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Journal of Geometric Mechanics

2013 , Volume 5 , Issue 1

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Continuous and discrete embedded optimal control problems and their application to the analysis of Clebsch optimal control problems and mechanical systems
Anthony M. Bloch, Peter E. Crouch and Nikolaj Nordkvist
2013, 5(1): 1-38 doi: 10.3934/jgm.2013.5.1 +[Abstract](223) +[PDF](681.4KB)
Abstract:
In this paper we define ``embedded optimal control problems" which prescribe parametrized families of well defined associated optimal control problems. We show that the extremal generating Hamiltonian equations for an embedded optimal control problem and any associated optimal control problem are simply related by a projection. Furthermore normal extremals project to normal extremals and similarly for abnormal extremals. An interesting class of embedded optimal control problems consists of Clebsch optimal control problems. We provide necessary conditions for a Clebsch optimal control problem to describe a variational problem and thereby a mechanical system. There may be many advantages to analyzing an embedded optimal control problem instead of a particular associated optimal control problem, for example the former being defined on a linear space and the latter on a nonlinear space. The continuous analysis is paralleled by a similar discrete analysis. We define a discrete embedded/Clebsch optimal control problem along with associated discrete optimal control problems and we show results that are analogous to the continuous results. We apply the theory, both in the continuous and the discrete setting, to two example systems: mechanical systems on matrix Lie groups and mechanical systems on $n$-spheres.
Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups
François Gay-Balmaz, Cesare Tronci and Cornelia Vizman
2013, 5(1): 39-84 doi: 10.3934/jgm.2013.5.39 +[Abstract](261) +[PDF](716.4KB)
Abstract:
We formulate Euler-Poincaré equations on the Lie group $Aut(P)$ of automorphisms of a principal bundle $P$. The corresponding flows are referred to as EP$Aut$ flows. We mainly focus on geodesic flows associated to Lagrangians of Kaluza-Klein type. In the special case of a trivial bundle $P$, we identify geodesics on certain infinite-dimensional semidirect-product Lie groups that emerge naturally from the construction. This approach leads naturally to a dual pair structure containing $\delta\text{-like}$ momentum map solutions that extend previous results on geodesic flows on the diffeomorphism group (EPDiff). In the second part, we consider incompressible flows on the Lie group $Aut_{vol}(P)$ of volume-preserving bundle automorphisms. In this context, the dual pair construction requires the definition of chromomorphism groups, i.e. suitable Lie group extensions generalizing the quantomorphism group.
Vector fields with distributions and invariants of ODEs
BronisŁaw Jakubczyk and Wojciech Kryński
2013, 5(1): 85-129 doi: 10.3934/jgm.2013.5.85 +[Abstract](262) +[PDF](655.6KB)
Abstract:
We study dynamic pairs $(X,V)$ where $X$ is a vector field on a smooth manifold $M$ and $V\subset TM$ is a vector distribution, both satisfying certain regularity conditions. We construct basic invariants of such objects and solve the equivalence problem. In particular, we assign to $(X,V)$ a canonical connection and a canonical frame on a certain frame bundle. We compute the curvature and torsion. The results are applied to the problem of time scale preserving equivalence of ordinary differential equations and of Veronese webs. The framework of dynamic pairs $(X,V)$ is shown to include sprays, control-affine systems, mechanical control systems, Veronese webs and other structures.
Computing metamorphoses between discrete measures
Casey L. Richardson and Laurent Younes
2013, 5(1): 131-150 doi: 10.3934/jgm.2013.5.131 +[Abstract](321) +[PDF](1329.6KB)
Abstract:
Metamorphosis is a mathematical framework for diffeomorphic pattern matching in which one defines a distance on a space of images or shapes. In the case of image matching, this distance involves computing the energetically optimal way in which one image can be morphed into the other, combining both smooth deformations and changes in the image intensity. In [12], Holm, Trouvé and Younes studied the metamorphosis of more singular deformable objects, in particular measures. In this paper, we present results on the analysis and computation of discrete measure metamorphosis, building upon the work in [12]. We show that, when matching sums of Dirac measures, minimizing evolutions can include other singular distributions, which complicates the numerical approximation of such solutions. We then present an Eulerian numerical scheme that accounts for these distributions, as well as some numerical experiments using this scheme.

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