arXiv subjects

Fabio Bagagiolo

Publications and source records attributed to Fabio Bagagiolo.

At least 19 recordsLinked to original sources

arXiv2026

The petrol station game: the regional average price. A mean field analysis

In January 2023, the obligation for petrol stations to display the average fuel price calculated on a regional basis was introduced by the Italian Government. A mean field game model is here proposed to describe the evolution of the fuel price. In particular, each petrol station decides its own price, for the next day, taking account of an estimate of the average price, to be communicated by the Local Authority. The existence and the evolution of a mean field equilibrium are analytically investigated. Some simulations are compared with real data from the Autonomous Province of Trento, Italy.

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arXiv2026

State-constrained optimal control of the continuity equation and infinite-dimensional viscosity solutions

We study a finite horizon optimal control problem for the continuity equation under a weighted integral state constraint on the mass outside a fixed set. The model is cast in a Hilbert framework for densities. On a suitable invariant compact subset, we prove that the value function is Lipschitz continuous and satisfies, by dynamic programming, the associated infinite dimensional constrained Hamilton Jacobi Bellman equation in viscosity sense (subsolution in the interior, supersolution up to the boundary). We finally prove a comparison principle and uniqueness in the Lipschitz class.

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arXiv2025

Wave-front tracking for a quasi-linear scalar conservation law with hysteresis II: the case of Preisach

We consider the Cauchy problem for the quasi-linear scalar conservation law \[u_t+\mathcal{F}(u)_t+u_x=0,\] where $\mathcal{F}$ is a specific hysteresis operator. Hysteresis models a rate-independent memory relationship between the input $u$ and its output, giving a non-local feature to the equation. In a previous work the authors studied the case when $\mathcal{F}$ is the Play operator. In the present article, we extend the analysis to the case of Preisach operator, which is probably the most versatile mathematical model to describe hysteresis in the applications, especially for the presence of some kind of internal variables. This fact has required a new analysis of the equation. Starting from the Riemann problem, we address the so-called wave-front tracking method for a solution to the Cauchy problem with bounded variation initial data. An entropy-like condition is also exploited for uniqueness.

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arXiv2025

Flows of vector fields and the Kalman Theorem

We give two proofs of the Kalman Theorem, alternative to the most common ones, which infer such a classical result of Control Theory using just very basic facts on flows of vector fields. These proofs are apt to be generalised in diverse directions -- in fact one of them has been already generalised, yielding new criteria for local controllability of non-linear real analytic controlled systems.

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arXiv2025

A zero-sum differential game for two opponent masses

We investigate an infinite dimensional partial differential equation of Isaacs' type, which arises from a zero-sum differential game between two masses. The evolution of the two masses is described by a controlled transport/continuity equation, where the control is given by the vector velocity field. Our study is set in the framework of the viscosity solutions theory in Hilbert spaces, and we prove the uniqueness of the value functions as solutions of the Isaacs equation.

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arXiv2025

A single player and a mass of agents: a pursuit evasion-like game

We study a finite-horizon differential game of pursuit-evasion like, between a single player and a mass of agents. The player and the mass directly control their own evolution, which for the mass is given by a first order PDE of transport equation type. Using also an adapted concept of non-anticipating strategies, we derive an infinite dimensional Isaacs equation, and by dynamic programming techniques we prove that the value function is the unique viscosity solution on a suitable invariant subset of a Hilbert space.

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arXiv2024

Wave-front tracking for a quasi-linear scalar conservation law with hysteresis

In this article we deal with the Cauchy problem for the quasi-linear scalar conservation law \[u_t+ {\cal F}(u)_t+u_x=0,\] where ${\cal F}$ is a specific hysteresis operator, namely the Play operator. Hysteresis models a rate-independent memory relationship between the input $u$ and its output. Its presence in the partial differential equation gives a particular non-local feature to the latter allowing us to capture phenomena that may arise in some application fields. Riemann problems and the interactions between shock lines are studied and via the so-called Wave-Front Tracking method a solution to the Cauchy problem with bounded variation initial data is constructed. The solution found satisfies an entropy-like condition, making it the unique solution in the class of entropy admissible ones.

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arXiv2023

A time-dependent switching mean-field game on networks motivated by optimal visiting problems

Motivated by an optimal visiting problem, we study a switching mean-field game on a network, where both a decisional and a switching time-variable is at disposal of the agents for what concerns, respectively, the instant to decide and the instant to perform the switch. Every switch between the nodes of the network represents a switch from $0$ to $1$ of one component of the string $p = (p_1,\ldots, p_n)$ which, in the optimal visiting interpretation, gives information on the visited targets, being the targets labeled by $i=1,\ldots, n$. The goal is to reach the final string $(1, \ldots, 1)$ in the final time $T$, minimizing a switching cost also depending on the congestion on the nodes. We prove the existence of a suitable definition of an approximated $\varepsilon$-mean-field equilibrium and then address the passage to the limit when $\varepsilon$ goes to 0.

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arXiv2022

The orienteering problem: a hybrid control formulation

In the last years, a growing number of challenging applications in navigation, logistics, and tourism were modeled as orienteering problems. This problem has been proposed in relation to a sport race where certain control points must be visited in a minimal time. In a certain kind of these competitions, the choice of the number and the order for the control points are left to the competitor. We propose an original approach to solve the orienteering problem based on hybrid control. The continuous state of the system contains information about the navigation toward the next control point. In contrast, the discrete state keeps track of the already visited ones. The control problem is solved using non-standard dynamical programming techniques.

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arXiv2022

Hybrid control for optimal visiting problems for a single player and for a crowd

In an optimal visiting problem, we want to control a trajectory that has to pass as close as possible to a collection of target points or regions. We introduce a hybrid control-based approach for the classic problem where the trajectory can switch between a group of discrete states related to the targets of the problem. The model is subsequently adapted to a mean-field framework to study viability and crowd fluxes to model a multitude of indistinguishable players.

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arXiv2022

A hybrid control framework for an optimal visiting problem

The optimal visiting problem is the optimization of a trajectory that has to touch or pass as close as possible to a collection of target points. The problem does not verify the dynamic programming principle, and it needs a specific formulation to keep track of the visited target points. In this paper, we introduce a hybrid approach by adding a discontinuous part of the trajectory switching between a group of discrete states related to the targets. Then, we show the well-posedness of the related Hamilton-Jacobi problem, by reformulating the optimal visiting as a collection of time-dependent optimal stopping problems.

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arXiv2020

Origin-to-destination network flow with path preferences and velocity controls: a mean field game-like approach

In this paper we consider a mean field approach to modeling the agents flow over a transportation network. In particular, beside a standard framework of mean field games, with controlled dynamics by the agents and costs mass-distribution dependent, we also consider a path preferences dynamics obtained as a generalization of the so-called noisy best response dynamics. Such a preferences dynamics says the agents choose their path having access to global information about the network congestion state and based on the observation of the decision of the agents that have preceded. We prove the existence of a mean field equilibrium obtained as a fixed point of a map over a suitable set of time-varying mass-distributions, defined edge by edge in the network. We also address the case where the admissible set of controls is suitably bounded depending on the mass-distribution on the edge itself.

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arXiv2020

Hysteresis and controllability of affine driftless systems: some case studie

We investigate the controllability of some kinds of driftless affine systems where hysteresis effects are taken into account, both in the realization of the control and in the state evolution. In particular we consider two cases: the one when hysteresis is represented by the so-called play operator, and the one when it is represented by a so-called delayed relay. In the first case we prove that, under some hypotheses, whenever the corresponding non-hysteretic system is controllable, then we can also, at least approximately, control the hysteretic one. This is obtained by some suitably constructed approximations for the inputs in the hysteresis operator. In the second case we prove controllability for a generic hysteretic delayed switching system. Finally, we investigate some possible connections between the two cases.

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arXiv2020

A hybrid differential game with switching thermostatic-type dynamics and cost

In this paper we consider an infinite horizon zero-sum differential game where the dynamics of each player and the running cost are also depending on the evolution of some discrete (switching) variables. In particular, such switching variables evolve according to the switching law of a so-called thermostatic delayed relay, applied to the players' states. We first address the problem of the continuity of both lower and upper value function. Then, by a suitable representation of the problem as a coupling of several exit-time differential games, we characterize those value functions as, respectively, the unique solution of a coupling of several Dirichlet problems for Hamilton-Jacobi-Isaacs equations. The concept of viscosity solutions and a suitable definition of boundary conditions in the viscosity sense is used in the paper. Finally, we give some sufficient conditions for the existence of an equilibrium.

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arXiv2019

A differential game with exit costs

We study a differential game where two players separately control their own dynamics, pay a running cost, and moreover pay an exit cost (quitting the game) when they leave a fixed domain. In particular, each player has its own domain and the exit cost consists of three different exit costs, depending whether either the first player only leaves its domain, or the second player only leaves its domain, or they both simultaneously leave their own domain. We prove that, under suitable hypotheses, the lower and upper value are continuous and are, respectively, the unique viscosity solution of a suitable Dirichlet problem for a Hamilton-Jacobi-Isaacs equation. The continuity of the values relies on the existence of suitable non-anticipating strategies respecting the domain-constraint. This problem is also treated in this work.

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arXiv2019

A mean field approach to model flows of agents with path preferences over a network

In this paper, we address the problem of modeling the traffic flow of a heritage city whose streets are represented by a network. We consider a mean field approach where the standard forward backward system of equations is also intertwined with a path preferences dynamics. The path preferences are influenced by the congestion status on the whole network as well as the possible hassle of being forced to run during the tour. We prove the existence of a mean field equilibrium as a fixed point, over a suitable set of time-varying distributions, of a map obtained as a limit of a sequence of approximating functions. Then, a bi-level optimization problem is formulated for an external controller who aims to induce a specific mean field equilibrium.

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arXiv2019

Hybrid Thermostatic Approximations of Junctions for some Optimal Control Problems on Networks

We study some optimal control problems on networks with junctions, approximate the junctions by a switching rule of delay-relay type and study the passage to the limit when $\varepsilon$, the parameter of the approximation, goes to zero. First, for a twofold junction problem we characterize the limit value function as viscosity solution and maximal subsolution of a suitable Hamilton-Jacobi problem. Then, for a threefold junction problem we consider two different approximations, recovering in both cases some uniqueness results in the sense of maximal subsolution.

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arXiv2019

Optimal control of the mean field equilibrium for a pedestrian tourists' flow model

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 he/she may visit. We prove the existence of a mean field equilibrium. A mean field 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 equilibrium.

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