arXiv subjects

Benedetto Piccoli

Publications and source records attributed to Benedetto Piccoli.

At least 55 records · Page 3Linked to original sources

arXiv2020

Multiscale control of generic second order traffic models by driver-assist vehicles

We study the derivation of generic high order macroscopic traffic models from a follow-the-leader particle description via a kinetic approach. First, we recover a third order traffic model as the hydrodynamic limit of an Enskog-type kinetic equation. Next, we introduce in the vehicle interactions a binary control modelling the automatic feedback provided by driver-assist vehicles and we upscale such a new particle description by means of another Enskog-based hydrodynamic limit. The resulting macroscopic model is now a Generic Second Order Model (GSOM), which contains in turn a control term inherited from the microscopic interactions. We show that such a control may be chosen so as to optimise global traffic trends, such as the vehicle flux or the road congestion, constrained by the GSOM dynamics. By means of numerical simulations, we investigate the effect of this control hierarchy in some specific case studies, which exemplify the multiscale path from the vehicle-wise implementation of a driver-assist control to its optimal hydrodynamic design.

physics.soc-ph↗

arXiv2020

A multiscale model for traffic regulation via autonomous vehicles

Autonomous vehicles (AVs) allow new ways of regulating the traffic flow on road networks. Most of available results in this direction are based on microscopic approaches, where ODEs describe the evolution of regular cars and AVs. In this paper, we propose a multiscale approach, based on recently developed models for moving bottlenecks. Our main result is the proof of existence of solutions for open-loop controls with bounded variation.

math.AP↗

arXiv2020

Metabolic graphs, LIFE method and the modeling of drug action on Mycobacterium tuberculosis

This paper serves as a framework for designing advanced models for drug action on metabolism. Drug treatment may affect metabolism by either enhancing or inhibiting metabolic reactions comprising a metabolic network. We introduce the concept of \textit{metabolic graphs}, a generalization of hypergraphs having specialized features common to metabolic networks. Linear-in-flux-expression (briefly LIFE) is a methodology for analyzing metabolic networks and simulating virtual patients. We extend LIFE dynamics to be compatible with metabolic graphs, including the more complex interactions of enhancer and inhibitor molecules that affect biochemical reactions. We discuss results considering network structure required for existence and uniqueness of equilibria on metabolic graphs and show simulations of drug action on \textit{Mycobacterium tuberculosis} (briefly MTB)

q-bio.MN↗

arXiv2019

Model-based assessment of the impact of driver-assist vehicles using kinetic theory

In this paper we consider a kinetic description of follow-the-leader traffic models, which we use to study the effect of vehicle-wise driver-assist control strategies at various scales, from that of the local traffic up to that of the macroscopic stream of vehicles. We provide a theoretical evidence of the fact that some typical control strategies, such as the alignment of the speeds and the optimisation of the time headways, impact on the local traffic features (for instance, the speed and headway dispersion responsible for local traffic instabilities) but have virtually no effect on the observable macroscopic traffic trends (for instance, the flux/throughput of vehicles). This unobvious conclusion, which is in very nice agreement with recent field studies on autonomous vehicles, suggests that the kinetic approach may be a valid tool for an organic multiscale investigation and possibly design of driver-assist algorithms.

nlin.AO↗

arXiv2019

A Wasserstein norm for signed measures, with application to nonlocal transport equation with source term

We introduce the optimal transportation interpretation of the Kantorovich norm on thespace of signed Radon measures with finite mass, based on a generalized Wasserstein distancefor measures with different masses.With the formulation and the new topological properties we obtain for this norm, we proveexistence and uniqueness for solutions to non-local transport equations with source terms, whenthe initial condition is a signed measure.

math.AP↗

arXiv2019

Generalized Dynamic Programming Principle and Sparse Mean-Field Control Problems

In this paper we study optimal control problems in Wasserstein spaces, which are suitable to describe macroscopic dynamics of multi-particle systems. The dynamics is described by a parametrized continuity equation, in which the Eulerian velocity field is affine w.r.t. some variables. Our aim is to minimize a cost functional which includes a control norm, thus enforcing a \emph{control sparsity} constraint. More precisely, we consider a nonlocal restriction on the total amount of control that can be used depending on the overall state of the evolving mass. We treat in details two main cases: an instantaneous constraint on the control applied to the evolving mass and a cumulative constraint, which depends also on the amount of control used in previous times. For both constraints, we prove the existence of optimal trajectories for general cost functions and that the value function is viscosity solution of a suitable Hamilton-Jacobi-Bellmann equation. Finally, we discuss an abstract Dynamic Programming Principle, providing further applications in the Appendix.

math.OC↗

arXiv2019

Are commercially implemented adaptive cruise control systems string stable?

In this article, we assess the string stability of seven 2018 model year adaptive cruise control (ACC) equipped vehicles that are widely available in the US market. Seven distinct vehicle models from two different vehicle makes are analyzed using data collected from more than 1,200 miles of driving in car-following experiments with ACC engaged by the follower vehicle. The resulting dataset is used to identify the parameters of a linear second order delay differential equation model that approximates the behavior of the black box ACC systems. The string stability of the data-fitted model associated with each vehicle is assessed, and the main finding is that all seven vehicle models have string unstable ACC systems. For one commonly available vehicle model that offers ACC as a standard feature on all trim levels, we validate the string stability finding with a multi-vehicle platoon experiment in which all vehicles are the same year, make, and model. In this test, an initial disturbance of 6 mph is amplified to a 25 mph disturbance, at which point the last vehicle in the platoon is observed to disengage the ACC. The data collected in the driving experiments is made available, representing the largest publicly available comparative driving dataset on ACC equipped vehicles.

eess.SY↗

arXiv2019

Stability of Metabolic Networks via Linear-In-Flux-Expressions

The methodology named LIFE (Linear-in-Flux-Expressions) was developed with the purpose of simulating and analyzing large metabolic systems. With LIFE, the number of model parameters is reduced by accounting for correlations among the parameters of the system. Perturbation analysis on LIFE systems results in less overall variability of the system, leading to results that more closely resemble empirical data. These systems can be associated to graphs, and characteristics of the graph give insight into the dynamics of the system. This work addresses two main problems: 1. for fixed metabolite levels, find all fluxes for which the metabolite levels are an equilibrium, and 2. for fixed fluxes, find all metabolite levels which are equilibria for the system. We characterize the set of solutions for both problems, and show general results relating stability of systems to the structure of the associated graph. We show that there is a structure of the graph necessary for stable dynamics. Along with these general results, we show how stability analysis from the fields of network flows, compartmental systems, control theory and Markov chains apply to LIFE systems.

math.OC↗

arXiv2018

Measure dynamics with Probability Vector Fields and sources

We introduce a new formulation for differential equation describing dynamics of measures on an Euclidean space, that we call Measure Differential Equations with sources. They mix two different phenomena: on one side, a transport-type term, in which a vector field is replaced by a Probability Vector Field, that is a probability distribution on the tangent bundle; on the other side, a source term. Such new formulation allows to write in a unified way both classical transport and diffusion with finite speed, together with creation of mass. The main result of this article shows that, by introducing a suitable Wasserstein-like functional, one can ensure existence of solutions to Measure Differential Equations with sources under Lipschitz conditions. We also prove a uniqueness result under the following additional hypothesis: the measure dynamics needs to be compatible with dynamics of measures that are sums of Dirac masses.

math.AP↗

arXiv2018

Open canals flow with fluvial to torrential phase transitions on networks

Network flows and specifically open canal flows can be modeled by systems of balance laws defined on topological graphs. The shallow water or Saint-Venant system of balance laws is one of the most used model and present two phases: fluvial or sub-critical and torrential or super critical. Phase transitions may occur within the same canal but transitions related to networks are less investigated. In this paper we provide a complete characterization of possible phase transitions for a simple network with two canals and one junction. Our analysis allows the study of more complicate scenarios. Moreover, we provide some numerical simulations to show the theory at work.

math.NA↗

arXiv2018

Sparse control of Hegselmann-Krause models: Black hole and declustering

This paper elaborates control strategies to prevent clustering effects in opinion formation models. This is the exact opposite of numerous situations encountered in the literature where, on the contrary, one seeks controls promoting consensus. In order to promote declustering, instead of using the classical variance that does not capture well the phenomenon of dispersion, we introduce an entropy-type functional that is adapted to measuring pairwise distances between agents. We then focus on a Hegselmann-Krause-type system and design declustering sparse controls both in finite-dimensional and kinetic models. We provide general conditions characterizing whether clustering can be avoided as function of the initial data. Such results include the description of black holes (where complete collapse to consensus is not avoidable), safety zones (where the control can keep the system far from clustering), basins of attraction (attractive zones around the clustering set) and collapse prevention (when convergence to the clustering set can be avoided).

math.OC↗

arXiv2018

Well-posedness for scalar conservation laws with moving flux constraints

We consider a strongly coupled ODE-PDE system representing moving bottlenecks immersed in vehicular traffic. The PDE consists of a scalar conservation law modeling the traffic flow evolution and the ODE models the trajectory of a slow moving vehicle. The moving bottleneck influences the bulk traffic flow via a point flux constraint, which is given by an inequality on the flux at the slow vehicle position. We prove uniqueness and continuous dependence of solutions with respect to initial data of bounded variation. The proof is based on a new backward in time method established to capture the values of the norm of generalized tangent vectors at every time.

math.AP↗

arXiv2017

Measure differential equations

A new type of differential equations for probability measures on Euclidean spaces, called Measure Differential Equations (briefly MDEs), is introduced. MDEs correspond to Probability Vector Fields, which map measures on an Euclidean space to measures on its tangent bundle. Solutions are intended in weak sense and existence, uniqueness and continuous dependence results are proved under suitable conditions. The latter are expressed in terms of the Wasserstein metric on the base and fiber of the tangent bundle. MDEs represent a natural measure-theoretic generalization of Ordinary Differential Equations via a monoid morphism mapping sums of vector fields to fiber convolution of the corresponding Probability Vector Fields. Various examples, including finite-speed diffusion and concentration, are shown, together with relationships to Partial Differential Equations. Finally, MDEs are also natural mean-field limits of multi-particle systems, with convergence results extending the classical Dubroshin approach.

math.OC↗

arXiv2017

Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments

Traffic waves are phenomena that emerge when the vehicular density exceeds a critical threshold. Considering the presence of increasingly automated vehicles in the traffic stream, a number of research activities have focused on the influence of automated vehicles on the bulk traffic flow. In the present article, we demonstrate experimentally that intelligent control of an autonomous vehicle is able to dampen stop-and-go waves that can arise even in the absence of geometric or lane changing triggers. Precisely, our experiments on a circular track with more than 20 vehicles show that traffic waves emerge consistently, and that they can be dampened by controlling the velocity of a single vehicle in the flow. We compare metrics for velocity, braking events, and fuel economy across experiments. These experimental findings suggest a paradigm shift in traffic management: flow control will be possible via a few mobile actuators (less than 5%) long before a majority of vehicles have autonomous capabilities.

eess.SY↗

arXiv2017

A Collapsed Generalized Aw-Rascle-Zhang Model and Its Model Accuracy

This work presents a collapsed generalized Aw-Rascle-Zhang (CGARZ) model, which fits into a generic second order model (GSOM) framework. GSOMs augment the evolution of the traffic density by a second state variable characterizing a property of vehicles or drivers. A cell transmission model for the numerical solution of GSOMs is derived, which is based on analyzing the sending and receiving functions of the traffic density and total property. The predictive accuracy of the CGARZ model is then compared to the classical first-order LWR and four second-order models. To that end, a systematic approach to calibrate model parameters from sensor flow-density data is introduced and applied to all models studied. The comparative model validation is conducted using two types of field data: vehicle trajectory data, and loop detector data. It is shown that the CGARZ model provides an intriguing combination of simple model dynamics in the free-flow regime and a representation of the spread of flow-density data in the congested regime, while possessing a competitive prediction accuracy.

physics.soc-ph↗

arXiv2017

A convex formulation of traffic dynamics on transportation networks

This article proposes a numerical scheme for computing the evolution of vehicular traffic on a road network over a finite time horizon. The traffic dynamics on each link is modeled by the Hamilton-Jacobi (HJ) partial differential equation (PDE), which is an equivalent form of the Lighthill-Whitham-Richards PDE. The main contribution of this article is the construction of a single convex optimization program which computes the traffic flow at a junction over a finite time horizon and decouples the PDEs on connecting links. Compared to discretization schemes which require the computation of all traffic states on a time-space grid, the proposed convex optimization approach computes the boundary flows at the junction using only the initial condition on links and the boundary conditions of the network. The computed boundary flows at the junction specify the boundary condition for the HJ PDE on connecting links, which then can be separately solved using an existing semi-explicit scheme for single link HJ PDE. As demonstrated in a numerical example of ramp metering control, the proposed convex optimization approach also provides a natural framework for optimal traffic control applications.

physics.soc-ph↗

arXiv2017

Mean-Field Sparse Jurdjevic--Quinn Control

We consider nonlinear transport equations with non-local velocity, describing the time-evolution of a measure, which in practice may represent the density of a crowd. Such equations often appear by taking the mean-field limit of finite-dimensional systems modelling collective dynamics. We first give a sense to dissipativity of these mean-field equations in terms of Lie derivatives of a Lyapunov function depending on the measure. Then, we address the problem of controlling such equations by means of a time-varying bounded control action localized on a time-varying control subset with bounded Lebesgue measure (sparsity space constraint). Finite-dimensional versions are given by control-affine systems, which can be stabilized by the well known Jurdjevic--Quinn procedure. In this paper, assuming that the uncontrolled dynamics are dissipative, we develop an approach in the spirit of the classical Jurdjevic--Quinn theorem, showing how to steer the system to an invariant sublevel of the Lyapunov function. The control function and the control domain are designed in terms of the Lie derivatives of the Lyapunov function, and enjoy sparsity properties in the sense that the control support is small. Finally, we show that our result applies to a large class of kinetic equations modelling multi-agent dynamics.

math.OC↗

arXiv2016

Control of reaction-diffusion equations on time-evolving manifolds

Among the main actors of organism development there are morphogens, which are signaling molecules diffusing in the developing organism and acting on cells to produce local responses. Growth is thus determined by the distribution of such signal. Meanwhile, the diffusion of the signal is itself affected by the changes in shape and size of the organism. In other words, there is a complete coupling between the diffusion of the signal and the change of the shapes. In this paper, we introduce a mathematical model to investigate such coupling. The shape is given by a manifold, that varies in time as the result of a deformation given by a transport equation. The signal is represented by a density, diffusing on the manifold via a diffusion equation. We show the non-commutativity of the transport and diffusion evolution by introducing a new concept of Lie bracket between the diffusion and the transport operator. We also provide numerical simulations showing this phenomenon.

math.OC↗