Control Theory from the Geometric Viewpoint

This book presents some facts and methods of the Mathematical Control Theory treated from the geometric point of view. The book is mainly based on graduate courses given by the first coauthor in the years 2000-2001 at the International School for Advanced Studies, Trieste, Italy. Mathematical prerequisites are reduced to standard courses of Analysis and Linear Algebra plus some basic Real and Functional Analysis. No preliminary knowledge of Control Theory or Differential Geometry is required. What this book is about? The classical deterministic physical world is described by smooth dynamical systems: the future in such a system is com pletely determined by the initial conditions. Moreover, the near future changes smoothly with the initial data. If we leave room for "free will" in this fatalistic world, then we come to control systems. We do so by allowing certain param eters of the dynamical system to change freely at every instant of time. That is what we routinely do in real life with our body, car, cooker, as well as with aircraft, technological processes etc. We try to control all these dynamical systems! Smooth dynamical systems are governed by differential equations. In this book we deal only with finite dimensional systems: they are governed by ordi nary differential equations on finite dimensional smooth manifolds. A control system for us is thus a family of ordinary differential equations. The family is parametrized by control parameters." Cover Preface Contents 1 - Vector Fields and Control Systems on Smooth Manifolds 2 - Elements of Chronological Calculus 3 - Linear Systems 4 - State Linearizability of Nonlinear Systems 5 - The Orbit Theorem and its Applications 6 - Rotations of the Rigid Body 7 - Control of Configurations 8 - Attainable Sets 9 - Feedbakc and State Equivalence of Control Systems 10 - Optimal Control Problem 11 - Elements of Exterior Calculus and Symplectic Geometry 12 - Pontryagin Maximum Principle 13 - Examples of Optimal Control Problems 14 - Hamiltonian Systems with Convex Hamiltonians 15 - Linear Time-Optimal Problem 16 - Linear-Quadratic Problem 17 - Sufficient Optimality Conditions, Hamilton-Jacobi Equation, and Dynamic Programming 18 - Hamiltonian Systems for Geometric Optimal Control Problems 19 - Examples of Optimal Control Problems on Compact Lie Groups 20 - Second Order Optimality Conditions 21 - Jacobi Equation 22 - Reduction 23 - Curvature 24 - Rolling Bodies A. Appendix References List of Figures Index

This book presents some facts and methods of Mathematical Control Theory treated from the geometric viewpoint. It is devoted to finite-dimensional deterministic control systems governed by smooth ordinary differential equations. The problems of controllability, state and feedback equivalence, and optimal control are studied.

Some of the topics treated by the authors are covered in monographic or textbook literature for the first time while others are presented in a more general and flexible setting than elsewhere.

Although being fundamentally written for mathematicians, the authors make an attempt to reach both the practitioner and the theoretician by blending the theory with applications. They maintain a good balance between the mathematical integrity of the text and the conceptual simplicity that might be required by engineers. It can be used as a text for graduate courses and will become most valuable as a reference work for graduate students and researchers.