My research interests mainly lies in the modeling and the solving of optimal control problems. I do have a paricular focus on the development of practical numerical solving using indirect methods, mainly in aerospace.

Current research projects

Currently (October 2015), my research projects are:

Aerospace

With Emmanuel Télat, we are developping a new approach to the solving of the bi-boost continuous orbit transfer. The idea is to use an adapted shooting method for which the trick resides mainly in the way we construct the initialization. This initialization is based on a proof of convergence of the continuous transfer toward a well chosen impulse transfer. Then, using the geometric meaning of the Pontryagin Maximum Principle’s adjoint vector, we can use the impulse transfer solution to pick a good initial guess for the shooting method.

Again with Emmanuel Télat, we are advising the thesis research of Maxime Chupin on the subject of Design of optimal trajectories using the invariant manifolds spawning from the libration points of a 3 body problem.

With Monique Chyba and Geoff Patterson, we are studying the optimal control problem of a rendez-vous with a mini-moon (a temprorary captured orbiter near the Earth). The aim is to design minimum time and minimum fuel interception missions starting from a neighborhood of the L₂ libration point of the Earth-Moon 3 body problem.

Acoustic Tomography

In the ANR project AVENTURES, initiated by Maïtine Bergounioux, we are interested in the modeling and solving of the Photo and Thermo Acoustic Tomography of soft tissues. The main idea is to model this imaging problem as an optimal control problem in which the control is the acoustic property that we want to recover.

Méthods in optimal control

In collaboration with Emmanuel Télat, we have some ideas regarding some subjects in optimal control theory, namelly (to name a few):

• Regularization/penalization of optimal control problem with state constraints. The main idea is to view state constraint optimal control problem as a modified optimal control problem on stratified domain and obtain convergence results of a regularization of some kind. The main tagrte is to get convergence results similar to the ones in Convergence results for smooth regularizations of hybrid nonlinear optimal control problems.
• Using continuation techniques and the behavior of abnormal/singular/conjugated optimal control, we aim at designing a method to build for local optimal synthesis.

Past projects

Parmis mes projets passés, dont certains bénéficient toujours d’une suite, on peut lister.

Regularization of optimal control problem on stratified domain

We consider a class of hybrid nonlinear optimal control problems having a discontinuous dynamics ruled by a partition of the state space. For this class of problem, some hybrid versions of the usual Pontryagin Maximum Principle are known. We introduce general regularization procedures, parameterized by a small parameter, smoothing the previous hybrid problems to standard smooth optimal control problems, for which we can apply the usual Pontryagin Maximum Principle. We investigate the question of the convergence of the resulting extremals as the regularization parameter tends to zero. Under some general assumptions, we prove that smoothing regularization procedures converge, in the sense that the solution of the regularized problem (as well as its extremal lift) converges to the solution of the initial hybrid problem. To illustrate our convergence result, we apply our approach to the minimal time low-thrust coplanar orbit transfer with eclipse constraint. See Convergence results for smooth regularizations of hybrid nonlinear optimal control problems for more details.

Orbital transfer

In this article (Continuation from a flat to a round Earth model in the coplanar orbit transfer problem) we focus on the problem of minimization of the fuel consumption for the coplanar orbit transfer problem. This problem is usually solved numerically by a shooting method, based on the application of the Pontryagin Maximum Principle, however the shooting method is known to be hard to initialize and the convergence is difficult to obtain due to discontinuities of the optimal control. Several methods are known in order to overcome that problem, however in this article we introduce a new approach based on the following preliminary remark. When considering a 2D flat Earth model with constant gravity, the optimal control problem of passing from an initial configuration to some final configuration by minimizing the fuel consumption can be very efficiently solved, and the solution leads to a very efficient algorithm. Based on that, we propose a continuous deformation from this flat Earth model to a modified flat Earth model that is diffeomorphic to the usual round Earth model. The resulting numerical continuation process thus provides a new way to solve the problem of minimization of the fuel consumption for the coplanar orbit transfer problem.

(2004-2008) Design of efficient trajectories for an autonomous underwater vehicle

This research was done during my post-doctoral fellowship at the University of Hawaii at Manoa, Mathematics Department, under the supervision of Monique Chyba.
One subject of this research is the study of the controllability of the AUV (Autonomous Underwater Vehicle) ODIN (Omni Directional Intelligent Navigator) of ASL (Autonomous System Laboratory). Along with the study of this specific vehicle, we also aimed at extending our results to more general mechanical system which dynamics results from a Lagrangian of the form Kinetic minus Potential energies. This controllability study led to the construction of admissible trajectopries linking 2 configurations. We called those trajectories pure motion trajectories.
After this controllability study, we focused on the numerical solving of the optimal control problem of linking 2 configurations of ODIN with minimum time. The solutions of this problem are of the bang-singular type, which makes an indirect method very tricky to use. This explain why we resorted to a direct method, where we rewrote the optimal control problem as a nonlinear parameteric optimization problem. Using such a direct method and constraining the minimum time optimal solution to be usable in practice (meaning no singular arc, no discontinuity of the control, etc…), we were able to use the optimal solution on our test-bed vehicle in a pool.
Then, we focused on other criterion, mainly the minimum energy one which is probably the most relevant since it maximizes the autonomy of the AUV (because the AUV carries with it all its power sources). In order to cope with the rather long computing time required to solve a direct transcription of the minimum energy (or time) optimal control problem, we developped the so-called Switching Time Parameter Problem (STPP) where we look for a solution in the set of control having no more than a given number of switchings. We proved the existence of a solution of this problem provided there is at least has many switchings as there is coordinates to change between the initial and the final configurations. This method proved to be very effective in terms of execution time and of applicability to our test-bed vehicle.

(2001-2004) Low thrust minimum fuel orbital transfer

This is my PhD thesis research, done at the Institut Polytechnique de Toulouse, under the supervision of Joseph Noailles and Joseph Gergaud.
This study deals with a problem of space mechanics. This problem is a low thrust orbital transfer around the Earth with maximization of the final mass. The difficulty comes from the discontinuities of the optimal command and from the lack of knowledge concerning the number and the location of those discontinuities. Since the single shooting method is very sensitive with respect to the initialization that, we parameterize the criteria in order to link the minimization of the energy to the minimization of the consumption. The parameterized shooting function defines a homotopy on which we apply a differential continuation method in order to follow the zero path. A second homotopy, namely a discrete one, free our method from any a priori knowledge about the optimal control structure. This is very interesting as the optimal control exhibits more than 1000 switchings for a thrust of 0.1 N and an initial mass of 1500 kg. This method is implemented in a software and is successfully applied to our problem. The results allow us to outline some empiric laws as the independency of the final mass with respect to the thrust.