Fixed-term researcher on the GEO-ellipse distance in low-thrust rendezvous – M/F

Job Description

Missions

The aim of this project is to study, using the tools of real algebraic geometry and polynomial optimisation, the geometric safety of rendezvous manoeuvres in geostationary orbit (GEO) performed by low-thrust electric propulsion.
When thrust is cut off, the satellite follows a ballistic Keplerian ellipse one of whose foci is the centre of the Earth. Safe rendezvous imposes two opposing constraints on this final ellipse: it must stay clear of a non-collision tube around the geostationary ring (minimum distance greater than or equal to a given tolerance), while remaining close enough to that ring for the remaining propulsion budget to close the gap (maximum distance less than or equal to a second tolerance).
For a given final orbit, this problem is well understood: the minimum distance is the classical "minimum orbital intersection distance" (MOID), and the maximum distance is obtained at no additional cost from the same critical-point computation.
The open research question at the heart of this project is the parametric version: to characterise, once and for all, the set of final orbits satisfying the safety bounds, viewed as a region of the five-dimensional space of orbital elements. This "safe set" is a semi-algebraic object whose boundary is formed by explicit polynomial walls. The task is to formulate, compute and certify these walls, which identify the directions in which safety is broken — structural information that no numerical MOID code can provide.

Activity

The research activities are organised around the following lines:
– Formulate the safety condition as a quantifier-elimination problem in the theory of real closed fields, posed over the space of orbital elements.
– Exploit the symmetries of the problem: reduction of the geostationary variable by radial projection, restriction to slices with fixed orbital plane, then coplanar specialisation leading to an explicit two-dimensional picture.
– Establish and manipulate the polynomial critical-point system (Karush-Kuhn-Tucker conditions, bi-normal characterisation) associated with the distance function between the ballistic ellipse and the geostationary circle.
– Compute the boundary of the safe set as a discriminant locus: determine the polynomial equations of the walls (tolerance-tangency walls and bi-normal-coalescence walls).
– Implement and adapt the tools of certified real solving and effective algebraic geometry: Gröbner bases (F4/F5 algorithms, FGLM change of ordering, rational univariate representation), the critical-point method, discriminant varieties, as well as the msolve and RAGlib libraries.
– Certify the results (sum-of-squares positivity certificates, certified isolation of real roots) and, where appropriate, carry out numerical validation by homotopy continuation.
– Write up the results as articles and present them at specialised conferences and seminars.

Your Profil

Skills

The candidate will have a solid background in applied mathematics, ideally with several of the following skills:
– Real algebraic geometry, effective algebraic geometry and computer algebra (Gröbner bases, polynomial system solving, quantifier elimination).
– Polynomial optimisation and semidefinite programming, moment hierarchies and sums of squares.
– Command of at least one scientific and symbolic computing environment (Maple, Julia, Python, SageMath) and a taste for careful implementation; experience with the msolve, RAGlib, FGb libraries or equivalents would be appreciated.
– Knowledge of celestial mechanics, astrodynamics or optimal control would be an asset, without being a prerequisite.
– Mathematical rigour, autonomy, and the ability to carry a research project through to the writing-up of its results.
– Good command of scientific English, written and spoken; knowledge of French is not required.

Your Work Environment

The work will take place at LAAS-CNRS in Toulouse, within a team specialising in methods and algorithms for optimisation and applied algebraic geometry. The topic lies at the interface of two mathematically mature but historically distinct communities: celestial mechanics and astrodynamics on the one hand, effective real algebraic geometry and computer algebra on the other.
The project draws its applied motivation from the space sector — making rendezvous manoeuvres in geostationary orbit by electric propulsion safe — and is set within an international, collaborative research environment, in connection with leading teams in certified polynomial solving and optimisation.
The recruited person will have access to the laboratory's computing resources, will take part in the seminars of the team and of the wider community, and will be encouraged to present their work at the field's international conferences. Toulouse, the European capital of aeronautics and space, offers a particularly favourable scientific environment and quality of life.

Constraints and risks

-

Compensation and benefits

Compensation

Gross monthly salary ranging from €3,041.58 to €3,467.33.

Annual leave and RTT

44 jours

Remote Working practice and compensation

Pratique et indemnisation du TT

Transport

Prise en charge à 75% du coût et forfait mobilité durable jusqu’à 300€

About the offer

About the CNRS

The CNRS is a major player in fundamental research on a global scale. The CNRS is the only French organization active in all scientific fields. Its unique position as a multi-specialist allows it to bring together different disciplines to address the most important challenges of the contemporary world, in connection with the actors of change.

CNRS

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