0000000001011322
AUTHOR
Bernard Bonnard
Mécanique céleste et contrôle de systèmes spatiaux
Mécanique céleste et contrôle de systèmes spatiaux
Geometric analysis of minimum time Keplerian orbit transfers
The minimum time control of the Kepler equation is considered. The typical application is the transfer of a satellite from an orbit around the Earth to another one, both orbits being elliptic. We recall the standard model to represent the system. Its Lie algebraic structure is first analyzed, and controllability is established for two different single-input subsystems, the control being oriented by the velocity or by the orthoradial direction. In both cases, a preliminary analysis of singular and regular extremals is also given, using the usual concept of order to classify the contacts. Moreover, the singularity of the multi-input model---which is a particular case of a subriemannian system…
The transcendence needed to compute the sphere and wave front in Martinet sub-Riemannian geometry
Consider a \it{sub-Riemannian geometry} $(U,D,g)$ where $U$ is a neighborhood of $O$ in $\mathbb{R}^3$, $D$ is a \it{Martinet type distribution} identified to $Ker \,\omega$, $\omega =dz-\f{y^2}{2}dx$, $q=(x,y,z)$ and $g$ is a \it{metric on $D$} which can be taken in the normal form : \mbox{$a(q)dx^2+c(q)dy^2$}, \mbox{$a=1+yF(q)$}, \mbox{$c=1+G(q)$}, \mbox{$G_{|x=y=0}=0$}. In a previous article we analyzed the \it{flat case} : \mbox{$a=c=1$} ; we showed that the set of geodesics is integrable using \it{elliptic integrals} of the \it{first and second kind} ; moreover we described the sphere and the wave front near the abnormal direction using the \it{\mbox{exp-log} category}. The objective o…
Geometric and numerical techniques in optimal control of the two and three-body problems
The objective of this article is to present geometric and numerical techniques developed to study the orbit transfer between Keplerian elliptic orbits in the two-body problem or between quasi-Keplerian orbits in the Earth-Moon transfer when low propulsion is used. We concentrate our study on the energy minimization problem. From Pontryagin's maximum principle, the optimal solution can be found solving the shooting equation for smooth Hamiltonian dynamics. A first step in the analysis is to find in the Kepler case an analytical solution for the averaged Hamiltonian, which corresponds to a Riemannian metric. This will allow to compute the solution for the original Kepler problem, using a nume…
Working Notes on the Time Minimal Saturation of a Pair of Spins and Application in Magnetic Resonance Imaging
In this article, we analyse the time minimal control for the saturation of a pair of spins of the same species but with inhomogeneities of the applied RF-magnetic field, in relation with the contrast problem in Magnetic Resonance Imaging. We make a complete analysis based on geometric control to classify the optimal syntheses in the single spin case to pave the road to analyze the case of two spins. The Bocop software is used to determine local minimizers for physical test cases and Linear Matrix Inequalities approach is applied to estimate the global optimal value and validate the previous computations. This is complemented by numerical computations combining shooting and continuation meth…
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.
Smooth homotopies for single-input time optimal orbital transfer
International audience
Smooth approximations of single-input controlled Keplerian trajectories: homotopies and averaging
International audience
Optimal control with state constraints and the space shuttle re-entry problem
In this article, we initialize the analysis under generic assumptions of the small \textit{time optimal synthesis} for single input systems with \textit{state constraints}. We use geometric methods to evaluate \textit{the small time reachable set} and necessary optimality conditions. Our work is motivated by the \textit{optimal control of the atmospheric arc for the re-entry of a space shuttle}, where the vehicle is subject to constraints on the thermal flux and on the normal acceleration. A \textit{multiple shooting technique} is finally applied to compute the optimal longitudinal arc.
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.
Second order optimality conditions in optimal control with applications
The aim of this article is to present the algorithm to compute the first conjugate point along a smooth extremal curve. Under generic assumptions, the trajectory ceases to be optimal at such a point. An implementation of this algorithm, called \texttt{cotcot}, is available online and based on recent developments in geometric optimal control. It is applied to analyze the averaged optimal transfer of a satellite between elliptic orbits.