1,720,970 research outputs found
Attainability property for a probabilistic target in wasserstein spaces
Full text linkIn this paper we establish an attainability result for the minimum time function of a control problem in the space of probability measures endowed with Wasserstein distance. The dynamics is provided by a suitable controlled continuity equation, where we impose a nonlocal nonholonomic constraint on the driving vector field, which is assumed to be a Borel selection of a given set-valued map. This model can be used to describe at a macroscopic level a so-called multiagent system made of several possible interacting agents
STOCHASTIC EQUILIBRIUM SOLUTION FOR A DEBT MANAGEMENT PROBLEM WITH CURRENCY DEVALUATION
No full textConsider a model of debt management, where a sovereign state trades some bonds to service the debt with a pool of risk-neutral competitive foreign investors. At each time, the government decides which fraction of the gross domestic product (GDP) should be used to repay the debt, and how much to devaluate its currency. Both these operations have the effect to reduce the actual size of the debt, but have a social cost in terms of welfare sustainability. When the debt-to-GDP ratio reaches a given size x*, bankruptcy instantly occurs. Moreover, at any time the sovereign state can declare bankruptcy by paying a correspondent bankruptcy cost. To offset the possible loss of part of their investment, the foreign investors buy bonds at a discounted price which is not given a priori. This leads to a nonstandard optimal control problem. For a given bankruptcy threshold x*, we show that the optimization problem admits an equilibrium solution. The paper also studies properties of optimal feedback strategies, and the asymptotic behaviour of the expected total cost to the borrower as x* is pushed to infinity
Compatibility of state constraints and dynamics for multiagent control systems
Full text linkThis study concerns the problem of compatibility of state constraints with a multiagent control system. Such a system deals with a number of agents so large that only a statistical description is available. For this reason, the state variable is described by a probability measure on Rd representing the density of the agents and evolving according to the so-called continuity equation which is an equation stated in the Wasserstein space of probability measures. The aim of the paper is to provide a necessary and sufficient condition for a given constraint (a closed subset of the Wasserstein space) to be compatible with the controlled continuity equation. This new condition is characterized in a viscosity sense as follows: the distance function to the constraint set is a viscosity supersolution of a suitable Hamilton–Jacobi–Bellman equation stated on the Wasserstein space. As a byproduct and key ingredient of our approach, we obtain a new comparison theorem for evolutionary Hamilton–Jacobi equations in the Wasserstein space
Constrained Mean Field Games Equilibria as Fixed Point of Random Lifting of Set-Valued Maps
Full text linkWe introduce an abstract framework for the study of general mean field game and mean field control problems. Given a multiagent system, its macroscopic description is provided by a time-depending probability measure, where at every instant of time the measure of a set represents the fraction of (microscopic) agents contained in it. The trajectories available to each of the microscopic agents are affected also by the overall state of the system. By using a suitable concept of random lift of set-valued maps, together with fixed point arguments, we are able to derive properties of the macroscopic description of the system from properties of the set-valued map expressing the admissible trajectories for the microscopical agents. We apply the results in the case in which the admissible trajectories of the agents are the minimizers of a suitable integral functional depending also from the macroscopic evolution of the system. Copyright (C) 2022 The Authors
On a class of modified Wasserstein distances induced by concave mobility functions defined on bounded intervals
Full text linkWe study a new class of distances between Radon measures similar to those studied in J. Dolbeault, B. Nazaret, G. Savaré, "A new class of transport distances between measures",Calc. Var. Partial Differential Equations, 34 (2009), pp. 193--231. These distances (more correctly pseudo-distances because can assume the value ) are defined generalizing the dynamical formulation of the Wasserstein distance by means of a concave mobility function. We are mainly interested in the physical interesting case (not considered in D.-N.-S.) of a concave mobility function defined in a bounded interval. We state the basic properties of the space of measures endowed with this pseudo-distance. Finally, we study in detail two cases: the set of measures defined in with finite moments and the set of measuresdefined in a bounded convex set. In the two cases we give sufficient conditions for the convergence of sequences with respect to the distance and we prove a property of boundedness
Measure-theoretic Lie brackets for nonsmooth vector fields
Full text linkIn this paper we prove a generalization of the classical notion of commutators of vector fields in the framework of measure theory, providing an extension of the set-valued Lie bracket introduced by Rampazzo-Sussmann for Lipschitz continuous vector fields. The study is motivated by some applications to control problems in the space of probability measures, modeling situations where the knowledge of the state is probabilistic, or in the framework of multi-agent systems, for which only a statistical description is available. Tools of optimal transportation theory are used
Anisotropic tempered diffusion equations
Full text linkWe introduce a functional framework which is specially suited to formulate several classes of anisotropic evolution equations of tempered diffusion type. Under an amenable set of hypothesis involving a very natural potential function, these models can be shown to belong to the entropy solution framework devised by Andreu et al. (2005), therefore ensuring well-posedness. We connect the properties of this potential with those of the associated cost function, thus providing a link with optimal transport theory and a supply of new examples of relativistic cost functions. Moreover, we characterize the anisotropic spreading properties of these models and we determine the Rankine-Hugoniot conditions that rule the temporal evolution of jump hypersurfaces under the given anisotropic flows
Optimality conditions and regularity results for time optimal control problems with differential inclusions
No full textWe study the time optimal control problem with a general target S for a class of differential inclusions that satisfy mild smoothness and controllability assumptions. In particular, we do not require Petrov’s condition at the boundary of S. Consequently, the minimum time function T(·) fails to be locally Lipschitz— never mind semiconcave—near S. Instead of such a regularity, we use an exterior sphere condition for the hypograph of T(·) to develop the analysis. In this way, we obtain dual arc inclusions which we apply to show the constancy of the Hamiltonian along optimal trajectories and other optimality conditions in Hamiltonian form. We also prove an upper bound for the Hausdorff measure of the set of all non-Lipschitz points of T(·) which implies that the minimum time function is of special bounded variation (SBV)
- …
