Publication

Optimal control of geodesics in Riemannian manifolds

Klaus-Dieter Semmler
2004
Journal Articles

Abstract

Summary: We present a method based on an optimal control technique for numerical computations of geodesic paths between two fixed points of a Riemannian manifold under the assumption of existence. In this method, the control variable is the tangent vector to the geodesic we are looking for. Defining a cost function corresponding to the requested control, we explain how to derive the optimal control algorithm by the use of an adjoint state method for the calculation of the gradient of that cost function. We then give a geometrical interpretation of the adjoint state. After having introduced the discrete optimal control algorithm, we show an application to wooden roof design.

Official source

https://infoscience.epfl.ch/entities/publication/4bd3df7d-cc98-47db-84ba-e81379c6da51

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Klaus-Dieter Semmler
2004
Journal Articles

Abstract

Official source

https://infoscience.epfl.ch/entities/publication/4bd3df7d-cc98-47db-84ba-e81379c6da51

About this result

Related concepts (32)

Riemannian manifold

In differential geometry, a Riemannian manifold or Riemannian space (M, g), so called after the German mathematician Bernhard Riemann, is a real, smooth manifold M equipped with a positive-definite inner product gp on the tangent space TpM at each point p. The family gp of inner products is called a Riemannian metric (or Riemannian metric tensor). Riemannian geometry is the study of Riemannian manifolds. A common convention is to take g to be smooth, which means that for any smooth coordinate chart (U, x) on M, the n2 functions are smooth functions.

Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike.

Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point. Riemann introduced an abstract and rigorous way to define curvature for these manifolds, now known as the Riemann curvature tensor. Similar notions have found applications everywhere in differential geometry of surfaces and other objects. The curvature of a pseudo-Riemannian manifold can be expressed in the same way with only slight modifications.

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Optimal control of geodesics in Riemannian manifolds

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