[PDF] Optimal Control Of Random Sequences In Problems With Constraints eBook

Author : A.B. Piunovskiy
Publisher : Springer
Page : 348 pages
File Size : 13,48 MB
Release : 2011-10-13
Category : Mathematics
ISBN : 9789401155090

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Controlled stochastic processes with discrete time form a very interest ing and meaningful field of research which attracts widespread attention. At the same time these processes are used for solving of many applied problems in the queueing theory, in mathematical economics. in the theory of controlled technical systems, etc. . In this connection, methods of the theory of controlled processes constitute the every day instrument of many specialists working in the areas mentioned. The present book is devoted to the rather new area, that is, to the optimal control theory with functional constraints. This theory is close to the theory of multicriteria optimization. The compromise between the mathematical rigor and the big number of meaningful examples makes the book attractive for professional mathematicians and for specialists who ap ply mathematical methods in different specific problems. Besides. the book contains setting of many new interesting problems for further invf'stigatioll. The book can form the basis of special courses in the theory of controlled stochastic processes for students and post-graduates specializing in the ap plied mathematics and in the control theory of complex systf'ms. The grounding of graduating students of mathematical department is sufficient for the perfect understanding of all the material. The book con tains the extensive Appendix where the necessary knowledge ill Borel spaces and in convex analysis is collected. All the meaningful examples can be also understood by readers who are not deeply grounded in mathematics.

Author : A.B. Piunovskiy
Publisher : Springer Science & Business Media
Page : 355 pages
File Size : 29,33 MB
Release : 2012-12-06
Category : Mathematics
ISBN : 9401155089

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Controlled stochastic processes with discrete time form a very interest ing and meaningful field of research which attracts widespread attention. At the same time these processes are used for solving of many applied problems in the queueing theory, in mathematical economics. in the theory of controlled technical systems, etc. . In this connection, methods of the theory of controlled processes constitute the every day instrument of many specialists working in the areas mentioned. The present book is devoted to the rather new area, that is, to the optimal control theory with functional constraints. This theory is close to the theory of multicriteria optimization. The compromise between the mathematical rigor and the big number of meaningful examples makes the book attractive for professional mathematicians and for specialists who ap ply mathematical methods in different specific problems. Besides. the book contains setting of many new interesting problems for further invf'stigatioll. The book can form the basis of special courses in the theory of controlled stochastic processes for students and post-graduates specializing in the ap plied mathematics and in the control theory of complex systf'ms. The grounding of graduating students of mathematical department is sufficient for the perfect understanding of all the material. The book con tains the extensive Appendix where the necessary knowledge ill Borel spaces and in convex analysis is collected. All the meaningful examples can be also understood by readers who are not deeply grounded in mathematics.

Author : Kok Lay Teo
Publisher : Springer Nature
Page : 581 pages
File Size : 46,96 MB
Release : 2021-05-24
Category : Mathematics
ISBN : 3030699137

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The aim of this book is to furnish the reader with a rigorous and detailed exposition of the concept of control parametrization and time scaling transformation. It presents computational solution techniques for a special class of constrained optimal control problems as well as applications to some practical examples. The book may be considered an extension of the 1991 monograph A Unified Computational Approach Optimal Control Problems, by K.L. Teo, C.J. Goh, and K.H. Wong. This publication discusses the development of new theory and computational methods for solving various optimal control problems numerically and in a unified fashion. To keep the book accessible and uniform, it includes those results developed by the authors, their students, and their past and present collaborators. A brief review of methods that are not covered in this exposition, is also included. Knowledge gained from this book may inspire advancement of new techniques to solve complex problems that arise in the future. This book is intended as reference for researchers in mathematics, engineering, and other sciences, graduate students and practitioners who apply optimal control methods in their work. It may be appropriate reading material for a graduate level seminar or as a text for a course in optimal control.

Author : Eitan Altman
Publisher : CRC Press
Page : 260 pages
File Size : 35,98 MB
Release : 1999-03-30
Category : Mathematics
ISBN : 9780849303821

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This book provides a unified approach for the study of constrained Markov decision processes with a finite state space and unbounded costs. Unlike the single controller case considered in many other books, the author considers a single controller with several objectives, such as minimizing delays and loss, probabilities, and maximization of throughputs. It is desirable to design a controller that minimizes one cost objective, subject to inequality constraints on other cost objectives. This framework describes dynamic decision problems arising frequently in many engineering fields. A thorough overview of these applications is presented in the introduction. The book is then divided into three sections that build upon each other. The first part explains the theory for the finite state space. The author characterizes the set of achievable expected occupation measures as well as performance vectors, and identifies simple classes of policies among which optimal policies exist. This allows the reduction of the original dynamic into a linear program. A Lagranian approach is then used to derive the dual linear program using dynamic programming techniques. In the second part, these results are extended to the infinite state space and action spaces. The author provides two frameworks: the case where costs are bounded below and the contracting framework. The third part builds upon the results of the first two parts and examines asymptotical results of the convergence of both the value and the policies in the time horizon and in the discount factor. Finally, several state truncation algorithms that enable the approximation of the solution of the original control problem via finite linear programs are given.

Author : Vu Anh Huynh
Publisher :
Page : 143 pages
File Size : 14,97 MB
Release : 2014
Category :
ISBN :

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Controlling dynamical systems in uncertain environments is fundamental and essential in several fields, ranging from robotics, healthcare to economics and finance. In these applications, the required tasks can be modeled as continuous-time, continuous-space stochastic optimal control problems. Moreover, risk management is an important requirement of such problems to guarantee safety during the execution of control policies. However, even in the simplest version, finding closed-form or exact algorithmic solutions for stochastic optimal control problems is comuputationally challenging. The main contribution of this thesis is the development of theoretical foundations, and provably-correct and efficient sampling-based algorithms to solve stochastic optimal control problems in the presence of complex risk constraints. In the first part of the thesis, we consider the mentioned problems without risk constraints. We propose a novel algorithm called the incremental Markov Decision Process (iMDP) to compute incrementally any-time control policies that approximate arbitrarily well an optimal policy in terms of the expected cost. The main idea is to generate a sequence of finite discretizations of the original problem through random sampling of the state space. At each iteration, the discretized problem is a Markov Decision Process that serves as am incrementally refined model of the original problem. We show that the iMDP algorithm guarantees asymptotic optimality while maintaining low computational and space complexity. In the second part of the thesis, we consider risk constraints that are expressed as either bounded trajectory performance or bounded probabilities of failure. For the former, we present the first extended iMDP algorithm to approximate arbitrarily well an optimal feedback policy of the constrained problem. For the latter, we present a martingale approach that diffuses a risk constraint into a martingale to construct time-consistent control policies. The martingale stands for the level of risk tolerance that is contingent on available information over time. By augmenting the system dynamics with the martingale, the original risk-constrained problem is transformed into a stochastic target problem. We present the second extended iMDP algorithm to approximate arbitrarily well an optimal feedback policy of the original problem by sampling in the augmented state space and computing proper boundary values for the reformulated problem. In both cases, sequences of policies returned from the extended algorithms are both probabilistically sound and asymptotically optimal. The effectiveness of these algorithms is demonstrated on robot motion planning and control problems in cluttered environments in the presence of process noise.

Author : Gennadii E. Kolosov
Publisher : CRC Press
Page : 424 pages
File Size : 12,2 MB
Release : 2020-08-27
Category : Mathematics
ISBN : 1000146758

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"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 : Jane Kehoe Cullum
Publisher :
Page : 202 pages
File Size : 42,16 MB
Release : 1966
Category :
ISBN :

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The first part is devoted to proving that for a control problem satisfying the proper differentiability hypotheses and in which the optimization is made over a set of trajectories of the associated differential system that are in a fixed closed subset A of E(n), if an optimal solution exists such that the optimal trajectory is on the boundary of A and such that in a neighborhood of this trajectory, the boundary of A is the C(2)- diffeomorphic image of an open set in E(n-1), then this optimal solution satisfies a modified version of Pontryagin's maximum principle. The proof presented is direct and uses only the constructions used in the proof of Pontryagin's principle. If a C(2)- diffeomorphism exists, it is proved that the problems considered by Gamkrelidze are included in the problems considered in this paper. The restriction made by Gamkrelidze that the controls be piecewise smooth is removed, and the condition that the control sets be regular is relaxed. In the second part, three types of approximations of sequences of trajectories paired with their controls to a trajectory and its control are defined. The first type involves only the convergence of the trajectories, the second and third types add the convergence of the corresponding controls in the weak L(2)-topology and the strong L(2)-topology respectively. Next penalty functions are introduced and the problems generated perturbed; it is proved that the preceding results still hold for this new family of problems. Finally, results involving controllability hypotheses are obtained, and a specialized theorem involving approximations of type three is proved.

Author : Riccardo Zoppoli
Publisher : Springer Nature
Page : 532 pages
File Size : 46,26 MB
Release : 2019-12-17
Category : Technology & Engineering
ISBN : 3030296938

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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.

Author : A. E. Bryson
Publisher : CRC Press
Page : 500 pages
File Size : 10,97 MB
Release : 1975-01-01
Category : Technology & Engineering
ISBN : 9780891162285

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This best-selling text focuses on the analysis and design of complicated dynamics systems. CHOICE called it “a high-level, concise book that could well be used as a reference by engineers, applied mathematicians, and undergraduates. The format is good, the presentation clear, the diagrams instructive, the examples and problems helpful...References and a multiple-choice examination are included.”

Author : Daniel Liberzon
Publisher : Princeton University Press
Page : 255 pages
File Size : 18,77 MB
Release : 2012
Category : Mathematics
ISBN : 0691151873

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This textbook offers a concise yet rigorous introduction to calculus of variations and optimal control theory, and is a self-contained resource for graduate students in engineering, applied mathematics, and related subjects. Designed specifically for a one-semester course, the book begins with calculus of variations, preparing the ground for optimal control. It then gives a complete proof of the maximum principle and covers key topics such as the Hamilton-Jacobi-Bellman theory of dynamic programming and linear-quadratic optimal control. Calculus of Variations and Optimal Control Theory also traces the historical development of the subject and features numerous exercises, notes and references at the end of each chapter, and suggestions for further study. Offers a concise yet rigorous introduction Requires limited background in control theory or advanced mathematics Provides a complete proof of the maximum principle Uses consistent notation in the exposition of classical and modern topics Traces the historical development of the subject Solutions manual (available only to teachers) Leading universities that have adopted this book include: University of Illinois at Urbana-Champaign ECE 553: Optimum Control Systems Georgia Institute of Technology ECE 6553: Optimal Control and Optimization University of Pennsylvania ESE 680: Optimal Control Theory University of Notre Dame EE 60565: Optimal Control