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For more than forty years, the equation y'(t) = Ay(t) + u(t) in Banach spaces has been used as model for optimal control processes described by partial differential equations, in particular heat and diffusion processes. Many of the outstanding open problems, however, have remained open until recently, and some have never been solved. This book is a survey of all results know to the author, with emphasis on very recent results (1999 to date). The book is restricted to linear equations and two particular problems (the time optimal problem, the norm optimal problem) which results in a

Control theory. --- Calculus of variations. --- Linear control systems. --- Mathematical optimization. --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Automatic control --- Isoperimetrical problems --- Variations, Calculus of --- Dynamics --- Machine theory

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Variational arguments are classical techniques whose use can be traced back to the early development of the calculus of variations and further. Rooted in the physical principle of least action they have wide applications in diverse fields. This book provides a concise account of the essential tools of infinite-dimensional first-order variational analysis illustrated by applications in many areas of analysis, optimization and approximation, dynamical systems, mathematical economics and elsewhere. The book is aimed at both graduate students in the field of variational analysis and researchers who use variational techniques, or think they might like to. Large numbers of (guided) exercises are provided that either give useful generalizations of the main text or illustrate significant relationships with other results. Jonathan M. Borwein, FRSC is Canada Research Chair in Collaborative Technology at Dalhousie University. He received his Doctorate from Oxford in 1974 and has been on faculty at Waterloo, Carnegie Mellon and Simon Fraser Universities. He has published extensively in optimization, analysis and computational mathematics and has received various prizes both for research and for exposition. Qiji J. Zhu is a Professor in the Department of Mathematics at Western Michigan University. He received his doctorate at Northeastern University in 1992. He has been a Research Associate at University of Montreal, Simon Fraser University and University of Victoria, Canada.

Mathematics. --- Functional analysis. --- Mathematical optimization. --- Calculus of variations. --- Optimization. --- Calculus of Variations and Optimal Control; Optimization. --- Functional Analysis. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- Simulation methods --- System analysis --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Math --- Science

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This work develops the methodology according to which classes of discontinuous functions are used in order to investigate a correctness of boundary-value and initial boundary-value problems for the cases with elliptic, parabolic, pseudoparabolic, hyperbolic, and pseudohyperbolic equations and with elasticity theory equation systems that have nonsmooth solutions, including discontinuous solutions. With the basis of this methodology, the monograph shows a continuous dependence of states, namely, of solutions to the enumerated boundary-value and initial boundary-value problems (including discontinuous states) and a dependence of solution traces on distributed controls and controls at sectors of n-dimensional domain boundaries and at n–1-dimensional function-state discontinuity surfaces (i.e., at mean surfaces of thin inclusions in heterogeneous media). Such an aspect provides the existence of optimal controls for the mentioned systems with J.L. Lions’ quadratic cost functionals. Besides this, the authors consider some new systems, for instance, the ones described by the conditionally correct Neumann problems with unique states on convex sets, and such states admit first-order discontinuities. These systems are also described by quartic equations with conjugation conditions, by parabolic equations with constraints that contain first-order time state derivatives in the presence of concentrated heat capacity, and by elasticity theory equations. In a number of cases, when a set of feasible controls coincides with corresponding Hilbert spaces, the authors propose to use the computational algorithms for the finite-element method. Such algorithms have the increased order of the accuracy with which optimal controls are numerically found. Audience This book is intended for specialists in applied mathematics, scientific researchers, engineers, and postgraduate students interested in optimal control of heterogeneous distributed systems with states described by boundary-value and initial boundary-value problems.

Distributed operating systems (Computers) --- Operating systems (Computers) --- Computer operating systems --- Computers --- Disk operating systems --- Systems software --- Operating systems --- Mathematical optimization. --- Differential equations, partial. --- Calculus of Variations and Optimal Control; Optimization. --- Partial Differential Equations. --- Optimization. --- Partial differential equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- Calculus of variations. --- Partial differential equations. --- Isoperimetrical problems --- Variations, Calculus of

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This book discusses a new discipline, variational analysis, which contains the calculus of variations, differential calculus, optimization, and variational inequalities. To such classic branches of mathematics, variational analysis provides a uniform theoretical base that represents a powerful tool for the applications. The contributors are among the best experts in the field. Audience The target audience of this book includes scholars in mathematics (especially those in mathematical analysis), mathematical physics and applied mathematics, calculus of variations, optimization and operations research, industrial mathematics, structural engineering, and statistics and economics.

Variational inequalities (Mathematics) --- Mathematical optimization --- Inequalities, Variational (Mathematics) --- Calculus of variations --- Differential inequalities --- Mathematical optimization. --- Global analysis (Mathematics). --- Differential Equations. --- Mathematics. --- Calculus of Variations and Optimal Control; Optimization. --- Analysis. --- Optimization. --- Ordinary Differential Equations. --- Applications of Mathematics. --- Math --- Science --- Analysis, Global (Mathematics) --- Differential topology --- Functions of complex variables --- Geometry, Algebraic --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Maxima and minima --- Operations research --- Simulation methods --- System analysis --- 517.91 Differential equations --- Differential equations --- Calculus of variations. --- Mathematical analysis. --- Analysis (Mathematics). --- Differential equations. --- Applied mathematics. --- Engineering mathematics. --- Engineering --- Engineering analysis --- 517.1 Mathematical analysis --- Isoperimetrical problems --- Variations, Calculus of --- Mathematics --- Calculus. --- Analysis (Mathematics) --- Fluxions (Mathematics) --- Infinitesimal calculus --- Limits (Mathematics) --- Functions --- Geometry, Infinitesimal

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Award-winning monograph of the Ferran Sunyer i Balaguer Prize 2004. This book studies regularity properties of Mumford-Shah minimizers. The Mumford-Shah functional was introduced in the 1980s as a tool for automatic image segmentation, but its study gave rise to many interesting questions of analysis and geometric measure theory. The main object under scrutiny is a free boundary K where the minimizer may have jumps. The book presents an extensive description of the known regularity properties of the singular sets K, and the techniques to get them. Some time is spent on the C^1 regularity theorem (with an essentially unpublished proof in dimension 2), but a good part of the book is devoted to applications of A. Bonnet's monotonicity and blow-up techniques. In particular, global minimizers in the plane are studied in full detail. The book is largely self-contained and should be accessible to graduate students in analysis.The core of the book is composed of regularity results that were proved in the last ten years and which are presented in a more detailed and unified way.

Calculus of variations. --- Geometric measure theory. --- Manifolds (Mathematics) --- Boundary value problems. --- Differential equations, Partial. --- Isoperimetrical problems --- Variations, Calculus of --- Maxima and minima --- Partial differential equations --- Boundary conditions (Differential equations) --- Differential equations --- Functions of complex variables --- Mathematical physics --- Initial value problems --- Geometry, Differential --- Topology --- Measure theory --- Mathematical optimization. --- Functional analysis. --- Differential equations, partial. --- Calculus of Variations and Optimal Control; Optimization. --- Functional Analysis. --- Partial Differential Equations. --- Functional calculus --- Calculus of variations --- Functional equations --- Integral equations --- Optimization (Mathematics) --- Optimization techniques --- Optimization theory --- Systems optimization --- Mathematical analysis --- Operations research --- Simulation methods --- System analysis --- Partial differential equations.