1 Vector Fields and Control Systems on Smooth Manifolds 1 1.1 Smooth Manifolds 1 1.2 Vector Fields on Smooth Manifolds 4 1.3 Smooth Differential Equations and Flows on Manifolds 8 1.4 Control Systems 12 2 Elements of Chronological Calculus 21 2.1 Points, Diffeomorphisms, and Vector Fields 21 2.2 Seminorms and $C^{\infty }(M)$-Topology 25 2.3 Families of Functionals and Operators 26 2.4 Chronological Exponential 28 2.5 Action of Diffeomorphisms on Vector Fields 37 2.6 Commutation of Flows 40 2.7 Variations Formula 41 2.8 Derivative of Flow with Respect to Parameter 43 3 Linear Systems 47 3.1 Cauchy's Formula for Linear Systems 47 3.2 Controllability of Linear Systems 49 4 State Linearizability of Nonlinear Systems 53 4.1 Local Linearizability 53 4.2 Global Linearizability 57 5 The Orbit Theorem and its Applications 63 5.1 Formulation of the Orbit Theorem 63 5.2 Immersed Submanifolds 64 5.3 Corollaries of the Orbit Theorem 66 5.4 Proof of the Orbit Theorem 67 5.5 Analytic Case 72 5.6 Frobenius Theorem 74 5.7 State Equivalence of Control Systems 76 6 Rotations of the Rigid Body 81 6.1 State Space 81 6.2 Euler Equations 84 6.3 Phase Portrait 88 6.4 Controlled Rigid Body: Orbits 90 7 Control of Configurations 97 7.1 Model 97 7.2 Two Free Points 100 7.3 Three Free Points 101 7.4 Broken Line 104 8 Attainable Sets 109 8.1 Attainable Sets of Full-Rank Systems 109 8.2 Compatible Vector Fields and Relaxations 113 8.3 Poisson Stability 116 8.4 Controlled Rigid Body: Attainable Sets 118 9 Feedback and State Equivalence of Control Systems 121 9.1 Feedback Equivalence 121 9.2 Linear Systems 123 9.3 State-Feedback Linearizability 131 10 Optimal Control Problem 137 10.1 Problem Statement 137 10.2 Reduction to Study of Attainable Sets 138 10.3 Compactness of Attainable Sets 140 10.4 Time-Optimal Problem 143 10.5 Relaxations 143 11 Elements of Exterior Calculus and Symplectic Geometry 145 11.1 Differential 1-Forms 145 11.2 Differential $k$-Forms 147 11.3 Exterior Differential 151 11.4 Lie Derivative of Differential Forms 153 11.5 Elements of Symplectic Geometry 157 12 Pontryagin Maximum Principle 167 12.1 Geometric Statement of PMP and Discussion 167 12.2 Proof of PMP 172 12.3 Geometric Statement of PMP for Free Time 177 12.4 PMP for Optimal Control Problems 179 12.5 PMP with General Boundary Conditions 182 13 Examples of Optimal Control Problems 191 13.1 The Fastest Stop of a Train at a Station 191 13.2 Control of a Linear Oscillator 194 13.3 The Cheapest Stop of a Train 197 13.4 Control of a Linear Oscillator with Cost 199 13.5 Dubins Car 200 14 Hamiltonian Systems with Convex Hamiltonians 207 15 Linear Time-Optimal Problem 211 15.1 Problem Statement 211 15.2 Geometry of Polytopes 212 15.3 Bang-Bang Theorem 213 15.4 Uniqueness of Optimal Controls and Extremals 215 15.5 Switchings of Optimal Control 218 16 Linear-Quadratic Problem 223 16.1 Problem Statement 223 16.2 Existence of Optimal Control 224 16.3 Extremals 227 16.4 Conjugate Points 229 17 Sufficient Optimality Conditions, Hamilton-Jacobi Equation,Dynamic Programming 235 17.1 Sufficient Optimality Conditions 235 17.2 Hamilton-Jacobi Equation 242 17.3 Dynamic Programming 244 18 Hamiltonian Systems for Geometric Optimal Control Problems 247 18.1 Hamiltonian Systems on Trivialized Cotangent Bundle 247 18.2 Lie Groups 255 18.3 Hamiltonian Systems on Lie Groups 260 19 Examples of Optimal Control Problems on Compact Lie Groups 265 19.1 Riemannian Problem 265 19.2 A Sub-Riemannian Problem 267 19.3 Control of Quantum Systems 271 19.4 A Time-Optimal Problem on $SO(3)$ 284 20 Second Order Optimality Conditions 293 20.1 Hessian 293 20.2 Local Openness of Mappings 297 20.3 Differentiation of the Endpoint Mapping 304 20.4 Necessary Optimality Conditions 309 20.5 Applications 318 20.6 Single-Input Case 321 21 Jacobi Equation 333 21.1 Regular Case: Derivation of Jacobi Equation 334 21.2 Singular Case: Derivation of Jacobi Equation 338 21.3 Necessary Optimality Conditions 342 21.4 Regular Case: Transformation of Jacobi Equation 343 21.5 Sufficient Optimality Conditions 346 22 Reduction 355 22.1 Reduction 355 22.2 Rigid Body Control 358 22.3 Angular Velocity Control 359 23 Curvature 363 23.1 Curvature of 2-Dimensional Systems 363 23.2 Curvature of 3-Dimensional Control-Affine Systems 373 24 Rolling Bodies 377 24.1 Geometric Model 377 24.2 Two-Dimensional Riemannian Geometry 379 24.3 Admissible Velocities 383 24.4 Controllability 384 24.5 Length Minimization Problem 387 A Appendix 393 A.1 Homomorphisms and Operators in $C^{\infty }(M)$ 393 A.2 Remainder Term of the Chronological Exponential 395 References 399 List of Figures 407 Index 409