Supervised learning for kinetic consensus control

In this paper, how to successfully and efficiently condition a target population of agents towards consensus is discussed. To overcome the curse of dimensionality, the mean field formulation of the consensus control problem is considered. Although such formulation is designed to be independent of the number of agents, it is feasible to solve only for moderate intrinsic dimensions of the agents space. For this reason, the solution is approached by means of a Boltzmann procedure, i.e. quasi-invariant limit of controlled binary interactions as approximation of the mean field PDE. The need for an efficient solver for the binary interaction control problem motivates the use of a supervised learning approach to encode a binary feedback map to be sampled at a very high rate. A gradient augmented feedforward neural network for the Value function of the binary control problem is considered and compared with direct approximation of the feedback law

Gradient-augmented supervised learning of optimal feedback laws using state-dependent Riccati equations

A supervised learning approach for the solution of large-scale nonlinear stabilization problems is presented. A stabilizing feedback law is trained from a dataset generated from State-dependent Riccati Equation solvers. The training phase is enriched by the use of gradient information in the loss function, which is weighted through the use of hyperparameters. High-dimensional nonlinear stabilization tests demonstrate that real-time sequential large-scale Algebraic Riccati Equation solvers can be substituted by a suitably trained feedforward neural network

Control with uncertain data of socially structured compartmental epidemic models

The adoption of containment measures to reduce the amplitude of the epidemic peak is a key aspect in tackling the rapid spread of an epidemic. Classical compartmental models must be modified and studied to correctly describe the effects of forced external actions to reduce the impact of the disease. In addition, data are often incomplete and heterogeneous, so a high degree of uncertainty must naturally be incorporated into the models. In this work we address both these aspects, through an optimal control formulation of the epidemiological model in presence of uncertain data. After the introduction of the optimal control problem, we formulate an instantaneous approximation of the control that allows us to derive new feedback controlled compartmental models capable of describing the epidemic peak reduction. The need for long-term interventions shows that alternative actions based on the social structure of the system can be as effective as the more expensive global strategy. The importance of the timing and intensity of interventions is particularly relevant in the case of uncertain parameters on the actual number of infected people. Simulations related to data from the recent COVID-19 outbreak in Italy are presented and discussed

Structure preserving schemes for mean-field equations of collective behavior

In this paper we consider the development of numerical schemes for mean-field equations describing the collective behavior of a large group of interacting agents. The schemes are based on a generalization of the classical Chang-Cooper approach and are capable to preserve the main structural properties of the systems, namely nonnegativity of the solution, physical conservation laws, entropy dissipation and stationary solutions. In particular, the methods here derived are second order accurate in transient regimes whereas they can reach arbitrary accuracy asymptotically for large times. Several examples are reported to show the generality of the approach.Comment: Proceedings of the XVI International Conference on Hyperbolic Problem

Uncertainty quantification for kinetic models in socio-economic and life sciences

Kinetic equations play a major rule in modeling large systems of interacting particles. Recently the legacy of classical kinetic theory found novel applications in socio-economic and life sciences, where processes characterized by large groups of agents exhibit spontaneous emergence of social structures. Well-known examples are the formation of clusters in opinion dynamics, the appearance of inequalities in wealth distributions, flocking and milling behaviors in swarming models, synchronization phenomena in biological systems and lane formation in pedestrian traffic. The construction of kinetic models describing the above processes, however, has to face the difficulty of the lack of fundamental principles since physical forces are replaced by empirical social forces. These empirical forces are typically constructed with the aim to reproduce qualitatively the observed system behaviors, like the emergence of social structures, and are at best known in terms of statistical information of the modeling parameters. For this reason the presence of random inputs characterizing the parameters uncertainty should be considered as an essential feature in the modeling process. In this survey we introduce several examples of such kinetic models, that are mathematically described by nonlinear Vlasov and Fokker--Planck equations, and present different numerical approaches for uncertainty quantification which preserve the main features of the kinetic solution.Comment: To appear in "Uncertainty Quantification for Hyperbolic and Kinetic Equations

Kinetic models for optimal control of wealth inequalities

We introduce and discuss optimal control strategies for kinetic models for wealth distribution in a simple market economy, acting to minimize the variance of the wealth density among the population. Our analysis is based on a finite time horizon approximation, or model predictive control, of the corresponding control problem for the microscopic agents' dynamic and results in an alternative theoretical approach to the taxation and redistribution policy at a global level. It is shown that in general the control is able to modify the Pareto index of the stationary solution of the corresponding Boltzmann kinetic equation, and that this modification can be exactly quantified. Connections between previous Fokker-Planck based models and taxation-redistribution policies and the present approach are also discussed

Optical spectroscopic variability of Herbig Ae/Be stars

We analysed 337 multi-epoch optical spectra of 38 Herbig Ae/Be (HAeBe) stars to gain insights into the variability behaviour of the circumstellar (CS) atomic gas. Equivalent widths (EWs) and line fluxes of the Halpha, [OI]6300, HeI5876 and NaID lines were obtained for each spectrum; the Halpha line width at 10% of peak intensity (W10) and profile shapes were also measured and classified. The mean line strengths and relative variabilities were quantified for each star. Simultaneous optical photometry was used to estimate the line fluxes. We present a homogeneous spectroscopic database of HAeBe stars. The lines are variable in practically all stars and timescales, although 30 % of the objects show a constant EW in [OI]6300, which is also the only line that shows no variability on timescales of hours. The HeI5876 and NaID EW relative variabilities are typically the largest, followed by those in [OI]6300 and Halpha. The EW changes can be larger than one order of magnitude for the HeI5876 line, and up to a factor 4 for Halpha. The [OI]6300 and Halpha EW relative variabilities are correlated for most stars in the sample. The Halpha mean EW and W10 are uncorrelated, as are their relative variabilities. The Halpha profile changes in 70 % of the objects. The massive stars in the sample usually show more stable Halpha profiles with blueshifted self-absorptions and less variable 10% widths. Our data suggest multiple causes for the different line variations, but the [OI]6300 and Halpha variability must share a similar origin in many objects. The physical mechanism responsible for the Halpha line broadening does not depend on the amount of emission; unlike in lower-mass stars, physical properties based on the Halpha luminosity and W10 would significantly differ. Our results provide additional support to previous works that reported different physical mechanisms in Herbig Ae and Herbig Be stars.Comment: 10 pages, 5 figures, 2 appendixe