Fabio Bagagiolo

Mean-Field Games and Dynamic Demand Management in Power Grids

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 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.

Singular Perturbation of a Finite Horizon Problem with State-Space Constraints

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.

Hysteresis in Filtration through Porous Media

We study an evolution problem for filtration through porous media, accounting for hysteresis in the saturation versus pressure constitutive relation. We provide a weak formulation of the problem, assuming that the memory effect in the constitutive relation consists not only of a rate-independent component but also of a rate-dependent one. We prove an existence result, which also applies to the case where the hysteresis operator is of Preisach-type.

Opinion dynamics and stubbornness through mean-field games

This paper provides a mean field game theoretic interpretation of opinion dynamics and stubbornness. The model describes a crowd-seeking homogeneous population of agents, under the influence of one stubborn agent. The game takes on the form of two partial differential equations, the Hamilton-Jacobi-Bellman equation and the Kolmogorov-Fokker-Planck equation for the individual optimal response and the population evolution, respectively. For the game of interest, we establish a mean field equilibrium where all agents reach ε-consensus in a neighborhood of the stubborn agents opinion.

Minimum Time for a hybrid system with thermostatic switchings

In this paper we study a minimum time problem for a hybrid system subject to thermostatic switchings. We apply the Dynamic Programming method and the viscosity solution theory of Hamilton-Jacobi equations. We regard the problem as a suitable coupling of two minimum-time/exit-time problems. Under some controllability conditions, we prove that the minimum time function is the unique bounded below continuous function which solves a system of two Hamilton-Jacobi equations coupled via the boundary conditions.

Mean-Field Game Modeling the Bandwagon Effect with Activation Costs

This paper provides a mean-field game theoretic model of the bandwagon effect in social networks. This effect can be observed whenever individuals tend to align their own opinions to a mainstream opinion. The contribution is threefold. First, we describe the opinion propagation as a mean-field game with local interactions. Second, we establish mean-field equilibrium strategies in the case where the mainstream opinion is constant. Such strategies are shown to have a threshold structure. Third, we extend the use of threshold strategies to the case of time-varying mainstream opinion and study the evolution of the macroscopic system.

Bandwagon effect in mean-field games

This paper provides a mean-field game theoretic model of the bandwagon effect in social networks. The latter phenomenon can be observed whenever individuals tend to align their own opinions to a mainstream opinion. The contribution is three-fold. First, we provide a mean-field games framework that describes the opinion propagation under local interaction. Second, we establish mean-field equilibrium strategies in the case where the mainstream opinion is stationary. Such strategies are shown to have a threshold structure. Third, we study conditions under which a given opinion distribution is stationary if agents implement optimal non-idle and threshold strategies.

Robust optimality of linear saturated control in uncertain linear network flows

We propose a novel approach that, given a linear saturated feedback control policy, asks for the objective function that makes robust optimal such a policy. The approach is specialized to a linear network flow system with unknown but bounded demand and politopic bounds on controlled flows. All results are derived via the Hamilton-Jacobi-Isaacs and viscosity theory.

Game Theoretic Decentralized Feedback Controls in Markov Jump Processes

This paper studies a decentralized routing problem over a network, using the paradigm of mean-field games with large number of players. Building on a state-space extension technique, we turn the problem into an optimal control one for each single player. The main contribution is an explicit expression of the optimal decentralized control which guarantees the convergence both to local and to global equilibrium points. Furthermore, we study the stability of the system also in the presence of a delay which we model using an hysteresis operator. As a result of the hysteresis, we prove existence of multiple equilibrium points and analyze convergence conditions. The stability of the system is illustrated via numerical studies.

Non-memoryless Pedestrian Flow in a Crowded Environment with Target Sets

This work deals with the problem of managing the excursionist flow in historical cities. Venice is considered as a case study. There, in high season, thousands of excursionists arrive by train in the morning, spend the day visiting different sites, reach again the train station in late afternoon, and leave. With the idea of avoiding congestion by directing excursionists along different routes, a mean field model is introduced. Network/switching is used to describe the excursionists costs as a function of their position taking into consideration whether they have already visited a site or not, i.e., allowing excursionists to have memory of the past when making decisions. In particular, we analyze the model in the framework of Hamilton-Jacobi/transport equations, as it is standard in mean field games theory. Finally, to provide a starting datum for iterative solution algorithms, a second model is introduced in the framework of mathematical programming. For this second approach, some numerical experiments are presented.