Optimal control of ordinary differential equations

3rd cycl

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

Singular arcs in the generalized Goddard's Problem

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 report, 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

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

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

Characterization of a local quadratic growth of the Hamiltonian for control constrained optimal control problems

International audienceWe consider an optimal control problem with inequality control constraints given by smooth functions satisfying the hypothesis of linear independence of gradients of active constraints. For this problem, we formulate a generalization of strengthened Legendre condition and prove that this generalization is equivalent to the condition of a local quadratic growth of the Hamiltonian subject to control constraints.Nous considérons un problème de commande optimale avec inégalités sur la commande définies par des fonctions lisses satisfaisant l'hypothèse d'indépendance linéaire des gradients des contraintes actives. Pour ce problème, nous formulons une généralisation de la condition de Legendre forte, et prouvons que cette généralisation est équivalente à la croissance quadratique du hamiltonien soumise aux contraintes sur la commande

Bocop - A collection of examples

In this document we present a collection of classical optimal control problems which have been implemented and solved with Bocop. We recall the main features of the problems and of their solutions, and describe the numerical results obtained

On the convergence of the Sakawa-Shindo algorithm in stochastic control

International audienceWe analyze an algorithm for solving stochastic control problems,based on Pontryagin's maximum principle, due to Sakawa andShindo in the deterministic case and extended to the stochasticsetting by Mazliak. We assume that either the volatility is anaffine function of the state, or the dynamics are linear. We obtain a monotone decrease of the cost functions as well as,in the convex case, the fact that the sequence of controls is minimizing, and converges to an optimal solution if it is bounded. In a specific case we interpret the algorithm as the gradient plus projection method and obtain a linear convergence rate to the solution

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