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Geometric Control Theory and Sub-Riemannian Geometry
اطلاعات کتابشناختی
Geometric Control Theory and Sub-Riemannian Geometry
Author :
Publisher :
Springer International Publishing,
Pub. Year :
2014
Subjects :
Geometry, Riemannian. Control theory. Mathematics.
Call Number :
QA 649 .G4686 2014
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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)
