Control Strategies for the Fokker-Planck Equation

Using a projection-based decoupling of the Fokker-Planck equation, control strategies that allow to speed up the convergence to the stationary distribution are investigated. By means of an operator theoretic framework for a bilinear control system, two different feedback control laws are proposed. Projected Riccati and Lyapunov equations are derived and properties of the associated solutions are given. The well-posedness of the closed loop systems is shown and local and global stabilization results, respectively, are obtained. An essential tool in the construction of the controls is the choice of appropriate control shape functions. Results for a two dimensional double well potential illustrate the theoretical findings in a numerical setup

Optimal Control for a Class of Infinite Dimensional Systems Involving an L∞L^\infty-term in the Cost Functional

An optimal control problem with a time-parameter is considered. The functional to be optimized includes the maximum over time-horizon reached by a function of the state variable, and so an $L^\infty$-term. In addition to the classical control function, the time at which this maximum is reached is considered as a free parameter. The problem couples the behavior of the state and the control, with this time-parameter. A change of variable is introduced to derive first and second-order optimality conditions. This allows the implementation of a Newton method. Numerical simulations are developed, for selected ordinary differential equations and a partial differential equation, which illustrate the influence of the additional parameter and the original motivation.Comment: 21 pages, 8 figure

Mean field optimization problems: stability results and Lagrangian discretization

We formulate and investigate a mean field optimization (MFO) problem over a set of probability distributions $\mu$ with a prescribed marginal $m$. The cost function depends on an aggregate term, which is the expectation of $\mu$ with respect to a contribution function. This problem is of particular interest in the context of Lagrangian potential mean field games (MFGs) and their discretization. We provide a first-order optimality condition and prove strong duality. We investigate stability properties of the MFO problem with respect to the prescribed marginal, from both primal and dual perspectives. In our stability analysis, we propose a method for recovering an approximate solution to an MFO problem with the help of an approximate solution to an MFO with a different marginal $m$, typically an empirical distribution. We combine this method with the stochastic Frank-Wolfe algorithm of a previous publication of ours to derive a complete resolution method

A mesh-independent method for second-order potential mean field games

This article investigates the convergence of the Generalized Frank-Wolfe (GFW) algorithm for the resolution of potential and convex second-order mean field games. More specifically, the impact of the discretization of the mean-field-game system on the effectiveness of the GFW algorithm is analyzed. The article focuses on the theta-scheme introduced by the authors in a previous study. A sublinear and a linear rate of convergence are obtained, for two different choices of stepsizes. These rates have the mesh-independence property: the underlying convergence constants are independent of the discretization parameters

Two methods of pruning Benders' cuts and their application to the management of a gas portfolio

In this article, we describe a gas portfolio management problem, which is solved with the SDDP (Stochastic Dual Dynamic Programming) algorithm. We present some improvements of this algorithm and focus on methods of pruning Benders' cuts, that is to say, methods of picking out the most relevant cuts among those which have been computed. Our territory algorithm allows a quick selection and a great reduction of the number of cuts. Our second method only deletes cuts which do not contribute to the approximation of the value function, thanks to a test of usefulness. Numerical results are presented.Dans cet article, nous décrivons un problème de gestion d'un portefeuille gazier, résolu avec l'algorithme SDDP (Stochastic Dual Dynamic Programming). Nous présentons quelques améliorations de cette algorithme et nous nous concentrons sur des méthodes d'élagage des coupes de Benders, c'est-à-dire, des méthodes pour sélectionner les coupes les plus pertinentes parmi celles déjà calculées. Notre algorithme des territoires permet une sélection rapide et une grande réduction du nombre de coupes. Notre seconde méthode ne supprime que les coupes qui ne contribuent pas à l'approximation de la fonction valeur, à l'aide d'un test d'utilité. Nous présentons des résultats numériques

Sensitivity analysis for the outages of nuclear power plants

International audienceNuclear power plants must be regularly shut down in order to perform refueling and maintenance operations. The scheduling of the outages is the first problem to be solved in electricity production management. It is a hard combinatorial problem for which an exact solving is impossible. Our approach consists in modelling the problem by a two-level problem. First, we fix a feasible schedule of the dates of the outages. Then, we solve a low-level problem of optimization of elecricity production, by respecting the initial planning. In our model, the low-level problem is a deterministic convex optimal control problem. Given the set of solutions and Lagrange multipliers of the low-level problem, we can perform a sensitivity analysis with respect to dates of the outages. The approximation of the value function which is obtained could be used for the optimization of the schedule with a local search algorithm.Les centrales nucléaires doivent être régulièrement arrêtées afin de réaliser des opérations de maintenance et de rechargement en combustible nucléaire. La planification de ces arrêts constitue le premier problème à résoudre en gestion de la production d'électricité. C'est un problème combinatoire difficile qui ne peut être résolu exactement. Notre approche consiste à modéliser ce problème par un problème à deux niveaux. Tout d'abord, nous fixons un calendrier admissible des dates des arrêts des centrales. Puis, nous résolvons un sous-problème de production d'électricité, en respectant le calendrier initial. Dans notre modèle, ce sous-problème est un problème de contrôle optimal déterministe et convexe. Etant donnés les solutions et multiplicateurs de Lagrange du sous-problème, nous pouvons réaliser une analyse de sensibilité par rapport aux dates des arrêts. L'approximation de la fonction valeur que nous obtenons devrait permettre de mettre en place un algorithme de recherche locale pour l'optimisation de ces dates d'arrêts