the presence of relative noise with magnitude as high as 1%. The methods in [CM94a, CM95] are versions of a class of dual space methods that have been proposed for solving inverse scattering problems in the frequency domain for an inhomogeneous body [CM94b, CK92]. A succinct summary of Herglotz kernel based versions of this approach can be found in [AMP94]. Among other fundamental efforts in the frequency domain are those of Oughstun and Sherman [SO81, OS88] who employed improved asymptotic analysis to extend the classical theory of Sommerfeld and Brillouin on propagation of electromagnetic pulses in dispersive Lorentz media. A problem directly motivated by an application is given in [RGKM97], where time domain models with polarization and conductivity operators were formulated for the penetration of radar waves in soil. The ground is represented by a stratified semi-infinite slab and electromagnetic signals are generated by a circular loop of current. Under stringent regularity conditions on parameters and kernels, unique determination of dielectric parameters is guaranteed for a sufficiently rich set of observations. Finally, we note that inverse scattering problems involving the the geometrical reconstruction of a body with known dielectric properties have been widely

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Banks, Buksas, and Lin

studied (e.g., see [BKP97, vdBCK95] and the references therein) using integral equation methods connecting incident and reflected wave fields. In this monograph we work exclusively in the time domain and formulate both forward and inverse problems for planar waves incident on layered strata. Using the full time varying Maxwell's equations, we incorporate very general polarization constitutive laws. These convolution formulations are sufficiently general to include all known or hypothetical polarization mechanisms as special cases. In our model, we employ a source term that can represent an antenna producing a point source spatial input consisting of pulsed microwaves. In a variational formulation suitable for both theoretical and computational analysis we investigate two types of inverse problems. In the first we assume that a layer of dielectric material with unknown dielectric parameters and geometry (thickness) has a back boundary of supraconductive (e.g., metal) material. The inverse problem consists of estimating dielectric properties and geometry from reflections of incident microwave pulses. In the second problem, the supraconductive back boundary is replaced by a second layer of known dielectric properties and an acoustic reflector or mirror. A chapter is devoted to a rather thorough theoretical treatment of the problems with supraconductive back boundary in a variational formulation. Existence, uniqueness, and continuous dependence results for the forward problems are given. A related semigroup formulation is discussed. For inverse problems we present theoretical foundations (including a convergence framework for approximation methods) which lead to existence and continuous dependence (on data) of estimates. We develop computational methods based on finite elements in space and finite differences in time, and we use these to demonstrate that microwave pulses from the antenna point source can produce Brillouin or Brillouin and Sommerfeld precursors in Debye or Lorentz media, respectively, similar to those found using asymptotic analysis or frequency domain techniques. In a series of computational examples, we investigate the feasibility of estimating dielectric parameters and geometry using observations of electric field reflections from dispersive layers. Emphasis is on estimating conductivity and polarization constitutive laws (Debye, Lorentz, or more general media represented by a general dielectric response function, i.e., a displacement susceptibility convolution kernel). We develop and test a two-step algorithm in which reflections from the first interface in a stratified media are used to obtain initial estimates of dielectric parameters. Reflections of the media propagated field from a supraconductive back interface are used to estimate layer thickness and to refine the initial estimates of the dielectric parameters for the layer. A modification of the inverse problem wherein one replaces the supraconductive back interface by a stationary standing acoustic wave is then investigated. It is demonstrated that this acoustic wave can be tuned to enhance reflections from the second interface to permit useful application of the two-step algorithm.

Introduction

5

In the closing chapter we discuss the importance of and potential applications (biomedical, civilian, and military) for the techniques developed in this monograph. In closing this introductory chapter, we note the differences between the physical models and the solution methods employed in some of the literature cited above and our discussions which follow. A number of these contributions differ from ours in that they deal only with dissipative materials. This eliminates the need to reconstruct parameters which are functions of frequency or convolution kernels which are functions of time. Some do consider materials which are inhomogeneous in the space variable, however, and while the model formulation is in the time domain, the forward and inverse problems in many are solved by wave-splitting and invariant imbedding techniques. A number of these papers do consider physical problems similar to ours, in which planar electromagnetic waves impinge normally on slabs of material. Among the references cited which employ dispersive models, many differ from ours by using a frequency domain approach which is best suited for physical problems involving time harmonic solutions. Among those treatments which use time domain models, most again use the wave splitting approach popular with the dissipative materials to formulate the forward and inverse problems. Only one contribution [HFL96] uses an optimization approach and formulates an inverse problem using the time domain data itself. While this paper and others consider inhomogeneous materials, the geometry of the material slab is considered to be known. This differs fundamentally from our problems in that we do not assume the thickness of the material slab is known a priori and attempt to identify this dimension along with the parameters describing the electromagnetic material properties.

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Chapter 2

Problem Formulation and Physical Modeling 2.1

Motivation

The ability to interrogate the interior of tissues and other materials has wide ranging applications to medical imaging, the early detection of anomalies, and other problems in nondestructive interrogation of sensitive materials. Noninvasive interrogating techniques are most valuable in determining substructure in biological tissues due to the fact that they usually result in much less discomfort in subjects. Low amplitude or low energy microwaves (electromagnetic waves in the frequency range of 3 to 300 GHz) can pass through many media without causing any known damage. On the other hand, chemical and physical changes in biological tissue can result in changes in its electromagnetic characteristics such as electric and magnetic polarization mechanisms and conductivity. Furthermore, it is hoped that the in vivo electromagnetic characteristics of tissues and organs can be correlated with metabolic functioning. Hence the accurate determination of these dielectric properties can be employed in the evaluation of the functional integrity of tissues and organs in subjects. Other applications in biology and the environmental sciences of the use of microwaves in noninvasive interrogation procedures can be found in a recently published review article [AMP94]. Additional potential applications for the interrogation ideas discussed below include nondestructive damage detection in aircraft, mine, ordinance and camouflage detection, and subsurface and atmospheric environmental modeling. Use of ultrasonic waves is another popular technique employed in noninvasive interrogation of media in both industrial and medical applications. It has been well known since 1922 [Bri22] that electromagnetic and sound waves can interact in a medium and influence each other's propagation. This interaction has been 7

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Figure 2.1: Schematic diagram of geometry. the subject of substantial investigation in acoustooptics [DD91, Kor97, XS92], and numerous acoustooptic devices have been developed in many applications in industry—neural nets, optical excision, and fiber optics, to name just a few. The goals of electromagnetic interrogation as presented here are the determination of both the geometry and dielectric properties of the materials under investigation. We consider the generalized problem depicted schematically in Figure 2.1. The domain £7 of the object under consideration has both a known and an unknown portion of the boundary. The unknown portion F(g) is presumed to be backed by a supraconductive material with an effectively infinite conductivity. On this boundary with outer normal n, we thus have E x n — 0 and B • n = 0. Note that the unknown nature of the boundary is represented by its dependence on a set of parameters q which are to be determined to establish the geometry of the object. Alternatively, we may suppose that the supraconductive backing material is replaced by an interface with an acoustically excited medium. Reflections from the interior of the domain £7 would then be created by changes in material properties induced by the acoustic wave. The location of these changes in the material properties would then be described in analogous fashion by the parameters q. The electric and magnetic fields inside i7 and exterior to 17 (this region will be denoted 17o) are governed by the macroscopic Maxwell's equations [Jac75, Bal89,

Problem Formulation and Physical Modeling

9

Str41, E1193]. To describe the electromagnetic behavior of complex materials, we express Maxwell's equations in a general form which includes terms for electric and magnetic polarization. We have

The vector-valued functions E and H represent the strengths of the electric and magnetic fields, respectively, while D and B are the electric and magnetic flux densities, respectively. The two current contributions are denoted by Jc, the conduction current density, and Js, a source current density. The electric and magnetic polarizations are represented by P and M, respectively. The scalar quantity p represents the density of free electric charges unaccounted for in the electric polarization. The three quantities M, P, and Jc embody the behavior of the material in response to the electromagnetic fields. Additional material dependent equations (constitutive laws) are required to determine their dependence on the components of the fields E and H. The dependence of these equations on the material is reflected both in the choice of a mathematical model and in the parameters (possibly operators) appearing in the model. Estimation of these parameters or operators are the goals in the inverse problems we formulate below. The region OQ external to the medium is treated as empty space and is devoid of conductivity or polarization effects, hence M = 0, P = 0, and Jc = 0 in Slo, and all of the necessary parameters for the determination of the fields are assumed known in this domain. The source current density term Js will also be nonzero only at points in QQ. The presence of any time varying vector valued current source will generate the electromagnetic waves in this domain which illuminate the target medium fi. We make certain assumptions about the material which are reflected in the constitutive relations in the domain Q. For the media of interest to us, we can neglect magnetic effects; we also assume that Ohm's law governs the electric conductivitv. Hence for x & £1

For dispersive media, it is generally recognized [AMP94, APM89, Bal89, Jac75, E1193] that frequency dependence of conductivity a as well as the dielectric permittivity e (in a displacement constitutive law D = eE to be dis-

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cussed momentarily) is important. This is often [APM89, Jac75, HFL96, BBOO] treated by assuming a nonlocality in time through introduction of conductive and electric susceptibility kernels in convolution relationships for Jc — JC[E] and P = P[E]. We will do precisely this below in our treatment of electric polarization. However, following arguments in [APM89], one can establish that introducing frequency dependence via susceptibility kernels in the polarization automatically results in frequency dependence of the conductivity even if one uses the simple form of Ohm's law given above (i.e., assuming instantaneous or local-in-time dependence of Jc on the electric field E). We therefore shall not, in our treatment here, assume a more complicated relationship for conductivity even though the ideas and methods presented in this monograph could readily be used to do so. This is not necessary to obtain the desired frequency dependence of conductivity and it would add to the computational burden in the inverse problems we treat here. Moreover, it is not at all clear that one could separately estimate frequency dependence of conductivity and polarization using observations of the macroscopic electric field (i.e., additional nonuniqueness of parameters in the inverse or parameter estimation problems would be introduced). Thus, with no real loss of generality and for simplicity, we restrict ourselves to the usual Ohmic conductivity in our discussion in this monograph. To describe the behavior of the media's macroscopic electric polarization P, we employ a general integral equation model in which the polarization explicitly depends on the past history of the electric field. This model is sufficiently general to include microscopic polarization mechanisms such as dipole or orientational polarization as well as ionic and electronic polarization [Bal89, Fr658, E1193] and other frequency dependent polarization mechanisms. The resulting constitutive law can be given in terms of a polarization or displacement susceptibility kernel g (which we shall sometimes refer to as the dielectric response function or DRF)

by

We note that this model presupposes that P(0, x] = 0. In the electromagnetic literature (e.g., see [Jac75, BK85, APM89, E1193]), the relationship is often expressed as

in the case of E(t,x] = 0 for t < 0, which is of interest here. These are both related to our formulation by the simple change of variables: s = t — £. We prefer our form of the equation since then any time derivatives of P are borne

Problem Formulation and Physical Modeling

11

by the kernel function g and not the variable E. Specifically, under (2.1) the term Q^-(t,x) — P(t,x), which will appear in subsequent equations, is given by

while the more traditional representation leads to

The presence of E under the integral term in (2.3) complicates the analysis and solution of the Maxwell's equations considerably. Although the first formulation leads to the additional terms g(0, x)E(t, x) and g(0, x)E(t, x), we shall see below that these terms cause no increase in the complexity of the problem analysis or computation. We note that an attempt to include a component of the polarization which depends on the instantaneous value of the electric field would add a delta function in the time variable to the DRF g(s, x). This introduces some mathematical complexities which, for simplicity, we avoid by treating instantaneous polarization when it arises in a different, but completely equivalent, manner. Allowing the instantaneous component of the polarization to be related to the electric field by a dielectric constant so that P;n = eoxE and denoting the remainder of the electric polarization with P, we find

where er — I + x > 1 is a relative permittivity. The parameter er can be treated as a spatially dependent parameter to allow for instantaneous effects on displacement in Q due to the electric field originating in QQThe constitutive law in (2.1) is also sufficiently general to include models based on differential equations and systems of differential equations or delay differential equations (see [BJ70]) whose solutions can be expressed through fundamental solutions (in general variation-of-parameters representations). For example, the choice of kernel function g(i] = e~t/T6o(es — £oo)/ r m ^ corresponds to the differential equation of the Debye model for orientational or dipolar polarization in fi given by

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Banks, Buksas, and Lin

Here, es is the static relative permittivity (sometimes denoted esr, e.g., see [Bal89, p. 50]). The presence of instantaneous polarization is accounted for in this case by the coefficient €00 in the electric flux equation. That is, er = e^ in fi, er = I in f^o- The remainder of the electric polarization is seen to be a decaying exponential, driven by the electric field, less the part included in the instantaneous polarization. This model was first proposed by Debye in [Deb29, vH54] to model the behavior of materials whose molecules possess permanent dipole moments. The magnitude of the polarization term P represents the degree of alignment of these individual moments. The choice of coefficients in (2.4) gives a physical interpretation toes and eoo as the relative permittivities of the medium in the limit of the static field and very high frequencies respectively. In the static case, we have P = 0, so P = eo(es ~~ eoo)E and D = ese0E. For very high frequencies, rP dominates P so P w 0 and D — e^vE. We will also consider the Lorentz model for electronic polarization which, in differential form, is represented with the second order equation:

The so-called plasma frequency is defined to be up = ujQ^/es — e^. A simple variation of constants solution yields the correct kernel function

where Z/Q = For more complex dielectric materials, a simple Debye or Lorentz polarization model is often not adequate to characterize the dispersive behavior of the material. One can then turn to a combination of multiples of Debye, Lorentz, or even more general nth order mechanisms. In the time domain, such an nth order model is given analytically by

which in the frequency domain takes the form

where q(s) is the polynomial q(s] = Y^=oajs^- ^o model multiple (e.g., N) Debye, Lorentz, or nth order polarization mechanisms one would then have (in

Problem Formulation and Physical Modeling

13

the frequency domain)

where the degree of Qnum is less than the degree of Qden. In the time domain one obtains the equivalent polarization law

where k i ( N ) > K2(N) corresponding to a stable medium. To see that (2.6) is included as a special case of the general polarization law (2.1), we first observe that in the time domain (2.6) becomes P — ^-j Pi, where

These equations can be written as first order n^ dimensional vector systems for the variables

where Mi = diag(l,..., 1,0:^.), [/^Q] = col(0,... ,0,/30), and Ai is the n; x n; matrix with 1's on the superdiagonal, (—a l 0 , — a | , . . . , —a^._ 1 ) in the n, row and zeros elsewhere. We note that this matrix representation requires some care in its interpretation, as the elements of the vector Pi and the quantity E are all vector-valued functions themselves. The matrices Mj and Ai should be viewed as the matrix representations of linear transformations acting on a vector of unknowns, in which the transformations are expressed through scalar-vector multiplication of the scalar coefficients of Mi and Ai and the vector elements of

A.

14

Banks, Buksas, and Lin

The variation of parameters representation can be used (assuming -Pi(O) = 0) with Gi(t] = exp(M~lAit] to obtain

so that the first component Pi of Pi has the representation

we are thus led to the expression

This has the same form as (2.1), where the susceptibility kernel is the sum of kernels deriving from impulse response solutions of general nth order polarization models (including first and second order Debye and Lorentz models, respectively). Using a standard variation-of-parameters representation for systems with delays (see [BJ70]), one can also argue that polarization models with intrinsic explicit delays of the form

are also contained as special cases of (2.1). Polarization effects are dependent on physical variables such as mass density, temperature, material inhomogeneity, etc. (This dependence on physical quantities is usually expressed through changes in the coefficients in the mathematical models of the previous discussions.) This fact can be used to produce "acoustic gratings" to act as electromagnetic reflecting interfaces. In particular, the introduction of an acoustic wave will change the density of the fluid. (Indeed, acoustic waves are simply pressure waves which involve density variations.) This in turn will affect electromagnetic properties, such as the refraction index, of the fluid. This is known as the acoustooptic effect. Consequently, any electromagnetic wave transmitted into a material with an acoustic field will be modulated by the field. At the same time, the material electrostriction caused by the electromagnetic waves will also affect the propagation of the pressure wave in the fluid [MI68]. This produces a fully coupled nonlinear model with equations for both the electromagnetic and acoustic pressure waves (see [BL] and [MI68, p. 825]). In our initial efforts, we focus on the effects of variations in

Problem Formulation and Physical Modeling

15

acoustic pressure as a reflector of electromagnetic waves. We ignore the effect of electromagnetic forces in the acoustic equation under the tacit assumption that the effect is weak. To demonstrate the effect of the acoustic wave on the electromagnetic wave, we begin with a common assumption [Kor97] that the electric susceptibility is an afhne function of the acoustic pressure p(t, z):

Then we have Note that we have returned to an instantaneous model of polarization, rather than a DRF model such as Debye. Since it plays an important role in the subsequent analysis and computations, we compute

More generally, we may assume that the fluid in the acoustically affected part of the domain obeys a generalized pressure dependent polarization rule [Dan67, Chapter 9]:

To further simplify this preliminary investigation, we take

which we note is not a special case of (2.7). However, we shall see below that for typical values of dielectric parameters, the contributions of the /o(p) and f i (p) terms are several orders of magnitude smaller than that of the /2 (p) term (and the other terms in (2.19) below). Hence a reasonable first approximation to (2.7) is to ignore the /o, /i terms and take as the polarization assumption in the acoustic region the approximation

2.2

Estimation Methodology

Adopting a rather standard approach, we propose to identify the unknown parameters in a given model of polarization and a geometric representation by attempting to minimize the difference between simulations and observations of time-domain data. The data are measurements of the electric field at points in the exterior domain Q,Q at discrete times. The simulation is a computed

16

Banks, Buksas, and Lin

solution to Maxwell's equations with the constitutive laws for polarization, using candidate values of the geometric and material parameters. The criterion for optimization is a least-squares measurement of the difference between the simulation and the observed data given by

The Ei are measurements of the electric field taken at specific locations and times. The E(ti,£i;q) are solutions evaluated at the same locations and times from the simulation using the full set of parameter values q. We note that use of least squares formulations in inverse problems in electromagnetic scattering as well as in geophysical (seismic reservoir) problems is well established (e.g., see [PSS84, GW65, Bor99, BK89b] and the numerous references contained therein). We note two nontrivial difficulties with this approach and propose solutions, the efficacy of which will be demonstrated in the particular implementation discussed in this monograph. The unknown location of part of the boundary creates computational challenges. During the course of an iterative optimization procedure, simulations will be repeated many times for different locations of the unknown part of the boundary of 0. That is, iterative-based methods generally will involve changing domains and hence changing discretization grids in the usual finite element or finite difference approximation schemes. Any associated computational scheme (with domain changing with each iterative step) will be prohibitive in effort and time. We address this difficulty by employing the "method of mappings" [BK89a, Pir83, BKW90] and transforming the problem on £1 (J fio with unknown geometry to one with known geometry (a reference domain fi) at the expense of introducing additional unknowns into the equations that must be solved on this new domain. This technique, developed earlier [Pir83] in connection with general problems in shape design and optimization, has been used successfully in thermal inverse problems [BK89a, BKW90] related to the determination of unknown thermal boundaries and interfaces defined by material defects. Conceptually, one transforms a parameter dependent domain T>(q) = QQ U Q(q) to a reference domain fi which is independent of the parameterization q of the unknown part of the boundary (see Figure 2.2) through a parameter dependent transformation of the independent variable. That is, one defines a map T(q} : ^oUO(g) i—> fi so that X = T(q)X, where X = (x,y,z) and X = (x,y, z) are the independent coordinates in V = OQ U £l(q) and fi, respectively. On the original domain fio U &( H V*, where H = L 2 (0,1). We assume throughout this section that the slab region fi = [zi, 22] has been mapped to [zi, 1] so that our domain of interest is Q = [0,1] with 7, /?, and a(t) bounded on [0,1] and vanishing outside [zi,l]. Thus, in (3.1) the inner products should be properly interpreted in the sense of (2.17), (2.18). However, as explained in the last section, one can equivalently treat the well-posedness questions of this section using the unweighted L2 inner product in (3.1), and we shall do this. From a general theory presented in [BIW95], [BSW96, Chapter 4], one sees that (3.1) differs from the usual lightly damped second order systems of [BSW96] only by the presence of the terms cE(t, 0)0(0) due to the absorbing left boundary 27

28

Banks, Buksas, and Lin

condition and { J0 a(t — s)E(s)ds, 0) resulting from the convolution representation for the polarization. (The (@E, 0} term is of no additional consequence in any well-posedness analysis since it is readily handled in developing a priori estimates.) From the general theory one might expect to seek solutions of (3.1) in the sense of L2(Q,T;V)* ~ L 2 (0,T;F*) with E G L 2 (0,T;\0,£ e L 2 (0,T;#) and E G L 2 (0,T; V*} for appropriate interpretation of the (-,-) in (3.1), i.e., the duality product {•,•)}/*, v which reduces to the H = L2 inner product in all terms of (3.1) except the first and last. We recall from Section 2.3 that the input J(t] results from a point source (antenna) at z — 0 and hence J(i) has the form g(t)6(z) for a windowed time signal g ( t ) . This motivates our desire for results allowing J(t) values in V* . In addition to differences one might encounter due to the polarization term, if one obtains as usual E G I/ 2 (0, T; H}, where H = Z/ 2 (0, 1), then questions arise concerning the interpretation of the boundary term cE(t, 0)0(0) which has the appearance of pointwise evaluation of an L 2 (0, 1) "function" at z = 0. Correct interpretation of this term will result from our arguments below. We follow the general approach using sesquilinear forms as in [BIW95, BSW96] which are standard in the research literature [LioTl, Wlo87]. We first rewrite (3.1) by adding a term (kE,(f))fj to both sides of the equation. The positive constant k is chosen so that (3 = k + /3 satisfies fi > e > 0 on [0, 1] for some constant e; this is possible since by assumption (3 G L°°(0, 1). We define a sesquilinear form a\ : V x V —> C by

Equation (3.1) can then be rewritten as

for all 0 G V, where it is readily seen that a\ is ^-continuous and F-elliptic. That is, there are positive constants Ci,c 2 such that

To establish existence of solutions to (3.2), (3.4), where $ G V, \& G H, we follow the ideas in [BSW96] and choose a subset {wi}^! spanning V. (Without loss of generality we may assume linear independence of these elements.) Let Vm = span{tui,... ,wm} and define Galerkin "approximates"

Well-Posedness

29

where the {e™ (t}}™^ are determined by substitution into (3.4) and requiring this system of ordinary differential equations to hold for 0 = Wi, i = 1, 2, . . . , m. This m-dimensional s s t e m is solved with initial conditions

where the 3>m, $m are chosen in Vm so that $m —> $ in V, ^m —> \I> in #. We thus find that Em(t) satisfies (3.4) with 0 = Em(t) G V so that for each £ we have the system

This system will allow us to obtain bounds for {Em}, {Em}, and {Em(-, 0)} that are independent of m. Using the fact that

we may rewrite (3.6) as

Integration along with use of the V ellipticity of a\ yields

where

30

Banks, Buksas, and Lin Assuming that a is bounded on [0,T] x [0, 1] we have for T £ [0, t]

Thus we find

We also have

Finally, to consider the term Ta, we use (assuming that

We obtain

Combining (3.7), (3.8), (3.9), and (3.10), we obtain

Well-Posedness

31

Assuming that J E Hl(Q,T; V*} and using the boundedness of {Em(Q)} in V and {Em(0}} in H (which follows from the convergences of {$m} and {^m}, respectively), we may employ Gronwall's inequality along with the inequality (3.11) to conclude that {Em} is bounded in C(0,T; H}, {Em} is bounded in C(0, T; V) and {Em(; 0)} is bounded in L 2 (0, T). Thus we find (extracting subsequences and reindexing as usual) there exist E E L2(0, T; V), .E E L2(0, T; #), and EL E L 2 (0,T) such that

The limit function E is a candidate for solution of (3.2), (3.4) and we must verify that E = E, EL = E(-,Q) in some sense and that we may pass to the limit in the version of (3.4) for Em to obtain (3.4) for the limit function. First we note that for each m

and

Passing to the limit (in the weak H sense in (3.12)) we obtain

We find that (3.14) holds in the H sense for each t € [0, T] and hence E — E while (3.15) yields that jE?(t,0) exists and is continuous in t. In actuality E(t,Q) is absolutely continuous with E(t,Q) — Ei(t) for almost every t. We note, in fact, that the same arguments used in [BGS97, Lemma 5.1(b)] can be used to establish that Em also converges weakly in C(0, T; H ) to E so that E E C(0,T; #) n L 2 (0,T; V). Thus we have that our candidate E for solution of (3.2), (3.4) satisfies

We must show that E satisfies (3.4). For this we follow directly the arguments of [BSW96, pp. 100-101]. Taking V e Cl[Q,T] with ^(T) = 0 and

32 choosing i^j(t) = i(j(t}wj where the {'Wj}°^1 fixing ji, that for all m> j, Em must satisfy

Banks, Buksas, and Lin 1areas

chosen before, we have,

Integrating by parts in the first term and then taking the limit as ra —* oo, with the convergences of (3.16)-(3.18) we obtain

It follows that for every Wj we have in the L 2 (0, T) sense (except in the first term which is in the distributional sense in t)

Since {wj}(^.l was chosen total in V we thus obtain that E G L2(0,T; V*) and that E satisfies (3.4). From (3.14) we know that E(0) = $ and the arguments that E = * follow exactly as those in [BSW96, p. 101]. Hence we find that E is a solution of (3.2), (3.4). Continuous dependence of solutions to (3.2), (3.4) on $,^, and J follow readily from the inequality (3.11) and some standard arguments. Noting that I • \H < (J\ • \v for some constant /^ and letting

we observe that (3.11) implies

Well-Posedness

33

for some positive constant v independent of ra. Using Gronwall's inequality again, we obtain

Recalling that E m (0) = $m -» $ in V and £ m (0) = ^ m -» $ in # so that from (3.19) we have limtf m < K, where A" = |*|^ + (c2 + l)|$|v + ^\^\2Hl(o T-V*)> we may use weak lower semicontinuity of norms, the convergences of (3.16) and (3.17), and (3.20) to conclude that

Since the mapping ($, ^, J) -> (E, E) is linear from F x F x Hl(0,T; V*} to L 2 (0,T;V) x L 2 (0,T;7J) we see that (3.21) yields continuous dependence of solutions (E,E) of (3.2), (3.4) on initial data ($,*) and input J. For uniqueness of solutions to (3.2), (3.4), we again follow the standard arguments given in [BSW96, pp. 102-103]. In this case the details are tedious but rather straightforward. As usual, it suffices to show that the only solution of (3.4) corresponding to zero initial data ($ = \I/ = 0 in (3.2)) and zero input (J = 0) is the trivial solution. Let E be a solution corresponding to $ = \I> = J — 0 and for arbitrary s in (0, T) define

so that i/>s(T) = 0 and i^s(t] 6 F for each t. We then find that

Hence, choosing (f) — ijjs-(t) in (3.4) and integrating over t from 0 to s, we have

Observing that

34

Banks, Buksas, and Lin

and

we may use (3.22) to obtain

It follows immediately that

From the definition of TJJS we have for each

so that

Using (3.25) and arguments exactly like those behind the estimate (3.8) for TI (T) we find for t < s

Well-Posedness

35

Using (3.26) and (3.27) we thus obtain

or

for arbitrary s E (0,Tj. Invoking the Gronwall inequality once again, we conclude that E(£) = 0 on (0,T) and solutions of (3.2), (3.4) are unique. Summarizing our discussions in this section, we see that we have proved the following result. Theorem 1. Suppose that J E Hl(0,T;V*), 7,/5 E L°°(0,l), 00 a E L°°(0,T;L (0,1)) with a, (3,7 vanishing outside [zi,l]. Then for $ E V = #^(0,1),^ E H = L 2 (0,1) ; we have that solutions to (3.2), (3.4) exist and are unique. These solutions satisfy E E L 2 (0, T; V) n C(0, T; H), E E I/ 2 (0, T; #), and E E L 2 (0,T;y*). Moreover, t —* E ( t , Q ) is absolutely continuous with E(-,Q) E L 2 (0,T). The solutions depend continuously on (3>,\I>, i 7) as maps /rom I/ x H x tf^O,!1; V*) to L 2 (0,T;1/) x L 2 (0,T;//).

3.2

A Semigroup Formulation

While the variational formulation of the previous section provides adequate wellposedness results for our subsequent discussions, it is of some mathematical interest to determine whether the integro-partial differential system (3.1), (3.2) has a semigroup based formulation. For the sake of completeness, we present such a formulation next while noting that for this particular problem, improved regularity results over our theorem in Section 3.1 will not be obtained using semigroup methods. For this section we assume that as before 7, j3 6 L°°(0,1) while a E L°°((0, T) x (0,1)) and a,/3,7 vanish outside Q. We further assume that t t—> a(t, •) is positive in O and and monotone decreasing to zero so that a(t, •) < 0 whenever a E Hl(Q,T). This monotonicity assumption is typical of the usual assumptions in displacement susceptibility kernels (e.g., see [BloSl, p. 102] or [Hop77]). We further assume for this section that a(t, •) is constant in £1 so that a(t,z) = In(z)a(t} for some monotone decreasing function a. We note that a(t,z) = —I^(z)g(i) for the Debye and Lorentz polarization laws for g given in Section 2.1 satisfies this separation assumption. Moreover the Debye DRF also satisfies the required monotonicity assumption. Unfortunately, due to its oscillatory nature, the Lorentz DRF does not satisify the monotonicity condition.

36

Banks, Buksas, and Lin We consider the term (tacitly assuming E(s) = 0 for s < 0)

from (3.1) and note that it can be equivalently written

where G(f ) = r. We observe at this point that G(£) > 0 with G(f ) > 0 on (— r, 0]. As in the previous section, we take V = -H^(0, 1), H ~ L2(0, 1) and assume that fi = [zi , 1] C [0, 1] is the region of interest. We shall have use of H = L2(17) and shall denote the restriction of functions (f) in L 2 (0, 1) to J7 again by 0 and write G L 2 (fi) whenever no confusion will result. Motivated by the "strong" form of (3.1), i.e., see (2.13) with er = 1, and using the above definitions and approximating, we may write (3.1) as

where, of course, the derivatives must be interpreted in a weak or distributional sense. Following [BFW88, BFW89, FI90] and [BMZ96], we define an auxiliary variable w(t) in W = L2G(-r, 0; H) by w(t](9] = E(t] - E(t + 9), -r < 9 < 0. Since G(9,z) = G(9,z] > 0 for 9 6 (— r, 0],z G £1 and G is constant in fi, we may take as an inner product for W the weighted L2 inner product

under which W is a Hilbert space. We note that by our notational convention explained above, we have w(t) G W for any E(t,z) with E(-, •) G LQ(— r, Q;H). Using a standard shift notation, we may write w(t) = E(t) — E(t + 6) = E(t) — £*(0), where El(0) = E(t + 0) for -r < d < 0. Adding and subtracting appropriate terms in (3.30), we find

Well-Posedness

37

or, equivalently,

where Gn = f_r G(t)d£ and w(t}(£) = E(t) - £?'(£).

For our semigroup formulation, we consider (3.31) in the state space Z = V x H x W = tfjj(0,l) x £ 2 (0,1) x L^(-r,0,H) with states (,^,ry) = (E(t),E(t),w(t)} = (E(t),E(t),E(t) - £'(•))• To define an infinitesimal generator, we begin by defining a fundamental set of component operators. Let A&C(V,V*) be denned by

where ^o is the Dirac operator SQIJJ = ^(0). Then we find

so that it is readily seen that a\ : V x V \—> C defined by

is symmetric, V continuous, and V coercive (i.e., ci||y — AQ| 0). We also define operators B <E L(V, V] and K e £(W, F) by

so that

Since G(£, z] — 0 for z e (0, l)\fl, we abuse notation and write this as

even though, strictly speaking, TJ(^, 2) is only defined for z e £2. With these definitions and notations, (3.31) can then be written as

38

Banks, Buksas, and Lin

or

We rewrite (3.38) as a first order system in the state £(£) = (-B(t), £"(£), w(i)), where w(t) = E(t) — El. To aid in this we introduce another operator

defined on dom D =

We then observe that w(t) — E(i) — El satisfies

Thus we may formally rewrite (3.38) as a first order system and adjoin to it the equation We then obtain the first order system for £(£) given by

where A given by

is defined on

that is, ^.$ = (,A + BV' + A'ry, ^ + -Dry) for $ = (0,^,77). The forcing function ^" in (3.40) is given by T = 001(0,^7,0). To argue that A is the infinitesimal generator of a Co-semigroup, we actually consider the system (3.40) on an equivlaent space Z\ — V\ x H x W, where Vj is the space V with equivalent inner product {^i,^}^ = A £ V and ZA E #, so we can again apply the arguments for Theorem 1 to establish the existence of a unique solution WA to the problem (3.51)-(3.52) so that We next verify that WA = -^A- To assist in this, we define

It is clear that Integrating (3.51) over (0, £) and using (3.52), we obtain

Using (3.52), integration by parts and the consistency condition, we can readily argue that the expression in brackets in (3.54) vanishes. Comparing equations (3.53)-(3.54) with equations (3.48)-(3.49) and using the uniqueness of solutions to equations (3.48)-(3.49) we then find that Thus the desired regularity expressed in (3.50) follows from that of WAWe next formally differentiate both sides of (3.51) with respect to t and introduce a variable w& for WA to obtain

with

Well-Posedness

45

and

We combine (3.55) with the initial conditions

where YA = -(a(0)$A + /^A + 7^A - c2\&A). We tnen argue as abt the solution w& of (3.55) is the same as WA, and

In these arguments we use the conditions 7(0) = cr(0) = (3(0) = 0 as well as the consistency condition ^'(0) = c$"(Q). We next provide uniform (in A) bounds on EA- Recall UA = £A- Integrating (3.51) over (0, t) and then choosing 0 = EA (this is possible due to the regularity (3.56)), we obtain

After an integration by parts the second term in the right side of (3.57) can be written as

Using this in (3.57) along with the Cauchy-Schwarz inequality, we obtain directly

46

Banks, Buksas, and Lin

where we have used &(0) = 0. Here the constant C depends only on k, T, and the bounds 7 = |7|L°°, & — max{|a|L°°, H£,°O), and j3c = min{c2,/3}. Using the bounds for ^ A ,