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Smooth and broken Hamiltonian curves in optimal control

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Abstract

In optimal control, minimizing trajectories are projections on the ambient manifold of Hamiltonian curves on the cotangent bundle ("extremal curves"). These curves may be smooth and we report on results on the geodesic flow of almost-Riemannian metrics on the 2-sphere. Such metrics have singularities that can be suitably dealt with in a Hamiltonian framework. We show in particular that their caustics are given in terms of a billiard in the Poincaré disk. In general though, the relevant Hamiltonian is only C0 and minimizing trajectories are projections of broken curves (Lipschitz but not C1). An important case for the control of mechanical systems is the case of two competing Hamil- tonians. Under suitable assumptions, neighbouring extremals are all broken and there is still a good notion of caustic. A more subtle situation arises for time minimization as neighbouring Hamiltonian curves of a broken extremal may be smooth or broken. In a well-chosen blow-up, the singularity of the extremal can be interpreted as a heteroclinic connection between two hyperbolic equilibria, resulting in a logarithmic singularity of the Hamiltonian flow.
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Dates and versions

hal-01956030 , version 1 (17-12-2018)

Identifiers

  • HAL Id : hal-01956030 , version 1

Cite

Jean-Baptiste Caillau. Smooth and broken Hamiltonian curves in optimal control. Recent advances in Hamiltonian dynamics and symplectic topology, 2018, Padova, France. ⟨hal-01956030⟩
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