Commands is a team with a global view on dynamic optimization in its various aspects: trajectory optimization, geometric control, deterministic and stochastic optimal control, stochastic programming, dynamic programming and HamiltonJacobiBellman approach.
Our aim is to derive new and powerful algorithms for solving numerically these various 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 wellposedness and accuracy raises interesting and difficult theoretical questions, such as, for trajectory optimization: qualification conditions and secondorder optimality condition, wellposedness of the shooting algorithm, estimates for discretization errors; for the HamiltonJacobiBellman approach: accuracy estimates, strong uniqueness principles when state constraints are present, for stochastic programming problems: sensitivity w.r.t. the probability laws, formulation of risk measures.
At the same time we are deeply involved in applications of critical importance, in particular: trajectories of space vehicles (in collaboration with CNES, the French space agency), as well as management, storage and trading of energy resources (in collaboration with EDF, GDF and TOTAL).
We have studied the advantage of having a singular arc for real world launcher trajectories. Although no such phenomenon seems to occur with the present Ariane V launcher, we have shown the possibility of having one when the launcher has bigger aerodynamic reference area and specific impulsion. The principal investigator for this study was P. Martinon. See the report .
We have developed a fast numerical code (for dimensions 2 to 4) on sparse grids for the resolution of HamiltonJacobiBellman equations whose solutions take only values 0 and 1. This code computes, in a few seconds, capture basins and associated time optimal trajectories. We are currently aiming to use this code for solving a climbing problem (for aircrafts) with maximal final mass, under a structural constraint on dynamic pressure.
For deterministic optimal control we will distinguish two approaches, trajectory optimization, in which the object under consideration is a single trajectory, and the HamiltonJacobiBellman approach, based on dynamic principle, in which a family of optimal control problems is solved.
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 optimizationreally 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 . Substantial improvements were also obtained by using tools of differential geometry, which concern a precise understanding of optimal syntheses in low dimension for large classes of nonlinear control systems, see Bonnard, Faubourg and Trélat .
Overviews of numerical methods for trajectory optimization are provided in Pesch , Betts . We follow here the classical presentation that distinguishes between direct and indirect methods.
Dynamic programmingwas 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 HamiltonJacobi 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 firstorder PDE appears to be wellposed in the framework of viscosity solutionsintroduced 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 CapuzzoDolcetta .
A interesting byproduct of the HJB approach is an expression of the optimal control in feedback form. Also it reaches the global optimum, whereas trajectory optimization algorithms are of
local nature. A major difficulty when solving the HJB equation is the high cost for a large dimension
nof the state (complexity is exponential with respect to
n).
The socalled direct methodsconsist 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.
Many authors prefer to have a coarse discretization for the control variables (typically constant or piecewiselinear on each time step) and a higher order discretization for the state equation. The idea is both to have an accurate discretization of dynamics (since otherwise the numerical solution may be meaningless) and to obtain a smallscale resulting nonlinear programming problem. See e.g. Kraft . A typical situation is when a few dozen of timesteps are enough and there are no more than five controls, so that the resulting NLP has at most a few hundreds of unknowns and can be solved using full matrices software. On the other hand, the error order (assuming the problem to be unconstrained) is governed by the (poor) control discretization. Note that the integration scheme does not need to be specified (provided it allows to compute functions and gradients with enough precision) and hence general Ordinary Differential Equations integrators may be used.
On the other hand, a full discretization (i.e. in a context of RungeKutta methods, with different values of control for each inner substep of the scheme) allows to obtain higher orders that can be effectively computed, see Hager , Bonnans , being related to the theory of partitioned RungeKutta schemes, Hairer et al. . In an interiorpoint 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. .
For large horizon problems integrating from the initial time to the final time may be impossible (finding a feasible point can be very hard !). Analogously to the indirect method of multiple shooting algorithm, a possibility is to add (to the control variables), as optimization parameters, the state variables for a given set of times, subject of course to “sticking” constraint. Note that once more the integration scheme does not need to be specified. Integration of the ODE can be performed in parallel. See Bock .
Recent proposals were made of methods based on a reformulation of the problem based on (possibly flat) output variables. By the definition, control and state variables are combinations of derivatives of these output variables. When the latter are represented on a basis of smooth functions such as polynomials, their derivatives are given linear combinations of the coefficients, and so the need for integration is avoided. One must of course take care of the possibly complicated expression of constraints that can make numerical difficulties. The numerical analysis of these methods seems largely open. See on this subject Petit, Milam and Murray .
The collocation approach for solving an ODE consists in a polynomial interpolation of the dynamic variable, the dynamic equation being enforced only at limited number of points (equal to the degree of the polynomial). Collocation can also be performed on each time step of a onestep method; it can be checked than collocation methods are a particular case of RungeKutta methods.
It is known that the polynomial interpolation with equidistant points is unstable for more than about 20 points, and that the Tchebycheff points should be preferred, see e.g. Section 5.2.6. Nevertheless, several papers suggested the use of pseudospectral methods Ross and Fahroo in which a single (over time) highorder polynomial approximation is used for the control and state. Therefore pseudospectral methods should not be used in the case of nonsmooth (e.g. discontinuous) control.
In view of model and data uncertainties there is a need for robust solutions. Robust optimization has been a subject of increasing importance in recent years see BenTal and Nemirovski . For dynamic problems taking the worstcase of the perturbation at each timestep may be too conservative. Specific remedies have been proposed in specific contexts, see BenTal et al. , Diehl and Björnberg .
A relatively simple method taking into account robustness, applicable to optimal control problems, was proposed in Diehl, Bock and Kostina .
The dominant ones (for optimal control problems as well as for other fields) have been successively the augmented Lagrangian approach (1969, due to Hestenes and Powell , see also Bertsekas ) successive quadratic programming (SQP: late seventies, due to , , and interiorpoint algorithms since 1984, Karmarkar . See the general textbooks on nonlinear programming , , .
When ordered by time the optimality system has a “band structure”. One can take easily advantage of this with interiorpoint algorithms whereas it is not so easy for SQP methods; see Berend et al. . There exist some very reliable SQP softwares SNOPT, some of them dedicated to optimal control problems, Betts , as well as robust interiorpoint software, see Morales et al. , Wächter and Biegler , and for application to optimal control Jockenhövel et al. .
We have developed a general SQP algorithm , for sparse nonlinear programming problems, and the associated software for optimal control problems; it has been applied to atmospheric reentry problems, in collaboration with CNES .
More recently, in collaboration with CNES and ONERA, we have developed a sparse interiorpoint algorithm with an embedded refinement procedure. The resulting TOPAZE code has been applied to various space trajectory problems , , . The method takes advantage of the analysis of discretization errors, is wellunderstood for unconstrained problems .
The indirect approach eliminates control variables using Pontryagin's maximum principle, and solves the twopoints 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
IR^{n}(
nis the number of state variables), and a methodology for finding an initial point.
The choice of the discretization scheme for the numerical integration of the boundary value problem can have a great impact on the convergence of the method. First, the integration itself can be tricky. If the state equation is stiff (the linearized system has fast modes) then the statecostate has both fast and unstable modes. Also, discontinuities of the control or its derivative, due to commutations or changes in the set of active constraints, lead to the use of sophisticated variable step integrators and/or switching detection mechanisms, see Hairer et al. , . Another point is the computation of gradients for the Newton method, for which basic finite differences can give inaccurate results with variable step integrators (see Bock ). This difficulty can be treated in several ways, such as the socalled “internal differentiation” or the use of variational equations, see Gergaud and Martinon .
Most optimal control problems include control and state constraints. In that case the formulation of the TPBVP must take into account entry and exit times of boundary arcs for these constraints, and (for state constraints of order at least two) times of touch points (isolated contact points). In addition for state constrained problems, the socalled “alternative formulation” (that allows to eliminate the “algebraic variables, i.e. control and state, from the algebraic constraints) has to be used, see Hartl, Sethi and Vickson .
Another interesting point is the presence of singular arcs, appearing for instance when the control enters in the system dynamics and cost function in a linear way, which is common in practical applications. As for state contraints, the formulation of the boundary value problem must take into account these singular arcs, over which the expression of the optimal control typically involves higher derivatives of the Hamiltonian, see Goh and Robbins .
As mentioned before, finding a suitable initial point can be extremely difficult for indirect methods, due to the small convergence radius of the Newton type method used to solve the boundary value problem. Homotopy methods are an effective way to address this issue, starting from the solution of an easier problem to obtain a solution of the target problem (see Allgower and Georg ). It is sometimes possible to combine the homotopy approach with the Newton method used for the shooting, see Deuflhard .
With a given value of the initial costate are associated (through in integration of the reduced statecostate system) a control and a state, and therefore a cost function. The latter can therefore be minimized by adhoc minimization algorithms, see Dixon and BartholomewBiggs . The advantage of this point of view is the possibility to use the various descent methods in order to avoid convergence to a local maximum or saddlepoint. The extension of this approach to constrained problems (especially in the case of state constraints) is an open and difficult question.
We have recently clarified under which properties shooting algorithms are wellposed in the presence of state constraints. The (difficult) analysis was carried out in , . A related homotopy algorithm, restricted to the case of a single firstorder state constraint, has been proposed in .
We also conducted a study of optimal trajectories with singular arcs for space launcher problems. The results obtained for the generalized threedimensional Goddard problem (see ) have been successfully adapted for the numerical solution of a realistic launcher model (Ariane 5 class).
Furthermore, we continue to investigate the effects of the numerical integration of the boundary value problem and the accurate computation of Jacobians on the convergence of the shooting method. As initiated in , we focus more specifically on the handling of discontinuities, with ongoing work on the geometric integration aspects (Hamiltonian conservation).
Geometric approaches succeeded in giving a precise description of the structure of optimal trajectories, as well as clarifying related questions. For instance, there have been many works
aiming to describe geometrically the set of attainable points, by many authors (Krener, Schättler, Bressan, Sussmann, Bonnard, Kupka, Ekeland, Agrachev, Sigalotti, etc). It has been proved,
in particular, by [
KrenerSchättler, SICON, 1989], that, for generic singleinput controlaffine systems in
IR^{3},
, where the control satisfies the constraint

u
1, the boundary of the accessible set in small time consists of the surfaces
generated by the trajectories
x_{+}x_{}and
x_{}x_{+}, where
x_{+}(resp.
x_{}) is an arc corresponding to the control
u= 1(resp.
u= 1); moreover, every point inside the accessible set can be reached with a trajectory of the form
x_{}x_{+}x_{}or
x_{+}x_{}x_{+}. It follows that minimal time trajectories of generic singleinput controlaffine systems in
IR^{3}are locally of the form
x_{}x_{+}x_{}or
x_{+}x_{}x_{+}, i.e., are bangbang with at most two switchings.
This kind of result has been slightly improved recently by AgrachevSigalotti, although they do not take into account possible state constraints.
In , we have extended that kind of result to the case of state constraints: we described a complete classification, in terms of the geometry (Lie configuration) of the system, of local minimal time syntheses, in dimension two and three. This theoretical study was motivated by the problem of atmospheric reentry posed by the CNES, and in , we showed how to apply this theory to this concrete problem, thus obtaining the precise structure of the optimal trajectory.
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.
From the dynamic programming principle, we derive a characterization of the value function as being a solution (in viscosity sense) of an HamiltonJacobiBellman equation, wich 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).
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 SemiLagrangian 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.
In a realistic optimal control problem, there are often constraints on the state (reaching a target, restricting the state of the system in an acceptable domain, ...). When some controlability assumptions are not satisfied, the value function associated to such a problem is discontinuous and the region of discontinuity is of great importance since it separates the zone of admissible trajectories and the nonadmissible zone.
In this case, it is not reasonable to use the usual numerical schemes (based on finite differences) for solving the HJB equation. Indeed, these schemes provide poor approximation quality because of the numerical diffusion.
Discrete approximations of the HamiltonJacobi equation for an optimal control problem of a differentialalgebraic system were studied in .
Numerical methods for the HJB equation in a bilevel optimization scheme where the upperlevel variables are design parameters were used in . The algorithm has been applied to the parametric optimization of hybrid car engines.
Within the framework of the thesis of N. Megdich, we have studied new antidiffusive schemes for HJB equations with discontinuous data , . One of these schemes is based on the Ultrabee algorithm proposed, in the case of advection equation with constant velocity, by Roe and recently revisited by DesprésLagoutière , . 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 nonrisky assets, superreplication with uncertain volatility, management of power resources (dams, gas). Air traffic control is another example of such problems.
By stochastic programming we mean stochastic optimal control in a discrete time (or even static) setting; see the overview by Ruszczynski and Shapiro . The static and single recourse cases are essentially wellunderstood; by contrast the truly dynamic case (multiple recourse) presents an essential difficulty Shapiro , Shapiro and Nemirovski . So we will speak only of the latter, assuming decisions to be measurable w.r.t. a certain filtration (in other words, all information from the past can be used).
In the standard case of minimization of an expectation (possibly of a utility function) a dynamic programming principleholds. Essentially, this says that the decision is a function of the present state (we can ignore the past) and that a certain reversetime induction over the associated values holds. Unfortunately a straighforward resolution of the dynamic programming principle based on a discretization of the state space is out of reach (again this is the curse of dimensionality). For convex problems one can build lower convex approximations of the value function: this is the Stochastic dual dynamic programming(SDDP) approach, Pereira and Pinto . Another possibility is a parametric approximation of the value function; however determining the basis functions is not easy and identifying (or, we could say in this context, learning) the best parameters is a nonconvex problem, see however Bertsekas and J. Tsitsiklis , Munos .
A popular approach is to sample the uncertainties in a structured way of a tree(branching occurs typically at each time). Computational limits allow only a small number of branching, far less than the amount needed for an accurate solution Shapiro and Nemirovski . Such a poor accuracy may nevertheless (in the absence of a more powerful approach) be a good way for obtaining a reasonable policy. Very often the resulting programs are linear, possibly with integer variables (onoff switches of plants, investment decisions), allowing to use (possibly dedicated) mixed integer linear programming codes. The tree structure (coupling variables) can be exploited by the numerical algorithms, see Dantzig and Wolfe , Kall and Wallace .
By Monte Carlo we mean here sampling a given number of independent trajectories (of uncertainties). In the special case of optimal stopping (e.g., American options) it happens that the state space and the uncertainty space coincide. Then one can compute the transition probabilities of a Markov chain whose law approaches the original one, and then the problem reduces to the one of a Markov chain, see . Let us mention also the quantization approach, see .
In the general case a useful possibility is to compute a tree by agregating the original sample, as done in .
Maximizing the expectation of gains can lead to a solution with a too high probability of important losses (bankruptcy). In view of this it is wise to make a compromise between expected gains and risk of high losses. A simple and efficient way to achieve that may be to maximize the expectation of a utility function; this, however, needs an adhoc tuning. An alternative is the meanvariance compromise, presented in the case of portfolio optimization in Markowitz . A useful generalization of the variance, uncluding dissymetric functions such as semideviations, is the theory of deviation measures, Rockafellar et al. .
Another possibility is to put a constraint on the level of gain to be obtained with a high probability value say at least 99%. The corresponding concept of valueatrisk leads to difficult nonconvex optimization problems, although convex relaxations may be derived, see Shapiro and Nemirovski .
Yet the most important contribution of the recent years is the axiomatized theory of risk measures Artzner et al. , satisfying the properties of monotonicity and possibly convexity.
In a dynamic setting, risk measures (over the total gains) are not coherent (they do not obey a dynamic programming principle). The theory of coherent risk measuresis an answer in which risk measures over successive time steps are inductively applied; see Ruszczyński and Shapiro . Their drawback is to have no clear economic interpretation at the moment. Also, associated numerical methods still have to be developed.
The study of relations between chance constraints (constraints on the probability of some event) and robust optimization is the subject of intense research. The idea is, roughly speaking, to solve a robust optimization (some classes of which are tractable in the sense of algorithmic complexity). See the recent work by BenTal and Teboulle .
The case of continuoustime can be handled with the Bellman dynamic programming principle, which leads to obtain a characterization of the value function as solution of a second order HamiltonJacobiBellman equation , .
Sometimes this 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 , .
The numerical discretization of second order HJB equations was the subject of several contributions. The book of KushnerDupuis gives a complete synthesis on the chain Markov schemes (i.e Finite Differences, semiLagrangian, 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).
In the framework of the thesis of R. Apparigliato (that will finish at the end of 2007) we have studied the robust optimization approach to stochastic programming problems, in the case of hydroelectric production, for one valley. The main difficulty lies with both the dynamic character of the system and the large number of constraints (capacity of each dam). We have also studied the simplified electricity production models for respecting the “margin” constraint. In the framework of the thesis of G. Emiel and in collaboration with CEPEL, we have studied largescale bundle algorithms for solving (through a dual “price decomposition” method) stochastic problems for the Brazilian case.
For solving stochastic control problems, we studied the socalled 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 SternBrocot tree . The GFD scheme was used as a basis for the approximation of an HJB equation coming from a superreplication problem. The problem was motivated by a study conducted in collaboration with Société Générale, see .
Within the framework of the thesis of Stefania Maroso, we also contributed to the study of the error estimate of the approximation of Isaac equation associated to a differential game with one player , and also for the approximation of HJB equation associated with the impulse problem .
The field has been strongly influenced by the work of J.L. Lions, who started its systematic study of optimal control problems for PDEs in , in relation with singular perturbation problems , and illposed problems . A possible direction of research in this field consists in extending results from the finitedimensional case such as Pontryagin's principle, secondorder conditions, structure of bangbang controls, singular arcs and so on. On the other hand PDEs have specific features such as finiteness of propagation for hyperbolic systems, or the smoothing effect of parabolic sytems, so that they may present qualitative properties that are deeply different from the ones in the finitedimensional case.
The study of controllability properties for infinite dimensional systems is an illustration of this point. In view of the finiteness of propagation for hyperbolic systems, instantaneous controllability is not possible (as it is in the finitedimensional case) and specific tools have to be developped. The theme has been very active since the eighties, see e.g. . When analyzing whether the solution of the PDE can be driven to a given final target by means of a finite energy control applied say on a part of the boundary of the domain, it follows from functional analysis that the desired surjectivity property is equivalent to “strong” injectivity of the adjoint mapping, and it can be shown easily that the latter is nothing but an observability property (similarly to the finitedimensional case). The Hilbert Uniqueness Method consists in determining the solution with minimal energy (solution of some linear quadratic optimal control). It is by now well known that, for hyperbolic equations such as wavelike equations, the discretization of the HUM method fails for most numerical schemes, due to to high frequency spurious solutions; specific remedies such as Tychonoff regularization, multigrid methods, mixed finite elements, numerical viscosity and filtering of high frequencies are described in , . Observability/controllability properties depend in a very sensitive way on the class of PDE under consideration. The heat and wave equations behave in a significantly different way, because of their different behavior with respect to time reversal.
Unilateral systems in mechanics, plasticity theory, multiphases heat (Stefan) equations, etc. are described by inequalities; see Duvaut and Lions , Kinderlehrer and Stampacchia . For an overview in a finite dimensional setting, see Harker and Pang . Optimizing such sytems often needs dedicated schemes with specific regularization tools, see Barbu , Bermúdez and Saguez . Nonconvex variational inequalities can be dealt as well in Controllability of such systems is discussed in Brogliato et al. .
As for finitedimensional problems, but with additional difficulties, there is a need for a better understanding of stability and sensitivity issues, in relation with the convergence of numerical algorithms. The secondorder analysis for optimal control problems of PDE's in dealt with in e.g. , . No much is known in the case of state constraints. At the same time the convergence of numerical algorithms is strongly related to this secondorder analysis.
Many models in control problems couple standard finite dimensional control dynamics with partial differential equations (PDE's). For instance, a well known but difficult problem is to optimize trajectories for planes landoff, so as to minimize, among others, noise pollution. Noise propagation is modeled using wave like equations, i.e., hyperbolic equations in which the signal propagates at a finite speed. By contrast when controlling furnaces one has to deal with the heat equation, of parabolic type, which has a smooting effect. Optimal control laws have to reflect such strong differences in the model.
Let us mention some applications where optimal control of PDEs occurs. One can study the atmospheric reentry problem with a model for heat diffusion in the vehicle. Another problem is the one of traffic flow, modeled by hyperbolic equations, with control on e.g. speed limitations. Of course control of beams, thin structures, furnaces, are important.
An overview of sensitivity analysis of optimization problems in a Banach space setting, with some applications to the control of PDEs of elliptic type, is given in the book . See also .
We studied various regularization schemes for solving optimal control problems of variational inequalities: see Bonnans and D. Tiba , Bonnans and E. Casas , Bergounioux and Zidani . The wellposed of a “nonconvex” variational inequality modelling some mechanical equilibrium is considered in Bonnans, Bessi and Smaoui .
In Coron and Trélat , , we prove that it is possible, for both heat like and wave like equations, to move from any steadystate to any other by means of a boundary control, provided that they are in the same connected component of the set of steadystates. Our method is based on an effective feedback procedure which is easily and efficiently implementable. The first work was awarded SIAM Outstanding Paper Prize (2006).
Our techniques are generic, but we have especially in view the fields of Aerospace trajectories (rockets, planes), automotive industry (car design), chemical engineering (optimization of transient phases, batch processes).
There are also important economic applications such as the optimization of storage and management, especially of natural and power resources, portfolio optimization.
COTCOT (Conditions of Order Two and COnjugate Times). Freeware developed by B. Bonnard, J.B. Caillau and E. Trélat in 20042005. This package of routines, callable from MATLAB, generates, for a given optimal control problem, the equations of the maximum principle using automatic differentiation (based on ADIFOR). A FORTRAN code is generated and mex files are created for MATLAB. Numerical integration of the underlying differential equations is led, and the solution of the associated shooting problem is then obtained using FORTRAN codes interfaced with MATLAB.
This software, motivated by our studies for the CNES, was used to solve the atmospheric reentry problem and the orbit transfer problem. A more specific use of that software is currently of interest for EADS. The latter supports a study implementing a specific tool for the orbit transfer.
SIMPLICIAL software for indirect shooting
LAUNCHER software for space launcher trajectory optimization.Based on the Simplicialpackage, this software has been specifically developed for the CNES by P. Martinon for the study of singular arcs in realistic launcher problems, in the framework of a contract with the CNES within the OPALE group.
TOPAZE code for trajectory optimization.Developed in the framework of the PhD Thesis of J. LaurentVarin, supported by CNES and ONERA. Implementation of an interiorpoint algorithm for multiarc trajectory optimization, with builtin refinement. Applied to several academic, launcher and reentry problems.
SOHJB code for second order HJB equations. Developped since 2004 in C++. It solves the stochastic HJB equations in dimension 2. The code based on the Generalized Finite Differences, include a step of decomposition of the covariance matrices in elementary diffusions pointing towards grid points. The implementation is very fast and was mainly tested on academic examples.
Sparse HJBUltrabee. Developped by N. Megdich, in collaboration with O. Bokanowski (Paris 6 & 7), in Scilab to deal with optimal control problems with 2 or 3 state variables. A specific software dedicated to space problems is currently developped, with E. Cristiani, in C++, in the framework of a contract with the CNES.
The reference
gives stability results for nonlinear optimal control problems subject to a regular state constraint of
secondorder. The strengthened LegendreClebsch condition is assumed to hold, and no assumption on the structure of the contact set is made. Under a weak secondorder sufficient condition
(taking into account the active constraints), we show that the solutions are Lipschitz continuous w.r.t. the perturbation parameter in the
L^{2}norm, and Hölder continuous in the
norm. We use a generalized implicit function theorem in metric spaces by Dontchev and Hager
. The difficulty is that multipliers associated with secondorder state constraints have a low regularity
(they are only bounded measures). We obtain Lipschitz stability of a “primitive” of the state constraint multiplier.
We have an ongoing work on the characterization of either “bounded strong” or “Pontryagin mimina” satisfying a quadratic growth condition, for optimal control problems of ordinary differential equations with constraints on initialfinal state and control. No ClebschLegendre condition is assumed. This extends previous work by A. Milyutin and N. Osmolovskii where the control constraints were assumed to be uniformly linearly independent.
With J. LaurentVarin (Direction des lanceurs, CNES Evry). We started in Fall 2006 a study of the multidimensional singular arc that can occur in the atmospheric flight of a launcher. The physical reason for not having a bangbang control (despite the fact that the hamiltonian function is affine w.r.t. the control), is that aerodynamic forces may make a high speed ineffective. We investigate variants of Goddard's problems for nonvertical trajectories. The control is the thrust force, and the objective is to maximize a certain final cost, typically, the final mass. In this article, performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal trajectories may involve singular arcs (along which the norm of the thrust is neither zero nor maximal), that are computed and characterized. Numerical simulations are carried out, both with direct and indirect methods, demonstrating the relevance of taking into account singular arcs in the control strategy. The indirect method we use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method. The homotopic approach leads to a quadratic regularization of the problem and is a way to tackle with the problem of nonsmoothness of the optimal control.
With J. LaurentVarin (Direction des lanceurs, CNES Evry). In the frame of a research contract with the CNES (french space agency), we started in Fall 2006 a study of the singular arcs that can occur in the atmospheric climbing phase of a space launcher. These singular arcs correspond to time intervals where the optimal thrust level of the launcher is neither maximal nor zero, contrary to the usual bangbang command law (which comes from the fact that the Hamiltonian of the system is affine with respect to the control). The physical reason behind this phenomenon is that aerodynamic forces may make high speed ineffective (namely the drag term, proportional to the speed squared).
We first investigate threedimensional (i.e. nonvertical) variants of the Goddard problem, with the thrust force as the control and typically maximizing the final mass as the cost. Performing an analysis based on the Pontryagin Maximum Principle, we prove that optimal trajectories may involve singular arcs, that are computed and characterized. Numerical experimentations are carried out, both with indirect and direct methods, demonstrating the relevance of taking into account the singular arcs in the control strategy. The indirect method we primarily use is based on our previous theoretical analysis and consists in combining a shooting method with an homotopic method (continuation). The homotopic approach involves a quadratic regularization of the problem and is a way to handle the nonsmoothness of the optimal control, providing both sufficient information on the singular structure of the control and a suitable initial point for the shooting method. These results are published in .
Then we tackle a heavy multistage launcher problem (an Ariane V flight to the geostationary transfer orbit) with a realistic physical model for the thrust and drag forces. As a preliminary result, we first solve 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. Once again, we primarily use an indirect shooting method (based on Pontryagin's Maximum Principle), coupled to a continuation (homotopy) approach. We introduce a new way to determine the singular control, the usual conditions based on the derivatives of the Hamiltonian with respect to the control being unusable due to the presence of tabulated data in the physical model. The solutions we obtain are confirmed by some additional experiments with a direct method. 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 or without singular arcs. Future developments include the study of next generation launchers, reusable and/or aerolifted.
This work led to the development of two numerical codes for solving optimal control problems with singular arcs by a combined shootinghomotopy method. The first one is for the Goddard problem ( simplicial, freely available), and the second is specifically designed for the CNES heavy launcher problems ( launcher).
With Y. Chitour (U. Paris XI) and F. Jean (ENSTA). When applying methods of optimal control to motion planning or stabilization problems, some theoretical or numerical difficulties may arise, due to the presence of specific trajectories, namely, minimizing singular trajectories of the underlying optimal control problem. In this article, we provide characterizations for singular trajectories of controlaffine systems. We prove that, under generic assumptions, such trajectories share nice properties, related to computational aspects; more precisely, we show that, for a generic system – with respect to the Whitney topology –, all nontrivial singular trajectories are of minimal order and of corank one. These results, established both for driftless and for controlaffine systems, extend our previous results. As a consequence, for generic controlaffine systems (with or without drift) defined by more than two vector fields, and for a fixed cost, there do not exist minimizing singular trajectories. Besides, we prove that, given a controlaffine system satisfying the Lie algebra rank condition, singular trajectories are strictly abnormal, generically with respect to the cost. We then show how these results can be used to derive regularity results for the value function and in the theory of HamiltonJacobi equations, which in turn have applications for stabilization and motion planning, both from the theoretical and implementation issues.
With L. Rifford (U. Nice). Let
Mbe a smooth connected and complete manifold of dimension
n, and
be a smooth nonholonomic distribution of rank
mnon
Min this article. We prove that, if there exists a smooth Riemannian metric on
for which no nontrivial singular path is minimizing, then there exists a smooth repulsive stabilizing section of
on
M. Moreover, in dimension three, the assumption of the absence of singular minimizing horizontal paths can be dropped in the Martinet case. The proofs are based on the study, using
specific results of nonsmooth analysis, of an optimal control problem of Bolza type, for which we prove that the corresponding value function is semiconcave and is a viscosity solution of a
HamiltonJacobi equation, and establish fine properties of optimal trajectories.
With J.B. Lasserre, D. Henrion and C. Prieur, LAAS, Toulouse. We consider in the class of nonlinear optimal control problems (OCP) with polynomial data, i.e., the differential equation, state and control constraints and cost are all described by polynomials, and more generally for OCPs with smooth data. In addition, state constraints as well as state and/or action constraints are allowed. We provide a simple hierarchy of LMI (linear matrix inequality)relaxations whose optimal values form a nondecreasing sequence of lower bounds on the optimal value. Under some convexity assumptions, the sequence converges to the optimal value of the OCP. Preliminary results show that good approximations are obtained with few moments.
With P. Lotito, U. Tandil (Argentina). In the framework of the STIC AmSud “Energetic Optimization” project and of the internship of S. Aronna, we have established some qualitative properties of optimal controls for a continuoustime hydrothermal electricity production model. We show that in the case of several “parallel” dams, nonuniqueness of optimal trajectories may occur, and we characterize them in some cases. Then we discuss singular arcs using a reformulation of controls that allows to separate the “linear” and “nonlinear controls.
In collaboration with O. Bokanowski (Lab. JLL, Paris 7), we have continued the study of antidiffusive numerical schemes for HJB equations coming from stateconstrained optimal control problems (RDV problems, target problems, capture basin).
In
, we prove the convergence of a nonmonotonous scheme for a onedimensional first order
HamiltonJacobiBellman equation of the form
,
v(0,
x) =
v_{0}(
x). The scheme is related to the HJBUltrabee scheme suggested in
, which has an antidiffusive behavior, but which convergence was not proved. We show for general
discontinuous initial data a firstorder convergence of the scheme, in
L^{1}norm towards the viscosity solution, we also derive an error estimate (in
L^{1}norm).
Recently, P. Jaisson has started to extend these results to the convergence proof of the Nbee scheme also proposed in .
The computation of the capture basins can be reduced to the resolution of an HJB equation whose solution takes only values 0 and 1 (0 in the area of admissible trajectories and 1 in the nonadmissible area). For this situation, the HJBultrabee scheme is particularly adapted and allows an accurate localization of the interface (front) between the zone of 0 and that of 1. Taking into account this nice property, we developed numerical codes (for dimensions 2 and 3) on sparse grids allowing to store only the meshs crossed by the front. This implementation leads to a better management of the storage capacity and an important gain of computing time. Several numerical examples were performed validating the method (see the forthcoming thesis of N. Megdich). Moreover, we currently work on an optimized code in C++. Our objective being to carry out a fast code in 4d to deal with the climbing problem of a space shuttle (current contract with CNES).
In connection with the recruitment of E. Cristiani et N. Forcadel (postdoc positions), we have started to work on the generalization of the FastMarching Methods (FMM) for solving general HJB equations.
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 assets.
In the particular case of an obstacle problem of the form
min(
A
x
b,
x
g) = 0where
Ais an
N×
Nmatrix satisfying a monotonicity assumption, the convergence of Howard's algorithm may be achieved in no more than
Niterations, instead of the usual
2
^{N}bound. Still in the case of obstacle problem, we established the equivalence between Howard's algorithm and a primaldual active set algorithm
.
We also propose an Howardtype algorithm for a "doubleobstacle" problem of the form
max(min(
A
x
b,
x
g),
x
h) = 0.
We finally illustrate the algorithms on the discretization of nonlinear PDE's arising in the context of mathematical finance (American option, and Merton's portfolio problem), and for the doubleobstacle problem.
In collaboration with O. Bokanowski
In , we study an approximation scheme for the HJB equation based on the generalized finite differences algorithm introduced in , . We prove the existence, uniqueness of a bounded discrete solution. We also verify the monotonicity and stability of the scheme. Moreover, we give a consistance error approximation. Then, by using the same arguments as in , we prove the convergence of the discrete solutions towards the value function , when the discretization step size tends to 0. Moreover, we have perfomed several numerical tests to validate the theoretical convergence result.
Within a framework of Internship (PFE) of Saad Serghini (from EMI School, Rabat), we studied the pricing of AmerAsian options. This pricing leads to a second order variational inequations whose diffusion term (matrix of covariance) may be degenerated and is dominated by an advection term. In order to limit the numerical diffusion, in particular when covariance is cancelled, we tested and compared several antidiffusive schemes (VanLeer, Nbee, Discrete Galerkin). See the report .
With C. Sagastizábal, CEPEL, Rio de Janeiro. In the framework of G. Emiel's PhD thesis we study possible evolutions of nonsmooth optimization algorithms when dealing with large scale problems. One of the main motivations is the resolution of stochastic optimization problems through Lagrangian decomposition. Those problems arise in particular in midterm production planning. The past year focused on two approaches : dynamic Lagrangian relaxation, and incremental resolution.
The Lagrangian relaxation framework is commonly used in the resolution process of complex optimization problems. For example, it allows to generate bounds for mixedinteger linear programming problems. However, when the number of dualized constraints is very large (exponential in the dimension of the primal problem), explicit dualization is not possible. To reduce the dual dimension, heuristics were proposed in the literature that involve a separation procedure to dynamically select a restricted set of constraints to be dualized along the iterations. This relax–and–cut type approach has shown numerical efficiency. Belloni and Sagastizábal have obtained recently primaldual convergence when using an adapted bundle method for the dual step, under minimal assumptions on the separation procedure. In , we extend these results to the subgradient scheme, widely used in the mixedinteger literature.
When dealing with the midterm production planning problem, Lagrangian relaxation of coupling constraints yield a separable dual function. It can be decomposed in several dual subfunctions. The idea of an incremental resolution is to take full advantage of this structure by making a dual iteration after the evaluation of each subfunction. Indeed, computational times resulting from the subproblems resolutions are preponderant. Hence, we may achieve a significant amelioration by adopting such an algorithmic scheme. It has already been studied by Nedić and Bertsekas when using a subgradient algorithm for the dual phase. We applied this approach in conjunction with Bundle algorithms. The recent theory of approximated bundle methods provides a nice framework for the convergence study. Numerical results still need to be further investigated but seem promising.
The Unit Commitment Problem (UCP) consists of defining the minimalcost power generation schedule for a given set of power plants. Due to many complex constraints, the deterministic UCP, even in its deterministic version, is a challenging largesize, nonconvex, nonlinear optimization problem, but there exist nowadays efficient tools to solve it. For a very short term horizon, the deterministic UCP is satisfactory; it is currently used for the daily scheduling in an industrial way. For the two/fourweek time horizon which we are concerned with, uncertainty becomes significant and cannot be ignored anymore, making it necessary to treat the UCP as a stochastic problem. Dealing with uncertainty introduces a level of complexity that is of an order of magnitude higher than in the deterministic case. Thus, there is a need to design new stochastic optimization techniques and models, that are implementable in an industrial context.
The first part of the PhD of R. Apparigliato focusses on the problem of hydraulic management. One of the principal uncertainty factors, affecting hydro production system, is reservoir inflows. Indeed, these are highly variable. We can forecast these inflows on the two next weeks and estimate the maximal distances on the associated trend. Optimal decisions obtained with a deterministic approach aren't robust for inflow realizations and, so, lead to constraints violations. In this context, we suggest approaching this problem with the help of robust optimization, in order to obtain robust production decisions. This choice is justified first of all by the necessity of implementing a new approach and making a comparison with respect to existing techniques. Furthermore it is justified due to the important economical interests in the definition of a robust hydraulic management on the weekly horizon. Such an approach was tested on a representative valley of French system. The valley is managed on 7 days (84 time steps) and consists on 3 reservoirs and 6 production groups.
Third year: The third year of the PhD served for finalizing this study, for making a summary of the works of the first two years on this subject. The robust approaches were compared with a deterministic one, the so called Deterministic with Periodic Revision (DPR), close to current practices in exploitation. The main results show that robust models allow us to decrease strongly the volume constraints violations (by about 75  95% in our example) until making them almost nil. The costs are only increased by 0,5%. The size of the model is almost the same as the original deterministic one. The results obtained by the robust approach are thus satisfactory and allow us to eventually implement it in an operational context, thus taking uncertainty into account. A large number of oral communications have been given on this subject. An internal report which introduces robust optimization was published as well as an application of the hydraulic problem . An article is in preparation.
Abstract: In the deregulation context of the electric markets, the weekly unit commitment could play a leading role, in complement of its current role of making scheduling for the daily horizon: the management of the production margin, defined as the difference between the total offer and the total demand, taking into account stochasticity affecting the electrical system. This problem consists of determining which optimal decisions, according to a certain economical criteria, satisfy:
hedge against supply shortage risk in order to satisfy demand in 99% of the situations,
sell on markets the surplus of production with regard to a safety margin, as far as the legal and commercial measures allow it.
At present, there is no operational tool to satisfy needs relative to the role of management of the production margin and the risk. This established fact doesn't allow us to approach the problems of the risk measure, hedging or even that of trading. The second year of the PhD was partially dedicated to the formulation and to the resolution of the problem of the active management of the margin. From an analysis, a first formulation taking into account the interruption contracts and market products was made with the help of chance constraints and implemented for tests. This formulation is for the moment only time dependent (open loop).
Third year: Several research axes were examined in order to improve this openloop formulation: formulation of the problem allowing the modification maintenance schedules of thermal units, improvement of the computation of the production margin,... However, the effort was mainly focussed on the application of a formulation in closed loop, that is a control dependent on the stochastic process and of time. It could permit us to make decisions function of the uncertainty history. We propose to apply a new approach: stochastic programming with step decision rules (SPSDR). Multistage stochastic programming is known to suffer from an exponential increase of the number of variables (decisions and states) when the number of steps increases linearly. The SPSDR approach tries to approximate the solution of such problems by combining several techniques. The first idea is to work with independent experts . Every expert works on the basis of small pool of scenarios, drawn randomly in the original set of scenarios, according to the initial stochastic process. The second idea is that experts work with decision rules, combining step functions. Optimal decision rules are then combined to build the final decision rule. This last one is then applied on a large number of scenarios. First results are relatively encouraging. The determination of the margin process is described in a EDF note . The formulations of the problem in open and closed loop have lead to several oral communications.
With J.M. Coron (U. ParisSud) and R. Vazquez (U. Sevilla). We consider the problem of generating and tracking a trajectory between two arbitrary parabolic profiles of a periodic 2D
channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poiseuille flow, this problem is frequently cited as a paradigm for transition to turbulence. Our
procedure consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. Using a backstepping method, we then design boundary control
laws guaranteeing that the error between the state and the trajectory decays exponentially in
L^{2},
H^{1}, and
H^{2}norms. The result is first proved for the linearized Stokes equations, then shown to hold locally for the nonlinear NavierStokes system. see the report
.
With R. Abraham and M. Bergounioux, U. Orléans. We propose in this work a variational method for tomographic reconstruction of blurred and noised binary images based on a penalization process of a minimization problem settled in the space of bounded variation functions. We prove existence and/or uniqueness results and derive an optimality system, both for the minimization problem and its penalized version. Numerical simulations are provided to demonstrate the relevance of the approach. See .
With G. Carbou (U. Bordeaux) and S Labbé (IMAG, Grenoble). We investigate the problem of controlling the magnetic moment in a ferromagnetic nanowire submitted to an external magnetic field in the direction of the nanowire. The system is modeled with the one dimensional LandauLifschitz equation. In the absence of control, there exist particular solutions, which happen to be relevant for practical issues, called travelling walls. In this paper, we prove that it is possible to move from a given travelling wall profile to any other one, by acting on the external magnetic field. The control laws are simple and explicit, and the resulting trajectories are shown to be stable. See .
An important part of the work described below relies on recent developments on model theoretic structures (ominimal structures) which generalize axiomatically the qualitative properties
of semialgebraic sets (see van den Dries, Shiota). Semialgebraic sets are subsets of
IR^{n}defined by a finite number of polynomial equalities and inequalities. Finite union/intersection and complement of semialgebraic sets are semialgebraic; more importantly linear
projections of semialgebraic sets remain semialgebraic (Tarski Seideberg principle). These facts yield remarkable stability properties as well as a kind of “finiteness of pathologies”
principle. An illustration of these considerations could be as follows: take a bounded semialgebraic set
and a polynomial function
P:
IR^{n}×
I
R
^{m}
I
R, then the
nonsmoothfunction
has a semialgebraic graph (stability) and is smooth everywhere save perhaps on an finite union of manifolds of low dimension (finitess/tameness of pathologies). Keeping in mind the
semialgebraic model, an ominimal structure over
IRis a boolean collection
of subsets of
enjoying (in particular) two major properties: the family is stable with respect to linear projection and “onedimensional” sets are exactly finite union of intervals. A
function/pointtoset mapping is said to belong to such a structure if its graph belongs to
. This “theoretical” extensions of real algebraic geometry could seem useless if the only example of ominimal structure was given by the collection of semialgebraic sets. Two major
breakthroughs by Gabrielov (globally subanalytic sets) and Wilkie (logexp structure) have shown that ominimal structure are numerous. This fact is, to our opinion, of high importance for
applied mathematics.
The striking stability results enjoyed by such structures can be indeed used to show that many finitedimensional optimization problems are (or could be) formulated within this setting. This was the starting point for the use of such a theory in variational analysis and for the study of some related optimization algorithms. This being said, a general idea to understand what could be obtained in this framework, is to think “ominimal” problems which are often qualified in a more vivid way as “tame” as problems which are generically wellposed or wellbehaved.
More specifically the works we present here were developed in view of the study of two following general problems
convergence of gradient methods in a nonsmooth and nonconvex setting,
convergence of Newton's method either smooth or nonsmooth.
Particular attention was dedicated to the analysis of convergence rate of such methods and to what is usually called global convergence. This last term means that the sequence/curve generated by the algorithm/dynamical system converges to a specific equilibrium despite the fact that a continuum of critical points may be involved.
Superlinear convergence of the Newton method for nonsmooth equations requires a “semismoothness" assumption. In
we have proved that locally Lipschitz functions definable in an ominimal structure (in particular
semialgebraic or globally subanalytic functions) are semismooth. Semialgebraic, or more generally, globally subanalytic mappings present the special interest of being
order semismooth, where
is a
positiveparameter. As an application of this new estimate, we prove that the error at the
kth step of the Newton method behaves like
.
INRIAEDF (PhD of R. Apparigliato), 20052007. Application of recourse optimization for risk management in short term power planning. Involved researchers: F. Bonnans.
INRIAEDF (PhD of G. Emiel), 20052008. Solving large scale problems for middle term power planning. Involved researchers: F. Bonnans.
ENSTACNES (OPALE pole framework), 2007. HJB approach for the atmospheric reentry problem. Involved researchers: F. Bonnans, H. Zidani.
ENSTADGA, 20072008. Study of HJB equations associated to motion planning. Involved researchers: N. Forcadel, H. Zidani.
Univ. OrléansCNES (OPALE pole framework), 2007. Singular arcs in the launchers problem. Involved researchers: F. Bonnans, P. Martinon (INRIA), E. Trélat.
In the setting of the STIC AmSud project on “Energy Optimization” we have a collaboration with P. Lotito (U. Tandil) on deterministic continuoustime models for the optimization of hydrothermal electricity.
With Claudia Sagastizábal, CEPEL, Rio de Janeiro : we are currently analysing some approaches for stochasting programming, with application to the production of electricity.
With Felipe Alvarez (CMM, Universidad de Chile, Santiago) we study the logarithmic penalty approach for optimal control problems.
Italy: U. Roma (Sapienza). With M. Falcone: numerical methods for the resolution of HJB equations.
Portugal: U. Aveiro. With D.F.M. Torres. Invariant laws and optimal syntheses.
Longterm visits: N. Osmolovskii (6 months, SRI, Warsaw, Poland).
Shortterm visits: invited professors, Claudia Sagastizábal and Mikhail Solodov (1 week, IMPA, Rio de Janeiro, Brazil), Hector RamirezCabrera (1 week, CMM and U. Chile, Santiago), Pablo Lotito (2 weeks, Argentina).
Student internships: Soledad Aronna, U. Rosario, Argentina, 3 months, and Saad Serghini, EMI, Rabat, 3 months.
F. Bonnans is one of the three Corresponding Editors of “ESAIM:COCV” (Control, Optimisation and Calculus of Variations), and Associate Editor of “Applied Mathematics and Optimization”, and “Optimization, Methods and Software”.
F. Bonnans is member of the CouncilatLarge of the Mathematical Programming Society (20062009), member of the Optimal Control Technical Committee 2.4 of IFAC (International Federation of Automatic Control), 20052008, and member of the board of the SMAIMODE group (20072010). He is one of the organizers of SPO (Séminaire Parisien d'Optimisation, IHP, Paris).
E. Trélat is Associate Editor of “ESAIM:COCV” (Control, Optimisation and Calculus of Variations) and of "International Journal of Mathematics and Statistics".
F. Bonnans
Associate Professor, Ecole Polytechnique (Courses on Operations Research and Optimal Control, 50 h), and Course on Continuous Optimization, Mastere de Math. et Applications, Filière "OJME", Optimisation, Jeux et Modélisation en Economie, Université Paris VI (18 h).
Introduction to stochastic programming. I Escuela Franco LatinoAmericana de Optimizacion Energetica. Pergamino (Argentina), 8 h, April 2328, 2007.
N. Forcadel
Numerical methods in finance, 12h (third year of ENSTA M2 MMMEF, U. Paris 1).
Minicourse Fast Marching Method, Automn School on “Introduction to numerical methods for moving boundaries”, Ensta, November 1214, 2007.
A. Hermant
ENSTA: Quadratic Optimization(first year, 12h), Dynamical systems and introduction to automatic control(first year, 12h).
ENSMP (Mines): Mathematics (integration), first year, 26h.
P. Jaisson
Course on partial differential equations, L3 (third year), University of Versailles Saint Quentin (UVSQ), 36h.
E. Trélat
Professor of Mathematics, University of Orléans where he teaches continuous optimization, numerical analysis, automatic, and optimal control. He is responsible of the Master of Mathematics which involves in particular automatic and control.
Courses on Optimal Controlin ENSTA (22 h) and in the Master ATSI of University ParisSud (12h).
H. Zidani  Professeur at ENSTA (70h)
Courses at ENSTA: Quadratic Optimization(first year, 21h), "Front propagation" course (third year), HamiltonJacobiBellman approach to Optimal Control(third year, 21 h), “Numerical methods for front propagation” (third year, 21 h),
`Courses in the Master “Ingénierie Mathématiques”, U. of ParisSud Orsay.
ATS of University ParisSud : `Optimal control” (30 h).
N. Megdich
As duty of the half ATER position: Automatic control(Lessons, Master 1, 30h, Supervision of personal work, Master 1, and Exercices, Master 2, 30h).
Autumn School, Course Introduction to numerical methods for moving boundaries, Ensta, 1214 Novembre, 2007. N. Forcadel and H. Zidani, members of organizing commitee.
CEAEDFINRIA Course Optimal control: Algorithms and Applications, Rocquencourt, 30 May1st June 2007. F. Bonnans, organizer.
Journées CODE 07 (Conférence de la SMAI sur l'Optimisation et la Décision), Institut Henri Poincaré, 1820 avril 2007. F. Bonnans, member of organizing commitee.
CIFA 2008, Bucharest. E. Trélat, in charge of the stream “Optimization and Control of nonlinear systems”.
E. Trélat : ANR SICOMAF Thematic day “Control and Ferromagnetism”, Université ParisSud 11 (Orsay), June 21, 2007.
E. Trélat: CzechFrenchGerman Conference on Optimization, Heidelberg, Sept. 1721, 2007, and Conference SMAI 2007, PrazsurArly, June 48.
F. Bonnans: EuroptOMS joint meeting: 2nd Conference on Optimization Methods and Software and 6th EUROPT Workshop "Advances in Continuous Optimization", Prague, July 47, 2007.
R. Apparigliato
Andrieu, L. and Apparigliato, R. and Lisser, A. and Tang, A.: Stochastic Optimization under Risk Constraints: Application to Hedging Problem in Electrical Industry. Power Systems Management Conference, Athens, 58 June 2007.
Andrieu, L. and Apparigliato, R. and Lisser, A. and Tang, A.: Stochastic Optimization under Risk Constraints: Application to Hedging Problem in Electrical Industry. 11th International Conference on Stochastic Programming, Vienna, 2731 August 2007.
Apparigliato, R. and Thénié, J. and Vial, J.P.: Step decision rules for multistage stochastic programming: Application on a Hedging Problem in Electrical Industry. 11th International Conference on Stochastic Programming, Vienna, 2731 August 2007.
Apparigliato, R. and Vial, J.P. and Zorgati, R.: Gestion hebdomadaire d'une vallée hydraulique par optimisation robuste. Conférence conjointe FRANCOROROADEF, Grenoble, 2023 Feb. 2007.
Apparigliato, R. and Vial, J.P. and Zorgati, R.: Weekly management of a hydraulic valley by robust optimization. 3ème Cycle Romand de Recherche Opérationnelle, Zinal (Suisse), 48 March 2007
J. Bolte
Characterizations of KurdykaLojasiewicz inequality. International Conference on "Nonconvex Programming, Local and Global Approaches. Theory, Algorithms and Applications". INSA, Rouen, Dec. 1721.
F. Bonnans
S. Aronna, F. Bonnans, P. Lotito: Optimal control techniques for hydropower production. International Conference on "Nonconvex Programming, Local and Global Approaches. Theory, Algorithms and Applications". INSA, Rouen, Dec. 1721.
G. Emiel
G. Emiel, C. Sagastizabal: Dynamic subgradient methods. European Conference on Operational Research EURO XXII, Prague, July 2007
G. Emiel: Modeling dependency for Credit Risk. Mathematics and Finance : Research in Options, Rio de Janeiro, Oct. 2007.
A. Hermant
Conditions d'optimalité du secondordre pour les problèmes de commande optimale avec contraintes sur l'état. Application au tir. Journée des Doctorants du CMAP, Ecole Polytechnique, March 7, 2007.
Conditions d'optimalité du secondordre pour les problèmes de commande optimale avec contraintes vectorielles sur l'état ; application au tir. Conférence sur l'Optimisation et la Décision CODE 2007, Paris, April 1820, 2007.
Stability and sensitivity analysis for optimal control problems with a firstorder state constraint and application to continuation methods. 23rd IFIP Conference on System Modeling and Optimization Cracow, Poland, July 2327, 2007.
Stability and sensitivity analysis for optimal control problems with a firstorder state constraint and application to continuation methods. 13th CzechFrenchGerman Conference on Optimization Heidelberg, Sept. 1721, 2007.
N. Megdich
A fast antidissipative method for the minimum time problem. Application to atmospheric reentry. Conférence sur l'Optimisation et la Décision CODE 2007, Paris, April 1820, 2007.
P. Martinon
Détection de commutations et utilisation des équations variationnelles pour la méthode de tir. Conférence sur l'Optimisation et la Décision CODE 2007, Paris, April 1820, 2007.
SNumerical conservation of first integrals for an optimal control problem with control discontinuities. 13th CzechFrenchGerman Conference on Optimization Heidelberg, Sept. 1721, 2007.
E. Ottenwaelter
Schémas numériques de résolution de l'équation de HamiltonJacobiBellman de la commande optimale stochastique. Journée des Doctorants du CMAP, Ecole Polytechnique, March 7, 2007.
E. Trélat
R. Abraham, M. Bergounioux, E. Trélat: A penalization approach for tomographic reconstruction of binary axially symmetric objects. International Conference on "Nonconvex Programming, Local and Global Approaches. Theory, Algorithms and Applications". INSA, Rouen, Dec. 1721.
R. Apparigliato: Gestion hebdomadaire d'une vallée hydraulique par optimisation robuste. Séminaire scientifique de l'Institut Français du Pétrole, RueilMalmaison, 26 Oct. 2007.
N. Forcadel: Résultats d'homogénéisation pour la dynamique des dislocations et pour certains systèmes de particules. Séminaire d'analyse appliquée, Université, Nov.26, 2007.
Résultats d'homogénéisation pour la dynamique des dislocationsGroupe de travail Mécanique des fluides, U.P.S. Toulouse, Oct. 25, and Groupe de travail d'homogénéisation, U. Paris, Nov. 19, 2007, and Séminaire d'analyse numérique, Université de Rennes, Dec. 20, 2007.
E. Trélat: Regularity of the value function in optimal control; applications to viscosity solutions and to stabilization. Séminaire d'Analyse Appliquée, Univ. Brest (Jan. 16), Seminar of Mathematics, U. Orléans, (Feb 1st)., “Image day”, U. Orléans, (Oct. 25).
H. Zidani: Numerical Methods for HamiltonJacobi Equations. Séminaire du Lab. JacquesLouis Lions, Université ParisVI, Paris (March 8).