Part I. Foundations: 1. Random variables 2. Probability measures 3. Stochastic processes 4. The stochastic integral Part II. Existence and Uniqueness: 5. Linear equations with additive noise 6. Linear equations with multiplicative noise 7. Existence and uniqueness for nonlinear equations 8. Martingale solutions Part III. Properties of Solutions: 9. Markov properties and Kolmogorov equations 10. Absolute continuity and Girsanov's theorem 11. Large time behaviour of solutions 12. Small noise asymptotic.

Stochastic Equations in Infinite Dimensions

Time-delay systems: an overview of some recent advances and open problems

In this paper, we study the local discontinuous Galerkin (LDG) methods for nonlinear, time-dependent convection-diffusion systems. These methods are an extension of the Runge--Kutta discontinuous Galerkin (RKDG) methods for purely hyperbolic systems to convection-diffusion systems and share with those methods their high parallelizability, high-order formal accuracy, and easy handling of complicated geometries for convection-dominated problems. It is proven that for scalar equations, the LDG methods are L2-stable in the nonlinear case. Moreover, in the linear case, it is shown that if polynomials of degree k are used, the methods are kth order accurate for general triangulations; although this order of convergence is suboptimal, it is sharp for the LDG methods. Preliminary numerical examples displaying the performance of the method are shown.

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The Local Discontinuous Galerkin Method for Time-Dependent Convection-Diffusion Systems

Differential-Difference Equations. By Richard Bellman and Kenneth L. Cooke. Pp. xvi, 465. 1963. (Academic Press)

Numerical Initial Value Problems in Ordinary Differential Equations

Preface to the Second Edition Preface to Volume I of the First Edition Preface to Volume II of the First Edition List of Figures Introduction Part I. Finite Dimensional Linear Control of Dynamical Systems Control of Linear Finite Dimensional Differential Systems Linear Quadratic Two-Person Zero-Sum Differential Games Part II. Representation of Infinite Dimensional Linear Control Dynamical Systems Semi-groups of Operators and Interpolation Variational Theory of Parabolic Systems Semi-group Methods for Systems with Unbounded Control and Observation Operators Differential Systems with Delays Part III. Qualitative Properties of Linear Control Dynamical Systems Controllability and Observability for a Class of Infinite Dimensional Systems Part IV. Quadratic Optimal Control: Finite Time Horizon Systems with Bounded Control Operators: Control Inside the Domain Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary Part V. Quadratic Optimal Control: Infinite Time Horizon Systems with Bounded Control Operators: Control Inside the Domain Systems with Unbounded Control Operators: Parabolic Equations with Control on the Boundary Systems with Unbounded Control Operators: Hyperbolic Equations with Control on the Boundary Appendix A. An Isomorphism Result References Index

Representation and Control of Infinite Dimensional Systems

This book provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is no longer a set of parameters or functions but the shape or the structure of a geometric object. Shapes and Geometries: Analysis, Differential Calculus, and Optimization presents the extensive, recently developed theoretical foundation to shape optimization in a form that can be used by the engineering community. It also clearly explains the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field.

Shapes and Geometries: Analysis, Differential Calculus, and Optimization

There is disclosed a charged particle beam apparatus capable of adjusting the convergence angle of the charged particle probe independent of the current in the probe. The apparatus comprises a charged particle gun producing a charged particle beam, a first condenser lens, a second condenser lens, an objective lens, an aperture, and an auxiliary condenser lens. The first condenser lens, the second condenser lens, and the objective lens which are arranged in this order act to sharply focus the beam to form an charged particle probe impinging on a specimen. The aperture is installed between the second condenser lens and the objective lens. The auxiliary lens is located between the objective lens and the aperture to control the convergence angle of the probe. The passage of the charged particle beam which extends from the aperture to the specimen is wide enough to prevent the current in the charged particle probe striking the specimen from varying if the strength of the auxiliary lens and the strength of the objective lens are varied.

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Representation and Control of Infinite Dimensional Systems, 2 nd edition

Many flexible structures consist of a large number of components coupled end to end in the form of a chain. In this paper, we consider the simplest type of such structures which is formed by N seri...

Modeling stabilization and control of serially connected beams

This considerably enriched new edition provides a self-contained presentation of the mathematical foundations, constructions, and tools necessary for studying problems where the modeling, optimization, or control variable is the shape or the structure of a geometric object. Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization, Second Edition presents the latest ground-breaking theoretical foundation to shape optimization in a form that can be used by the engineering and scientific communities. It also clearly explains the state-of-the-art developments in a mathematical language that will attract mathematicians to open questions in this important field. A series of generic examples has been added to the introduction and special emphasis has been put on the construction of important metrics. Advanced engineers in various application areas use basic ideas of shape optimization, but often encounter difficulties due to the sophisticated mathematical foundations for the field. This new version of the book challenges these difficulties by showing how the mathematics community has made extraordinary progress in establishing a rational foundation for shape optimization. This area of research is very broad, rich, and fascinating from both theoretical and numerical standpoints. It is applicable in many different areas such as fluid mechanics, elasticity theory, modern theories of optimal design, free and moving boundary problems, shape and geometric identification, image processing, and design of endoprotheses in interventional cardiology. Audience: This book is intended for applied mathematicians and advanced engineers and scientists, but the book is also structured as an initiation to shape analysis and calculus techniques for a broader audience of mathematicians. Some chapters are self-contained and can be used as lecture notes for a minicourse. The material at the beginning of each chapter is accessible to a broad audience, while the subsequent sections may sometimes require more mathematical maturity. Contents: List of Figures; Preface; Chapter 1: Introduction: Examples, Background, and Perspectives; Chapter 2: Classical Descriptions of Geometries and Their Properties; Chapter 3: Courant Metrics on Images of a Set; Chapter 4: Transformations Generated by Velocities; Chapter 5: Metrics via Characteristic Functions; Chapter 6: Metrics via Distance Functions; Chapter 7: Metrics via Oriented Distance Functions; Chapter 8: Shape Continuity and Optimization; Chapter 9: Shape and Tangential Differential Calculuses; Chapter 10: Shape Gradients under a State Equation Constraint; Elements of Bibliography; Index of Notation; Index.

Michel C. Delfour

Papers

Representation and Control of Infinite Dimensional Systems

Shapes and Geometries: Analysis, Differential Calculus, and Optimization

Representation and Control of Infinite Dimensional Systems, 2 nd edition

Modeling stabilization and control of serially connected beams

Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization