Communications in Mathematical Sciences

Volume 11 (2013)

Number 1

Geometric dynamics of optimization

Pages: 163 – 231



François Gay-Balmaz (California Institute of Technology, Pasadena, Calif.; Laboratoire de Météorologie Dynamique, ´Ecole Normale Supérieure, Paris, France)

Darryl D. Holm (Department of Mathematics, Imperial College London, United Kingdom)

Tudor S. Ratiu (Section de Mathématiques, École Polytechnique Fédérale de Lausanne, Switzerland)


This paper investigates a family of dynamical systems arising from an evolutionary re-interpretation of certain optimal control and optimization problems. We focus particularly on the application in image registration of the theory of metamorphosis. Metamorphosis is a means of tracking the optimal changes of shape that are necessary for registration of images with various types of data structures, without requiring that the transformations of shape be diffeomorphisms, but penalizing them if they are not. The possibilities of this approach are just beginning to be developed. In particular, metamorphosis and its related variants in the geometric approach to control and optimization can be expected to produce many exciting opportunities for new applications and analysis in geometric dynamics.


optimal control, optimization, geometric mechanics, symmetry

2010 Mathematics Subject Classification

34H05, 49J20, 49J27

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