Optimal Control and Geometry: Integrable Systems

Author: Velimir Jurdjevic

Publisher: Cambridge University Press

ISBN: 1107113881

Category: Mathematics

Page: 400

View: 4741

The synthesis of symplectic geometry, the calculus of variations and control theory offered in this book provides a crucial foundation for the understanding of many problems in applied mathematics. Focusing on the theory of integrable systems, this book introduces a class of optimal control problems on Lie groups, whose Hamiltonians, obtained through the Maximum Principle of optimality, shed new light on the theory of integrable systems. These Hamiltonians provide an original and unified account of the existing theory of integrable systems. The book particularly explains much of the mystery surrounding the Kepler problem, the Jacobi problem and the Kovalevskaya Top. It also reveals the ubiquitous presence of elastic curves in integrable systems up to the soliton solutions of the non-linear Schroedinger's equation. Containing a useful blend of theory and applications, this is an indispensable guide for graduates and researchers in many fields, from mathematical physics to space control.
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Geometry and Complexity Theory

Author: J. M. Landsberg

Publisher: Cambridge University Press

ISBN: 110819141X

Category: Computers

Page: N.A

View: 6288

Two central problems in computer science are P vs NP and the complexity of matrix multiplication. The first is also a leading candidate for the greatest unsolved problem in mathematics. The second is of enormous practical and theoretical importance. Algebraic geometry and representation theory provide fertile ground for advancing work on these problems and others in complexity. This introduction to algebraic complexity theory for graduate students and researchers in computer science and mathematics features concrete examples that demonstrate the application of geometric techniques to real world problems. Written by a noted expert in the field, it offers numerous open questions to motivate future research. Complexity theory has rejuvenated classical geometric questions and brought different areas of mathematics together in new ways. This book will show the beautiful, interesting, and important questions that have arisen as a result.
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Special Functions and Orthogonal Polynomials

Author: Richard Beals,Roderick Wong

Publisher: Cambridge University Press

ISBN: 1107106982

Category: Mathematics

Page: 500

View: 3417

A comprehensive graduate-level introduction to classical and contemporary aspects of special functions.
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Integrable Hamiltonian Systems on Complex Lie Groups

Author: Velimir Jurdjevic

Publisher: American Mathematical Soc.

ISBN: 0821837648

Category: Mathematics

Page: 133

View: 5130

This paper is a study of the elastic problems on simply connected manifolds $M_n$ whose orthonormal frame bundle is a Lie group $G$. Such manifolds, called the space forms in the literature on differential geometry, are classified and consist of the Euclidean spaces $\mathbb{E}^n$, the hyperboloids $\mathbb{H}^n$, and the spheres $S^n$, with the corresponding orthonormal frame bundles equal to the Euclidean group of motions $\mathbb{E}^n\rtimes SO_n(\mathbb{R})$, the rotation group $SO_{n+1}(\mathbb{R})$, and the Lorentz group $SO(1,n)$. The manifolds $M_n$ are treated as the symmetric spaces $G/K$ with $K$ isomorphic with $SO_n(R)$. Then the Lie algebra $\mathfrak{g}$ of $G$ admits a Cartan decomposition $\mathfrak{g}=\mathfrak{p}+\mathfrak{k}$ with $\mathfrak{k}$ equal to the Lie algebra of $K$ and $\mathfrak{p}$ equal to the orthogonal comlement $\mathfrak{k}$ relative to the trace form. The elastic problems on $G/K$ concern the solutions $g(t)$ of a left invariant differential systems on $G$ $$\frac{dg}{dt}(t)=g(t)(A_0+U(t)))$$ that minimize the expression $\frac{1}{2}\int_0^T (U(t),U(t))\,dt$ subject to the given boundary conditions $g(0)=g_0$, $g(T)=g_1$, over all locally bounded and measurable $\mathfrak{k}$ valued curves $U(t)$ relative to a positive definite quadratic form $(\,, \,)$ where $A_0$ is a fixed matrix in $\mathfrak{p}$. These variational problems fall in two classes, the Euler-Griffiths problems and the problems of Kirchhoff. The Euler-Griffiths elastic problems consist of minimizing the integral $$\tfrac{1}{2}\int_0^T\kappa^2(s)\,ds$$ with $\kappa (t)$ equal to the geodesic curvature of a curve $x(t)$ in the base manifold $M_n$ with $T$ equal to the Riemannian length of $x$. The curves $x(t)$ in this variational problem are subject to certain initial and terminal boundary conditions. The elastic problems of Kirchhoff is more general than the problems of Euler-Griffiths in the sense that the quadratic form $(\,, \,)$ that defines the functional to be minimized may be independent of the geometric invariants of the projected curves in the base manifold. It is only on two dimensional manifolds that these two problems coincide in which case the solutions curves can be viewed as the non-Euclidean versions of L. Euler elasticae introduced in 174. Each elastic problem defines the appropriate left-invariant Hamiltonian $\mathcal{H}$ on the dual $\mathfrak{g}^*$ of the Lie algebra of $G$ through the Maximum Principle of optimal control. The integral curves of the corresponding Hamiltonian vector field $\vec{\mathcal{H}}$ are called the extremal curves. The paper is essentially concerned with the extremal curves of the Hamiltonian systems associated with the elastic problems. This class of Hamiltonian systems reveals a remarkable fact that the Hamiltonian systems traditionally associated with the movements of the top are invariant subsystems of the Hamiltonian systems associated with the elastic problems. The paper is divided into two parts. The first part of the paper synthesizes ideas from optimal control theory, adapted to variational problems on the principal bundles of Riemannian spaces, and the symplectic geometry of the Lie algebra $\mathfrak{g},$ of $G$, or more precisely, the symplectic structure of the cotangent bundle $T^*G$ of $G$. The second part of the paper is devoted to the solutions of the complexified Hamiltonian equations induced by the elastic problems. The paper contains a detailed discussion of the algebraic preliminaries leading up to $so_n(\mathbb{C})$, a natural complex setting for the study of the left invariant Hamiltonians on real Lie groups $G$ for which $\mathfrak{g}$ is a real form for $so_n(\mathbb{C})$. It is shown that the Euler-Griffiths problem is completely integrable in any dimension with the solutions the holomorphic extensions of the ones obtained by earlier P. Griffiths. The solutions of the elastic problems of Kirchhoff are presented in complete generality on $SO_3(\mathbb{C})$ and there is a classification of the integrable cases on $so_4(\mathbb{C})$ based on the criteria of Kowalewski-Lyapunov in their study of the mechanical tops. Remarkably, the analysis yields essentially only two integrables cases analogous to the top of Lagrange and the top of Kowalewski. The paper ends with the solutions of the integrable complex Hamiltonian systems on the $SL_2(\mathbb{C})\times SL_2(\mathbb{C})$, the universal cover of $SO_4(\mathbb{C})$.
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Unsolved Problems in Mathematical Systems and Control Theory

Author: Vincent D. Blondel,Alexandre Megretski

Publisher: Princeton University Press

ISBN: 1400826152

Category: Mathematics

Page: 352

View: 5405

This book provides clear presentations of more than sixty important unsolved problems in mathematical systems and control theory. Each of the problems included here is proposed by a leading expert and set forth in an accessible manner. Covering a wide range of areas, the book will be an ideal reference for anyone interested in the latest developments in the field, including specialists in applied mathematics, engineering, and computer science. The book consists of ten parts representing various problem areas, and each chapter sets forth a different problem presented by a researcher in the particular area and in the same way: description of the problem, motivation and history, available results, and bibliography. It aims not only to encourage work on the included problems but also to suggest new ones and generate fresh research. The reader will be able to submit solutions for possible inclusion on an online version of the book to be updated quarterly on the Princeton University Press website, and thus also be able to access solutions, updated information, and partial solutions as they are developed.
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Mathematical Control Theory

Author: John B. Baillieul,J.C. Willems

Publisher: Springer Science & Business Media

ISBN: 1461214165

Category: Mathematics

Page: 360

View: 7970

This volume on mathematical control theory contains high quality articles covering the broad range of this field. The internationally renowned authors provide an overview of many different aspects of control theory, offering a historical perspective while bringing the reader up to the very forefront of current research.
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Geometric Control Theory

Author: Velimir Jurdjevic

Publisher: Cambridge University Press

ISBN: 9780521495028

Category: Mathematics

Page: 492

View: 8288

A modern version of the calculus of variations, encompassing geometric mechanics, differential geometry, and optimal control.
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Control and Nonlinearity

Author: Jean-Michel Coron

Publisher: American Mathematical Soc.

ISBN: 0821849182

Category: Commande non linéaire

Page: 426

View: 7683

This book presents methods to study the controllability and the stabilization of nonlinear control systems in finite and infinite dimensions. The emphasis is put on specific phenomena due to nonlinearities. In particular, many examples are given where nonlinearities turn out to be essential to get controllability or stabilization. Various methods are presented to study the controllability or to construct stabilizing feedback laws. The power of these methods is illustrated by numerous examples coming from such areas as celestial mechanics, fluid mechanics, and quantum mechanics. The book is addressed to graduate students in mathematics or control theory, and to mathematicians or engineers with an interest in nonlinear control systems governed by ordinary or partial differential equations.
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Nonholonomic Mechanics and Control

Author: A.M. Bloch

Publisher: Springer

ISBN: 1493930176

Category: Science

Page: 565

View: 1156

This book explores connections between control theory and geometric mechanics. The author links control theory with a geometric view of classical mechanics in both its Lagrangian and Hamiltonian formulations, and in particular with the theory of mechanical systems subject to motion constraints. The synthesis is appropriate as there is a rich connection between mechanics and nonlinear control theory. The book provides a unified treatment of nonlinear control theory and constrained mechanical systems that incorporates material not available in other recent texts. The book benefits graduate students and researchers in the area who want to enhance their understanding and enhance their techniques.
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The American Mathematical Monthly

The Official Journal of the Mathematical Association of America

Author: N.A

Publisher: N.A

ISBN: N.A

Category: Mathematicians

Page: N.A

View: 4654

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Geometric Dynamics

Author: C. Udriste

Publisher: Springer Science & Business Media

ISBN: 9401141878

Category: Mathematics

Page: 395

View: 7787

Geometric dynamics is a tool for developing a mathematical representation of real world phenomena, based on the notion of a field line described in two ways: -as the solution of any Cauchy problem associated to a first-order autonomous differential system; -as the solution of a certain Cauchy problem associated to a second-order conservative prolongation of the initial system. The basic novelty of our book is the discovery that a field line is a geodesic of a suitable geometrical structure on a given space (Lorentz-Udri~te world-force law). In other words, we create a wider class of Riemann-Jacobi, Riemann-Jacobi-Lagrange, or Finsler-Jacobi manifolds, ensuring that all trajectories of a given vector field are geodesics. This is our contribution to an old open problem studied by H. Poincare, S. Sasaki and others. From the kinematic viewpoint of corpuscular intuition, a field line shows the trajectory followed by a particle at a point of the definition domain of a vector field, if the particle is sensitive to the related type of field. Therefore, field lines appear in a natural way in problems of theoretical mechanics, fluid mechanics, physics, thermodynamics, biology, chemistry, etc.
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Analysis and Geometry in Control Theory and its Applications

Author: Piernicola Bettiol,Piermarco Cannarsa,Giovanni Colombo,Monica Motta,Franco Rampazzo

Publisher: Springer

ISBN: 3319069179

Category: Mathematics

Page: 235

View: 5307

Since the 1950s control theory has established itself as a major mathematical discipline, particularly suitable for application in a number of research fields, including advanced engineering design, economics and the medical sciences. However, since its emergence, there has been a need to rethink and extend fields such as calculus of variations, differential geometry and nonsmooth analysis, which are closely tied to research on applications. Today control theory is a rich source of basic abstract problems arising from applications, and provides an important frame of reference for investigating purely mathematical issues. In many fields of mathematics, the huge and growing scope of activity has been accompanied by fragmentation into a multitude of narrow specialties. However, outstanding advances are often the result of the quest for unifying themes and a synthesis of different approaches. Control theory and its applications are no exception. Here, the interaction between analysis and geometry has played a crucial role in the evolution of the field. This book collects some recent results, highlighting geometrical and analytical aspects and the possible connections between them. Applications provide the background, in the classical spirit of mutual interplay between abstract theory and problem-solving practice.
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Geometric and Numerical Optimal Control

Application to Swimming at Low Reynolds Number and Magnetic Resonance Imaging

Author: Bernard Bonnard,Monique Chyba,Jérémy Rouot

Publisher: Springer

ISBN: 3319947915

Category: Mathematics

Page: 108

View: 769

This book introduces readers to techniques of geometric optimal control as well as the exposure and applicability of adapted numerical schemes. It is based on two real-world applications, which have been the subject of two current academic research programs and motivated by industrial use – the design of micro-swimmers and the contrast problem in medical resonance imaging. The recently developed numerical software has been applied to the cases studies presented here. The book is intended for use at the graduate and Ph.D. level to introduce students from applied mathematics and control engineering to geometric and computational techniques in optimal control.
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Encyclopedia of mathematical physics

Author: Sheung Tsun Tsou

Publisher: Academic Pr

ISBN: 9780125126601

Category: Science

Page: 3500

View: 9445

The Encyclopedia of Mathematical Physics provides a complete resource for researchers, students and lecturers with an interest in mathematical physics. It enables readers to access basic information on topics peripheral to their own areas, to provide a repository of the core information in the area that can be used to refresh the researcher's own memory banks, and aid teachers in directing students to entries relevant to their course-work. The Encyclopedia does contain information that has been distilled, organised and presented as a complete reference tool to the user and a landmark to the body of knowledge that has accumulated in this domain. It also is a stimulus for new researchers working in mathematical physics or in areas using the methods originated from work in mathematical physics by providing them with focused high quality background information. * First comprehensive interdisciplinary coverage * Mathematical Physics explained to stimulate new developments and foster new applications of its methods to other fields * Written by an international group of experts * Contains several undergraduate-level introductory articles to facilitate acquisition of new expertise * Thematic index and extensive cross-referencing to provide easy access and quick search functionality * Also available online with active linking.
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Geometric Optimal Control

Theory, Methods and Examples

Author: Heinz Schättler,Urszula Ledzewicz

Publisher: Springer Science & Business Media

ISBN: 1461438349

Category: Mathematics

Page: 640

View: 2599

This book gives a comprehensive treatment of the fundamental necessary and sufficient conditions for optimality for finite-dimensional, deterministic, optimal control problems. The emphasis is on the geometric aspects of the theory and on illustrating how these methods can be used to solve optimal control problems. It provides tools and techniques that go well beyond standard procedures and can be used to obtain a full understanding of the global structure of solutions for the underlying problem. The text includes a large number and variety of fully worked out examples that range from the classical problem of minimum surfaces of revolution to cancer treatment for novel therapy approaches. All these examples, in one way or the other, illustrate the power of geometric techniques and methods. The versatile text contains material on different levels ranging from the introductory and elementary to the advanced. Parts of the text can be viewed as a comprehensive textbook for both advanced undergraduate and all level graduate courses on optimal control in both mathematics and engineering departments. The text moves smoothly from the more introductory topics to those parts that are in a monograph style were advanced topics are presented. While the presentation is mathematically rigorous, it is carried out in a tutorial style that makes the text accessible to a wide audience of researchers and students from various fields, including the mathematical sciences and engineering. Heinz Schättler is an Associate Professor at Washington University in St. Louis in the Department of Electrical and Systems Engineering, Urszula Ledzewicz is a Distinguished Research Professor at Southern Illinois University Edwardsville in the Department of Mathematics and Statistics.
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Geometry of Feedback and Optimal Control

Author: B. Jakubczyk,Witold Respondek

Publisher: Courier Corporation

ISBN: 9780824790684

Category: Mathematics

Page: 564

View: 4675

Gathering the most important and promising results in subfields of nonlinear control theory - previously available only in journals - this comprehensive volume presents the state of the art in geometric methods, their applications in optimal control, and feedback transformations. Supplemented with over 1200 references, equations, and drawings, this readily accessible resource is excellent for pure and applied mathematicians, analysts, and applied geometers specializing in control theory, differential equations, calculus of variations, differential geometry, and singularity theory, and graduate-level students in these disciplines.
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Learning Representation and Control in Markov Decision Processes

Author: Sridhar Mahadevan

Publisher: Now Publishers Inc

ISBN: 1601982380

Category: Computers

Page: 184

View: 5852

Provides a comprehensive survey of techniques to automatically construct basis functions or features for value function approximation in Markov decision processes and reinforcement learning.
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Introduction to Mechanics and Symmetry

A Basic Exposition of Classical Mechanical Systems

Author: J.E. Marsden,Tudor Ratiu

Publisher: Springer Science & Business Media

ISBN: 0387217924

Category: Science

Page: 586

View: 2431

A development of the basic theory and applications of mechanics with an emphasis on the role of symmetry. The book includes numerous specific applications, making it beneficial to physicists and engineers. Specific examples and applications show how the theory works, backed by up-to-date techniques, all of which make the text accessible to a wide variety of readers, especially senior undergraduates and graduates in mathematics, physics and engineering. This second edition has been rewritten and updated for clarity throughout, with a major revamping and expansion of the exercises. Internet supplements containing additional material are also available.
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