This paper applies mean field game theory to dynamic demand management. For a large population of electrical heating or cooling appliances (called agents), we provide a mean field game that guarantees desynchronization of the agents thus improving the power network resilience. Second, for the game at hand, we exhibit a mean field equilibrium, where each agent adopts a bang-bang switching control with threshold placed at a nominal temperature. At the equilibrium, through an opportune design of the terminal penalty, the switching control regulates the mean temperature (computed over the population) and the mains frequency around the nominal value. To overcome Zeno phenomena we also adjust the bang-bang control by introducing a thermostat. Third, we show that the equilibrium is stable in the sense that all agents' states, initially at different values, converge to the equilibrium value or remain confined within a given interval for an opportune initial distribution.
Art heritage cities are popular tourist destinations but for many of them overcrowding is becoming an issue. In this paper, we address the problem of modeling and analytically studying the flow of tourists along the narrow alleys of the historic center of a heritage city. We initially present a mean field game model, where both continuous and switching decisional variables are introduced to respectively describe the position of a tourist and the point of interest that it may visit. We prove the existence of a mean field game equilibrium. A mean field game equilibrium is Nash-type equilibrium in the case of infinitely many players. Then, we study an optimization problem for an external controller who aims to induce a suitable mean field game equilibrium.Keywords Tourist flow optimal control · mean field games · switching variables · dynamics on networks Mathematics Subject Classification (2010) 91A13 · 49L20 · 90B20 · 91A80
We study the singular perturbation of optimal control problems for nonlinear systems with constraints on the fast state variables and a cost functional either of Bolza type or involving the exit time of the system from a given domain. Under a controllability assumption on the fast variables, we show that these variables become controls in the limit problem. Our method consists of passing to the limit in the associated Hamilton-Jacobi-Bellman (HJB) equations by means of some tools in the theory of viscosity solutions.
We study an infinite horizon optimal control problem for a system with two state variables. One of them has the evolution governed by a controlled ordinary differential equation and the other one is related to the latter by a hysteresis relation, represented here by either a play operator or a Prandtl-Ishlinskii operator. By dynamic programming, we derive the corresponding (discontinuous) first order Hamilton-Jacobi equation, which in the first case is of finite dimension and in the second case is of infinite dimension. In both cases we prove that the value function is the only bounded uniformly continuous viscosity solution of the equation.2000 Mathematics Subject Classification: 47J40, 49J15, 49L20, 49L25.
In this paper we investigate different strategies to overcome the scallop theorem. We will show how to obtain a net motion exploiting the fluid's type change during a periodic deformation. We are interested in two different models: in the first one that change is linked to the magnitude of the opening and closing velocity. Instead, in the second one it is related to the sign of the above velocity. An interesting feature of the latter model is the introduction of a delay-switching rule through a thermostat. We remark that the latter is fundamental in order to get both forward and backward motion.
Abstract. We consider a model for the control of a linear network flow system with unknown but bounded demand and polytopic bounds on controlled flows. We are interested in the problem of finding a suitable objective function that makes robust optimal the policy represented by the so-called linear saturated feedback control. We regard the problem as a suitable differential game with switching cost and study it in the framework of the viscosity solutions theory for Bellman and Isaacs equations.Mathematics Subject Classification. 49L25, 49N90, 90C35.
This work deals with the problem of managing the excursionist flow in\ud
historic cities. Venice is considered as a case study. There, in high season, thousands\ud
of excursionists arrive by train in the morning; spend the day visiting different sites;\ud
reach again the train station in late afternoon and leave. With the idea of avoiding\ud
congestion by directing excursionists along different routes, a mean field model is\ud
introduced. Network/switching is used to describe the excursionists costs as a func-\ud
tion of their position, taking into consideration whether they have already visited\ud
a site or not, i.e. allowing excursionists to have memory of the past when making\ud
decisions. The problem is analized in the framework of Hamilton-Jacobi/transport\ud
equations, as it is standard in mean field games theory. In addition, to provide a\ud
starting datum for iterative solution algorithms, we introduce a second model in\ud
the framework of mathematical programming. For this second approach we present\ud
some numerical experiments
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