Control theory and optimization. 2, 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. Encyclopaedia of Mathematical Sciences......Page aa_0001.djvu Cover......Page aa_0003_0001.djvu Preface......Page aa_0006_0001.djvu Contents......Page aa_0010_0001.djvu 1.1 Smooth Manifolds......Page aa_0014_0001.djvu 1.2 Vector Fields on Smooth Manifolds......Page aa_0017_0001.djvu 1.3 Smooth Differential Equations and Flowson Manifolds......Page aa_0021_0001.djvu 1.4 Control Systems......Page aa_0025_0001.djvu 2.1 Points, Diffeomorphisms, and Vector Fields......Page aa_0033_0001.djvu 2.2 Seminorms and C∞(M)-Topology......Page aa_0037_0001.djvu 2.3 Families of Functionals and Operators......Page aa_0038_0001.djvu 2.4.1 ODEs with Discontinuous Right-Hand Side......Page aa_0040_0001.djvu 2.4.2 Definition of the Right Chronological Exponential......Page aa_0041_0001.djvu 2.4.3 Formal Series Expansion......Page aa_0042_0001.djvu 2.4.4 Estimates and Convergence of the Series......Page aa_0043_0001.djvu 2.4.5 Left Chronological Exponential......Page aa_0045_0001.djvu 2.4.7 Autonomous Vector Fields......Page aa_0047_0001.djvu 2.5 Action of Diffeomorphisms on Vector Fields......Page aa_0049_0001.djvu 2.6 Commutation of Flows......Page aa_0052_0001.djvu 2.7 Variations Formula......Page aa_0053_0001.djvu 2.8 Derivative of Flow with Respect to Parameter......Page aa_0055_0001.djvu 3.1 Cauchy's Formula for Linear Systems......Page aa_0058_0001.djvu 3.2 Controllability of Linear Systems......Page aa_0060_0001.djvu 4.1 Local Linearizability......Page aa_0063_0001.djvu 4.2 Global Linearizability......Page aa_0067_0001.djvu 5.1 Formulation of the Orbit Theorem......Page aa_0073_0001.djvu 5.2 Immersed Submanifolds......Page aa_0074_0001.djvu 5.3 Corollaries of the Orbit Theorem......Page aa_0076_0001.djvu 5.4 Proof of the Orbit Theorem......Page aa_0077_0001.djvu 5.5 Analytic Case......Page aa_0082_0001.djvu 5.6 Frobenius Theorem......Page aa_0084_0001.djvu 5.7 State Equivalence of Control Systems......Page aa_0086_0001.djvu 6.1 State Space......Page aa_0091_0001.djvu 6.2 Euler Equations......Page aa_0094_0001.djvu 6.3 Phase Portrait......Page aa_0098_0001.djvu 6.4 Controlled Rigid Body: Orbits......Page aa_0100_0001.djvu 6.4.1 Orbits of the 3-Dimensional System......Page aa_0101_0001.djvu 6.4.2 Orbits of the 6-Dimensional System......Page aa_0104_0001.djvu 7.1 Model......Page aa_0107_0001.djvu 7.2 Two Free Points......Page aa_0110_0001.djvu 7.3 Three Free Points......Page aa_0111_0001.djvu 7.4 Broken Line......Page aa_0114_0001.djvu 8.1 Attainable Sets of Full-Rank Systems......Page aa_0118_0001.djvu 8.2 Compatible Vector Fields and Relaxations......Page aa_0122_0001.djvu 8.3 Poisson Stability......Page aa_0125_0001.djvu 8.4 Controlled Rigid Body: Attainable Sets......Page aa_0127_0001.djvu 9.1 Feedback Equivalence......Page aa_0129_0001.djvu 9.2.1 Linear Systems with Scalar Control......Page aa_0131_0001.djvu 9.2.2 Linear Systems with Vector Control......Page aa_0134_0001.djvu 9.3 State-Feedback Linearizability......Page aa_0139_0001.djvu 10.1 Problem Statement......Page aa_0145_0001.djvu 10.2 Reduction to Study of Attainable Sets......Page aa_0146_0001.djvu 10.3 Compactness of Attainable Sets......Page aa_0148_0001_1.djvu 10.5 Relaxations......Page aa_0151_0001.djvu 11.1.2 Cotangent Bundle......Page aa_0153_0001.djvu 11.2 Differential k-Forms......Page aa_0155_0001.djvu 11.2.1 Exterior k-Forms......Page aa_0156_0001.djvu 11.2.2 Differential k-Forms......Page aa_0158_0001.djvu 11.3 Exterior Differential......Page aa_0159_0001.djvu 11.4 Lie Derivative of Differential Forms......Page aa_0161_0001.djvu 11.5.1 Liouville Form and Symplectic Form......Page aa_0165_0001.djvu 11.5.2 Hamiltonian Vector Fields......Page aa_0167_0001.djvu 11.5.3 Lagrangian Subspaces......Page aa_0173_0001.djvu 12.1 Geometric Statement of PMP and Discussion......Page aa_0175_0001.djvu 12.2 Proof of PMP......Page aa_0180_0001.djvu 12.3 Geometric Statement of PMP for Free Time......Page aa_0185_0001.djvu 12.4 PMP for Optimal Control Problems......Page aa_0187_0001.djvu 12.5 PMP with General Boundary Conditions......Page aa_0190_0001.djvu 13.1 The Fastest Stop of a Train at a Station......Page aa_0198_0001.djvu 13.2 Control of a Linear Oscillator......Page aa_0201_0001.djvu 13.3 The Cheapest Stop of a Train......Page aa_0204_0001.djvu 13.4 Control of a Linear Oscillator with Cost......Page aa_0206_0001.djvu 13.5 Dubins Car......Page aa_0207_0001.djvu 14. Hamiltonian Systems with Convex Hamiltonians......Page aa_0214_0001.djvu 15.1 Problem Statement......Page aa_0217_0001.djvu 15.2 Geometry of Polytopes......Page aa_0218_0001.djvu 15.3 Bang-Bang Theorem......Page aa_0219_0001.djvu 15.4 Uniqueness of Optimal Controls and Extremals......Page aa_0221_0001.djvu 15.5 Switchings of Optimal Control......Page aa_0224_0001.djvu 16.1 Problem Statement......Page aa_0229_0001.djvu 16.2 Existence of Optimal Control......Page aa_0230_0001.djvu 16.3 Extremals......Page aa_0233_0001.djvu 16.4 Conjugate Points......Page aa_0235_0001.djvu 17.1 Sufficient Optimality Conditions......Page aa_0241_0001.djvu 17.1.1 Integral Invariant......Page aa_0242_0001.djvu 17.1.2 Problem with Fixed Time......Page aa_0244_0001.djvu 17.1.3 Problem with Free Time......Page aa_0246_0001.djvu 17.2 Hamilton-Jacobi Equation......Page aa_0248_0001.djvu 17.3 Dynamic Programming......Page aa_0250_0001.djvu 18.1.1 Motivation......Page aa_0252_0001.djvu 18.1.2 Trivialization of T*M......Page aa_0253_0001.djvu 18.1.3 Symplectic Form on E×M......Page aa_0254_0001.djvu 18.1.4 Hamiltonian System on E×M......Page aa_0256_0001.djvu 18.2 Lie Groups......Page aa_0260_0001.djvu 18.2.1 Examples of Lie Groups......Page aa_0261_0001.djvu 18.2.2 Lie's TheoreIll for Linear Lie Groups......Page aa_0262_0001.djvu 18.2.3 Abstract Lie Groups......Page aa_0264_0001.djvu 18.3.1 Trivialization of the Cotangent Bundle of a Lie Group......Page aa_0265_0001.djvu 18.3.2 Hamiltonian System on M*×M......Page aa_0266_0001.djvu 18.3.3 Cmnpact Lie Groups......Page aa_0267_0001.djvu 19.1 Riemannian Problem......Page aa_0270_0001.djvu 19.2 A Sub-Riemannian Problem......Page aa_0272_0001.djvu 19.3 Control of Quantum Systems......Page aa_0276_0001.djvu 19.3.1 Elimination of the Drift......Page aa_0278_0001.djvu 19.3.2 Lifting of the Problems to Lie Groups......Page aa_0280_0001.djvu 19.3.4 Extremals......Page aa_0283_0001.djvu 19.3.5 Transversality Conditions......Page aa_0284_0001.djvu 19.3.6 OptiIllal Geodesics Upstairs and Downstairs......Page aa_0285_0001.djvu 19.4 A Time-Optimal Problem on SO(3)......Page aa_0289_0001.djvu 20.1 Hessian......Page aa_0297_0001.djvu 20.2.1 Critical Points of Corank One......Page aa_0301_0001.djvu 20.2.2 Critical Points of Arbitrary Corank......Page aa_0304_0001.djvu 20.3 Differentiation of the Endpoint Mapping......Page aa_0308_0001.djvu 20.4.1 Legendre Condition......Page aa_0313_0001.djvu 20.4.2 Regular Extremals......Page aa_0315_0001.djvu 20.4.3 Singular Extremals......Page aa_0316_0001.djvu 20.4.4 Necessary Conditions......Page aa_0321_0001.djvu 20.5.1 Abnormal Sub-Riemannian Geodesics......Page aa_0322_0001.djvu 20.5.2 Local Controllability of Bilinear SysteIll......Page aa_0324_0001.djvu 20.6 Single-Input Case......Page aa_0325_0001.djvu 21. Jacobi Equation......Page aa_0336_0001.djvu 21.1 Regular Case: Derivation of Jacobi Equation......Page aa_0337_0001.djvu 21.2 Singular Case: Derivation of Jacobi Equation......Page aa_0341_0001.djvu 21.3 Necessary Optimality Conditions......Page aa_0345_0001.djvu 21.4 Regular Case: Transformation of Jacobi Equation......Page aa_0346_0001.djvu 21.5 Sufficient Optimality Conditions......Page aa_0349_0001.djvu 22.1 Reduction......Page aa_0358_0001.djvu 22.2 Rigid Body Control......Page aa_0361_0001.djvu 22.3 Angular Velocity Control......Page aa_0362_0001.djvu 23.1 Curvature of 2-Dimensional Systems......Page aa_0365_0001.djvu 23.1.1 Moving Frame......Page aa_0366_0001.djvu 23.1.2 Jacobi Equation in Moving Frame......Page aa_0371_0001.djvu 23.2 Curvature of 3-Dimensional Control-Affine Systems......Page aa_0375_0001.djvu 24.1 Geometric Model......Page aa_0379_0001.djvu 24.2.1 Riemannian Geodesics......Page aa_0381_0001.djvu 24.2.2 Levi-Civita Connection......Page aa_0382_0001.djvu 24.3 Admissible Velocities......Page aa_0385_0001.djvu 24.4 Controllability......Page aa_0386_0001.djvu 24.5.1 Problem Statement......Page aa_0389_0001.djvu 24.5.3 Abnormal Extremals......Page aa_0390_0001.djvu 24.5.4 Normal Extremals......Page aa_0393_0001.djvu A.1 Homomorphisms and Operators in C∞(M)......Page aa_0395_0001.djvu A.2 Remainder Term of the Chronological Exponential......Page aa_0397_0001.djvu Books on control theory and nonholonomic geometry......Page aa_0401_0001.djvu Chronological calculus and related topics......Page aa_0402_0001.djvu Controllability......Page aa_0403_0001.djvu Geometric optimal control problems......Page aa_0405_0001.djvu Second variation and related topics......Page aa_0407_0001.djvu Other references......Page aa_0408_0001.djvu 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. TOC: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. Feedback and State Equivalence of Control Systems.- 10. Optimal Control Problem.- 11. Elements of Exterior Calculus and Symplectic Geometry.- 12. Pontryagin Masimum 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, 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.- Appendix.- Bibliography.- Index 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 comƯ 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 paramƯ 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 ordiƯ 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

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.