Covers developments in bilinear systems theory Focuses on the control of open physical processes functioning in a non-equilibrium mode Emphasis is on three primary disciplines: modern differential geometry, control of dynamical systems, and optimization theory Includes applications to the fields of quantum and molecular computing, control of physical processes, biophysics, superconducting magnetism, and physical information science
Author(s): Panos M. Pardalos, Vitaliy Yatsenko
Edition: 1
Year: 2008
Language: English
Pages: 374
Cover......Page 1
Series: Springer Optimization and Its Applications VOLUME 11......Page 3
OPTIMIZATION AND CONTROL OF BILINEAR SYSTEMS: Theory, Algorithms, and Applications......Page 4
Copyright - ISBN: 0387736689......Page 5
Preface......Page 14
Contents......Page 8
1. SYSTEM-THEORETICAL DESCRIPTION OF OPEN PHYSICAL PROCESSES ......Page 29
1.1 Equivalence of Control Systems ......Page 31
1.2 Lie Algebras, Lie Groups, and Representations ......Page 32
1.3 Selection of Mathematical Models ......Page 34
1.4 Bilinear Logic-Dynamical Realization of Nonlinear Control Systems ......Page 38
2 Global Bilinearization of Nonlinear Systems ......Page 40
3 Identification of Bilinear Control Systems ......Page 47
4.1 Systems on Lie Groups ......Page 48
4.2 Bilinear Realization of Nonlinear Systems ......Page 50
4.3 Approximation of Nonlinear Systems by Bilinear Systems ......Page 51
5 Controllability of Bilinear Systems ......Page 53
6.1 Observability and Lie Groups ......Page 55
6.2 Algorithms of Observability ......Page 61
6.3 Examples ......Page 64
6.4 Decoupling Problems ......Page 66
7.1 Right-Invariant Control Systems ......Page 68
7.2 Invertibility of Right-Invariant Systems ......Page 71
7.3 Left-Inverses for Bilinear Systems ......Page 77
8.1 Discrete Bilinear Systems and Invertability ......Page 84
8.2 Construction of Inverse Systems ......Page 85
8.3 Controllability of Inverse Systems ......Page 86
9.1 General Characteristics of Versal Models ......Page 87
9.2 Algorithms ......Page 88
10 Notes and Sources ......Page 92
2. CONTROL OF BILINEAR SYSTEMS ......Page 93
1.1 Optimal Control Problem ......Page 94
1.2 Reduction of Control Problem to Equivalent Problem for Bilinear Systems ......Page 95
1.3 Optimal Control of Bilinear Systems ......Page 98
1.4 On the Solution of the Euler–Lagrange Equation ......Page 100
2 Stability of Bilinear Systems ......Page 102
2.1 Normed Vector Space ......Page 103
2.2 Continuous Bilinear Systems ......Page 105
2.3 Discrete Bilinear Systems ......Page 107
3.1 Control of Fixed Points ......Page 110
3.2 Control of Limit Cycles ......Page 116
3.3 Variations in the Control Dynamics ......Page 117
4 Notes and Sources ......Page 119
3. BILINEAR SYSTEMS AND NONLINEAR ESTIMATION THEORY ......Page 121
1.1 Filtration Problems ......Page 122
1.2 Problem Statement ......Page 125
1.3 Preliminaries on Nonlinear and Bilinear Lattice Models ......Page 126
1.4 Adaptive Filter for Lattice Systems ......Page 128
1.5 Identification of Bilinear Lattice Models ......Page 131
1.6 A Generalization for Nonlinear Lattice Models ......Page 137
1.7 Estimation of the State Vector of CA3 Region ......Page 139
1.8 Detection and Prediction of Epileptic Seizures ......Page 143
2.1 Estimation Problem ......Page 146
2.2 Invertibility of Continuous MS and Estimation of Signal Parameters ......Page 147
2.3 Estimation of Parameters of an Almost Periodic Signal Under Discrete Measurements ......Page 152
2.4 Neural Network Estimation of Signal Parameters ......Page 155
2.5 Finite-Dimensional Bilinear Adaptive Estimation ......Page 157
2.6 Example ......Page 158
3.1 Lattice Systems and DMZ Equations ......Page 159
3.2 Structure of Estimation Algebra ......Page 163
4 Notes and Sources ......Page 166
4. CONTROL OF DYNAMICAL PROCESSES AND GEOMETRICAL STRUCTURES ......Page 167
1 Geometric Structures ......Page 169
1.1 Metric Spaces ......Page 170
1.2 Optimal Control ......Page 171
1.3 Identification of Nonlinear Agents and Yang-Mills Fields ......Page 173
1.4 The Estimation Algebra of Nonlinear Filtering Systems ......Page 174
1.5 Estimation Algebra and Identification Problems ......Page 175
2 Lie Groups and Yang-Mills Fields ......Page 177
3 Control of Multiagent Systems and Yang-Mills Representation ......Page 180
4 Dynamic Systems, Information, and Fiber Bundles ......Page 182
5 Fiber Bundles, Multiple Agents, and Observability ......Page 192
5.1 Smooth Nonlinear Systems ......Page 194
5.2 Minimality and Observability ......Page 196
6 Notes and Sources ......Page 204
1 Introduction ......Page 205
2 Stability and Levitation ......Page 207
3 Dynamics of Magnetically Levitated Systems ......Page 210
4.1 Statement of the Problem ......Page 219
4.2 Optimal Synthesis of Chaotic Dynamics ......Page 221
4.3 Chaotic Dynamics of Levitated Probes ......Page 223
4.4 Asymptotic Stability of Measurements ......Page 224
4.5 Synthesizing the Adaptive Filter ......Page 226
4.7 Numerical Analysis of the Estimation Model ......Page 228
4.8 Construction of the Sensor ......Page 230
5 Nonlinear Dynamics and Chaos ......Page 231
6 Notes and Sources ......Page 233
6. OPTIMIZATION AND CONTROL OF QUANTUM-MECHANICAL PROCESSES ......Page 235
1.1 Evolution of Quantum Systems ......Page 238
1.2 Finite Control of Quantum Systems ......Page 242
1.3 Amplitude-Frequency Control ......Page 244
1.4 Resonance Control of a Three-Level System ......Page 246
2 Simulation of Quantum Control Systems ......Page 247
2.1 Mathematical Models of Quantum Objects ......Page 248
2.2 Dynamics of Quantum Systems and Control ......Page 249
2.3 Physical Constraints ......Page 251
2.4 Hierarchy of Time Scales ......Page 252
3 Representation of the Interaction ......Page 254
3.1 Approximation of the Model ......Page 256
3.2 Quantum Bilinear Dynamics ......Page 257
3.3 Hamiltonian Dynamics ......Page 260
4 The Bellman Principle and Quantum Systems ......Page 261
4.1 Deterministic Optimal Control ......Page 262
4.2 The Bellman-Hamilton-Jacobi Theory and Differential Forms ......Page 264
4.3 Stochastic Optimal Control and Schr¨odinger Equations ......Page 267
5 Classical and Quantum Controlled Lattices: Self-Organization, Optimization and Biomedical Applications......Page 269
5.1 Hamiltonian Models of Cellular Dynamatons ......Page 271
5.2 Self-Organization of Neural Networks ......Page 275
5.3 Bilinear Lattices and Epileptic Seizures ......Page 280
5.4 Quantum Model of Neural Networks ......Page 285
6 Notes and Sources ......Page 287
7. MODELING AND GLOBAL OPTIMIZATION IN BIOMOLECULAR SYSTEMS ......Page 289
1.1 Mathematical Models ......Page 290
1.2 Kolmogorov Equations and Bilinear Dynamical Systems ......Page 292
1.3 Modeling and Experimental Results ......Page 301
2 Bilinear Models of Biological Membranes ......Page 306
2.1 Controlled Model of the Channel ......Page 308
2.2 Generalized Equation of Diffusion ......Page 315
2.3 Structure of a Functioning Channel ......Page 318
3.1 Ecological Monitoring and Living Objects ......Page 324
3.2 Experimental Results ......Page 325
3.3 Identification of a Bilinear Sensitive Element ......Page 331
3.4 Separation of Pollutant Characteristics by Neural Chips ......Page 335
4 Notes and Sources ......Page 340
8. MODELING AND ANALYSIS OF BILINEAR SYSTEMS ......Page 341
1.1 Modeling without Hidden Variables ......Page 342
1.2 Modeling with Hidden Variables ......Page 344
1.3 Controlling Chaos ......Page 350
2.1 Non-Gaussian Signals and Backscattering Process ......Page 352
2.2 Sea Clutter Attractor ......Page 353
2.3 Mathematical Model of Sea Clutter ......Page 355
3.1 Nonparametric Models for Epilepsy Data ......Page 356
3.2 Reconstruction of the Parameter Spaces of the Human Brain ......Page 357
4.1 Nonlinear Dynamics and Epilepsy ......Page 366
4.2 Reconstructing Equations of the Epileptic Brain from Experimental Data ......Page 367
4.3 Quadratic Programming Problem ......Page 369
5.1 Experimental Data ......Page 371
5.2 Methods for the Analysis of Time Series ......Page 373
5.3 Numerical Results ......Page 376
6 Notes and Sources ......Page 379
References ......Page 381
Index ......Page 397
