# Boundary Estimation Problems Arising in Thermal Tomography

01 Nov 1989-

TL;DR: In this paper, the identification of two-dimensional spatial domains arising in the detection and characterization of structural flaws in materials is considered for a thermal diffusion system with external boundary input, observations of the temperature on the surface are used in a output least squares approach.

Abstract: Problems on the identification of two-dimensional spatial domains arising in the detection and characterization of structural flaws in materials are considered. For a thermal diffusion system with external boundary input, observations of the temperature on the surface are used in a output least squares approach. Parameter estimation techniques based on the method of mappings are discussed and approximation schemes are developed based on a finite element Galerkin approach. Theoretical convergence results for computational techniques are given and the results are applied to experimental data for the identification of flaws in the thermal testing of materials.

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TL;DR: In this paper, the authors considered the inverse Dirichlet problem, where the role of the unknown is played by an inaccessible part of the boundary, and the role is assumed to be played by overdetermined boundary data for the remaining, accessible, part.

Abstract: In this paper we study a class of inverse problems associated to elliptic boundary value problems. More precisely, those inverse problems in which the role of the unknown is played by an inaccessible part of the boundary and the role of the data is played by overdetermined boundary data for the elliptic equation assigned on the remaining, accessible, part of the boundary. We treat the case of arbitrary space dimension n > 2. Such problems arise in applied contexts of nondestructive testing of materials for either electric or thermal conductors, and are known to be ill-posed. In this paper we obtain essentially best possible stability estimates. Here, in the context of ill-posed problems, stability means the continuous dependence of the unknown upon the data when additional a priori information on the unknown boundary (such as its regularity) is available. Mathematics Subject Classification (2000): 35R30 (primary), 35R25, 35R35, 35B60, 31B20 (secondary). 1. Introduction In this paper we shall deal with two inverse boundary value problems. Suppose Q is a bounded domain in W with sufficiently smooth boundary a S2, a part of which, say I (perhaps some interior connected component of a S2 or some inaccessible portion of the exterior component of a03A9), is not known. This could be the case of an electrically conducting specimen, which is possibly defective due to the presence of interior cavities or of corroded parts, which are not accessible to direct inspection. See for instance [K-S-V]. The aim is to detect the presence of such defects by nondestructive methods collecting current and voltage measurements on the accessible part A of the boundary If we assume that the inaccessible part I of a 03A9 is electrically insulated, then, given a nontrivial function 1/1 on A, having zero average (which represents Work supported in part by MURST. Pervenuto alla Redazione il 21 settembre 1999 e in forma definitiva il 24 giugno 2000. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) Vol. XXIX (2000), pp. 755-806 756 the assigned current density on the accessible part A of 8Q), we have that the voltage potential u inside Q satisfies the following Neumann type boundary value problem Here, v is the exterior unit normal to aS2 and cr = denotes the known symmetric conductivity tensor ant it is assumed to satisfy a hypothesis of uniform ellipticity. Let us remark that the solution to ( 1.1 a )-( 1.1 c) is unique up to an undetermined additive constant. In order to specify a single solution, we shall assume, from now on, the following normalization condition Suppose, now, that E is an open subset of which is contained in A, and on which the voltage potential can be measured. Then, the inverse problem consists of determining I provided is known. This is the first object of our study and we shall refer to it as the Inverse Neumann Problem (Neumann case, for short). An allied problem is the one associated to the direct Dirichlet problem Here, as above, I, A are the inaccessible, respectively, accessible, parts of and ar is the conductivity tensor satisfying the same hypotheses. Our second object of study is the inverse problem consisting in the determination of I from the knowledge of orVM’ where E C A is as above. We shall refer to it as the Inverse Dirichlet Problem (Dirichlet case, for short). We believe that also this problem may be of interest for concrete applications of nondestructing testing, for instance in thermal imaging. In this case, the inaccessible boundary I could represent a priviledged isothermal surface, such as a solidification front. Of course, it should be kept in mind that, dealing with thermal processes, the evolutionary model based on parabolic, rather than elliptic, equations is in general more appropriate, for related issues see, for instance, [B-K-W], [Bi], [V 1 ] . However, we trust that also a preliminary study of a stationary model may be instructive. Such two problems, the Neumann and Dirichlet cases, are known to be ill-posed. Indeed there are examples that show that, under a priori assumptions on the unknown boundary I regarding its regularity (up to any finite order of differentiability), the continuous dependence (stability) of I from the measured data in the Neumann case, orVM’ VIE in the Dirichlet case) is, at best, of logarithmic type. See [A12] and also [Al-R]. 757 The main purpose of this paper is to prove stability estimates of logarithmic type (hence, best possible) for both the Neumann and Dirichlet cases, (Theorems 2.1, 2.2), when the space dimension n > 2 is arbitrary. We recall that, for the case n = 2, results comparable to ours have been found in [Be-V] when o~ is homogeneous and in [R], [Al-R] when o~ can be inhomogeneous and also discontinuous. Other related results for the case of dimension two can be found in [Bu-C-Yl], [Bu-C-Y2], [Bu-C-Y3], [Bu-C-Y4], [An-B-J]. Let us also recall that, typically, the above mentioned results are based on arguments related, in various ways, to complex analytic methods, which do not carry over the higher dimensional case. In the sequel of this Introduction, we shall illustrate the new tools we found necessary to develop and exploit when n > 2. But first, let us comment briefly on the connection with another inverse problem which has become quite popular in the last ten years, namely the inverse problem of cracks. On one hand there are similarities, in fact a crack can be viewed as a collapsed cavity, that is a portion of surface inside the conductor, such that a homogeneous Neumann condition like (I,lc) holds on the two sides of the surface. On the other hand there are differences, for the uniqueness in the crack problem at least two appropriate distinct measurements are necessary [F-V], whereas for our problems, either the Neumann or the Dirichlet case, any single nontrivial measurement suffices for uniqueness, see for instance [Be-V] for a discussion of the uniqueness issue. Let us also recall that for the crack problem in dimensions bigger than two, various basic problems regarding uniqueness are still unanswered. See, for the available results and for references [Al-DiB ] . It is therefore clear that a study of the stability for the crack problem in dimensions higher than two shall require new ideas. Nonetheless, the authors believe that the techniques developed here might be useful also in the treatment of the crack problem. The methods we use in this paper are based essentially on a single unifying theme: Quantitative Estimates of Unique Continuation, and we shall exploit it under various different facets, namely the following ones. (a) Stability Estimates of Continuation from Cauchy Data. Since we are given the Cauchy data on E for a solution u to (l.la), we shall need to evaluate how much a possible error on such Cauchy data can affect the interior values of u. Such stability estimates for Cauchy problems associated to elliptic equations have been a central topic of ill-posed problems since the beginning of their modem theory, [H], [Pul], [Pu2]. Here, since one of our underlying aims will be to treat our problems under possibly minimal regularity assumptions, we shall assume the conductivity cr to be Lipschitz continuous (this is indeed the minimal regularity ensuring the uniqueness for the Cauchy problem, [PI], [M]). Our present stability estimates (Propositions 3.1, 3.2, 4.1, 4.2) will elaborate on inequalities due to Trytten [T] who developed a method first introduced by Payne [Pal], [Pa2]. The additional difficulty encountered here will be that we shall need to compare solutions u 1, u 2 which are defined on possibly different domains Q2 758 whose boundaries are known to agree on the accessible part A only. Let us recall that a similar approach, but restricted to the topologically simpler two-dimensional setting, has already been used in [All], [Be-V]. We shall obtain that, if the error on the measurement on the Cauchy data is small, then for the Neumann case, also IVUII I is small, in an L2 average sense, on S21 B Q2, the part of S21 which exceeds Q2. And the same holds for on SZ2 BQI (Propositions 3.1, 3.2). In the Dirichlet case instead we shall prove that u 1 itself is small in S21 B Q2, and the same holds for u2 on S22 (Propositions 4.1, 4.2). (b) Estimates of Continuation from the Interior. We shall also need interior average lower bounds on u and on its gradient (Propositions 3.3, 4.3), on small balls contained inside S2. Bounds of this type have been introduced in [Al-Ros-S, Lemma 2.2] in the context of a different inverse boundary value problem. The tools here involve another form of quantitative unique continuation, namely the following. (c) Three Spheres Inequalities. Also this one is a rather classical theme in connection with unique continuation. Aside from the classical Hadamard’s three circles theorem, in the context of elliptic equations we recall Landis [La] and Agmon [Ag]. Under our assumptions of Lipschitz continuity on a, our estimates (see (5.47) below) shall refer to differential inequalities on integral norms originally due to Garofalo and Lin [G-L], later developed by Brummelhuis [Br] and Kukavica [Ku]. (d) Doubling Inequalities in the Interior. This rather recent tool has been introduced by Garofalo and Lin in the above mentioned paper [G-L]. It provides an efficient method of estimating the local average vanishing rate of a solution to (l.la). Let us recall that it also provides a remarkable bridge to the powerful theory of Muckenhoupt weights [C-F] and that this last connection has been crucially used in [Al-Ros-S] and also in [V2]. The last, fundamental, appearance of quantitative estimates of unique continuation is the following. (e) Doubling Inequalities at the Boundary. For our purposes it will be crucial to evaluate the vanishing rate of Vu (in the Neumann case) or of u (in the Dirichlet case) near the inaccessible boundary I. In particular, the fact that such a rate is not worse than polynomial (Propo

133 citations

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TL;DR: In this paper, a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation was proposed, and a theoretical foundation for this Newton method is given by establishing the differentiability of the initial value problem with respect to the interior boundary curve in the sense of a domain derivative.

Abstract: We consider the inverse problem of reconstructing the interior boundary curve of an arbitrary-shaped annulus from overdetermined Cauchy data on the exterior boundary curve. For the approximate solution of this ill-posed and nonlinear problem we propose a regularized Newton method based on a boundary integral equation approach for the initial boundary value problem for the heat equation. A theoretical foundation for this Newton method is given by establishing the differentiability of the initial boundary value problem with respect to the interior boundary curve in the sense of a domain derivative. Numerical examples indicate the feasibility of our method.

77 citations

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TL;DR: This paper examines uniqueness and stability results for an inverse problem in thermal imaging by developing and using an inversion procedure to study the stability of the inverse problem for various experimental configurations.

Abstract: This paper examines uniqueness and stability results for an inverse problem in thermal imaging. The goal is to identify an unknown boundary of an object by applying a heat flux and measuring the induced temperature on the boundary of the sample. The problem is studied both in the case in which one has data at every point on the boundary of the region and the case in which only finitely many measurements are available. An inversion procedure is developed and used to study the stability of the inverse problem for various experimental configurations.

71 citations

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TL;DR: Based on the method of fundamental solutions and discrepancy principle for the choice of location for source points, the authors extend the application of the computational method to determine an unknown free boundary of a Cauchy problem of parabolic-type equation from measured Dirichlet and Neumann data with noises.

Abstract: Based on the method of fundamental solutions and discrepancy principle for the choice of location for source points, we extend in this paper the application of the computational method to determine an unknown free boundary of a Cauchy problem of parabolic-type equation from measured Dirichlet and Neumann data with noises. The standard Tikhonov regularization technique with the L-curve method for an optimal regularized parameter is adopted for solving the resultant highly ill-conditioned system of linear equations. Both one-dimensional and two-dimensional numerical examples are given to verify the efficiency and accuracy of the proposed computational method. Copyright © 2007 John Wiley & Sons, Ltd.

57 citations

01 Jan 2003

TL;DR: A survey of several recent and emerging ideas and ideas on modeling and system interrogation in the presence of uncertainty that the authors feel have potential for applications related to bioterrorism.

Abstract: In this paper we present a survey of several recent and emerging ideas and e orts on modeling and system interrogation in the presence of uncertainty that we feel have signi cant potential for applications related to bioterrorism. The rst focuses on physiologically based pharmacokinetic (PBPK) type models and the e ects of drugs, toxins and viruses on tissue, organs, individuals and populations wherein both intraand inter-individual variability are present when one attempts to determine kinetic rates, susceptibility, eÆcacy of toxins, antitoxins, etc., in aggregate populations. Methods combining deterministic and stochastic concepts are necessary to formulate and computationally solve the associated estimation problems. Similar issues arise in the HIV infectious models we also present below. A second e ort concerns the use of remote electromagnetic interrogation pulses linked to dielectric properties of materials to carry out macroscopic structural imaging of bulk packages (drugs, explosives, etc.) as well as test for presence and levels of toxic chemical compounds in tissue. These techniques also may be useful in functional imaging (e.g., of brain and CNS activity levels) to determine levels of threat in potential adversaries via changes in dielectric properties and conductivity. The PBPK and cellular level virus infectious models we discuss are special examples of a much wider class of population models that one might utilize to investigate potential agents for use in attacks, such as viruses, bacteria, fungi and other chemical, biochemical or radiological agents. These include general epidemiological models such as SIR infectious

54 citations

##### References

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01 Jan 1978

TL;DR: This book presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines as well as specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting.

Abstract: This book is based on the author's experience with calculations involving polynomial splines. It presents those parts of the theory which are especially useful in calculations and stresses the representation of splines as linear combinations of B-splines. After two chapters summarizing polynomial approximation, a rigorous discussion of elementary spline theory is given involving linear, cubic and parabolic splines. The computational handling of piecewise polynomial functions (of one variable) of arbitrary order is the subject of chapters VII and VIII, while chapters IX, X, and XI are devoted to B-splines. The distances from splines with fixed and with variable knots is discussed in chapter XII. The remaining five chapters concern specific approximation methods, interpolation, smoothing and least-squares approximation, the solution of an ordinary differential equation by collocation, curve fitting, and surface fitting. The present text version differs from the original in several respects. The book is now typeset (in plain TeX), the Fortran programs now make use of Fortran 77 features. The figures have been redrawn with the aid of Matlab, various errors have been corrected, and many more formal statements have been provided with proofs. Further, all formal statements and equations have been numbered by the same numbering system, to make it easier to find any particular item. A major change has occured in Chapters IX-XI where the B-spline theory is now developed directly from the recurrence relations without recourse to divided differences. This has brought in knot insertion as a powerful tool for providing simple proofs concerning the shape-preserving properties of the B-spline series.

10,258 citations

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03 Mar 1971

TL;DR: In this paper, the authors consider the problem of minimizing the sum of a differentiable and non-differentiable function in the context of a system governed by a Dirichlet problem.

Abstract: Principal Notations.- I Minimization of Functions and Unilateral Boundary Value Problems.- 1. Minimization of Coercive Forms.- 1.1. Notation.- 1.2. The Case when ?: is Coercive.- 1.3. Characterization of the Minimizing Element. Variational Inequalities.- 1.4. Alternative Form of Variational Inequalities.- 1.5. Function J being the Sum of a Differentiable and Non-Differentiable Function.- 1.6. The Convexity Hypothesis on $$ {U_{ad}} $$.- 1.7. Orientation.- 2. A Direct Solution of Certain Variational Inequalities.- 2.1. Problem Statement.- 2.2. An Existence and Uniqueness Theorem.- 3. Examples.- 3.1. Function Spaces on ?.- 3.2. Function Spaces on ?.- 3.3. Subspaces of Hm(?).- 3.4. Examples of Boundary Value Problems.- 3.5. Unilateral Boundary Value Problems (I).- 3.6. Unilateral Boundary Value Problems (II).- 3.7. Unilateral Boundary Value Problems (III).- 3.8. Unilateral Boundary Value Problems Case of Systems.- 3.9. Elliptic Operators of Order Greater than Two.- 3.10. Non-differentiable Functionals.- 4. A Comparison Theorem.- 4.1. General Results.- 4.2. An Application.- 5. Non Coercive Forms.- 5.1. Convexity of the Set of Solutions.- 5.2. Approximation Theorem.- Notes.- II Control of Systems Governed by Elliptic Partial Differential Equations.- 1. Control of Elliptic Variational Problems.- 1.1. Problem Statement.- 1.2. First Remarks on the Control Problem.- 1.3. The Set of Inequalities Defining the Optimal Control.- 2. First Applications.- 2.1. System Governed by the Dirichlet Problem Distributed Control.- 2.2. The Case with No Constraints.- 2.3. System Governed by a Neumann Problem Distributed Control.- 2.4. System Governed by a Neumann Problem Boundary Control.- 2.5. Local and Global Constraints.- 2.6. System Governed by a Differential System.- 2.7. System Governed by a 4th Order Differential Operator.- 2.8. Orientation.- 3. A Family of Examples with N = 0 and $$ {U_{ad}} $$ Arbitrary.- 3.1. General Case.- 3.2. Application (I).- 3.3. Application (II).- 4. Observation on the Boundary.- 4.1. System Governed by a Dirichlet Problem (I).- 4.2. Some Results on Non-homogeneous Dirichlet Problems.- 4.3. System Governed by a Dirichlet Problem (II).- 4.4. System Governed by a Neumann Problem.- 5. Control and Observation on the Boundary. Case of the Dirichlet Problem.- 5.1. Orientation.- 5.2. Boundary Control in L2(?).- 5.3. A "Controllability-Like" Problem.- 5.4. Pointwise Control and Observation.- 6. Constraints on the State.- 6.1. Orientation.- 6.2. Control and Constraints on the Boundary.- 7. Existence Results for Optimal Controls.- 7.1. Orientation.- 7.2. Distributed Control.- 7.3. Singular Perturbation of the System.- 7.4. Boundary Control.- 7.5. Control of Systems Governed by Unilateral Problems.- 8. First Order Necessary Conditions.- 8.1. Statement of the Theorem.- 8.2. Proof of the Theorem.- 8.2.1. "Algebraic" Transformation.- 8.2.2. General Remarks on the Utilization of (8.13.).- 8.2.3. Proof that dj,?0.- Notes.- III Control of Systems Governed by Parabolic Partial Differential Equations.- 1. Equations of Evolution.- 1.1. Data.- 1.2. Evolution Problems.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Some Examples.- 1.6. Semi-groups.- 2. Problems of Control.- 2.1. Notation. Immediate Properties.- 2.2. Set of Inequalities Characterizing the Optimal Control.- 2.3. Case (i). Set of Inequalities.- 2.4. Case (ii). Set of Inequalities.- 2.5. Orientation.- 3. Examples.- 3.1. Mixed Dirichlet Problem for a Second Order Parabolic Equation.- 3.1.1. C = Injection Map of L2(0, T V)?L2(Q).- 3.1.2. C = Identity Map of L2(0, T V) into itself.- 3.1.3. Observation of the Final State.- 3.2. Mixed Neumann Problem for a Parabolic Equation of Second Order.- 3.2.1. Case (i).- 3.2.2. Case (ii).- 3.3. System of Equations and Equations of Higher Order.- 3.3.1. System of Equations.- 3.3.2. Higher Order Equations.- 3.4. Additional Results.- 3.5. Orientation.- 4. Decoupling and Integro-Differential Equation of Riccati Type (I).- 4.1. Notation and Assumptions.- 4.2. Operator P(t), Function r(t).- 4.3. Formal Calculations.- 4.4. The Finite Dimensional Case Approximation.- 4.5. Passage to the Limit.- 4.6. Integro-Differential Equation of Riccati Type.- 4.7. Connections with the Hamilton-Jacobi Theory.- 4.8. The Case where Constraints are Present.- 4.9. Various Remarks.- 4.9.1. Direct Study of the "Riccati Equation".- 4.9.2. Another Approach to the Direct Study of the "Riccati Equation".- 4.9.3. Yet Another Approach to the Direct Study of the "Riccati Equation".- 5. Decoupling and Integro-Differential Equation of Riccati Type (II).- 5.1. Application of the Schwartz-Kernel Theorem.- 5.2. Example of a Mixed Neumann Problem with Boundary Control.- 5.3. Example of a Mixed Neumann Problem with Observation of the Final State.- 5.4. Mixed Neumann Problem, Observation of the Final State and Constraints in a Vector Space.- 5.5. Remarks on Decoupling in the Presence of Constraints.- 6. Behaviour as T ? + ?.- 6.1. Orientation and Hypotheses.- 6.2. The Case T = ?.- 6.3. Passage to the Limit as T ? + ?.- 7. Problems which are not Necessarily Coercive.- 7.1. Distributed Observation.- 7.2. Observation of the Final State.- 7.3. Examples where N = 0 and $$ {U_{ad}} $$ is not Bounded.- 8. Other Observations of the State and other Types of Control.- 8.1. Pointwise Observation of the State.- 8.2. Pointwise Control.- 8.3. Control and Observation on the Boundary.- 9. Boundary Control and Observation on the Boundary or of the Final State for a System Governed by a Mixed Dirichlet Problem.- 9.1. Orientation and Problem Statement.- 9.2. Non Homogeneous Mixed Dirichlet Problem.- 9.3. Definition of $$ \frac{{\partial y}}{{\partial {v_A}}} $$ Observation.- 9.4. Cost Function Equations of Optimal Control.- 9.5. Regular Control.- 9.6. Observation of the Final State.- 9.7. Observation of the Final State, Second Order Parabolic Operator.- 10. Controllability.- 10.1. Problem Statement.- 10.2. Controllability and Uniqueness.- 10.3. Super-Controllability and Super-Uniqueness.- 11. Control via Initial Conditions Estimation.- 11.1. Problem Statement. General Results.- 11.2. Examples.- 11.3. Controllability.- 11.4. An Estimation Problem.- 12. Duality.- 12.1. General Remarks.- 12.2. Example.- 13. Constraints on the Control and the State.- 13.1. A General Result.- 13.2. Applications (I).- 13.3. Applications (II).- 14. Non Quadratic Cost Functions.- 14.1. Orientation.- 14.2. An Example.- 14.3. Remarks on Decoupling.- 15. Existence Results for Optimal Controls.- 15.1. Orientation.- 15.2. Non-linear Problem with Distributed Control (I).- 15.3. Non-linear Problem with Distributed Control. Singular Perturbation.- 15.4. Non-linear Problem. Boundary Control.- 15.5. Utilization of Convexity and the Maximum Principle for Second Order Parabolic Equations.- 15.6. Control of Systems Governed by Evolution Inequalities.- 16. First Order Necessary Conditions.- 16.1. Statement of the Theorem.- 16.2. Proof of Theorem 16.1.- 16.2.1. "Algebraic" Transformation.- 16.2.2. Utilization of (16.11.).- 16.2.3. Proof of (16.12.).- 16.3. Remarks.- 17. Time Optimal Control.- 17.1. Problem Statement.- 17.2. Existence Theorem.- 17.3. Bang-Bang Theorem.- 18. Miscellaneous.- 18.1. Equations with Delay.- 18.1.1. Definition of the State.- 18.1.2. Control Problem.- 18.2. Spaces which are not Normable.- Notes.- IV Control of Systems Governed by Hyperbolic Equations or by Equations which are well Posed in the Petrowsky Sense.- 1. Second Order Evolution Equations.- 1.1. Notation and Hypotheses.- 1.2. Problem Statement. An Existence and Uniqueness Result.- 1.3. Proof of Uniqueness.- 1.4. Proof of Existence.- 1.5. Examples (I).- 1.6. Examples (II).- 1.7. Orientation.- 2. Control Problems.- 2.1. Notation. Immediate Properties.- 2.2. Case (2.5.).- 2.3. Case (2.6.).- 2.4. Case (2.7.).- 2.5. Case (2.8.).- 3. Transposition and Applications to Control.- 3.1. Transposition of Theorem 1.1.- 3.2. Application (I).- 3.3. Application (II).- 3.4. Application (III).- 4. Examples.- 4.1. Examples of Hyperbolic Problems. Distributed Control, Distributed Observation.- 4.2. Examples of Hyperbolic Systems. Distributed Control, Observation of the Final State.- 4.3. Petrowsky Type Equation. Distributed Control. Distributed Observation.- 4.4. Petrowsky Type Equation. Distributed Control. Observation of the Final State.- 4.5. Orientation.- 5. Decoupling.- 5.1. Problem Statement. Rewriting as a System of First Order Equations.- 5.2. Rewriting of the Set of Equations Determining the Optimal Control.- 5.3. Decoupling.- 5.4. Riccati Integro-differential Equation.- 5.5. Another Optimal Control Problem. Decoupling.- 6. Control via Initial Conditions. Estimation.- 6.1. Problem Statement.- 6.2. Coercivity of J(?).- 6.3. System of Equations Determining the Optimal Control.- 7. Boundary Control (I).- 7.1. Problem Statement.- 7.2. Definition of the State of the System.- 7.3. Distributed Observation.- 7.4. Boundary Observation.- 8. Boundary Control (II).- 8.1. Problem Statement.- 8.2. Control ? Regular.- 8.3. Examples.- 9. Parabolic-Hyperbolic Systems.- 9.1. Recapitulation of Some General Results.- 9.2. Complement.- 9.3. Control Problems.- 9.4. Example (I).- 9.5. Example (II).- 9.6. Decoupling.- 10. Existence Theorems.- 10.1. Orientation.- 10.2. Example. Introduction of a "Viscosity" Term.- 10.3. Time Optimal Control.- Notes.- V Regularization, Approximation and Penalization.- 1. Regularization.- 1.1. Parabolic Regularization.- 1.2. Application to Optimal Control.- 1.3. Application to Decoupling.- 1.4. Various Remarks.- 1.5. Regularization of the Control.- 2. Approximation in Terms of Systems of Cauchy-Kowaleska Type.- 2.1. Evolution Equation on a Variety.- 2.2. Approximation by a System of Cauchy-Kowaleska Type.- 2.3. Linearized Navier-Stokes Equation.- 3. Penalization.- Notes.

3,539 citations

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01 Jan 1984

TL;DR: A new book enPDFd methods for solving incorrectly posed problems that can be a new way to explore the knowledge and get one thing to always remember in every reading time, even step by step is shown.

Abstract: Spend your time even for only few minutes to read a book. Reading a book will never reduce and waste your time to be useless. Reading, for some people become a need that is to do every day such as spending time for eating. Now, what about you? Do you like to read a book? Now, we will show you a new book enPDFd methods for solving incorrectly posed problems that can be a new way to explore the knowledge. When reading this book, you can get one thing to always remember in every reading time, even step by step.

1,158 citations

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12 Dec 1983TL;DR: The techniques of the calculus of variation and of optimization proved to be successful for several optimal shape design problems however these remain expensive both in the qualification of the engineers required to understand the method and in computing time.

Abstract: The techniques of the calculus of variation and of optimization proved to be successful for several optimal shape design problems however these remain expensive both in the qualification of the engineers required to understand the method and in computing time. However it seems difficult to do without such techniques for 3-dimensional optimization problems. The field is well studied from the mathematical point of view but still in its beginnings from the industrial implementation side.

1,002 citations

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TL;DR: In this article, a differentiation with respect to the domain in boundary value problems is presented, where the domain is defined as the domain of the boundary value problem and boundary value is defined.

Abstract: (1980). Differentiation with Respect to the Domain in Boundary Value Problems. Numerical Functional Analysis and Optimization: Vol. 2, No. 7-8, pp. 649-687.

394 citations