Control Theory from the Geometric Viewpoint

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.

Vector Fields and Control Systems on Smooth Manifolds -- Elements of Chronological Calculus -- Linear Systems -- State Linearizability of Nonlinear Systems -- The Orbit Theorem and its Applications -- Rotations of the Rigid Body -- Control of Configurations -- Attainable Sets -- Feedback and State Equivalence of Control Systems -- Optimal Control Problem -- Elements of Exterior Calculus and Symplectic Geometry -- Pontryagin Maximum Principle -- Examples of Optimal Control Problems -- Hamiltonian Systems with Convex Hamiltonians -- Linear Time-Optimal Problem -- Linear-Quadratic Problem -- Sufficient Optimality Conditions, Hamilton-Jacobi Equation, Dynamic Programming -- Hamiltonian Systems for Geometric Optimal Control Problems -- Examples of Optimal Control Problems on Compact Lie Groups -- Second Order Optimality Conditions -- Jacobi Equation -- Reduction -- Curvature -- Rolling Bodies -- Appendix -- Bibliography -- Index.