Commands is a team devoted to dynamic optimization, both for deterministic and stochastic systems. This includes the following approaches: trajectory optimization, deterministic and stochastic optimal control, stochastic programming, dynamic programming and Hamilton-Jacobi-Bellman equation.
Our aim is to derive new and powerful algorithms for solving numerically these problems, with applications in several industrial fields. While the numerical aspects are the core of our approach it happens that the study of convergence of these algorithms and the verification of their well-posedness and accuracy raises interesting and difficult theoretical questions, such as, for trajectory optimization: qualification conditions and second-order optimality condition, well-posedness of the shooting algorithm, estimates for discretization errors; for the Hamilton-Jacobi-Bellman approach: accuracy estimates, strong uniqueness principles when state constraints are present, for stochastic programming problems: sensitivity analysis.
For many years the team members have been deeply involved in various industrial applications, often in the framework of PhD theses or of postdocs. The Commands team itself has dealt since its foundation in 2007 with several types of applications:
Space vehicle trajectories, in collaboration with CNES, the French space agency.
Aeronautics, in collaboration with the startup Safety Line.
Production, management, storage and trading of energy resources (in collaboration with EDF, GDF and TOTAL).
Energy management for hybrid vehicles (in collaboration with Renault).
We give more details in the Bilateral contracts section.
In collaboration with L. Giraldi and M. Zopello, we started in 2013 to study the optimal swimming strategies for micro-swimmers. Our approach allows us to solve the optimal control problem without making restrictive assumptions on the shape of the swimming movements. The first numerical results on the 3-link swimmer indicate the existence of a periodic stroke with a better displacement speed than the canonical stroke presented by Purcell in 1977. Further directions include optimal design of micro-swimmers and comparing our simulations to the movement of live micro-organisms.
In collaboration with CNES, a trajectory optimization problem for Ariane 5 was studied and analyzed by HJB approach. In this study, the flight model is considered in dimension 6 without simplification. The problem consists in maximizing the payload to steer the launcher from the launch base (Kourou) to the GEO orbit. The mission includes ballistic phases and the optimization also encompasses the intermediate GTO orbit parameters. The optimization criterion is the mass of the payload to be injected on the GEO.
Finally, the team completed 3 PhD and 4 patents in 2013.
The roots of deterministic optimal control are the “classical” theory of the calculus of variations, illustrated by the work of Newton, Bernoulli, Euler, and Lagrange (whose famous multipliers were introduced in ), with improvements due to the “Chicago school”, Bliss during the first part of the 20th century, and by the notion of relaxed problem and generalized solution (Young ).
Trajectory optimization really started with the spectacular achievement done by Pontryagin's group during the fifties, by stating, for general optimal control problems, nonlocal optimality conditions generalizing those of Weierstrass. This motivated the application to many industrial problems (see the classical books by Bryson and Ho , Leitmann , Lee and Markus , Ioffe and Tihomirov ). Since then, various theoretical achievements have been obtained by extending the results to nonsmooth problems, see Aubin , Clarke , Ekeland .
Dynamic programming was introduced and systematically studied by R. Bellman during the fifties. The HJB equation, whose solution is the value function of the (parameterized) optimal control problem, is a variant of the classical Hamilton-Jacobi equation of mechanics for the case of dynamics parameterized by a control variable. It may be viewed as a differential form of the dynamic programming principle. This nonlinear first-order PDE appears to be well-posed in the framework of viscosity solutions introduced by Crandall and Lions , , . These tools also allow to perform the numerical analysis of discretization schemes. The theoretical contributions in this direction did not cease growing, see the books by Barles and Bardi and Capuzzo-Dolcetta .
The so-called direct methods consist in an optimization of the trajectory, after having discretized time, by a nonlinear programming solver that possibly takes into account the dynamic structure. So the two main problems are the choice of the discretization and the nonlinear programming algorithm. A third problem is the possibility of refinement of the discretization once after solving on a coarser grid.
In the full discretization approach, general Runge-Kutta schemes with different values of control for each inner step are used. This allows to obtain and control high orders of precision, see Hager , Bonnans . In an interior-point algorithm context, controls can be eliminated and the resulting system of equation is easily solved due to its band structure. Discretization errors due to constraints are discussed in Dontchev et al. . See also Malanowski et al. .
In the indirect approach, the control is eliminated
thanks to Pontryagin's
maximum principle.
One has then to solve the two-points boundary value problem
(with differential variables state and costate) by a single or multiple shooting method.
The questions are here the choice of a discretization scheme for the integration of the boundary value problem, of a
(possibly globalized) Newton
type algorithm for solving the resulting finite dimensional problem
in
For state constrained problems or singular arcs, the formulation of the shooting function may be quite elaborate , , . As initiated in , we focus more specifically on the handling of discontinuities, with ongoing work on the geometric integration aspects (Hamiltonian conservation).
This approach consists in calculating the value function associated with the optimal control problem, and then synthesizing the feedback control and the optimal trajectory using Pontryagin's principle. The method has the great particular advantage of reaching directly the global optimum, which can be very interesting, when the problem is not convex.
Characterization of the value function From the dynamic programming principle, we derive a characterization of the value function as being a solution (in viscosity sense) of an Hamilton-Jacobi-Bellman equation, which is a nonlinear PDE of dimension equal to the number n of state variables. Since the pioneer works of Crandall and Lions , , , many theoretical contributions were carried out, allowing an understanding of the properties of the value function as well as of the set of admissible trajectories. However, there remains an important effort to provide for the development of effective and adapted numerical tools, mainly because of numerical complexity (complexity is exponential with respect to n).
Numerical approximation for continuous value function Several numerical schemes have been already studied to treat the case when the solution of the HJB equation (the value function) is continuous. Let us quote for example the Semi-Lagrangian methods , studied by the team of M. Falcone (La Sapienza, Rome), the high order schemes WENO, ENO, Discrete galerkin introduced by S. Osher, C.-W. Shu, E. Harten , , , , and also the schemes on nonregular grids by R. Abgrall , . All these schemes rely on finite differences or/and interpolation techniques which lead to numerical diffusions. Hence, the numerical solution is unsatisfying for long time approximations even in the continuous case.
One of the (nonmonotone) schemes for solving the HJB equation is based on the Ultrabee algorithm proposed, in the case of advection equation with constant velocity, by Roe and recently revisited by Després-Lagoutière , . The numerical results on several academic problems show the relevance of the antidiffusive schemes. However, the theoretical study of the convergence is a difficult question and is only partially done.
Optimal stochastic control problems occur when the dynamical system is uncertain. A decision typically has to be taken at each time, while realizations of future events are unknown (but some information is given on their distribution of probabilities). In particular, problems of economic nature deal with large uncertainties (on prices, production and demand). Specific examples are the portfolio selection problems in a market with risky and non-risky assets, super-replication with uncertain volatility, management of power resources (dams, gas). Air traffic control is another example of such problems.
Nonsmoothness of the value function. Sometimes the value function is smooth (e.g. in the case of Merton's portfolio problem, Oksendal ) and the associated HJB equation can be solved explicitly. Still, the value function is not smooth enough to satisfy the HJB equation in the classical sense. As for the deterministic case, the notion of viscosity solution provides a convenient framework for dealing with the lack of smoothness, see Pham , that happens also to be well adapted to the study of discretization errors for numerical discretization schemes , .
Numerical approximation for optimal stochastic control problems. The numerical discretization of second order HJB equations was the subject of several contributions. The book of Kushner-Dupuis gives a complete synthesis on the Markov chain schemes (i.e Finite Differences, semi-Lagrangian, Finite Elements, ...). Here a main difficulty of these equations comes from the fact that the second order operator (i.e. the diffusion term) is not uniformly elliptic and can be degenerated. Moreover, the diffusion term (covariance matrix) may change direction at any space point and at any time (this matrix is associated the dynamics volatility).
For solving stochastic control problems, we studied the so-called Generalized Finite Differences (GFD), that allow to choose at any node, the stencil approximating the diffusion matrix up to a certain threshold . Determining the stencil and the associated coefficients boils down to a quadratic program to be solved at each point of the grid, and for each control. This is definitely expensive, with the exception of special structures where the coefficients can be computed at low cost. For two dimensional systems, we designed a (very) fast algorithm for computing the coefficients of the GFD scheme, based on the Stern-Brocot tree .
Web page: http://
The Bocop project aims to develop an open-source toolbox for solving optimal control problems, with collaborations with industrial and academic partners. Optimal control (optimization of dynamical systems governed by differential equations) has numerous applications in transportation, energy, process optimization, and biology. The software reuses some packages from the COIN-OR library, in particular the well-known nonlinear programming solver Ipopt, features a user-friendly interface and can be deployed on Windows / Mac / Linux.
The project is supported by Inria with the recruitment of Vincent Grelard as developer in 2010-2012, and then Daphné Giorgi since October 2012. The first prototype was released at the end of 2011, Bocop is currently at version 1.1.4 and has been downloaded more than 700 times. The software was first succesfully tested on several academic problems, see available on http://bocop.org. Starting in 2012, several research collaborations were initiated in fields such as bio-reactors for energy production (, ), swimming micro-robots (), and quantum control for medical imaging (). Bocop was also featured during our participation in the Imatch "Optimisation and Control" in october, which resulted in a contract with the startup Safety Line (aeronautics).
Bocop auto-assessment according to Inria notice: A3up4, SO3, SM3, EM3up4, SDL4up5
This software simulates the evolution of controlled dynamical systems (possibly under uncertainties). The numerical algorithm here is based on HJB or viability approaches, and allows the design of optimal planning strategies (according to a criterion determined by the user: time, energy, ...). It also provides conflict resolution and avoidance of collisions with fixed or moving obstacles. So far, the software is used in collaboration with DGA for avoidance collision of UaVs, and by Volkswagen in some studies related to collision avoidance of cars.
This is a software for optimisation-based controller design for operating in different regimes or modes of operation. The software can be used, for example, to determine the optimal management for hybrid vehicles or hybrid engines with multiple energy sources. However, the methods used in software are still quite general and can be used in many applications.
Web page: http://www.ensta-paristech.fr/ zidani/BiNoPe-HJ
This project aims at developping sequential and parallel MPI/openMP C++ solvers for the approximation of Hamilton-Jacobi-Bellman (HJB) equations in a d-dimensional space. The main goal is to provide an HJB solvers that can work in dimension d (limited by the machine's capacity). The solver outputs can be visualized with Matlab or Paraview (via VTK files).
The development of the HJB Solver has been initiated under a partnership between COMMANDS and the SME HPC-project in the period between December 2009 to November 2011. Currently, it is still maintained and improved by COMMANDS.
In 2012, two versions were released:
HJB-SEQUENTIAL-REF: sequential version that can run on any machine
HJB-PARALLEL-REF: parallel version that can run only on multi-core architectures.
Shoot was designed for the resolution of optimal control problems via indirect methods (necessary conditions, Pontryagin's Maximum Principle). Such methods transform the original problem into finding a zero of a certain shooting function. The package offers several choices of integrators and solvers, and can handle control discontinuities. Features also include the use of variational equations to compute the Jacobian of the shooting function, as well as homotopy and grid shooting techniques for easier initialization. Shoot is an academic software, and was used during several research contracts with the CNES (french space agency).
In the paper , we state and prove first- and second-order necessary conditions in Pontryagin form for optimal control problems with pure state and mixed control-state constraints. We say that a Lagrange multiplier of an optimal control problem is a Pontryagin multiplier if it is such that Pontryagin's minimum principle holds, and we call optimality conditions in Pontryagin form those which only involve Pontryagin multipliers. Our conditions rely on a technique of partial relaxation, and apply to Pontryagin local minima.
In the paper , we consider sufficient conditions. More precisely, given a reference feasible trajectory of an optimal control problem, we say that the quadratic growth property for bounded strong solutions holds if the cost function of the problem has a quadratic growth over the set of feasible trajectories with a bounded control and with a state variable sufficiently close to the reference state variable. Our sufficient second-order optimality conditions in Pontryagin form ensure this property and ensure a fortiori that the reference trajectory is a bounded strong solution. Our proof relies on a decomposition principle, which is a particular second-order expansion of the Lagrangian of the problem.
In the paper we give an overview of the shooting technique for solving deterministic optimal control problems. This approach allows to reduce locally these problems to a finite dimensional equation. We first recall the basic idea, in the case of unconstrained or control constrained problems, and show the link with second-order optimality conditions and the analysis or discretization errors. Then we focus on two cases that are now better understood: state constrained problems, and affine control systems. We end by discussing extensions to the optimal control of a parabolic equation.
In the report we present new models, numerical simulations and rigorous analysis for the optimization of the velocity in a race. In a seminal paper, Keller , explained how a runner should determine his speed in order to run a given distance in the shortest time. We extend this analysis, based on the equation of motion and aerobic energy, to include a balance of anaerobic energy (or accumulated oxygen deficit) and an energy recreation term when the speed decreases. We also take into account that when the anaerobic energy gets too low, the oxygen uptake cannot be maintained to its maximal value. Using optimal control theory, we obtain a proof of Keller's optimal race, and relate the problem to a relaxed formulation, where the propulsive force represents a probability distribution rather than a value function of time. Our analysis leads us to introduce a bound on the variations of the propulsive force to obtain a more realistic model which displays oscillations of the velocity. Our numerical simulations qualitatively reproduce quite well physiological measurements on real runners. We show how, by optimizing over a period, we recover these oscillations of speed. We point out that our numerical simulations provide in particular the exact instantaneous anaerobic energy used in the exercise.
In the paper we discuss the optimal co-administration of two drugs for some acute myeloid leukemias (AML), and we are looking for in vitro protocols as a first step. This issue can be formulated as an optimal control problem. The dynamics of leukemic cell populations in culture is given by age-structured partial differential equations, which can be reduced to a system of delay differential equations, and where the controls represent the action of the drugs. The objective function relies on eigenelements of the uncontrolled model and on general relative entropy, with the idea to maximize the efficiency of the protocols. The constraints take into account the toxicity of the drugs. We present in this paper the modeling aspects, as well as theoretical and numerical results on the optimal control problem that we get.
In collaboration with team McTAO (Sophia), we studied in and the contrast imaging problem in nuclear magnetic resonance, modeled as Mayer problem in optimal control. The optimal solution can be found as an extremal, solution of the Maximum Principle and analyzed with the techniques of geometric control. A first synthesis of locally optimal solutions is given in the single-input case, with some preliminary results in the bi-input case. We conducted a comprehensive numerical investigation of the problem, using a combination of indirect shooting (HamPath software) and direct method (Bocop), with a moment-based (LMI) technique to estimate the global optimum.
In collaboration with the Inra-Inria team MODEMIC (Montpellier), we studied in a bio-reactor system describing the coupling of a culture of micro-algae and an anaerobic digester. Our aim is to optimize the production of methane in the digester during a certain number of days with respect to the dilution rate (the input flow of micro-algae in the digester). The mathematical model for the dynamics of the two reactors takes into account a periodic day-night model of the light in the culture of micro-algae, and a chemostat model for the digester. We first prove existence and attraction of periodic solutions for a one day period, and we apply Pontryagin's Maximum Principle (PMP) in order to characterize optimal controls. We provide numerical simulations for different light models, by a direct method that we refine using an indirect shooting. We also investigate the dependence of the optimal cost with respect to the ratio of the volumes of the two tanks. Finally, we investigate the optimal strategies over a large number of days without periodic constraints, and compared the mean cost to the optimal cost over one period.
In collaboration with team BIOCORE (Sophia), we investigated in techniques of Optimal Experiment Design for microalgae growth models. In order to have microalgae growth models that are useful for prediction and process optimization, reliable parameters need to be provided. This reliability implies a careful design of experiments that can be exploited for parameter estimation. OED techniques can provide guidelines for the design of experiments with high informative content that allow an accurate parameter estimation. We study a real experimental device devoted to evaluate the effect of temperature and light on microalgae growth. On the basis of a mathematical model of the experimental system, the optimal experiment design problem was solved as an optimal control problem. E-optimal experiments were obtained by using two discretization approaches, namely sequential and simultaneous. The results showed that an adequate parameterization of the experimental inputs provided optimal solutions very close to those provided by the simultaneous discretization. Simulation results showed the relevance of determining optimal experimental inputs for achieving an accurate parameter estimation.
In we focus on the N-link swimmer, a generalization of the classical Purcell swimmer. We use the simplification of the Resistive Force Theory to derive the motion equation for the swimmer in a fluid with a low Reynolds number. We prove that the swimmer is controllable in the whole plane when it is composed by more than 3 sticks and for almost every set of stick lengths. As a direct result, we show that there exists an optimal swimming strategy which leads to minimize the time to reach a desired configuration. Numerical experiments on the case of the Purcell swimmer suggest that the optimal strategy is periodic, i.e. composed of a sequence of identical strokes. Our results indicate that this candidate for an optimal stroke indeed gives a better speed than the classical Purcell stroke. Future directions for this work include the design of robotic micro-swimmers, as well as investigation of the movement of swimming micro-organisms.
The paper is concerned with stochastic optimal control for a running maximum cost. A direct approach based on dynamic programming techniques is studied leading to the characterization of the value function as the unique viscosity solution of a second order Hamilton-Jacobi-Bellman (HJB) equation with an oblique derivative boundary condition. A general numerical scheme is proposed and a convergence result is provided. Error estimates are obtained for the semi-Lagrangian scheme. These results can apply to the case of lookback options in finance. Moreover, optimal control problems with maximum cost arise in the characterization of the reachable sets for a system of controlled stochastic differential equations. Some numerical simulations on examples of reachable analysis are included to illustrate our approach.
An optimal finite-time horizon feedback control problem for (semi linear) wave equations is studied in . The feedback law can be derived from the dynamic programming principle and requires to solve the evolutionary Hamilton-Jacobi-Bellman (HJB) equation. Classical discretization methods based on finite elements lead to approximated problems governed by ODEs in high dimensional space which makes infeasible the numerical resolution by HJB approach. In the present paper, an approximation based on spectral elements is used to discretize the wave equation. The effect of noise is considered and numerical simulations are presented to show the relevance of the approach
The works , deal
with deterministic control problems where the dynamic and the
running cost can be completely different in two (or more)
complementary domains of the space
The goal of the paper is to study a singular perturbation problem in the framework of optimal control on multi-domains. We consider an optimal control problem in which the controlled system contains a fast and a slow variables. This problem is reformulated as an Hamilton-Jacobi-Bellman (HJB) equation. The main difficulty comes from the fact that the fast variable lives in a multi-domain. The geometric singularity of the multi-domains leads to the discontinuity of the Hamiltonian. Under a controllability assumption on the fast variables, the limit equation (as the velocity of the fast variable goes to infinity) is obtained via a PDE approach and by means of the tools of the control theory.
In , we consider the optimal control of solutions of first order Hamilton-Jacobi equations, where the Hamiltonian is convex with linear growth. This models the problem of steering the propagation of a front by constructing an obstacle. We prove existence of minimizers to this optimization problem as in a relaxed setting and characterize the minimizers as weak solutions to a mean field game type system of coupled partial differential equations. Furthermore, we prove existence and partial uniqueness of weak solutions to the PDE system. An interpretation in terms of mean field games is also discussed.
The paper gives a characterization of the uniform robust domain of attraction for a nite non-linear controlled system subject to perturbations and state constraints. We extend the Zubov approach to characterize this domain by means of the value function of a suitable in nite horizon state-constrained control problem which at the same time is a Lyapunov function for the system. We provide associated Hamilton-Jacobi-Bellman equations and prove existence and uniqueness of the solutions of these generalized Zubov equations.
In Laurent Pfeiffer's PhD, we study stochastic optimal control problems with a probability constraint on the final state. This constraint must be satisfied with a probability greater or equal than a given level. We analyse and compare two approaches for discrete-time problems: a first one based on a dynamic programming principle and a second one using Lagrange relaxation. These approaches can be used for continuous-time problems, for which we give numerical illustrations.
Following the “iMatch Contrôle Optimisation” event held at Inria Saclay on October 23rd (2012), a collaboration was initiated between COMMANDS and the startup Safety Line (http ://www.safety-line.fr), with a first contract on optimizing the ascent phase for commercial planes. A crucial aspect of this work is the identification of accurate and reliable models for the aerodynamic and thrust forces acting on the plane. For this study our partners at Safety Line provide us access to data recorded during several thousands of actual commercial flights, and COMMANDS recruited Stephan Maindrault as engineer to work on this project.
This contract between CNES and ENSTA lasted from February to December 2013, and was devoted to trajectory global optimization for an Ariane 5 launcher, using HJB techniques. The optimization was on the whole launch, including ballistic phases and the parameters of the intermediate GTO orbit, while maximizing the payload mass.
This project is a collaboration in the framework of a 3-year (2012-2015) research program funded by DGA. The title of the project is “Problèmes de commande optimale pour des systèmes non-linéaires en présence d’incertitudes et sous contraintes de probabilité de succès”.
The team is part of the collaborative project HJNet funded by the French National Research Agency (ANR-12-BS01-0008-01). It started in January 2013 and will end in December 2013. Website: http://hjnet.math.cnrs.fr
Instrument: Initial Training Network
Duration: January 2011 - December 2014
Coordinator: Inria
Partner: Univ. of Louvain, Univ. Bayreuth, Univ. Porto, Univ. Rome - La Sapienza, ICL, Astrium-Eads, Astos solutions, Volkswagen, Univ. Padova, Univ. Pierre et Marie Curie.
Inria contact: Hasnaa Zidani
Abstract: Optimisation-based control systems concern the determination of control strategies for complex, dynamic systems, to optimise some measures of best performance. It has the potential for application to a wide range of fields, including aerospace, chemical processing, power systems control, transportation systems and resource economics. It is of special relevance today, because optimization provides a natural framework for determining control strategies, which are energy efficient and respect environmental constraints. The multi-partner initial training network SADCO aims at: Training young researchers and future scientific leaders in the field of control theory with emphasis on two major themes sensitivity of optimal strategies to changes in the optimal control problem specification, and deterministic controller design; Advancing the theory and developing new numerical methods; Conveying fundamental scientific contributions within European industrial sectors.
Title: Optimization and control in network economics
Inria principal investigator: Frédéric Bonnans
International Partner (Institution - Laboratory - Researcher):
University of Chile (Chile) - Center for Mathematical Modeling - Joseph Frédéric Bonnans
Duration: 2012 - 2014
See also: http://www.cmm.uchile.cl/EA_OCONET
Limited resources in telecommunication, energy, gas and water supply networks, lead to multi-agent interactions that can be seen as games or economic equilibrium involving stochastic optimization and optimal control problems. Interaction occurs within a network, where decisions on what to produce, consume, trade or plan, are subject to constraints imposed by node and link capacities, risk, and uncertainty, e.g. the capacity of generators and transmission lines; capacity of pipeline in gas supply; switches and antennas in telecommunication. At the same time, nonlinear phenomena arise from price formation as a consequence of demand-supply equilibria or multi-unit auction processes in the case of energy and telecommunication. We will focus first in this project in electricity markets in which there are producers/consumers PCs, and an agent called ISO (Independent system operator) in charge of the management of the network. One major application we have in mind is the one of smart (electrical) grids, in view of the increased use of renewable energies, that is, a massive entry of wind, geothermal, solar in particular.
The team is involved in the "Energy Optimization" group of the Inria research center in Chile (CIRIC). Several visits to Chile were conducted in relation with this project.
Prof. B.S. Goh, Curtin University, Miri, Malaysia; two weeks in February.
M.S. Aronna, Rosario University, Argentina; one month (February and November).
Arthur Marly, second year student of ENS Lyon. Two months intership. Subject: optimal control of populations with state constraints. Supervisor: F. Bonnans.
F. Bonnans is Corresponding Editor of “ESAIM:COCV” (Control, Optimization and Calculus of Variations), and Associate Editor of “Applied Mathematics and Optimization”, “Optimization, Methods and Software”, and “Series on Mathematics and its Applications, Annals of The Academy of Romanian Scientists”.
F. Bonnans is chairman of the SMAI-MODE group (the optimization group of the French Applied Mathematics Society) until June 2013.
F. Bonnans: Optimal control, 7h, M2, Ensta, France.
F. Bonnans: Continuous Optimization, 18h, M2, Ecole Polytechnique and U. Paris 6, France.
F. Bonnans: Numerical analysis of partial differential equations arising in finance and stochastic control, 24h, M2, Ecole Polytechnique and U. Paris 6, France.
H. Zidani: Optimal control, 14h, M2, Ensta, France.
H. Zidani: Numerical methods for front propagation, 21h, M2, Ensta France
PhD: Xavier Dupuis, Optimal control with or witout memories. ended Nov. 2013, F. Bonnans.
PhD : Laurent Pfeiffer, Sensitivity analysis for optimal control problems; Stochastic optimal control with probability constraints. ended Nov. 2013, F. Bonnans.
PhD: Zhiping Rao, Hamilton-Jacobi equations with discontinuous coefficients. Ended Dec. 2013, H. Zidani and N. Forcadel.
PhD in progress : Imène Ben-Latifa, Optimal multiple stopping and valuation of swing options in jump models. Oct. 2010, F. Bonnans and M. Mnif (ENIT, Tunis).
PhD in progress: Athena Picarelli, First and Second Order Hamilton-Jacobi equations for State-Constrained Control Problems. Nov. 2011, O. Bokanowski and H. Zidani
PhD in progress: Cristopher Hermosilla, Feedback controls and optimal trajectories. Nov. 2011, H. Zidani.
PhD in progress: Mohamed Assellaou, Reachability analysis for stochastic controlled systems. Oct. 2011, O. Bokanowski and H.Zidani.
PhD in progress: Benjamin Heymann, Dynamic optimization with uncertainty; application to energy production. Oct. 2013, F. Bonnans.
Pierre Martinon presented an “Unithé ou Café” talk about optimal control on November 8th, entitled “Quand le mieux n'est pas l'ennemi du bien”.