L1L^1-Minimization for Mechanical Systems

Second order systems whose drift is defined by the gradient of a given potential are considered, and minimization of the $L^1$-norm of the control is addressed. An analysis of the extremal flow emphasizes the role of singular trajectories of order two [25,29]; the case of the two-body potential is treated in detail. In $L^1$-minimization, regular extremals are associated with controls whose norm is bang-bang; in order to assess their optimality properties, sufficient conditions are given for broken extremals and related to the no-fold conditions of [20]. An example of numerical verification of these conditions is proposed on a problem coming from space mechanics

Conjugate and cut loci in averaged orbital transfer

The objective of this Note is to describe the conjugate and cut loci associated with the averaged energy minimization problem in coplanar orbit transfer

Multi-phase averaging of time-optimal low-thrust transfers

International audienceAn increasing interest in optimal low-thrust orbital transfers was triggered in the last decade by technological progress in electric propulsion and by the ambition of efficiently leveraging on orbital perturbations to enhance the maneuverability of small satellites. The assessment of a control sequence that is capable of steering a satellite from a prescribed initial to a desired final state while minimizing a figure of interest is referred to as maneuver planning. From the dynamical point of view, the necessary conditions for optimality outlined by the infamous Pontryagin maximum principle (PMP) reveal the Hamiltonian nature of the system governing the joint motion of state and control variables. Solving the control problem via so-called indirect techniques, e.g., shooting method, requires the integration of several trajectories of the aforementioned Hamiltonian. In addition , PMP conditions exhibit very high sensitivity with respect to boundary values of the satellite longitude owing to the fast-oscillating nature of orbital motion. Hence, using perturbation theory to facilitate the numerical solution of the planning problem is appealing. In particular, averaging techniques were used since the early space age to gain understanding into the long-term evolution of perturbed satellite trajectories. However, it is not generally possible to treat low-thrust as any other perturbation (whose spectral content is well defined and predictable) because the control variables may introduce additional frequencies in the system. The talk focuses on time optimal maneuvers in a perturbed orbital environment, and it addresses two questions: (1) Is it possible to average the vector field of this problem? Optimal control Hamiltonians are not in the classical form of fast-oscillating systems. However, we demonstrate that averaged trajectories well approximate the original system if the ad-joint variables of the PMP (i.e., conjugate momenta associated to the enforcement of the equations of motion) are adequately transformed before integrating the averaged trajec-tory. We discuss this transformation in detail, and we emphasize fundamental differences with respect to well-known mean-to-osculating transformations of uncontrolled motion. (2) What is the impact of orbital perturbations and their frequencies on the controlled tra-jectory? We show that control variables are highly sensitive to small exogenous forces. Hence, even the crossing of a high-order resonance may trigger a dramatic divergence between trajectories of the averaged and original system. We then discuss how averaged resonant forms may be used to avoid this divergence. The methodology is finally applied to a deorbiting maneuver leveraging on solar radiation pressure. The presence of eclipses make the original planning problem highly challenging. Averaging with respect to satellite and Sun longitudes drastically simplifies the extremal flow yielding an averaged counterpart of the PMP conditions, which is reasonably easy to solve

A global optimality result with application to orbital transfer

The objective of this note is to present a global optimality result on Riemannian metrics ds^2=dr^2+(r^2/c^2)(G(\vphi)d\theta^2+d\vphi^2). This result can be applied to the averaged energy minimization coplanar orbit transfer problem

Geodesic flow of the averaged controlled Kepler equation

A normal form of the Riemannian metric arising when averaging the coplanar controlled Kepler equation is given. This metric is parameterized by two scalar invariants which encode its main properties. The restriction of the metric to \SS^2 is shown to be conformal to the flat metric on an oblate ellipsoid of revolution, and the associated conjugate locus is observed to be a deformation of the standard astroid. Though not complete because of a singularity in the space of ellipses, the metric has convexity properties that are expressed in terms of the aforementioned invariants, and related to surjectivity of the exponential mapping. Optimality properties of geodesics of the averaged controlled Kepler system are finally obtained thanks to the computation of the cut locus of the restriction to the sphere

Note on singular Clairaut-Liouville metrics

Computations on Clairaut-Liouville metrics on S^2 with a finite order singularity

Sufficient conditions for time optimality of systems with control on the disk

International audienceThe case of time minimization for affine control systems with control on the disk is studied. After recalling the standard sufficient conditions for local optimality in the smooth case, the analysis focusses on the specific type of singularities encountered when the control is prescribed to the disk. Using a suitable stratification, the regularity of the flow is analyzed, which helps to devise verifiable sufficient conditions in terms of left and right limits of Jacobi fields at a switching point. Under the appropriate assumptions, piecewise regularity of the field of extremals is obtained

Contribution à l'étude du contrôle en temps minimal des transferts orbitaux

Le contexte de ce travail est la mécanique spatiale. Plus précisément, on s'est intéressé, dans le cadre d'une collaboration avec le Centre National d'Etudes Spatiales, au problème du transfert orbital. Le modèle étudié est celui du contrôle en temps minimal d'un satellite que l'on souhaite insérer sur une orbite géostationnaire. Les contributions de cette thèse sont de trois ordres. Géométrique, tout d'abord, puisqu'on étudie la contrôlabilité du système ainsi que la géométrie des transferts (structure de la commande) à l'aide d'outils de contrôle géométrique. Sont ensuite présentées des méthodes de résolution spectrales et pseudo-spectrales utilisant les polynômes de Tchebycheff, puis des algorithmes basés sur un calcul adaptatif de discrétisation par ondelettes. Ces approches permettent de traiter numériquement le cas d'un satellite dont la poussée est forte à moyenne. Pour atteindre le domaine des poussées faibles, caractéristiques de la future propulsion électro-ionique, il faut finalement introduire de nouvelles techniques qui ont en commun d'être paramétriques (paramétrisation par la poussée ou par le critère). L'analyse des propriétés de ces méthodes se fait naturellement à l'aide de résultats de contrôle paramétrique. ABSTRACT : The context of this work is celestial mechanics. More precisely, in the framework of a collaboration with the French Space Agency, we have dealt with the orbit transfer problem. We study the minimum time control of a satellite that we want to reach a geostationnary orbit. The contributions of this thesis are of three kinds. Geometric, first, since we study the controllability of the system together with the geometry of the transfer (structure of the command) by means of geometric control tools. Then we present spectral and pseudo-spectral resolution methods, based on Chebyshev polynomials, as well as algorithms relying upon a wavelet-adaptive discretization. These approaches allow the numerical resolution of problems with strong or medium thrust satellites. In order to reach low thrusts, typical of the future electro-ionic propulsion, we finally need to introduce new techniques, namely parametric ones (parameterization by the thrust or by the criterion). The analysis of their properties is performed thanks to parametric control results

Formulary of geodesics of the projected averaged Kepler Hamiltonian

This short note gives the quadratures of the geodesics of the averaged Hamiltonian of the controlled Kepler equation (energy criterion) projected on \SS^2. The endpoints of the corresponding cut locus are also deduced, as well as the injectivity radius of the associated Riemannian metric on the $2$-sphere