Commande optimale

Article pour l'encyclopédie des sciences de l'ingénieurThe optimal control theory analyzes how to optimize dynamical systems with various criteria : reach a target in minimal time or minimal energy, maximize the efficiency of an industrial process for instance. This involves the optimization of both time independent parameters, and the control variables that are function of time. The article analyzes the first and second order optimality conditions, and the ways to solve them, by time discretization, the shooting algorithm, or dynamic programming.L'objet de la commande optimale est l'optimisation de systèmes dynamiques suivant différents objectifs : atteinte d'une cible en temps ou énergie minimale, maximisation du rendement d'un processus industriel par exemple. Pour cela on joue à la fois sur des paramètres indépendants du temps et sur les commandes qui, elles, dépendent du temps. L'article analyse les conditions d'optimalité du premier et second ordre, et leur résolution par discrétisation temporelle, algorithme de tir, ou programmation dynamique

First and second order necessary conditions for stochastic optimal control problems

International audienceIn this work we consider a stochastic optimal control problem with either convex control constraints or finitely many equality and inequality constraints over the final state. Using the variational approach, we are able to obtain first and second order expansions for the state and cost function, around a local minimum. This fact allows us to prove general first order necessary condition and, under a geometrical assumption over the constraint set, second order necessary conditions are also established. We end by giving second order optimality conditions for problems with constraints on expectations of the final state

Error estimates for the logarithmic barrier method in stochastic linear quadratic optimal control problems

International audienceWe consider a linear quadratic stochastic optimal control problem whith non-negativity control constraints. The latter are penalized with the classical logarithmic barrier. Using a duality argument and the stochastic minimum principle, we provide an error estimate for the solution of the penalized problem which is the natural extension of the well known estimate in the deterministic framework

Optimal control of PDEs in a complex space setting; application to the Schrödinger equation

International audienceIn this paper we discuss optimality conditions for abstract optimization problems over complex spaces. We then apply these results to optimal control problems with a semigroup structure. As an application we detail the case when the state equation is the Schrödinger one, with pointwise constraints on the "bilinear'" control. We derive first and second order optimality conditions and address in particular the case that the control enters the state equation and cost function linearly

A General Optimal Multiple Stopping Problem with an Application to Swing Options

International audienceIn their paper, Carmona and Touzi [8] studied an optimal multiple stopping time problem in a market where the price process is continuous. In this article, we generalize their results when the price process is allowed to jump. Also, we generalize the problem associated to the valuation of swing options to the context of jump diffusion processes. We relate our problem to a sequence of ordinary stopping time problems. We characterize the value function of each ordinary stopping time problem as the unique viscosity solution of the associated Hamilton–Jacobi–Bellman variational inequality

On the time discretization of stochastic optimal control problems: The dynamic programming approach

In this work, we consider the time discretization of stochastic optimal control problems. Under general assumptions on the data, we prove the convergence of the value functions associated with the discrete time problems to the value function of the original problem. Moreover, we prove that any sequence of optimal solutions of discrete problems is minimizing for the continuous one. As a consequence of the Dynamic Programming Principle for the discrete problems, the minimizing sequence can be taken in discrete time feedback form.Fil: Joseph Frédéric, Bonnans. Institut National de Recherche en Informatique et en Automatique; Francia. Centre National de la Recherche Scientifique; Francia. Université Paris-Saclay; FranciaFil: Gianatti, Justina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas. Universidad Nacional de Rosario. Centro Internacional Franco Argentino de Ciencias de la Información y de Sistemas; ArgentinaFil: Silva, Francisco J.. Centre National de la Recherche Scientifique; Francia. Universite de Limoges; Franci

Modelling and optimal control of a two-species bioproducing microbial consortium

Motivated by recent laboratory experiments, we study microbial populations with light-inducible genetic differentiation that generates a two-species microbial consortium relevant for bioproduction. First, we derive a hierarchy of models describing the evolution of the microbial populations, each with decreasing complexity. This sequential order reduction reveals the connections between several popular classes of models used in this context. Second, we demonstrate the analytical insight the order reduction provides by studying the optimal control of such a reduced-order system of nonlinear ordinary differential equations. Appealing to Pontryagin's maximum principle, we find different optimal control structures within different regions of the parameter space. Explicit solutions are obtained in a subset of parameter space, while, for the remainder of parameter space, closed-form solutions are obtained that depend on a scalar value that solves a particular transcendental equation. We show that a unique solution of the scalar equation exists and lies in a known compact interval, making its numerical approximation particularly easy. The analytical results are verified against direct numerical calculations

A Lagrangian approach for aggregative mean field games of controls with mixed and final constraints

The objective of this paper is to analyze the existence of equilibria for a class of deterministic mean field games of controls. The interaction between players is due to both a congestion term and a price function which depends on the distributions of the optimal strategies. Moreover, final state and mixed state-control constraints are considered, the dynamics being nonlinear and affine with respect to the control. The existence of equilibria is obtained by Kakutani's theorem, applied to a fixed point formulation of the problem. Finally, uniqueness results are shown under monotonicity assumptions