• Cover (1)
• Title Page (3)
• Copyright Page (4)
• Preface (6)
• Table of Contents (9)
• Some open problems (11)
  • 1 Singularities of time-optimal trajectories (11)
  • 2 Cutting the corners in sub-Riemannian spaces (13)
  • 3 “Morse–Sard theorem” for the endpoint maps (13)
  • 4 Unfolding the sub-Riemannian distance (14)
  • 5 Symmetries of vector distributions (15)
  • 6 Closed curves with a nondegenerate Frenet frame (16)
  • 7 Controllability of the Navier–Stokes equations controlled by a localized degenerate forcing (17)
  • 8 Diffusion along the Reeb field (18)
  • References (22)
• Geometry of Maslov cycles (24)
  • 1 Introduction (24)
  • 2 Lagrangian Grassmannian and universal Maslov cycles (26)
    • 2.1 The Lagrangian Grassmannian (26)
    • 2.2 Topology of Lagrangian Grassmannians (28)
    • 2.3 The universal Maslov cycle (28)
  • 3 Linear systems of quadrics (31)
    • 3.1 Local geometry and induced Maslov cylces (31)
    • 3.2 Linear systems of quadrics (33)
  • 4 Geometry of Gauss maps (35)
    • 4.1 Lagrange submanifolds of R2n (35)
    • 4.2 Lagrangian maps (36)
  • 5 Lagrange multipliers (36)
    • 5.1 Morse functions (38)
    • 5.2 Riemannian and sub-Riemannian geometry (40)
  • References (43)
• How to Run a Centipede: a Topological Perspective (45)
  • 1 Introduction (45)
    • 1.1 Setup (47)
    • 1.2 Conventions (47)
    • 1.3 Outline (48)
  • 2 Topology of AI and FbI (48)
    • 2.1 Topology of forbidden set (48)
    • 2.2 Feedback stabilization (49)
      • 2.2.1 Attractors (49)
    • 2.3 Vector fields and their basins (49)
  • 3 Universal cut (50)
    • 3.1 Main construction (50)
    • 3.2 Forbidden leg positions (52)
  • 4 Feedback stabilization on y (53)
    • 4.1 Rearranging the legs (53)
    • 4.2 Asymptotic stability (54)
  • 5 Further remarks and speculations (54)
  • Appendix (55)
  • A Discrete autonomous control (55)
    • A.1 Entrance-Base-Exit Flows (55)
    • A.2 Birational mappings (55)
  • References (59)
• Geometric and numerical techniques to compute conjugate and cut loci on Riemannian surfaces (60)
  • 1 Introduction (60)
  • 2 Riemannian metrics on surfaces of revolution (61)
    • 2.1 Generalities (61)
    • 2.2 Ellipsoids of revolution (65)
      • Oblate Case (66)
      • Prolate Case (66)
      • Conclusion (66)
  • 3 General Ellipsoids (66)
    • 3.1 Geometric Properties [7] (67)
    • 3.2 Geodesic Flow [7] (69)
    • 3.3 Results on the Conjugate and Cut Loci (70)
      • Numerical Computation of Conjugate and Cut Loci (71)
  • 4 Dynamics of spin particles (74)
    • Numerical Computation of Conjugate and Cut Loci (75)
  • References (79)
• On the injectivity and nonfocal domains of the ellipsoid of revolution (80)
  • Introduction (80)
  • 1 Preliminaries (81)
  • 2 Oblate case (84)
  • 3 Prolate case (87)
  • References (92)
• Null controllability in large time for the parabolic Grushin operator with singular potential (93)
  • 1 Introduction (93)
  • 2 Well-posedness and Fourier decomposition (95)
    • 2.1 Well-posedness of the Cauchy-problem (95)
    • 2.2 Fourier decomposition of the solution (97)
  • 3 Spectral analysis for the 1-D problem (99)
  • 4 A global Carleman inequality (101)
  • 5 Uniform observability (105)
  • 6 Open problems and perspectives (107)
  • References (107)
• The rolling problem: overview and challenges (109)
  • 1 Introduction (109)
  • 2 The early years: Mechanics and the new differential geometry (110)
    • 2.1 Chaplygin’s ball (111)
    • 2.3 Cartan’s development (113)
    • 2.2 Cartan’s “five variables” paper (112)
  • 3 A “forgotten” breakthrough (114)
  • 4 Revival: The two dimensional case and robotics (115)
    • 4.1 Rigidity of integral curves in Cartan’s distribution (115)
    • 4.2 Non-holonomy in robotics (117)
    • 4.3 Orbits and complete answer for controllability (118)
  • 5 Re-discovery of the higher dimensional case and interpolation (119)
    • 5.1 Sharpe’s definition (119)
    • 5.2 Applications to geometric interpolation (120)
  • 6 Nowadays: The coordinate-free approach (121)
    • 6.1 The controllability problem (122)
    • 6.2 Symmetries of the rolling problem (124)
    • 6.3 Generalizations and perspectives (126)
  • References (126)
• Optimal stationary exploitation of size-structured population with intra-specific competition (129)
  • 1 Introduction (129)
  • 2 Main results (131)
    • 2.1 Existence of a stationary solution (131)
    • 2.2 Optimal stationary solution (131)
    • 2.3 Necessary optimality condition (132)
  • 3 Proof of the theorems (133)
    • 3.1 Proof of Theorem 1 (133)
    • 3.2 Proof of Theorem 2 (135)
    • 3.3 Proof of Theorem 3 (136)
  • References (137)
• On geometry of affine control systems with one input (139)
  • 1 Introduction (139)
  • 2 Abnormal extremals of rank 2 distributions (143)
  • 3 Jacobi curves of abnormal extremals (144)
  • 4 Reduction to geometry of curves in projective spaces (146)
  • 5 Canonical projective structure on curves in projective spaces (147)
  • 6 Canonical frames for rank 2 distributions of maximal class (149)
  • 7 Canonical frames for rank 2 distributions of maximal class with distinguish parametrization on abnormal extremals (150)
  • 8 Symplectic curvatures for the structures under consideration (153)
  • 9 The maximally symmetric models (154)
  • References (157)
• Remarks on Lipschitz domains in Carnot groups (159)
  • 1 Introduction (159)
  • 2 Graphs and Lipschitz graphs (163)
  • 3 Intrinsic Lipschitz domains (165)
  • References (170)
• Differential-geometric and invariance properties of the equations of Maximum Principle (MP) (173)
  • 1 Introduction (173)
  • 2 MP for the time-optimal problem (174)
  • 3 The Pontryagin derivative PX (175)
  • 4 The Hamiltonian lift Vect M →h(T*M) U Vect T*M (176)
  • 5 Invariant representation of the sequence (4) (178)
  • 6 Identification of the Pontryagin derivatrive PX (179)
  • 7 Formulation of the final result (181)
• Curvature-dimension inequalities and Li-Yau inequalities in sub-Riemannian spaces (182)
  • 1 Introduction (182)
  • 2 From Riemannian to sub-Riemannian geometry (184)
  • 3 The curvature-dimension inequality CD(ρ, n) and the Ricci tensor (185)
  • 4 Li-Yau type estimates (188)
  • 5 The parabolic Harnack inequality for Ricci ≥ 0 (190)
  • 6 Off-diagonal Gaussian upper bounds for Ricci ≥ 0 (191)
  • 7 A sub-Riemannian Bonnet-Myers theorem (192)
  • 8 Global volume doubling when Ricci ≥ 0 (192)
  • 9 Sharp Gaussian bounds, Poincaré inequality and parabolic Harnack inequality (195)
  • 10 Negatively curved manifolds (197)
  • 11 Geometric examples (198)
    • 11.1 Riemannian manifolds (198)
    • 11.2 The three-dimensional Sasakian models (199)
    • 11.3 Sub-Riemannian manifolds with transverse symmetries (201)
    • 11.4 Carnot groups of step two (202)
    • 11.5 CR Sasakian manifolds (203)
  • References (203)
• Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds (205)
  • 1 Introduction (205)
  • 2 Basic notations (208)
    • 2.1 Hausdorff measures (208)
    • 2.2 Sub-Riemannian manifolds (209)
  • 3 Hausdorff dimensions and volumes of strongly equiregular submanifolds (210)
    • 3.1 Strongly equiregular submanifolds (210)
    • 3.2 Hausdorff volume (213)
  • 4 Hausdorff dimensions and volumes of analytic sub-Riemannian manifolds (216)
    • 4.1 Hausdorff dimension (216)
    • 4.2 Finiteness of the Hausdorff volume of balls (217)
    • 4.3 Examples (220)
  • References (221)
• The Delauney-Dubins Problem (223)
  • 1 Introduction (223)
  • 2 The Hybrid Maximum Principle and the Extremal curves (227)
  • 3 The Euclidean case (229)
    • 3.1 Extremals for the Euler-Griffiths problem (229)
    • 3.2 Extremals for the Delauney-Dubins problem (231)
    • 3.3 Integrals of motion and integrability (231)
  • 4 Non-Euclidean cases (237)
    • 4.1 Integrability (239)
  • References (243)
• On Local Approximation Theorem on Equiregular Carnot–Carathéodory Spaces (244)
  • 1 Introduction (244)
  • 2 Basic Definitions and Results (245)
  • 3 MainResults (255)
  • References (263)
• On curvature-type invariants for natural mechanical systems on sub-Riemannian structures associated with a principle G-bundle (266)
  • 1 Introduction (266)
    • 1.1 The extremals of the NMSR optimal control problems (267)
    • 1.2 Jacobi curves along normal extremals (267)
    • 1.3 Statement of the problem (268)
  • 2 Differential geometry of curves in Lagrange Grassmannian (269)
    • 2.1 Young diagrams (269)
    • 2.2 Normal moving frames (269)
    • 2.3 Consequences for NMSR optimal control problems (272)
  • 3 Algorithm for calculation of canonical splitting and (a, b)-curvature maps (274)
    • 3.1 Algorithm of normalization (275)
    • 3.2 Preliminary implementation of the algorithm (276)
  • 4 Calculus and the canonical splitting (280)
    • 4.1 Some useful formulas (280)
    • 4.2 Calculations of the canonical splitting (281)
  • 5 Curvature maps via the Riemannian curvature tensor and the tensor J on M (284)
  • References (288)
• On the Alexandrov Topology of sub-Lorentzian Manifolds (289)
  • 1 Introduction (289)
  • 2 Basic Concepts (291)
    • 2.1 Lorentzian Geometry (291)
    • 2.2 Sub-Lorentzian Manifolds (293)
  • 3 Reachable Sets, Causality and the Alexandrov Topology (296)
    • 3.1 Reachable Sets (296)
    • 3.2 The Alexandrov Topology (299)
    • 3.3 Links to Causality (300)
    • 3.4 The Alexandrov and Manifold Topology in sub-Lorentzian Geometry (304)
    • 3.5 The Open Causal Relations (307)
    • 3.6 Chronologically Open sub-Space-Times (308)
  • 4 The Time-Separation Topology (310)
  • References (312)
• The regularity problem for sub-Riemannian geodesics (314)
  • 1 Introduction (314)
  • 2 Basic facts (315)
  • 3 Known regularity results (318)
  • 4 Analysis of corner type singularities (320)
  • 5 Classification of abnormal extremals (322)
    • 5.1 Rank 2 distributions (322)
    • 5.2 Stratified nilpotent Lie groups (322)
  • 6 Some examples (324)
    • 6.1 Purely Lipschitz Goh extremals (324)
    • 6.2 A family of abnormal curves (325)
  • 7 An extremal curve with Hölder continuous first derivative (326)
  • 8 Final comments (332)
  • References (333)
• A case study in strong optimality and structural stability of bang–singular extremals (334)
  • 1 Introduction (334)
    • 1.1 Notation (337)
  • 2 Assumptions on the nominal problem (338)
    • 2.1 Pontryagin Maximum Principle and Regularity Assumptions (338)
    • 2.2 The extended second variation (340)
    • 2.3 Consequences of coercivity and controllability (341)
  • 3 Optimality in the nominal problem (342)
    • 3.1 Geometry near the singular arc (342)
    • 3.2 State–local optimality (344)
  • 4 Structural stability (347)
  • References (351)
• Approximate controllability of the viscous Burgers equation on the real line (352)
  • 1 Introduction (352)
  • 2 Main result and scheme of its proof (354)
  • 3 Cauchy problem (357)
    • 3.1 Existence, uniqueness, and regularity of a solution (357)
    • 3.2 Uniform continuity of the resolving operator in local norms (363)
  • 4 Proof of Theorem 2 (364)
    • 4.1 Extension: proof of Proposition 1 (364)
    • 4.2 Convexification: proof of Proposition 2 (364)
    • 4.3 Saturation (368)
    • 4.4 Large control space (368)
    • 4.5 Reduction to the case s = 0 (370)
  • References (371)
• Homogeneous affine line fields and affine lines in Lie algebras (372)
  • 1 Introduction (372)
    • 1.1 Local homogeneous subsets of the tangent bundle (372)
    • 1.2 Symmetry algebra sym (∑) (373)
    • 1.3 Construction of a local homogeneous subset of T Rn from an endowed n-dimensional Lie algebra (373)
    • 1.4 A general question on local homogeneous subsets of T Rn (373)
    • 1.5 Local homogeneous affine line fields. Main theorems (374)
    • 1.6 Plan of the paper (375)
  • 2 Tools (375)
    • 2.1 Splitting property of transitive Lie algebras (375)
    • 2.2 Proof of Theorem 1 for n = 2 (376)
    • 2.3 Classification of local homogeneous subsets of T Rn versus classification of endowed Lie algebras (377)
    • 2.4 Nagano principle (377)
    • 2.5 Finite dimensional transitive Lie algebras of vector fields (378)
  • 3 Proof of Theorem 1 (378)
    • 3.1 Proof of Lemma 1 (380)
    • 3.2 Proof of Lemma 3 (381)
  • 4 Complete classification of homogeneous affine line fields in T R3 (381)
  • 5 Classification of homogeneous bracket generating affine line fields in T R4 and proof of Theorem 2 (383)
  • References (385)