Showing papers in "Lecture Notes in Mathematics in 2008"

Input to State Stability: Basic Concepts and Results

Abstract: The analysis and design of nonlinear feedback systems has recently undergone an exceptionally rich period of progress and maturation, fueled, to a great extent, by (1) the discovery of certain basic conceptual notions, and (2) the identification of classes of systems for which systematic decomposition approaches can result in effective and easily computable control laws. These two aspects are complementary, since the latter approaches are, typically, based upon the inductive verification of the validity of the former system properties under compositions (in the terminology used in [62], the “activation” of theoretical concepts leads to “constructive” control). This expository presentation addresses the first of these aspects, and in particular the precise formulation of questions of robustness with respect to disturbances, formulated in the paradigm of input to state stability. We provide an intuitive and informal presentation of the main concepts. More precise statements, especially about older results, are given in the cited papers, as well as in several previous surveys such as [103] and [105] (of which the present paper represents an update), but we provide a little more detail about relatively recent work. Regarding applications and extensions of the basic framework, we give some pointers to the literature, but we do not focus on feedback design and specific engineering problems; for the latter we refer the reader to textbooks such as [43], [60], [58], [96], [66], [27], [44].

Finite Elements for the Stokes Problem

Abstract: 1 Dipartimento di Matematica “F. Casorati”, Universita degli studi di Pavia, Via Ferrata 1, 27100 Pavia, Italy daniele.boffi@unipv.it 2 Istituto Universitario di Studi Superiori (IUSS) and I.M.A.T.I.–C.N.R., Via Ferrata 3, 27100 Pavia Pavia, Italy brezzi@imati.cnr.it 3 Departement de Mathematiques et de Statistique, Pavillon Alexandre-Vachon, Universite Laval, 1045, Avenue de la Me decine, Quebec G1V 0A6, Canada mfortin@mat.ulaval.ca

Geometry of Optimal Control Problems and Hamiltonian Systems

Abstract: These notes are based on the mini-course given in June 2004 in Cetraro, Italy, in the frame of a C.I.M.E. school. Of course, they contain much more material that I could present in the 6 hours course. The main goal is to give an idea of the general variational and dynamical nature of nice and powerful concepts and results mainly known in the narrow framework of Riemannian Geometry. This concerns Jacobi fields, Morse's index formula, Levi Civita connection, Riemannian curvature and related topics. I tried to make the presentation as light as possible: gave more details in smooth regular situations and referred to the literature in more complicated cases.

Sliding Mode Control: Mathematical Tools, Design and Applications

Abstract: The sliding mode control approach is recognized as one of the efficient tools to design robust controllers for complex high-order nonlinear dynamic plant operating under uncertainty conditions. The research in this area were initiated in the former Soviet Union about 40 years ago, and then the sliding mode control methodology has been receiving much more attention from the international control community within the last two decades. The major advantage of sliding mode is low sensitivity to plant parameter variations and disturbances which eliminates the necessity of exact modeling. Sliding mode control enables the decoupling of the overall system motion into independent partial components of lower dimension and, as a result, reduces the complexity of feedback design. Sliding mode control implies that control actions are discontinuous state functions which may easily be implemented by conventional power converters with “on-off” as the only admissible operation mode. Due to these properties the intensity of the research at many scientific centers of industry and universities is maintained at high level, and sliding mode control has been proved to be applicable to a wide range of problems in robotics, electric drives and generators, process control, vehicle and motion control.

Topological Derivatives for Shape Reconstruction

Abstract: Topological derivative methods are used to solve constrained optimization reformulations of inverse scattering problems. The constraints take the form of Helmholtz or elasticity problems with different boundary conditions at the interface between the surrounding medium and the scatterers. Formulae for the topological derivatives are found by first computing shape derivatives and then performing suitable asymptotic expansions in domains with vanishing holes. We discuss integral methods for the numerical approximation of the scatterers using topological derivatives and implement a fast iterative procedure to improve the description of their number, size, location and shape.

On Continuity Properties of the Law of Integrals of Lévy Processes

Abstract: Let (ξ, η) be a bivariate Levy process such that the integral \(\int_0^\infty {e^{ - \xi _{t - } } d\eta _t }\)converges almost surely. We characterise, in terms of their Levy measures, those Levy processes for which (the distribution of) this integral has atoms. We then turn attention to almost surely convergent integrals of the form I := ∫ 0 ∞ g(ξ t ) dt, where g is a deterministic function. We give sufficient conditions ensuring that I has no atoms, and under further conditions derive that I has a Lebesgue density. The results are also extended to certain integrals of the form ∫ 0 ∞ g(ξ t ) dY t , where Y is an almost surely strictly increasing stochastic process, independent of ξ.

X-ray Tomography

Abstract: We give a survey on the mathematics of computerized tomography. We start with a short introduction to integral geometry, concentrating on inversion formulas, stability, and ranges. We then go over to inversion algorithms. We give a detailed analysis of the filtered backprojection algorithm in the light of the sampling theorem. We also describe the convergence properties of iterative algorithms. We shortly mention Fourier based algorithms and the recent progresses made in their accurate implementation. We conclude with the basics of algorithms for cone beam scanning which is the standard scanning mode in present days clinical practice.

A simple theory for the study of SDEs driven by a fractional Brownian motion, in dimension one

Abstract: We will focus — in dimension one — on the SDEs of the type dX t = σ(X t )dB t + b(X t )dt where B is a fractional Brownian motion. Our principal aim is to describe a simple theory — from our point of view — allowing to study this SDE, and this for any H∈(0,1). We will consider several definitions of solutions and, for each of them, study conditions under which one has existence and/or uniqueness. Finally, we will examine whether or not the canonical scheme associated to our SDE converges, when the integral with respect to fBm is defined using the Russo-Vallois synmetric integral.

Introduction to Random Walks on Noncommutative Spaces

Abstract: We introduce several examples of random walks on noncommutative spaces and study some of their probabilistic properties. We emphasize connections between classical potential theory and group representations.

Lecture Notes on Logically Switched Dynamical Systems

Abstract: The subject of logically switched dynamical systems is a large one which overlaps with many areas including hybrid system theory, adaptive control, optimal control, cooperative control, etc. Ten years ago we presented a lecture, documented in [1], which addressed several of the areas of logically switched dynamical systems which were being studied at the time. Since then there have been many advances in many directions, far to many too adequately address in these notes. One of the most up to date and best written books on the subject is the monograph by Liberzon [2] to which we refer the reader for a broad but incisive perspective as well an extensive list of references. In these notes we will deal with two largely disconnected topics, namely switched adaptive control (sometimes called supervisory control) and “flocking” which is about the dynamics of reaching a consensus in a rapidly changing environment. In the area of adaptive control we focus mainly on one problem which we study in depth. Our aim is to give a thorough analysis under realistic assumptions of the adaptive version of what is perhaps the most important design objective in all of feedback control, namely set-point control of a singleinput, single output process admitting a linear model. While the non-adaptive version the set-point control problem is very well understood and has been so for more than a half century, the adaptive version still is not because there is no credible counterpart in an adaptive context of the performance theories which address the non-adaptive version of the problem. In fact, even just the stabilization question for the adaptive version of the problem did not really get ironed out until ten years ago, except under unrealistic assumptions which ignored the effects of noise and/or un-modeled dynamics. As a first step we briefly discuss the problem of adaptive disturbance rejection. Although the switching logic we consider contains no logic or discrete

A Brief Review on Point Interactions

Abstract: We review properties and applications of point interaction Hamiltonians. This class of operators is first defined following a classical presentation and then generalized to cases in which some dynamical and/or geometrical parameters are varying with time. We recall their relations with smooth short range potentials.

Generalized Differentials, Variational Generators, and the Maximum Principle with State Constraints

Abstract: We present the technical background material for a version of the Pontryagin Maximum Principle with state space constraints and very weak technical hypotheses, based on a primal approach that uses generalized differentials and packets of needle variations. In particular, we give a detailed account of two theories of generalized differentials, the “generalized differential quotients” (GDQs) and the “approximate generalized differential quotients” (AGDQs), and prove the corresponding open mapping and separation theorems. We state—but do not prove—the resulting version of the Maximum Principle. The result does not require the time-varying vector fields corresponding to the various control values to be continuously differentiable, Lipschitz, or even continuous with respect to the state, since all that is needed is that they be “co-integrably bounded integrally continuous.” This includes the case of vector fields that are continuous with respect to the state, as well as large classes of discontinuous vector fields, containing, for example, rich sets of single-valued selections for almost semicontinuous differential inclusions. Uniqueness of trajectories is not required, since our methods deal directly with multivalued maps. The dynamical reference vector field and reference Lagrangian are only required to be “differentiable” along the reference trajectory in a very weak sense, namely, that of possessing suitable “variational generators.” This includes— but is much more general than—the conditions of the classical cases when the reference vector field and Lagrangian are differentiable with respect to the state and the variational generator can be taken to be the singleton of the classical differential, as well as the case when they are Lipschitz and the variational generator can be chosen to be the Clarke generalized Jacobian. In addition, for the Lagrangian one can chose the variational generator to be the Michel-Penot subdifferential. For the functions defining the state space constraints, all that is needed is the existence of a variational generator in a slightly different technical sense, which includes as a special case the object often referred to as ∂ x g in the literature, as well as many non-Lipschitz cases. The conclusion yields finitely additive measures, as in earlier work by other authors, and a Hamiltonian maximization inequality valid also at the jump times of the adjoint covector.

Proof of a Tanaka-like formula stated by J. Rosen in Séminaire XXXVIII

Abstract: Let B t be a one dimensional Brownian motion, and let α′ denote the derivative of the intersection local time of B t as defined by J. Rosen in [2]. The object of this paper is to prove the following formula $$\frac{1}{2}\alpha _t^\prime (x) + \frac{1}{2}\operatorname{sgn} (x)t = \int_0^t {L_s^{B_s - x} dB_s - \frac{1}{2}\int_0^t {\operatorname{sgn} (B_t - B_u - x)du} }$$ (1) which was given as a formal identity in [2] without proof.

Adjoint Fields and Sensitivities for 3D Electromagnetic Imaging in Isotropic and Anisotropic Media

Abstract: In this paper we give an overview of a recently developed method for solving an inverse Maxwell problem in environmental and geophysical imaging. Our main focus is on low-frequency cross-borehole electromagnetic induction tomography (EMIT), although related problems arise also in other applications in nondestructive testing and medical imaging. In typical applications (e.g. in environmental rernediation), the isotropic or anisotropic conductivity distribution in the earth needs to be reconstructed from surface-to-borehole electromagnetic data. Our method uses a back-propagation strategy (based on adjoint fields) for solving this inverse problem. The method works iteratively, and can be considered as a nonlinear generalization of the Algebraic Reconstruction Technique (ART) in X-ray tomography, or as a nonlinear Kaczmarz-type approach. We will also Propose a new regularization scheme for this method which is based on a proper choice of the function spaces for the inversion. A detailed sensitivity analysis for this problem is given, and a set of numerically calculated sensitivity functions for homogeneous isotropic media is presented.

Influence of Contact Modelling on the Macroscopic Plastic Response of Granular Soils Under Cyclic Loading

Abstract: An alternative to the use of continuous equations and constitutive models is the microscopic description of the material in terms of the grains themselves and the contacts (interactions) between them. This approach has been successfully applied in recent years to the study of many different problems in soil mechanics and granular physics. An open question is how realistic the microscopic model must be in order to accurately describe the macroscopic behavior observed in experiments. The objective of this contribution is to show the influence of different simple models of compacted granular soils on the overall elasto-plastic response of the system as a whole. We will focus our investigation on granular ratcheting, which is the persistent strain accumulation that a granular soil suffers under certain cyclic stress conditions. The direct influence of different models on the ratcheting response of the material will also help us to understand further this peculiar behavior of the system. The influence of particle shape will be also discussed.

Polarization-Based Optical Imaging

Abstract: In this study I present polarization effects resulting from the reflection and transmission of a narrow beam of light through biological tissues. This is done numerically with a Monte Carlo method based on a transport equation which takes into account polarization of light. We will show both time-independent and time-dependent computations, and we will discuss how polarization can be used in order to obtain better images. We consider biological tissues that can be modeled as continuous media varying randomly in space, containing inhomogeneities with no sharp boundaries.

Lectures on the Topological Vertex

Abstract: The theory of Gromov–Witten invariants was largely motivated by the study of string theory on Calabi–Yau manifolds, and has now developed into one of the most dynamic fields of algebraic geometry. During the last years there has been enormous progress in the development of the theory and of its computational techniques. Roughly speaking, and restricting ourselves to Calabi–Yau threefolds, we have the following mathematical approaches to the computation of Gromov–Witten invariants:

Introduction to Image Reconstruction

Abstract: The problem of reconstructing images from measurements at the boundary of a domain belong to the class of inverse problems. In practice, these measurements are incomplete and inaccurate leading to ill-posed problems. This means that 'exact' reconstructions are usually not possible. In this Introduction the reader will find some applications in which the main ideas about, stability and resolution in image reconstruction are discussed. We will see that although different applications or imaging modalities work under different physical principles and map different. physical parameters, they all share the same mathematical foundations and the tools used to create the images have a great deal in common. Current imaging problems deal with understanding the trade off between data size, the quality of the image and the computational tools used to create the image. In many cases. these tools represent the performance bottleneck due to the high operational count and the memory cost.

Time-Reversal and the Adjoint Imaging Method with an Application in Telecommunication

Abstract: These lecture notes provide a mathematical treatment of time-reversal experiments with a special emphasis on telecommunication. A direct link is established between time-reversal experiments and the adjoint imaging method. Based on this relationship. several iterative schemes are proposed for optimizing MIMO (multiple-input multiple-output) time-reversal systems in underwater acoustic and in wireless communication systems. Whereas in typical imaging applications these iterative schemes require the repeated solution of forward problems in a computer, the analogue in time-reversal communication schemes consists of a small number of physical time-reversal experiments and does not require exact knowledge of the environment in which the communication system operates. The discussion is put in the general framework of wave propagation by symmetric hyperbolic systems, with detailed discussions of the linear acoustic system for underwater communication and of the time-dependent system of Maxwell's equations for telecommunication. Moreover, in its general form as treated here, the theory will also apply for several other models of wave-propagation. such as for example linear elastic waves.

Floer Cohomology with Gerbes

Abstract: This is a written account of expository lectures delivered at the summer school on “Enumerative invariants in algebraic geometry and string theory” of the Centro Internazionale Matematico Estivo, held in Cetraro in June 2005. However, it differs considerably from the lectures as they were actually given. Three of the lectures in the series were devoted to the recent work of Donaldson–Thomas, Maulik–Nekrasov–Okounkov–Pandharipande, and Nakajima–Yoshioka. Since this is well documented in the literature, it seemed needless to write it up again. Instead, what follows is a greatly expanded version of the other lectures, which were a little more speculative and the least strictly confined to algebraic geometry. However, they should interest algebraic geometers who have been contemplating orbifold cohomology and its close relative, the so-called Fantechi–Gottsche ring, which are discussed in the final portion of these notes. Indeed, we intend to argue that orbifold cohomology is essentially the same as a symplectic cohomology theory, namely Floer cohomology. More specifically, the quantum product structures on Floer cohomology and on the Fantechi–Gottsche ring should coincide. None of this should come as a surprise, since orbifold cohomology arose chiefly from the work of Chen–Ruan in the symplectic setting, and since the differentials in both theories involve the counting of holomorphic curves. Nevertheless, the links between the two theories are worth spelling out. To illustrate this theme further, we will explain how both the Floer and orbifold theories can be enriched by introducing a flat U(1)-gerbe. Such a gerbe on a manifold (or orbifold) induces flat line bundles on its loop space and on its inertia stack, leading to Floer and orbifold cohomology theories with local coefficients. We will again argue that these two theories correspond. To explain all of this properly, an extended digression on the basic definitions and properties of gerbes is needed; it comprises the second of the three lectures.