A viability theory approach to a two-stage optimal control problem of technology adoption

A new technology adoption problem can be modelled as a two-stage control problem, in which model parameters ("technology") might be altered at some time. An optimal solution to utility maximisation for this class of problems needs to contain information on the time, at which the change will take place (0, finite or never), along with the optimal control strategies before and after the change. For the change, or switch, to occur the "new technology" value function needs to dominate the "old technology" value function, after the switch. We charaterise the value function using the fact that its hypograph is a viability kernel of an auxiliary problem and we study when the graphs can intersect. If they do not, the switch cannot occur at a positive time. Using this characterisation we analyse a technology adoption problem and showmodels, for which the switch will occur at time zero or never.technology adoption, value function, viability kernel, viscosity solutions

Numerical optimal control for HIV prevention with dynamic budget allocation

This paper is about numerical control of HIV propagation. The contribution of the paper is threefold: first, a novel model of HIV propagation is proposed; second, the methods from numerical optimal control are successfully applied to the developed model to compute optimal control profiles; finally, the computed results are applied to the real problem yielding important and practically relevant results.Comment: Submitted pape

A viability theory approach to a two-stage optimal control problem

A two-stage control problem is one, in which model parameters (“technology”) might be changed at some time. An optimal solution to utility maximisation for this class of problems needs to thus contain information on the time, at which the change will take place (0, finite or never) as well as the optimal control strategies before and after the change. For the change, or switch, to occur the “new technology” value function needs to dominate the “old technology” value function, after the switch. We charaterise the value function using the fact that its hypograph is a viability kernel of an auxiliary problem and study when the graphs can intersect and hence whether the switch can occur. Using this characterisation we analyse a technology switching problem

On Reflecting Boundary Problem for Optimal Control

International audienceThis paper deals with Mayer's problem for controlled systems with reflection on the boundary of a closed subset K. The main result is the characterization of the possibly discontinuous value function in terms of a unique solution in a suitable sense to a partial differential equation of Hamilton–Jacobi–Bellman type

Linearization techniques for L∞\mathbb{L}^{\infty}-control problems and dynamic programming principles in classical and L∞\mathbb{L}^{\infty}-control problems

International audienceThe aim of the paper is to provide a linearization approach to the $\mathbb{L}^{\infty}$-control problems. We begin by proving a semigroup-type behaviour of the set of constraints appearing in the linearized formulation of (standard) control problems. As a byproduct we obtain a linear formulation of the dynamic programming principle. Then, we use the $\mathbb{L}^{p}$ approach and the associated linear formulations. This seems to be the most appropriate tool for treating $\mathbb{L}^{\infty}$ problems in continuous and lower semicontinuous setting

Problème de contrôle réfléchis et de jeux différentiels avec coût de type sSupremum

Ce travail concerne la caractérisation des fonctions valeur irrégulières qui apparaissent dans différents problèmes de contrôle par des EDP et contient trois parties. Premièrement, nous étudions l'existence d'une valeur discontinue d'un jeu différentiel à somme nulle avec coût en supremum. La difficulté principale est la discontinuité du coût. Nous obtenons deux théorèmes qui répondent au problème d'existence d'une valeur semicontinue du jeu. Dans la deuxième partie, il s'agit de l'étude du problème de Mayer, dont la dynamique est donnée par un système contrôlé avec réflexion sur le bord d'un ensemble fermé de K. La fonction valeur associée est alors solution d'une EDP avec conditions au bord de type Neumann si K est C1 ou d'une inéquation variationnelle si K est un rétracte proximal. Enfin, nous montrons par des techniques de viabilité que l'épigraphe de la fonction valeur de type infinum et intégral en horizon infini est un noyau de viabilité.In this work divided in three parts, we characterize the discontinuous value functions appearing in several control by PDE. First, we study the existence of the value for a zero-sum diferential game with cost of supremum type. The major difficulty here is the fact that the payoff is discontinuous. We will obtain two theorems responding to the problem of the existence of a discontinuous value for the differential game. In the second we consider the Mayer problem with a dynamics given by a control system with reflection at the boundary of a non-empty closed set K. We prove that the corresponding value function is the unique solution in a approppriate sense of a variational inequality if k is a proximal retract and the viscosity solution of a PDE with Neumann type boundary conditions when K is C1. Finally, we show using viability techniques that the epigraph of the value function of infinum and integral type (with infinit horizon) is a viability kernel.BREST-BU Droit-Sciences-Sports (290192103) / SudocSudocFranceRomaniaFRR

Linearization Techniques for Controlled Piecewise Deterministic Markov Processes; Application to Zubov's Method

International audienceWe aim at characterizing domains of attraction for controlled piecewise deterministic processes using an occupational measure formulation and Zubov's approach. Firstly, we provide linear programming (primal and dual) formulations of discounted, infinite horizon control problems for PDMPs. These formulations involve an infinite-dimensional set of probability measures and are obtained using viscosity solutions theory. Secondly, these tools allow to construct stabilizing measures and to avoid the assumption of stability under concatenation for controls. The domain of controllability is then characterized as some level set of a convenient solution of the associated Hamilton-Jacobi integral-differential equation. The theoretical results are applied to PDMPs associated to stochastic gene networks. Explicit computations are given for Cook's model for gene expression

Existence of Solutions Belonging to a Tube for Non-Convex Sweeping Processes

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