Information path functional and informational macrodynamics

506 Pages·2010·4.363 MB·English

by  Lerner V.S.

I P F
NFORMATION ATH UNCTIONAL
I
AND NFORMATIONAL
MACRODYNAMICS

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I P F
NFORMATION ATH UNCTIONAL
I
AND NFORMATIONAL
MACRODYNAMICS

VLADIMIR S. LERNER

Nova
Nova Science Publishers, Inc.
New York
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LIBRARY OF CONGRESS CATALOGING-IN-PUBLICATION DATA

Lerner, Vladimir S., 1931-
Information path functional and informational macrodynamics / Vladimir S. Lerner.
p. cm.
ISBN 978-1-61470-092-0 (eBook)
1. Information theory. 2. Stochastic control theory. 3. Variational principles. 4. System
analysis--Mathematical models. I. Title.
QA402.37.L47 2009
629.8'312--dc22
2009002394

Published by Nova Science Publishers, Inc. (cid:30) New York
CONTENTS

Abstract xi
Preface xiii
Introduction xix
PART 1.
THE INFORMATION PATH FUNCTIONAL’S FOUNDATION 1
1.0. Introduction 3
1.1. The Initial Mathematical Models 7
1.1.1. Model of the Microlevel Process 7
1.1.2. Model of the Macrolevel Process 9
1.1.3. The Feedback Equation-Control Law 9
1.1.4. Model of the Programmable Trajectories (as a task) at Microlevel 10
1.1.5. Model of the Programmable Trajectories (as a task) at the Macrolevel 11
1.1.6. The Equations in Deviations 11
1.1.7. Model of Disturbances 13
1.1.8. The Microlevel Process’ Functional 13
1.1.9.The Jensen Inequality for Entropy Functional 18
1.2. Dynamic Approximation of a Random Information Functional
and the Path Functional 23
1.2.1. The Extremal Principle and the Problem Formulation 23
1.2.2. The Problem Solution. Information Path Functional 24
1.2.3. The Estimation of an Accuracy of the Probability’s Approximation 42
1.3. Variation Problem for the Information Path Functional and Its Solution 45
1.3.1. The Problem Statements 45
vi Contents
1.3.2. Solution to the Variation Problem 46
1.3.3. A Minimum Condition for the Microlevel’s Functional 60
1.3.4. The Optimal Control Synthesis 63
1.3.5. A Summary of the Information Path Functional Approach and IMD.
The Information Invariants 77
1.4. The IMD Information Space Distributed Macromodels 103
1.4.1. Introduction 103
1.4.2. The Variation Problem for Space Distributed Macromodel 105
1.4.3. The Invariant Conditions at the Transformation of the Space
Coordinates 107
1.4.4. The Parameters of the Space Transformation and the Distributed
Macromodels 114
1.4.5. The IMD Macromodel’s Singular Points and the Singular Trajectories 120
1.4.6. The IPF Natural Variation Problem, Singular Trajectories,
and the Field’s Invariants 128
1.5. The Cooperative Information Macromodels and Information Network 141
1.5.1. Introduction 141
1.5.2. The Time-Space Movement toward the Macromodel's Cooperation 142
1.5.3. The Consolidation And Aggregation of the Model Processes
in a Cooperative Information Network (IN) 164
1.5.4. The IN Dynamic Structure 173
1.5.5. Geometrical Structure of the Optimal Space Distributed Cooperative
Macromodel (OPMC). The IN Geometric Structure 182
1.6. The IMD Model’s Phenomena and an Information Code 195
1.6.1. The model Time Course and the Evaluation of the Information
Contributions into IN. The Triplet’s Genetic Code 195
1.6.2. The Model Information Geometry (IG), Its Specific, and the Structure 203
1.6.3. The Model Uncertainty Zone and Its Evaluation 211
1.6.4. Creation of the IN Geometry and a Genetic Code of the Information
Cellular Geometry 212
1.6.5. The Minimal Admissible Uncertainty and Its Connection to Physics 217
1.6.6. Information Structure of the Model Double Spiral’s (DSS)
Control Mechanism 221
Contents vii
1.6.7. Examples of the DSS Codes 224
1.6.8. A system’s Harmony, Regularities, and the VP 226
1.7. The Macrodynamic and Cooperative Complexities 229
1.7.1. Introduction 229
1.7.2. The Notion of Interactive and Cooperative Complexities
and Their Information Measures 231
1.7.3. The Information Indicator of a Cooperative Complexity 233
1.7.4. Illustration of Arising of the Information Cooperative Complexity
at Discrete Points of Applied Controls 239
1.7.5. The Complexity Invariant Measure in a Cooperative Dynamic
Process 245
1.7.6. The IN Cooperative Mechanism with the Complexity’s Measures 251
1.7.7. The Equations of the Spatial Information Cooperative Dynamics.
Information Attraction and Complexity 259
1.7.8. Connection to Kolmogorov’s Complexity 269
1.8. The Regularities of Evolutionary Dynamics and the Information
Law of Evolution 271
1.8.1. Introduction 271
1.8.2. The Equations Regularities and the Evolutionary Law 272
1.8.3. A mechanism of an Enhancement of the Acceptable Mutations 280
1.8.4. The Conditions of the Model’s Self-control, Adaptation,
and Self-organization 286
1.8.5. The Evolution of the Model Invariants and a Potential
Macroprocess’ Cyclicity 291
1.8.6. Requirements for the Model Self–organization.
The Results of Computer Simulations 297
1.8.7. Evaluation of Some Prognosis Parameters of the Evolutionary
Dynamics. Example 301
1.8.8. Information Mechanism of Assembling the Node's Frequencies
and Automatic Selection 302
1.8.9. The Functional Schema of the Evolutionary Informational
Mechanisms. A Summary 307
viii Contents
1.9. The Physical Analogies Related to the Information Path Functional 311
1.9.1. The Connection between the Information Path Functional
(IPF) and the Kolmogorov’s (K) Entropy of a Dynamic
System, and the Relations to Physics 311
1.9.2. An IPF Analogy with the Feynman Path Functional in Quantum
Mechanics and Information form of Schrödinger Equation 314
1.9.3. About the Invariant Transformation of the Model Imaginary
Eigenvalues and Information 321
1.9.4. The Superimposing Processes, Control and Asymmetry.
The IMD Relation to Nonequilibrium Thermodynamics (NT) 323
1.9.5. About the Notion of a System 330
PART 2.
THE INFORMATION PATH FUNCTIONAL’S AND IMD’S APPLICATIONS 335
2.1. Solution of the Control Problems for a Complex Object 335
2.1.1. The Control Problems for a Complex Object 335
2.1.2. Solving the Identification Problem 338
2.1.2.1. The Identification of the Concentrated Object's Models 338
2.1.2.2.The Identification of the Space Distributed Object's Models 342
2.1.3. Solving the Optimal Control Problem 349
2.1.3.1. A Joint Solution of the Optimal Control and Identification
Problems. The Basic Results 349
2.1.3.2. The Procedure of the Joint Identification, Optimal Control,
and Consolidation 353
2.1.3.3. Building the Object’s Cooperative Information Network 370
2.2. The Information Modeling of the Encoding-Decoding Processes
at Transformation of Biological and Cognitive Information 373
2.2.1. The Objective and Methodology 373
2.2.2. An Inner Information Structure of the IN with the Ranged
and Nonranged Sequences of the Starting Eigenvalues. The DSS Code 374
2.2.3 Mathematical Model of the IN with an Arbitrary Sequence
of the Starting Eigenvalues 378
Contents ix
2.2.4. The Procedure of Encoding, Compression, Synthesis, and Decoding
the IN Information 388
2.2.5. Summarized Results 394
2.2.6. About Other Related Applications 399
2.2.7. The Connections between Some Physical Neuronal Functions
and Mechanisms with Their IMD Information Analogies 402
2.3. Information Modeling and Control of Some Industrial Technology
Processes with Complex Superimposing Phenomena 409
2.3.1. Process Solidification and Its Application in Casting Technology 409
2.3.2. Some Electrotechnological Processes 420
A.2.3. Appendix: Additional Practical Results 421
2.4. An Elementary Information Macrodynamic Model of a Market
Economic System 427
2.4.1. About Information Systems Modeling of a Modern Economy.
The Objectives 427
2.4.2. An Elementary Local Production System (LP) 430
2.4.3. An Information Model of a Local Market 432
2.4.4. Managing the LP. A Bank and a Stock Market 437
2.4.5. Other Information Markets 441
2.4.6. Example 442
2.4.7. Summary 443
2.5. An Outline of the Computer Based Methodology 445
2.5.1. The Hierarchy of the Model’s Micro-and Macrovariables
and Their Identification 445
2.5.2. The Computer Restoration of the IMD Model 446
2.5.3. The Structure of the IMD Software Package 449
Conclusion 451
References 455
About the Author 471
Index 473