Optimization and Dynamical Systems

1. Front Matter

   Pages i-xiii

   PDF

2. Matrix Eigenvalue Methods

   • Uwe Helmke, John B. Moore

   Pages 1-42

3. Double Bracket Isospectral Flows

   • Uwe Helmke, John B. Moore

   Pages 43-80

4. Singular Value Decomposition

   • Uwe Helmke, John B. Moore

   Pages 81-100

5. Linear Programming

   • Uwe Helmke, John B. Moore

   Pages 101-124

6. Approximation and Control

   • Uwe Helmke, John B. Moore

   Pages 125-161

7. Balanced Matrix Factorizations

   • Uwe Helmke, John B. Moore

   Pages 163-200

8. Invariant Theory and System Balancing

   • Uwe Helmke, John B. Moore

   Pages 201-227

9. Balancing via Gradient Flows

   • Uwe Helmke, John B. Moore

   Pages 229-267

10. Sensitivity Optimization

    • Uwe Helmke, John B. Moore

    Pages 269-309

11. Back Matter

    Pages 311-403

    PDF

This work is aimed at mathematics and engineering graduate students and researchers in the areas of optimization, dynamical systems, control sys­ tems, signal processing, and linear algebra. The motivation for the results developed here arises from advanced engineering applications and the emer­ gence of highly parallel computing machines for tackling such applications. The problems solved are those of linear algebra and linear systems the­ ory, and include such topics as diagonalizing a symmetric matrix, singular value decomposition, balanced realizations, linear programming, sensitivity minimization, and eigenvalue assignment by feedback control. The tools are those, not only of linear algebra and systems theory, but also of differential geometry. The problems are solved via dynamical sys­ tems implementation, either in continuous time or discrete time , which is ideally suited to distributed parallel processing. The problems tackled are indirectly or directly concerned with dynamical systems themselves, so there is feedback in that dynamical systems are used to understand and optimize dynamical systems. One key to the new research results has been the recent discovery of rather deep existence and uniqueness results for the solution of certain matrix least squares optimization problems in geomet­ ric invariant theory. These problems, as well as many other optimization problems arising in linear algebra and systems theory, do not always admit solutions which can be found by algebraic methods.

• Dynamical System
• Dynamische Systeme
• Kontrolltheorie
• Lyapunov stability
• Numerische Lineare Algebra
• Optimisierung
• Signalverarbeitung
• control theory
• dynamical systems
• linear optimization
• numerical linear algebra
• optimization
• signal processing

• Department of Mathematics, University of Würzburg, Würzburg, Germany

  Uwe Helmke

• Department of Systems Engineering and Cooperative Research Centre for Robust and Adaptive Systems, Research School of Information Sciences and Engineering, Australian National University, Canberra, Australia

  John B. Moore

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