Constrained Mean Field Games Equilibria as Fixed Point of Random Lifting of Set-Valued Maps

We 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

Mini Max Wallpaper

Mini Max company formulated a problem for the automatic calculation of the number of wallpaper rolls necessary for decorating a room with wallpaper. The final goal is the development of a web-based calculator open for use to both Mini Max staff and the general public. We propose an approach for reducing the studied problem to the one-dimensional cutting-stock problem. We show this in details for the case of plain wallpapers as well as for the case of patterned wallpapers with straight match. The one-dimensional cutting-stock problem can be formulated as a linear integer programming problem. We develop an approach for calculating the needed number of wallpapers for relatively small problems, create an algorithm in a suitable graphical interface and make different tests. The tests show the efficiency of the proposed approach compared with the existent (available) wallpapers’ calculators

STOCHASTIC EQUILIBRIUM SOLUTION FOR A DEBT MANAGEMENT PROBLEM WITH CURRENCY DEVALUATION

Consider 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

A metric approach to plasticity via Hamilton-Jacobi equation

Thermodynamical consistency of plasticity models is usually written in terms of the so-called "maximum dissipation principle". In this paper, we discuss constitutive relations for dissipative materials written through suitable generalized gradients of a (possibly non-convex) metric. This new framework allows us to generalize the classical results providing an interpretation of the yield function in terms of HamiltonJacobi equations theory

Optimality conditions and regularity results for time optimal control problems with differential inclusions

We 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)

Set-Driven Evolution for Multiagent System

We consider the deterministic evolution in the Euclidean space of a multiagent system with a large number of agents (possibly infinitely many). At each instant of time, besides from time and its current position, the set of velocities available to each agent is influenced by the set described by the current position of all the other agents. The latter is in turn determined by the overall motion of the crowd of all the agents. The interplay to the microscopical point of view of each single agent, and the macroscopical one of the set-evolution yields a non-trivial dynamical system. This two-level multiagent system can be described either by the evolution of a probability measure-describing the instantaneous density of the crowd-or by the evolution of a set-describing the positions where there is at least one agent. In this paper, we precise the links between the two descriptions, providing also some quantitative estimates on the macroscopical admissible evolutions

Compatibility of state constraints and dynamics for multiagent control systems

This 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