This volume on mathematical control theory contains high quality articles covering the broad range of this field. The internationally renowned authors provide an overview of many different aspects of control theory, offering a historical perspective while bringing the reader up to the very forefront of current research.

1 Path Integrals and Stability.- 1.1 Introduction.- 1.2 Path Independence.- 1.3 Positivity of Quadratic Differential Forms.- 1.4 Lyapunov Theory for High-Order Differential Equations.- 1.5 The Bezoutian.- 1.5.1 The Routh Test.- 1.5.2 The Kharitonov Theorem.- 1.6 Dissipative Systems.- 1.7 Stability of Nonautonomous Systems.- 1.8 Conclusions.- 1.9 Appendixes.- 1.9.1 Appendix A: Notation.- 1.9.2 Appendix B: Linear Differential Systems.- 1.9.3 Appendix C: Proofs.- 2 The Estimation Algebra of Nonlinear Filtering Systems.- 2.1 Introduction.- 2.2 The Filtering Model and Background.- 2.3 Starting from the Beginning.- 2.4 Early Results on the Homomorphism Principle.- 2.5 Automorphisms that Preserve Estimation Algebras.- 2.6 BM Estimation Algebra.- 2.7 Structure of Exact Estimation Algebra.- 2.8 Structure of BM Estimation Algebras.- 2.9 Connection with Metaplectic Groups.- 2.10 Wei-Norman Representation of Filters.- 2.11 Perturbation Algebra and Estimation Algebra.- 2.12 Lie-Algebraic Classification of Maximal Rank Estimation Algebras.- 2.13 Complete Characterization of Finite-Dimensional Estimation Algebras.- 2.14 Estimation Algebra of the Identification Problem.- 2.15 Solutions to the Riccati P.D.E.- 2.16 Filters with Non-Gaussian Initial Conditions.- 2.17 Back to the Beginning.- 2.18 Acknowledgement.- 3 Feedback Linearization.- 3.1 Introduction.- 3.2 Linearization of a Smooth Vector Field.- 3.3 Linearization of a Smooth Control System by Change-of-State Coordinates.- 3.4 Feedback Linearization.- 3.5 Input-Output Linearization.- 3.6 Approximate Feedback Linearization.- 3.7 Normal Forms of Control Systems.- 3.8 Observers with Linearizable Error Dynamics.- 3.9 Nonlinear Regulation and Model Matching.- 3.10 Backstepping.- 3.11 Feedback Linearization and System Inversion.- 3.12 Conclusion.- 4 On the Global Analysis of Linear Systems.- 4.1 Introduction.- 4.2 The Geometry of Rational Functions.- 4.2.1 Spaces of Scalar-Input/Scalar-Output Linear Systems.- 4.2.2 The Deterministic Partial Realization Problem.- 4.3 Group Actions and the Geometry of Linear Systems.- 4.3.1 The Geometry of Matrix-Valued Rational Functions.- 4.3.2 Applications to Canonical Forms.- 4.3.3 A Signature Formula for the Maslov Index.- 4.4 The Geometry of Inverse Eigenvalue Problems.- 4.4.1 Inverse Eigenvalue Problems and the Hopf Degree.- 4.4.2 Pole Assignment by Output Feedback.- 4.5 Nonlinear Optimization on Spaces of Systems.- 4.5.1 A Classical Example: Uniqueness of Maximum Likelihood Estimates.- 4.5.2 The Partial Realization Problem Revisited.- 4.5.3 A Geometric Parameterization of Positive Rational Covariance Extensions.- 4.5.4 A Convex Optimization Scheme for Rational Covariance Extensions.- 5 Geometry and Optimal Control.- 5.1 Introduction.- 5.2 From Queen Dido to the Maximum Principle.- 5.3 Invariance, Covariance, and Lie Brackets.- 5.4 The Maximum Principle.- 5.5 The Maximum Principle as a Necessary Condition for Set Separation.- 5.6 Weakly Approximating Cones and Transversality.- 5.7 A Streamlined Version of the Classical Maximum Principle.- 5.8 Clarke's Nonsmooth Version and the ?ojasiewicz Improvement.- 5.9 Multidifferentials, Flows, and a General Version of the Maximum Principle.- 5.10 Three Ways to Make the Maximum Principle Intrinsic on Manifolds.- 5.11 Conclusion.- 6 Languages, Behaviors, Hybrid Architectures, and Motion Control.- 6.1 Introduction.- 6.2 MDLe: A Language for Motion Control.- 6.2.1 Performance Measure of a Plan.- 6.3 Hybrid Architecture.- 6.4 Application of MDLe to Path Planning with Nonholonomic Robots.- 6.4.1 Nonholonomic Constraints.- 6.5 PNMR: Path Planner for Nonholonomic Mobile Robots.- 6.5.1 Planning in the Obstacle-Free Disk.- 6.5.2 Tracing Boundaries.- 6.5.3 World Model Update.- 6.6 Conclusions.- 7 Optimal Control, Geometry, and Mechanics.- 7.1 Introduction.- 7.2 Variational Problems with Constraints and Optimal Control.- 7.3 Invariant Optimal Problems on Lie Groups.- 7.4 Sub-Riemannian Spheres-The Contact Case.- 7.5 Sub-Riemannian Systems on Lie Groups.- 7.6 Heavy Top and the Elastic Problem.- 7.7 Conclusion.- 8 Optimal Control, Optimization, and Analytical Mechanics.- 8.1 Introduction.- 8.2 Modeling Variational Problems in Mechanics and Control.- 8.2.1 Introduction.- 8.2.2 Variational Systems without External Forces.- 8.2.3 Mechanical Systems with External Forces.- 8.2.4 Relation to Optimal Control.- 8.2.5 Reduction.- 8.2.6 A Special Case.- 8.3 Optimization.- 8.4 Optimal Control Problems and Integrable Systems.- 8.4.1 Introduction.- 8.4.2 Optimal Control on Adjoint Orbits.- 8.4.3 Optimal Control on Symmetric Spaces.- 8.4.4 Optimal Control and the Toda Flow.- 9 The Geometry of Controlled Mechanical Systems.- 9.1 Introduction.- 9.2 Second-Order Generalized Control Systems.- 9.3 Flat Systems and Systems with Flat Inputs.- 9.4 Averaging Lagrangian and Hamiltonian Systems with Oscillatory Inputs.- 9.5 Stability and Flatness in Mechanical Systems with Oscillatory Inputs.- 9.6 Concluding Remarks.