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Structure Of Approximate Solutions Of Optimal Control Problems
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Author :Alexander J. Zaslavski Publisher :Springer Science & Business Media ISBN 13 :3319012401 Total Pages :133 pages Book Rating :4.3/5 (19 download)
Book Synopsis Structure of Approximate Solutions of Optimal Control Problems by : Alexander J. Zaslavski
Download or read book Structure of Approximate Solutions of Optimal Control Problems written by Alexander J. Zaslavski and published by Springer Science & Business Media. This book was released on 2013-08-04 with total page 133 pages. Available in PDF, EPUB and Kindle. Book excerpt: This title examines the structure of approximate solutions of optimal control problems considered on subintervals of a real line. Specifically at the properties of approximate solutions which are independent of the length of the interval. The results illustrated in this book look into the so-called turnpike property of optimal control problems. The author generalizes the results of the turnpike property by considering a class of optimal control problems which is identified with the corresponding complete metric space of objective functions. This establishes the turnpike property for any element in a set that is in a countable intersection which is open everywhere dense sets in the space of integrands; meaning that the turnpike property holds for most optimal control problems. Mathematicians working in optimal control and the calculus of variations and graduate students will find this book useful and valuable due to its presentation of solutions to a number of difficult problems in optimal control and presentation of new approaches, techniques and methods.
Book Synopsis Discrete-Time Optimal Control and Games on Large Intervals by : Alexander J. Zaslavski
Download or read book Discrete-Time Optimal Control and Games on Large Intervals written by Alexander J. Zaslavski and published by Springer. This book was released on 2017-04-03 with total page 402 pages. Available in PDF, EPUB and Kindle. Book excerpt: Devoted to the structure of approximate solutions of discrete-time optimal control problems and approximate solutions of dynamic discrete-time two-player zero-sum games, this book presents results on properties of approximate solutions in an interval that is independent lengthwise, for all sufficiently large intervals. Results concerning the so-called turnpike property of optimal control problems and zero-sum games in the regions close to the endpoints of the time intervals are the main focus of this book. The description of the structure of approximate solutions on sufficiently large intervals and its stability will interest graduate students and mathematicians in optimal control and game theory, engineering, and economics. This book begins with a brief overview and moves on to analyze the structure of approximate solutions of autonomous nonconcave discrete-time optimal control Lagrange problems.Next the structures of approximate solutions of autonomous discrete-time optimal control problems that are discrete-time analogs of Bolza problems in calculus of variations are studied. The structures of approximate solutions of two-player zero-sum games are analyzed through standard convexity-concavity assumptions. Finally, turnpike properties for approximate solutions in a class of nonautonomic dynamic discrete-time games with convexity-concavity assumptions are examined.
Book Synopsis Neural Approximations for Optimal Control and Decision by : Riccardo Zoppoli
Download or read book Neural Approximations for Optimal Control and Decision written by Riccardo Zoppoli and published by Springer Nature. This book was released on 2019-12-17 with total page 532 pages. Available in PDF, EPUB and Kindle. Book excerpt: Neural Approximations for Optimal Control and Decision provides a comprehensive methodology for the approximate solution of functional optimization problems using neural networks and other nonlinear approximators where the use of traditional optimal control tools is prohibited by complicating factors like non-Gaussian noise, strong nonlinearities, large dimension of state and control vectors, etc. Features of the text include: • a general functional optimization framework; • thorough illustration of recent theoretical insights into the approximate solutions of complex functional optimization problems; • comparison of classical and neural-network based methods of approximate solution; • bounds to the errors of approximate solutions; • solution algorithms for optimal control and decision in deterministic or stochastic environments with perfect or imperfect state measurements over a finite or infinite time horizon and with one decision maker or several; • applications of current interest: routing in communications networks, traffic control, water resource management, etc.; and • numerous, numerically detailed examples. The authors’ diverse backgrounds in systems and control theory, approximation theory, machine learning, and operations research lend the book a range of expertise and subject matter appealing to academics and graduate students in any of those disciplines together with computer science and other areas of engineering.
Book Synopsis Optimal Control Problems Arising in Mathematical Economics by : Alexander J. Zaslavski
Download or read book Optimal Control Problems Arising in Mathematical Economics written by Alexander J. Zaslavski and published by Springer Nature. This book was released on 2022-06-28 with total page 387 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of two large classes of discrete-time optimal control problems arising in mathematical economics. Nonautonomous optimal control problems of the first class are determined by a sequence of objective functions and sequence of constraint maps. They correspond to a general model of economic growth. We are interested in turnpike properties of approximate solutions and in the stability of the turnpike phenomenon under small perturbations of objective functions and constraint maps. The second class of autonomous optimal control problems corresponds to another general class of models of economic dynamics which includes the Robinson–Solow–Srinivasan model as a particular case. In Chap. 1 we discuss turnpike properties for a large class of discrete-time optimal control problems studied in the literature and for the Robinson–Solow–Srinivasan model. In Chap. 2 we introduce the first class of optimal control problems and study its turnpike property. This class of problems is also discussed in Chaps. 3–6. In Chap. 3 we study the stability of the turnpike phenomenon under small perturbations of the objective functions. Analogous results for problems with discounting are considered in Chap. 4. In Chap. 5 we study the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps. Analogous results for problems with discounting are established in Chap. 6. The results of Chaps. 5 and 6 are new. The second class of problems is studied in Chaps. 7–9. In Chap. 7 we study the turnpike properties. The stability of the turnpike phenomenon under small perturbations of the objective functions is established in Chap. 8. In Chap. 9 we establish the stability of the turnpike phenomenon under small perturbations of the objective functions and the constraint maps. The results of Chaps. 8 and 9 are new. In Chap. 10 we study optimal control problems related to a model of knowledge-based endogenous economic growth and show the existence of trajectories of unbounded economic growth and provide estimates for the growth rate.
Book Synopsis Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model by : Alexander J. Zaslavski
Download or read book Optimal Control Problems Related to the Robinson–Solow–Srinivasan Model written by Alexander J. Zaslavski and published by Springer Nature. This book was released on 2021-08-07 with total page 354 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of classes of optimal control problems arising in economic growth theory, related to the Robinson–Solow–Srinivasan (RSS) model. The model was introduced in the 1960s by economists Joan Robinson, Robert Solow, and Thirukodikaval Nilakanta Srinivasan and was further studied by Robinson, Nobuo Okishio, and Joseph Stiglitz. Since then, the study of the RSS model has become an important element of economic dynamics. In this book, two large general classes of optimal control problems, both of them containing the RSS model as a particular case, are presented for study. For these two classes, a turnpike theory is developed and the existence of solutions to the corresponding infinite horizon optimal control problems is established. The book contains 9 chapters. Chapter 1 discusses turnpike properties for some optimal control problems that are known in the literature, including problems corresponding to the RSS model. The first class of optimal control problems is studied in Chaps. 2–6. In Chap. 2, infinite horizon optimal control problems with nonautonomous optimality criteria are considered. The utility functions, which determine the optimality criterion, are nonconcave. This class of models contains the RSS model as a particular case. The stability of the turnpike phenomenon of the one-dimensional nonautonomous concave RSS model is analyzed in Chap. 3. The following chapter takes up the study of a class of autonomous nonconcave optimal control problems, a subclass of problems considered in Chap. 2. The equivalence of the turnpike property and the asymptotic turnpike property, as well as the stability of the turnpike phenomenon, is established. Turnpike conditions and the stability of the turnpike phenomenon for nonautonomous problems are examined in Chap. 5, with Chap. 6 devoted to the study of the turnpike properties for the one-dimensional nonautonomous nonconcave RSS model. The utility functions, which determine the optimality criterion, are nonconcave. The class of RSS models is identified with a complete metric space of utility functions. Using the Baire category approach, the turnpike phenomenon is shown to hold for most of the models. Chapter 7 begins the study of the second large class of autonomous optimal control problems, and turnpike conditions are established. The stability of the turnpike phenomenon for this class of problems is investigated further in Chaps. 8 and 9.
Book Synopsis Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems by : Alexander J. Zaslavski
Download or read book Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems written by Alexander J. Zaslavski and published by Springer. This book was released on 2014-08-20 with total page 114 pages. Available in PDF, EPUB and Kindle. Book excerpt: The structure of approximate solutions of autonomous discrete-time optimal control problems and individual turnpike results for optimal control problems without convexity (concavity) assumptions are examined in this book. In particular, the book focuses on the properties of approximate solutions which are independent of the length of the interval, for all sufficiently large intervals; these results apply to the so-called turnpike property of the optimal control problems. By encompassing the so-called turnpike property the approximate solutions of the problems are determined primarily by the objective function and are fundamentally independent of the choice of interval and endpoint conditions, except in regions close to the endpoints. This book also explores the turnpike phenomenon for two large classes of autonomous optimal control problems. It is illustrated that the turnpike phenomenon is stable for an optimal control problem if the corresponding infinite horizon optimal control problem possesses an asymptotic turnpike property. If an optimal control problem belonging to the first class possesses the turnpike property, then the turnpike is a singleton (unit set). The stability of the turnpike property under small perturbations of an objective function and of a constraint map is established. For the second class of problems where the turnpike phenomenon is not necessarily a singleton the stability of the turnpike property under small perturbations of an objective function is established. Containing solutions of difficult problems in optimal control and presenting new approaches, techniques and methods this book is of interest for mathematicians working in optimal control and the calculus of variations. It also can be useful in preparation courses for graduate students.
Author :Alexander J. Zaslavski Publisher :Springer Science & Business Media ISBN 13 :1461463874 Total Pages :121 pages Book Rating :4.4/5 (614 download)
Book Synopsis Structure of Solutions of Variational Problems by : Alexander J. Zaslavski
Download or read book Structure of Solutions of Variational Problems written by Alexander J. Zaslavski and published by Springer Science & Business Media. This book was released on 2013-02-11 with total page 121 pages. Available in PDF, EPUB and Kindle. Book excerpt: Structure of Solutions of Variational Problems is devoted to recent progress made in the studies of the structure of approximate solutions of variational problems considered on subintervals of a real line. Results on properties of approximate solutions which are independent of the length of the interval, for all sufficiently large intervals are presented in a clear manner. Solutions, new approaches, techniques and methods to a number of difficult problems in the calculus of variations are illustrated throughout this book. This book also contains significant results and information about the turnpike property of the variational problems. This well-known property is a general phenomenon which holds for large classes of variational problems. The author examines the following in relation to the turnpike property in individual (non-generic) turnpike results, sufficient and necessary conditions for the turnpike phenomenon as well as in the non-intersection property for extremals of variational problems. This book appeals to mathematicians working in optimal control and the calculus as well as with graduate students.
Book Synopsis Optimal Control Problems for Partial Differential Equations on Reticulated Domains by : Peter I. Kogut
Download or read book Optimal Control Problems for Partial Differential Equations on Reticulated Domains written by Peter I. Kogut and published by Springer Science & Business Media. This book was released on 2011-09-09 with total page 639 pages. Available in PDF, EPUB and Kindle. Book excerpt: In the development of optimal control, the complexity of the systems to which it is applied has increased significantly, becoming an issue in scientific computing. In order to carry out model-reduction on these systems, the authors of this work have developed a method based on asymptotic analysis. Moving from abstract explanations to examples and applications with a focus on structural network problems, they aim at combining techniques of homogenization and approximation. Optimal Control Problems for Partial Differential Equations on Reticulated Domains is an excellent reference tool for graduate students, researchers, and practitioners in mathematics and areas of engineering involving reticulated domains.
Book Synopsis Turnpike Theory for the Robinson–Solow–Srinivasan Model by : Alexander J. Zaslavski
Download or read book Turnpike Theory for the Robinson–Solow–Srinivasan Model written by Alexander J. Zaslavski and published by Springer Nature. This book was released on 2021-01-04 with total page 448 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of a class of optimal control problems arising in mathematical economics, related to the Robinson–Solow–Srinivasan (RSS) model. It will be useful for researches interested in the turnpike theory, infinite horizon optimal control and their applications, and mathematical economists. The RSS is a well-known model of economic dynamics that was introduced in the 1960s and as many other models of economic dynamics, the RSS model is determined by an objective function (a utility function) and a set-valued mapping (a technology map). The set-valued map generates a dynamical system whose trajectories are under consideration and the objective function determines an optimality criterion. The goal is to find optimal trajectories of the dynamical system, using the optimality criterion. Chapter 1 discusses turnpike properties for some classes of discrete time optimal control problems. Chapter 2 present the description of the RSS model and discuss its basic properties. Infinite horizon optimal control problems, related to the RSS model are studied in Chapter 3. Turnpike properties for the RSS model are analyzed in Chapter 4. Chapter 5 studies infinite horizon optimal control problems related to the RSS model with a nonconcave utility function. Chapter 6 focuses on infinite horizon optimal control problems with nonautonomous optimality criterions. Chapter 7 contains turnpike results for a class of discrete-time optimal control problems. Chapter 8 discusses the RSS model and compares different optimality criterions. Chapter 9 is devoted to the study of the turnpike properties for the RSS model. In Chapter 10 the one-dimensional autonomous RSS model is considered and the continuous time RSS model is studied in Chapter 11.
Book Synopsis Optimal Design of Control Systems by : Gennadii E. Kolosov
Download or read book Optimal Design of Control Systems written by Gennadii E. Kolosov and published by CRC Press. This book was released on 2020-08-27 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Covers design methods for optimal (or quasioptimal) control algorithms in the form of synthesis for deterministic and stochastic dynamical systems-with applications in aerospace, robotic, and servomechanical technologies. Providing new results on exact and approximate solutions of optimal control problems."
Author :Alexander J. Zaslavski Publisher :Springer Science & Business Media ISBN 13 :1461473780 Total Pages :382 pages Book Rating :4.4/5 (614 download)
Book Synopsis Nonconvex Optimal Control and Variational Problems by : Alexander J. Zaslavski
Download or read book Nonconvex Optimal Control and Variational Problems written by Alexander J. Zaslavski and published by Springer Science & Business Media. This book was released on 2013-06-12 with total page 382 pages. Available in PDF, EPUB and Kindle. Book excerpt: Nonconvex Optimal Control and Variational Problems is an important contribution to the existing literature in the field and is devoted to the presentation of progress made in the last 15 years of research in the area of optimal control and the calculus of variations. This volume contains a number of results concerning well-posedness of optimal control and variational problems, nonoccurrence of the Lavrentiev phenomenon for optimal control and variational problems, and turnpike properties of approximate solutions of variational problems. Chapter 1 contains an introduction as well as examples of select topics. Chapters 2-5 consider the well-posedness condition using fine tools of general topology and porosity. Chapters 6-8 are devoted to the nonoccurrence of the Lavrentiev phenomenon and contain original results. Chapter 9 focuses on infinite-dimensional linear control problems, and Chapter 10 deals with “good” functions and explores new understandings on the questions of optimality and variational problems. Finally, Chapters 11-12 are centered around the turnpike property, a particular area of expertise for the author. This volume is intended for mathematicians, engineers, and scientists interested in the calculus of variations, optimal control, optimization, and applied functional analysis, as well as both undergraduate and graduate students specializing in those areas. The text devoted to Turnpike properties may be of particular interest to the economics community.
Book Synopsis Optimal Control Problems Arising in Forest Management by : Alexander J. Zaslavski
Download or read book Optimal Control Problems Arising in Forest Management written by Alexander J. Zaslavski and published by Springer. This book was released on 2019-08-16 with total page 136 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of optimal control problems arising in forest management, an important and fascinating topic in mathematical economics studied by many researchers over the years. The volume studies the forest management problem by analyzing a class of optimal control problems that contains it and showing the existence of optimal solutions over infinite horizon. It also studies the structure of approximate solutions on finite intervals and their turnpike properties, as well as the stability of the turnpike phenomenon and the structure of approximate solutions on finite intervals in the regions close to the end points. The book is intended for mathematicians interested in the optimization theory, optimal control and their applications to the economic theory.
Book Synopsis The Construction of Algorithms for the Approximation of Certain Optimal Control Problems by : S. Ulm
Download or read book The Construction of Algorithms for the Approximation of Certain Optimal Control Problems written by S. Ulm and published by . This book was released on 1968 with total page 19 pages. Available in PDF, EPUB and Kindle. Book excerpt: The solution to certain optimum control problems can be reduced to problems of finding the absolute minima of certain functionals in function spaces. For an approximate solution to the latter problems use can be made of algorithms developed for the solution of nonlinear equations. In previous works by other authors Newton's method and a particular variant of the method of gradients were generalized for the solution of optimum control problems. On the basis of methods of functional analysis the present article gives a more general scheme for the construction of methods of this type for the solution of certain optimum control problems.
Book Synopsis Turnpike Conditions in Infinite Dimensional Optimal Control by : Alexander J. Zaslavski
Download or read book Turnpike Conditions in Infinite Dimensional Optimal Control written by Alexander J. Zaslavski and published by Springer. This book was released on 2019-07-23 with total page 570 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides a comprehensive study of turnpike phenomenon arising in optimal control theory. The focus is on individual (non-generic) turnpike results which are both mathematically significant and have numerous applications in engineering and economic theory. All results obtained in the book are new. New approaches, techniques, and methods are rigorously presented and utilize research from finite-dimensional variational problems and discrete-time optimal control problems to find the necessary conditions for the turnpike phenomenon in infinite dimensional spaces. The semigroup approach is employed in the discussion as well as PDE descriptions of continuous-time dynamics. The main results on sufficient and necessary conditions for the turnpike property are completely proved and the numerous illustrative examples support the material for the broad spectrum of experts. Mathematicians interested in the calculus of variations, optimal control and in applied functional analysis will find this book a useful guide to the turnpike phenomenon in infinite dimensional spaces. Experts in economic and engineering modeling as well as graduate students will also benefit from the developed techniques and obtained results.
Book Synopsis Turnpike Theory of Continuous-Time Linear Optimal Control Problems by : Alexander J. Zaslavski
Download or read book Turnpike Theory of Continuous-Time Linear Optimal Control Problems written by Alexander J. Zaslavski and published by Springer. This book was released on 2015-07-01 with total page 300 pages. Available in PDF, EPUB and Kindle. Book excerpt: Individual turnpike results are of great interest due to their numerous applications in engineering and in economic theory; in this book the study is focused on new results of turnpike phenomenon in linear optimal control problems. The book is intended for engineers as well as for mathematicians interested in the calculus of variations, optimal control and in applied functional analysis. Two large classes of problems are studied in more depth. The first class studied in Chapter 2 consists of linear control problems with periodic nonsmooth convex integrands. Chapters 3-5 consist of linear control problems with autonomous convex smooth integrands. Chapter 6 discusses a turnpike property for dynamic zero-sum games with linear constraints. Chapter 7 examines genericity results. In Chapter 8, the description of structure of variational problems with extended-valued integrands is obtained. Chapter 9 ends the exposition with a study of turnpike phenomenon for dynamic games with extended value integrands.
Book Synopsis Turnpike Phenomenon and Infinite Horizon Optimal Control by : Alexander J. Zaslavski
Download or read book Turnpike Phenomenon and Infinite Horizon Optimal Control written by Alexander J. Zaslavski and published by Springer. This book was released on 2014-09-04 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book is devoted to the study of the turnpike phenomenon and describes the existence of solutions for a large variety of infinite horizon optimal control classes of problems. Chapter 1 provides introductory material on turnpike properties. Chapter 2 studies the turnpike phenomenon for discrete-time optimal control problems. The turnpike properties of autonomous problems with extended-value integrands are studied in Chapter 3. Chapter 4 focuses on large classes of infinite horizon optimal control problems without convexity (concavity) assumptions. In Chapter 5, the turnpike results for a class of dynamic discrete-time two-player zero-sum game are proven. This thorough exposition will be very useful for mathematicians working in the fields of optimal control, the calculus of variations, applied functional analysis and infinite horizon optimization. It may also be used as a primary text in a graduate course in optimal control or as supplementary text for a variety of courses in other disciplines. Researchers in other fields such as economics and game theory, where turnpike properties are well known, will also find this Work valuable.
Book Synopsis Optimal Design of Control Systems by : Gennadii E. Kolosov
Download or read book Optimal Design of Control Systems written by Gennadii E. Kolosov and published by CRC Press. This book was released on 1999-06-01 with total page 424 pages. Available in PDF, EPUB and Kindle. Book excerpt: "Covers design methods for optimal (or quasioptimal) control algorithms in the form of synthesis for deterministic and stochastic dynamical systems-with applications in aerospace, robotic, and servomechanical technologies. Providing new results on exact and approximate solutions of optimal control problems."