8 research outputs found
Control problems on infinite horizon subject to time-dependent pure state constraints
In the last decades, control problems with infinite horizons and discount
factors have become increasingly central not only for economics but also for
applications in artificial intelligence and machine learning. The strong links
between reinforcement learning and control theory have led to major efforts
towards the development of algorithms to learn how to solve constrained control
problems. In particular, discount plays a role in addressing the challenges
that come with models that have unbounded disturbances. Although algorithms
have been extensively explored, few results take into account time-dependent
state constraints, which are imposed in most real-world control applications.
For this purpose, here we investigate feasibility and sufficient conditions for
Lipschitz regularity of the value function for a class of discounted infinite
horizon optimal control problems subject to time-dependent constraints. We
focus on problems with data that allow nonautonomous dynamics, and Lagrangian
and state constraints that can be unbounded with possibly nonsmooth boundaries
Partial Lipschitz regularity of the minimum time function for sub-Riemannian control systems
In Euclidean space of dimension 2 or 3, we study a minimum time problem
associated with a system of real-analytic vector fields satisfying
H\"ormander's bracket generating condition, where the target is a nonempty
closed set. We show that, in dimension 2, the minimum time function is locally
Lipschitz continuous while, in dimension 3, it is Lipschitz continuous in the
complement of a set of measure zero. In particular, in both cases, the minimum
time function is a.e. differentiable on the complement of the target. In
dimension 3, in general, there is no hope to have the same regularity result as
in dimension 2. Indeed, examples are known where the minimum time function
fails to be locally Lipschitz continuous.Comment: 18 pages, no figure
ProblĂšmes de contrĂŽle d'horizon infini sous contraintes d'Ă©tat et Ă©quations de Hamilton-Jacobi-Bellman
In this thesis we address infinite horizon control problems subject to state constraints. Partial and full sensitivity relations are obtained for nonautonomous optimal control problems in this setting, assuming the associated value function to be locally Lipschitz in the state. We also discuss sufficient conditions for the Lipschitz regularity of the value function. We focus on problems with cost functionals admitting a discount factor and allow time dependent dynamics and Lagrangians. Furthermore, state constraints may be unbounded and may have a nonsmooth boundary. Lipschitz regularity is recovered as a consequence of estimates on the distance of a given trajectory from the set of all its viable (feasible) trajectories, provided the discount rate is sufficiently large. We investigate as well the existence and uniqueness of weak solutions of nonautonomous Hamilton-Jacobi-Bellman equations on the domain (0, â) Ă A. The Hamiltonian is assumed to be merely measurable in time and the set A is closed. When state constraints arise, the classical analysis of the Hamilton-Jacobi-Bellman equation lacks an appropriate notion of solution because continuous solutions may not exist. In this work, we propose a notion of weak solution for which, under a suitable controllability assumption, existence and uniqueness theorems are valid in the class of lower semicontinuous functions vanishing at infinity. Finally, we study an autonomous Hamilton-Jacobi-Bellman equation, with Dirichlet boundary conditions, on a compact subset. We give semiconcavity results on its (unique) solution and sensitivity relations in terms of differential inclusions, extending a known result for the point-to-point sub-Riemannian distance when the Hörmander condition holds true.Dans cette thĂšse, nous abordons des problĂšmes de contrĂŽle optimal non autonomes Ă lâhorizon infini soumis Ă des contraintes dâĂ©tat. Des relations de sensibilitĂ©, partielle et totale, sont obtenues, en supposant que la fonction valeur associĂ©e soit localement Lipschitzienne par rapport Ă la variable dâĂ©tat. Nous discutons Ă©galement des conditions suffisantes pour la rĂ©gularitĂ© Lipschitz de la fonction valeur. Nous nous concentrons sur les problĂšmes liĂ©s aux fonctions de coĂ»t admettant un facteur dâactualisation, avec la dynamique et le Lagrangien dĂ©pendant du temps. De plus, les contraintes dâĂ©tat peuvent ĂȘtre non-bornĂ©s et peuvent avoir une frontiĂšre non lisse. La rĂ©gularitĂ© Lipschitz est obtenue Ă partir dâestimations sur la distance dâune trajectoire donnĂ©e de lâensemble de toutes les trajectoires viables, Ă condition que le taux dâactualisation soit suffisamment Ă©levĂ©. Nous Ă©tudions Ă©galement lâexistence et lâunicitĂ© des solutions faibles des Ă©quations non autonomes dâHamilton-Jacobi-Bellman sur un domaine de la forme (0, â)ĂA. LâHamiltonien est supposĂ© ĂȘtre uniquement mesurable par rapport au temps et lâensemble A est fermĂ©. En prĂ©sence de contraintes dâĂ©tat, (en gĂ©nĂ©ral) lâĂ©quation dâHamilton-Jacobi-Bellman nâadmet pas de solutions continues. Dans ce travail, nous proposons une notion de solution faible pour laquelle, sous une hypothĂšse de contrĂŽlabilitĂ© appropriĂ©e, les thĂ©orĂšmes dâexistence et dâunicitĂ© sont valides dans la classe des fonctions semicontinues infĂ©rieurement sâannulant Ă lâinfini. Enfin, nous Ă©tudions une Ă©quation autonome dâHamilton-Jacobi-Bellman sur un sous-ensemble compact, avec des conditions de Dirichlet sur la frontiĂšre. Dans ce contexte, nous obtenons des rĂ©sultats de semi-concavitĂ© de lâunique solution de lâĂ©quation et les relations de sensibilitĂ© sous la forme dâinclusions diffĂ©rentielles. Nous Ă©tendons ainsi un rĂ©sultat connu pour la distance sous-Riemannienne sous la condition dâHörmander
ProblĂšmes de contrĂŽle Ă lâhorizon infini soumis Ă des contraintes dâĂ©tat et Ă©quations de Hamilton-Jacobi-Bellman
Dans cette thĂšse, nous abordons des problĂšmes de contrĂŽle optimal non autonomes Ă lâhorizon infini soumis Ă des contraintes dâĂ©tat. Des relations de sensibilitĂ©, partielle et totale, sont obtenues, en supposant que la fonction valeur associĂ©e soit localement Lipschitzienne par rapport Ă la variable dâĂ©tat. Nous discutons Ă©galement des conditions suffisantes pour la rĂ©gularitĂ© Lipschitz de la fonction valeur. Nous nous concentrons sur les problĂšmes liĂ©s aux fonctions de coĂ»t admettant un facteur dâactualisation, avec la dynamique et le Lagrangien dĂ©pendant du temps. De plus, les contraintes dâĂ©tat peuvent ĂȘtre non-bornĂ©s et peuvent avoir une frontiĂšre non lisse. La rĂ©gularitĂ© Lipschitz est obtenue Ă partir dâestimations sur la distance dâune trajectoire donnĂ©e de lâensemble de toutes les trajectoires viables, Ă condition que le taux dâactualisation soit suffisamment Ă©levĂ©. Nous Ă©tudions Ă©galement lâexistence et lâunicitĂ© des solutions faibles des Ă©quations non autonomes dâHamilton-Jacobi-Bellman sur un domaine de la forme (0, â)ĂA. LâHamiltonien est supposĂ© ĂȘtre uniquement mesurable par rapport au temps et lâensemble A est fermĂ©. En prĂ©sence de contraintes dâĂ©tat, (en gĂ©nĂ©ral) lâĂ©quation dâHamilton-Jacobi-Bellman nâadmet pas de solutions continues. Dans ce travail, nous proposons une notion de solution faible pour laquelle, sous une hypothĂšse de contrĂŽlabilitĂ© appropriĂ©e, les thĂ©orĂšmes dâexistence et dâunicitĂ© sont valides dans la classe des fonctions semi-continues infĂ©rieurement sâannulant Ă lâinfini. Enfin, nous Ă©tudions une Ă©quation autonome dâHamilton-Jacobi-Bellman sur un sous-ensemble compact, avec des conditions de Dirichlet sur la frontiĂšre. Dans ce contexte, nous obtenons des rĂ©sultats de semi-concavitĂ© de lâunique solution de lâĂ©quation et les relations de sensibilitĂ© sous la forme dâinclusions diffĂ©rentielles. Nous Ă©tendons ainsi un rĂ©sultat connu pour la distance sous-Riemannienne sous la condition dâHörmander.In this thesis we address infinite horizon control problems subject to state constraints. Partial and full sensitivity relations are obtained for nonautonomous optimal control problems in this setting, assuming the associated value function to be locally Lipschitz in the state. We also discuss sufficient conditions for the Lipschitz regularity of the value function. We focus on problems with cost functionals admitting a discount factor and allow time dependent dynamics and Lagrangians. Furthermore, state constraints may be unbounded and may have a nonsmooth boundary. Lipschitz regularity is recov- ered as a consequence of estimates on the distance of a given trajectory from the set of all its viable (feasible) trajectories, provided the discount rate is sufficiently large. We investigate as well the existence and uniqueness of weak solutions of nonautonomous Hamilton-Jacobi-Bellman equations on the domain (0, â) Ă A. The Hamiltonian is assumed to be merely measurable in time and the set A is closed. When state constraints arise, the classical analysis of the Hamilton-Jacobi-Bellman equation lacks an appropriate notion of solution because continuous solutions may not exist. In this work, we propose a notion of weak solution for which, under a suitable controllability assumption, existence and uniqueness theorems are valid in the class of lower semicontinuous functions vanishing at infinity. Finally, we study an autonomous Hamilton-Jacobi-Bellman equation, with Dirichlet boundary conditions, on a compact subset. We give semiconcavity results on its (unique) solution and sensitivity relations in terms of differential inclusions, extending a known result for the point-to-point sub-Riemannian distance when the Hörmander condition holds true
Semiconcavity results and sensitivity relations for the sub-Riemannian distance
International audienceRegularity properties are investigated for the value function of the Bolza optimal control problem with affine dynamic and end-point constraints. In the absence of singular geodesics, we prove the local semiconcavity of the sub-Riemannian distance from a compact set Î â R n. Such a regularity result was obtained by the second author and L. Rifford in [Semiconcavity results for optimal control problems admitting no singular minimizing controls, Annales de l'IHP Analyse non linĂ©aire 25(4): 2008 ] when Î is a singleton. Furthermore, we derive sensitivity relations for time optimal control problems with general target sets Î, that is, without imposing any geometric assumptions on Î
Necessary conditions for infinite horizon optimal control problems with state constraints
International audiencePartial and full sensitivity relations are obtained for nonauto-nomous optimal control problems with infinite horizon subject to state constraints , assuming the associated value function to be locally Lipschitz in the state. Sufficient structural conditions are given to ensure such a Lipschitz regularity in presence of a positive discount factor, as it is typical of macroeconomics models