Control Theory from the Geometric Viewpoint

Agrachev, Andrei; Sachkov, Yu. L.

doi:10.1007/978-3-662-06404-7

Cited by 1,055 publications

(1,719 citation statements)

References 0 publications

Supporting

Mentioning

1,702

Contrasting

Unclassified

Order By: Relevance

“…The distribution ∆ is said to be nonholonomic (also called totally nonholonomic e.g. in [4]) if, for every x ∈ M , there is a m-tuple (f x 1 , . .…”

Section: Sub-riemannian Manifoldsmentioning

confidence: 99%

“…Since the distribution is nonholonomic on M , according to the ChowRashevsky Theorem (see [9,17,30,32,33]) the sub-Riemannian distance is finite and continuous 3 on M × M . Moreover, if the manifold M is a complete metric space 4 for the sub-Riemannian distance d SR , then, since M is connected, for every pair x, y of points of M there exists a horizontal path γ joining x to y such that d SR (x, y) = length g (γ).…”

Section: Sub-riemannian Manifoldsmentioning

confidence: 99%

See 1 more Smart Citation

Mass Transportation on Sub-Riemannian Manifolds

Figalli

Rifford

2010

Geom. Funct. Anal.

View full text Add to dashboard Cite

We study the optimal transport problem in sub-Riemannian manifolds where the cost function is given by the square of the sub-Riemannian distance. Under appropriate assumptions, we generalize Brenier-McCann's Theorem proving existence and uniqueness of the optimal transport map. We show the absolute continuity property of Wassertein geodesics, and we address the regularity issue of the optimal map. In particular, we are able to show its approximate differentiability a.e. in the Heisenberg group (and under some weak assumptions on the measures the differentiability a.e.), which allows to write a weak form of the Monge-Ampère equation.

show abstract

“…The distribution ∆ is said to be nonholonomic (also called totally nonholonomic e.g. in [4]) if, for every x ∈ M , there is a m-tuple (f x 1 , . .…”

Section: Sub-riemannian Manifoldsmentioning

confidence: 99%

Section: Sub-riemannian Manifoldsmentioning

confidence: 99%

Mass Transportation on Sub-Riemannian Manifolds

Figalli

Rifford

2010

Geom. Funct. Anal.

View full text Add to dashboard Cite

show abstract

“…Let W(t) ∈ so(3) be the variation vector field associated with a curve R(t) on SO(3) [2,28]. The vector field W(t) satisfies…”

Section: Continuous-time Resultsmentioning

confidence: 99%

Geometric structure-preserving optimal control of a rigid body

et al. 2009

View full text Add to dashboard Cite

Abstract. In this paper we study a discrete variational optimal control problem for the rigid body. The cost to be minimized is the external torque applied to move the rigid body from an initial condition to a pre-specified terminal condition. Instead of discretizing the equations of motion, we use the discrete equations obtained from the discrete Lagrange-d'Alembert principle, a process that better approximates the equations of motion. Within the discrete-time setting, these two approaches are not equivalent in general. The kinematics are discretized using a natural Lie-algebraic formulation that guarantees that the flow remains on the Lie group SO(3) and its algebra so(3). We use Lagrange's method for constrained problems in the calculus of variations to derive the discrete-time necessary conditions. We give a numerical example for a three-dimensional rigid body maneuver.

show abstract

“…Thus it is necessary to know the cost function in order to prove that there exists only one kind of lift. In the literature of optimal control, the proof of the Maximum Principle has been discussed taking into account varying hypotheses, [2,8,18,44,70,71,72]. Most authors believe and justify that the origin of this Principle is the calculus of variations, see [76] for instance.…”

Section: In Item (3a)mentioning

confidence: 99%

“…The use and the comprehension of this Principle does not always gather together. The understanding of this Maximum Principle never finishes as shows the continuous wide number of references in this topic [2,8,16,18,19,24,44,51,52,53,64,70,71,72] and references therein. We try to contribute to this process through a differential geometric approach.…”

Section: Introductionmentioning

confidence: 99%

Geometric Approach to Pontryagin’s Maximum Principle

Barbero-Lin͂án¹,

Muñoz-Lecanda²

2008

Acta Appl Math

View full text Add to dashboard Cite

Since the second half of the 20th century, Pontryagin's Maximum Principle has been widely discussed and used as a method to solve optimal control problems in medicine, robotics, finance, engineering, astronomy. Here, we focus on the proof and on the understanding of this Principle, using as much geometric ideas and geometric tools as possible. This approach provides a better and clearer understanding of the Principle and, in particular, of the role of the abnormal extremals. These extremals are interesting because they do not depend on the cost function, but only on the control system. Moreover, they were discarded as solutions until the nineties, when examples of strict abnormal optimal curves were found. In order to give a detailed exposition of the proof, the paper is mostly self-contained, which forces us to consider different areas in mathematics such as algebra, analysis, geometry.

show abstract

Control Theory from the Geometric Viewpoint

Cited by 1,055 publications

References 0 publications

Mass Transportation on Sub-Riemannian Manifolds

Mass Transportation on Sub-Riemannian Manifolds

Geometric structure-preserving optimal control of a rigid body

Geometric Approach to Pontryagin’s Maximum Principle

Contact Info

Product

Resources

About