Abstract. This article deals with the local sub-Riemannian g e ometry on R 3 (D g) w h e r e D is the distribution ker !, ! being the Martinet one-form: dz ; 1 2 y 2 dx and g is a Riemannian metric on D: We p r o ve that we can take g as a sum of squares adx 2 + cdy 2 : Then we analyze the at case where a = c = 1 : We parametrize the set of geodesics using elliptic integrals. This allows to compute the exponential mapping, the wave front, the conjugate and cut loci, and the sub-Riemannian sphere. A direct consequence of our computations is to show that the sphere is not sub-analytic. Some of these computations are generalized to a one parameter deformation of the at case.

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The objective of this article is to complete preliminary results from [4,13] concerning the time-minimal control of dissipative two-level quantum systems whose dynamics is governed by Lindblad equations. The extremal system is described by a 3D-Hamiltonian depending upon three parameters. We combine geometric techniques with numerical simulations to deduce the optimal solutions.

We present a study on the discoverability of temporarily captured orbiters (TCOs) by present day or near-term anticipated ground-based and space-based facilities. TCOs (Granvik et al. 2012) are potential targets for spacecraft rendezvous or human exploration (Chyba et al. 2014) and provide an opportunity to study the population of the smallest asteroids in the solar system. We find that present day ground-based optical surveys such as Pan-STARRS and ATLAS can discover the largest TCOs over years of operation. A targeted survey conducted with the Subaru telescope can discover TCOs in the 0.5 m to 1.0 m diameter size range in about 5 nights of observing. Furthermore, we discuss the application of space-based infrared surveys, such as NEOWISE, and ground-based meteor detection systems such as CAMS, CAMO and ASGARD in discovering TCOs.These systems can detect TCOs but at a uninteresting rate. Finally, we discuss the application of bi-static radar at Arecibo and Green Bank to discover TCOs.Our radar simulations are strongly dependent on the rotation rate distribution of the smallest asteroids but with an optimistic distribution we find that these systems have > 80% chance of detecting a > 10 cm diameter TCO in about 40 h of operation.

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SUMMARYFor general optimal control problems, Pontryagin's maximum principle gives necessary optimality conditions which are in the form of a Hamiltonian differential equation. For its numerical integration, symplectic methods are a natural choice. This article investigates to which extent the excellent performance of symplectic integrators for long-time integrations in astronomy and molecular dynamics carries over to problems in optimal control.Numerical experiments supported by a backward error analysis show that, for problems in low dimension close to a critical value of the Hamiltonian, symplectic integrators have a clear advantage. This is illustrated using the Martinet case in sub-Riemannian geometry. For problems like the orbital transfer of a spacecraft or the control of a submerged rigid body such an advantage cannot be observed. The Hamiltonian system is a boundary value problem and the time interval is in general not large enough so that symplectic integrators could benefit from their structure preservation of the flow.

Twelve years ago the Catalina Sky Survey discovered Earth's first known natural geocentric object other than the Moon, a few-meter diameter asteroid designated 2006 RH 120 . Despite significant improvements in ground-based telescope and detector technology in the past decade the asteroid surveys have not discovered another temporarily-captured orbiter (TCO; colloquially known as minimoons) but the all-sky fireball system operated in the Czech Republic as part of the European Fireball Network detected a bright natural meteor that was almost certainly in a geocentric orbit before it struck Earth's atmosphere. Within a few years the Large Synoptic Survey Telescope (LSST) will either begin to regularly detect TCOs or force a re-analysis of the creation and dynamical evolution of small asteroids in the inner solar system. The first studies of the provenance, properties, and dynamics of Earth's minimoons suggested that there should be a steady state population with about one 1-to 2-m diameter captured objects at any time, with the number of captured meteoroids increasing exponentially for smaller sizes. That model was then improved and extended to include the population of temporarily-captured flybys (TCFs), objects that fail to make an entire revolution around Earth while energetically bound to the Earth-Moon system. Several different techniques for discovering TCOs have been considered but their small diameters, proximity, and rapid motion make them challenging targets for existing ground-based optical, meteor, and radar surveys. However, the LSST's tremendous light gathering power and short exposure times could allow it to detect and discover many minimoons. We expect that if the TCO population is confirmed, and new objects are frequently discovered, they can provide new opportunities for (1) studying the dynamics of the Earth-Moon system, (2) testing models of the production and dynamical evolution of small asteroids from the asteroid belt, (3) rapid and frequent low delta-v missions to multiple minimoons, and (4) evaluating in-situ resource utilization techniques on asteroidal material. Here we review the past decade of minimoon studies in preparation for capitalizing on the scientific and commercial opportunities of TCOs in the first decade of LSST operations.