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Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite riemannian metric

Claudio Altafini (2004)

ESAIM: Control, Optimisation and Calculus of Variations

For a riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the riemannian exponential...

Reduction by group symmetry of second order variational problems on a semidirect product of Lie groups with positive definite Riemannian metric

Claudio Altafini (2010)

ESAIM: Control, Optimisation and Calculus of Variations

For a Riemannian structure on a semidirect product of Lie groups, the variational problems can be reduced using the group symmetry. Choosing the Levi-Civita connection of a positive definite metric tensor, instead of any of the canonical connections for the Lie group, simplifies the reduction of the variations but complicates the expression for the Lie algebra valued covariant derivatives. The origin of the discrepancy is in the semidirect product structure, which implies that the Riemannian exponential...

Régularité Lipschitzienne des Géodésiques Minimisantes pour Quelques Distributions Affines

Bensalem, Naceurdine (2008)

Serdica Mathematical Journal

2000 Mathematics Subject Classification: 49J15, 49J30, 53B50.In the context of sub-Riemannian geometry and the Lipschitzian regularity of minimizers in control theory, we investigate some properties of minimizing geodesics for certain affine distributions. In particular, we consider the case of a generalized H2-strong affine distribution and the case of an affine Plaff system of maximal class.

Resonance of minimizers for n-level quantum systems with an arbitrary cost

Ugo Boscain, Grégoire Charlot (2004)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem describing a laser-induced population transfer on a n -level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n = 2 and n = 3 ): instead of looking...

Resonance of minimizers for n-level quantum systems with an arbitrary cost

Ugo Boscain, Grégoire Charlot (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider an optimal control problem describing a laser-induced population transfer on a n-level quantum system. For a convex cost depending only on the moduli of controls (i.e. the lasers intensities), we prove that there always exists a minimizer in resonance. This permits to justify some strategies used in experimental physics. It is also quite important because it permits to reduce remarkably the complexity of the problem (and extend some of our previous results for n=2 and n=3): instead...

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