Control theory from the geometric viewpoint

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