Category Archives: Optimization and Control

Control with uncertain data of socially structured compartmental epidemic models

by Mattia Zanella

G. Albi, L. Pareschi, M. Zanella

Journal of Mathematical Biology, 82, 63, 2021. (Preprint arXiv)

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.

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

by Mattia Zanella

B. Piccoli, A. Tosin, M. Zanella

Zeitschrift für Angewandte Mathematik und Physik, 71:152, 2020.

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.

 

Uncertainty damping in kinetic traffic models by driver-assist controls

by Mattia Zanella

Andrea Tosin, Mattia Zanella

Mathematical Control and Related Fields, 11(3): 681-713, 2021. (Preprint arXiv)

In this paper, we propose a kinetic model of traffic flow with uncertain binary interactions, which explains the scattering of the fundamental diagram in terms of the macroscopic variability of aggregate quantities, such as the mean speed and the flux of the vehicles, produced by the microscopic uncertainty. Moreover, we design control strategies at the level of the microscopic interactions among the vehicles, by which we prove that it is possible to dampen the propagation of such an uncertainty across the scales. Our analytical and numerical results suggest that the aggregate traffic flow may be made more ordered, hence predictable, by implementing such control protocols in driver-assist vehicles. Remarkably, they also provide a precise relationship between a measure of the macroscopic damping of the uncertainty and the penetration rate of the driver-assist technology in the traffic stream.

 

Kinetic-controlled hydrodynamics for traffic models with driver-assist vehicles

by Mattia Zanella

Andrea Tosin, Mattia Zanella

Multiscale Modeling and Simulation, 17(2): 716-749, 2019. (Preprint arXiv)

We develop a hierarchical description of traffic flow control by means of driver-assist vehicles aimed at the mitigation of speed-dependent road risk factors. Microscopic feedback control strategies are designed at the level of vehicle-to-vehicle interactions and then upscaled to the global flow via a kinetic approach based on a Boltzmann-type equation. Then first and second order hydrodynamic traffic models, which naturally embed the microscopic control strategies, are consistently derived from the kinetic-controlled framework via suitable closure methods. Several numerical examples illustrate the effectiveness of such a hierarchical approach at the various scales.

 

Boltzmann games in heterogeneous consensus dynamics

by Mattia Zanella

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella

Journal of Statistical Physics, 175(1): 97–125, 2019. (Preprint arXiv)

We consider a constrained hierarchical opinion dynamics in the case of leaders’ competition and with complete information among leaders. Each leaders’ group tries to drive the followers’ opinion towards a desired state accordingly to a specific strategy. By using the Boltzmann-type control approach we analyze the best-reply strategy for each leaders’ population. Derivation of the corresponding Fokker-Planck model permits to investigate the asymptotic behaviour of the solution. Heterogeneous followers populations are then considered where the effect of knowledge impacts the leaders’ credibility and modifies the outcome of the leaders’ competition.

Control strategies for road risk mitigation in kinetic traffic modelling

by Mattia Zanella

Andrea Tosin, Mattia Zanella. 

IFAC-PapersOnLine, 51(9): 67-72, 2018. (Preprint arXiv)

In this paper we present a Boltzmann-type kinetic approach to the modelling of road traffic, which includes control strategies at the level of microscopic binary interactions aimed at the mitigation of speed-dependent road risk factors. Such a description is meant to mimic a system of driver-assist vehicles, which by responding locally to the actions of their drivers can impact on the large-scale traffic dynamics, including those related to the collective road risk and safety.

Recent advances in opinion modeling: control and social influence

by Mattia Zanella

Test3g_t5000Giacomo Albi, Lorenzo Pareschi, Giuseppe Toscani, Mattia Zanella                                                                         

Active Particles, Volulme 1. Advances in Theory, Models, and Applications, pp. 49-98, 2017. (Preprint arXiv)

We survey some recent developments on the mathematical modeling of opinion dynamics. After an introduction on opinion modeling through interacting multi-agent systems described by partial differential equations of kinetic type, we focus our attention on two major advancements: optimal control of opinion formation and influence of additional social aspects, like conviction and number of connections in social networks, which modify the agents’ role in the opinion exchange process.

Performance bounds for the mean-field limit of constrained dynamics

by Mattia Zanella

performance_boundplotMichael Herty, Mattia Zanella

Discrete and Continuous Dynamical Systems – Series A, 37(4): 2023-2043, 2017. (Preprint arXiv)

In this work we are interested in the mean-field formulation of kinetic models under control actions where the control is formulated through a model predictive control strategy (MPC) with varying horizon. The relation between the (usually hard to compute) optimal control and the MPC approach is investigated theoretically in the mean-field limit. We establish a computable and provable bound on the difference in the cost functional for MPC controlled and optimal controlled system dynamics in the mean-field limit. The result of the present work extends previous findings for systems of ordinary differential equations. Numerical results in the mean-field setting are given.

On the optimal control of opinion dynamics on evolving networks

by Mattia Zanella

immagine_sito

G. Albi, L. Pareschi, M. Zanella 

System Modeling and Optimization. CSMO 2015. IFIP Advances in Information and Communication Technology, vol 494, 2016. (Preprint arXiv)

In this work we are interested in the modelling and control of opinion dynamics spreading on a time evolving network with scale-free asymptotic degree distribution. The mathematical model is formulated as a coupling of an opinion alignment system with a probabilistic description of the network. The optimal control problem aims at forcing consensus over the network, to this goal a control strategy based on the degree of connection of each agent has been designed. A numerical method based on a model predictive strategy is then developed and different numerical tests are reported. The results show that in this way it is possible to drive the overall opinion toward a desired state even if we control only a suitable fraction of the nodes.

Uncertainty Quantification in Control Problems for Flocking Models

by Mattia Zanella

Giacomo Albi, Lorenzo Pareschi, Mattia Zanella

Mathematical Problems in Engineering, Vol. 2015, 2015.  (Preprint arXiv)

In this paper the optimal control of flocking models with random inputs is investigated from a numerical point of view. The effect of uncertainty in the interaction parameters is studied for a Cucker-Smale type model using a generalized polynomial chaos (gPC) approach. Numerical evidence of threshold effects in the alignment dynamic due to the random parameters is given. The use of a selective model predictive control permits to steer the system towards the desired state even in unstable regimes.

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