23 mai 2008 
INRIA  Saclay
IledeFrance 
Aerospatial dynamics and Optimal Control
 Dynamique Spatiale et Commande Optimale 
ENSTA /
UMA
32 Boulevard Victor, 75739 Paris cedex 15, FRANCE tél : (33) 1 45 52 54 01 Fax : (33) 1 45 52 54 82 see also : Venue 
This meeting is organized by
J.F. Bonnans
and
H. Zidani
(commands team).
The registration is free but mandatory. Lunch will be offered if you register before May 16.
Programme
Friday, May 23  Morning (Amphi Parmentier)
9h009h15 
Welcome 
9h1510h00 
The University of Colorado at Boulder
Optimal Feedback Control and Hamiltonian Dynamics Utilizing the Hamiltonian structure of the necessary conditions for optimal
control we present a generalization of the HamiltonJacobiBellmann
equation for the computation of optimal feedback controls. This
generalization decomposes the cost function solution of the HJB partial
differential equation into two terms, a purely dynamical term and one
involving boundary conditions only. Further, by applying classical
canonical transformation theory to the dynamical term we are able to find
solutions to variants of the HJB equation that have nonsingular terminal
conditions, allowing for improved algorithms for their computation. These
alternate solutions can be transformed into solutions for the cost function
of the HJB equation via a simple Legendre transformation. This work arises
out of our investigations into the solution of twopoint boundary value
problems in Hamiltonian dynamical systems. The general theory which we use
to solve these problems is first presented and then applied to the optimal
control problem. Finally, some novel applications of our optimal feedback
control laws are introduced and discussed.

10h0010h45 
CNES
Minimumfuel Deployment of Formation Flying Satellites  An Optimal Control Approach This talk focuses on the issue of minimumfuel deployment for satellite
formation flying. We address it as an optimal control problem, the
necessary optimality conditions of which are derived from Pontryagin's
Maximum Principle. These are numerically enforced by finding the root
of a socalled shooting function. However, optimal control laws for
minimumfuel problems are discontinuous and produce nonsmooth shooting
functions with singular Jacobian matrices. The resulting problems
cannot be solved easily and require the use of a regularization
technique. We extend here our previously developed
continuationsmoothing method to the multisatellite context by using an
adapted initialization procedure. Because realistic mission scenarios
may require it, our approach additionally offers the ability to
slightly modify a given maneuver strategy to balance the fuel
consumption among the satellites to a certain degree. A number of tests
concerning lowEarth orbits are carried out in the paper as examples.
Our model includes the J2 effect, which leads to numerical
difficulties. We show the efficiency of our method in this challenging
context: several maneuver strategies are detected and analyzed from the
space dynamics angle. We finally point out that beyond this
application, a whole class of deployment/reconfiguration problems may
be handled through this approach.

10h4511h05 
Break 
11h0511h50 
ONERA
Applications of optimal control to space transportation system design. This presentation gives an overview of different applications of optimal
control for early design studies of future space transportation systems.
The presentation addresses the context of these studies, and the specific
needs that have to be addressed by optimal control tools and methods. Two
families of applications are considered: launchers (expandable or reusable)
and reentry/aeroassisted vehicles. An overview of past and current work is
given; and the pros and cons of the optimal control methods employed are
discussed.

11h5012h35 
CNES
Activité R&T à la direction des lanceurs en optimisation de trajectoire 
12h3514h00 
Lunch 
Friday, May 23  Afternoon (Amphi Parmentier)
14h0014h45 
Univ. Bourgogne
Twobody control and applications The problem of finding minimum consumption trajectories of a spacecraft in
a 1/r^2 gravitational field is addressed. Applications arise from
collaborations with EADSAstrium Space Transportation and the French Space
Agency (CNES). As often in nonlinear problems, homotopy techniques are
ubiquitous in the methods developed. The presentation will try to
illustrate these aspects, both on the numerical (L^2L^1 homotopy) and
theoretical side (homotopy from the round metric on S^2 and averaging).
This is joint work with J. Gergaud (ENSEEIHT / Univ. Toulouse), C. Zayane
(EDF), B. Bonnard and G. Janin (Univ. Bourgogne).

14h4515h30 
INRIA Saclay IledeFrance
Optimal trajectories for space launcher problems We study optimal trajectories with singular arcs, i.e. flight phases with a
non maximal thrust, for a space launcher problem. We consider a flight to
the geostationary transfer orbit for a heavy multistage launcher (Ariane 5
class) and use a realistic physical model for the drag force and rocket
thrust. For preliminary result, we solve first the complete flight with
stage separations, at full thrust. Then we focus on the first atmospheric
climbing phase to investigate the possible existence of optimal
trajectories with singular arcs. We primarily use an indirect shooting
method based on Pontryagin's Minimum Principle, coupled with a continuation
(homotopy) approach. Additional experiments are made with a basic direct
method, which confirm the solutions obtained by the shooting. We study two
slightly different launcher models and observe that modifying parameters
such as the aerodynamic reference area and specific impulsion can indeed
lead to optimal trajectories with either full thrust or singular arcs.

15h3016h00 
Break 
16h0016h45 
Université Orsay  Paris Sud
Eight Lissajous Orbits in the EarthMoon system Euler and Lagrange proved the existence of five equilibrium points in the
circular restricted threebody problem. These equilibrium points are
known as Lagrange points (Euler points or libration points) and are
usually noted L1,...,L5. The existence of periodic orbits families
called halo orbits around those points, 3dimensional periodic orbits
isomorphic to ellipses is very well known. There exist other types of
periodic orbits around the Lagrange points. Indeed, Lyapunov orbits
(planar periodic orbits) also exist, as well as periodic orbits with
structure much more "complicated" known as Lissajous orbits. Among those
periodic orbits we have established analytically and numerically the
existence of families of Lissajous periodic orbits which are almost
vertical and have the shape of eight.
In this presentation, we prove the existence of these Eight Lissajous orbits and describe their properties. In particular, we show, using local Lyapunov exponents, that their invariant manifolds share nice global stability properties, which make them a great interest in space mission design. Finally, we show that the invariant manifolds of Eight Lissajous orbits can be used to visit almost all the Moon surface in the EarthMoon system. 
16h4517h30 
ENPC et ENSTA/OC
Robust approach for aerospatial optimal control problems The main purpose of this study is the planning of spatial rendezvous:
such problems can be formulated as optimal control problems with terminal
equality constraints. The cost function may reflect minimal terminal time
and/or minimal fuel consumption goals. However, the optimal control
derived from such a formulation is not "robust" in that the possibility
of satisfying the terminal constraints once a momentary breakdown of the
engine occurred, resulting in a deviation from the ideal trajectory, may
be very weak. We propose an alternative formulation in which the final
equality constraints must be satisfied with a certain prescribed
probability, given a stochastic model of breakdown occurrence and
duration. The final purpose is to redefine the planned trajectory in such
a way that the rendezvous can be successfully achieved despite engine
breakdowns with a certain probability. Of course, this modified
trajectory represents a certain loss of performance, as measured by the
cost function, with respect to the ideal optimal trajectory. The attempt
is thus to make this tradeoff "performance vs. safety" more explicit and
quantitative. In mathematical terms, this formulation of optimal control
problems with terminal equality constraints to be met in probability
raises several theoretical and algorithmic difficulties that will be
briefly described in the talk with the help of simple illustrative
examples. This resarch takes place under a contract with
ThalèsAleniaSpace and CNES Toulouse. Joint work of JeanPhilippe
Chancelier (CERMICSENPC), Pierre Carpentier (ENSTA/OC) and Guy Cohen
(CERMICSENPC).
