Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

LMMctrl

with Michael Herty, Lorenzo Pareschi

(23/07/2018, arxiv.1807.08547)

We are interested in high-order linear multistep schemes for time discretization of adjoint equations arising within optimal control problems. First we consider optimal control problems for ordinary differential equations and show loss of accuracy for Adams-Moulton and Adams-Bashford methods, whereas BDF methods preserve high–order accuracy. Subsequently we extend these results to semi–lagrangian discretizations of hyperbolic relaxation systems. Computational results illustrate theoretical findings

Posted in contributed article | Tagged , , | Comments Off on Linear multistep methods for optimal control problems and applications to hyperbolic relaxation systems

Pressureless Euler alignment system with control

blow-upGiacomo AlbiYoung-Pil ChoiAxel-Stefan Haeck

(31/01/2018,  arxiv:1801.10587)

We study a non-local hydrodynamic system with control. First we characterize the control dynamics as a sub-optimal approximation to the optimal control problem constrained to the evolution of the pressureless Euler alignment system. We then discuss the critical thresholds that leading to global regularity or finite-time blow-up of strong solutions in one and two dimensions. Finally we propose a finite volume scheme for numerical solutions of the controlled system. Several numerical simulations are shown to validate the theoretical and computational results of the paper.

Posted in preprint | Tagged , , , , | Comments Off on Pressureless Euler alignment system with control

Boltzmann games in heterogeneous consensus dynamics

Boltzmann_gameGiacomo Albi, Lorenzo Pareschi, Mattia Zanella

(07/12/17,  arxiv:1712.03224)

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.

 

Posted in preprint | Tagged , , | Comments Off on Boltzmann games in heterogeneous consensus dynamics

A Boltzmann approach to mean-field sparse feedback control

sparse_boltzmannGiacomo Albi, Massimo Fornasier, and Dante Kalise

(12/11/2016 arxiv.org/abs/1611.03988)

We study the synthesis of optimal control policies for large-scale multi-agent systems. The optimal control design induces a parsimonious control intervention by means of l-1, sparsity-promoting control penalizations. We study instantaneous and infinite horizon sparse optimal feedback controllers. In order to circumvent the dimensionality issues associated to the control of large-scale agent-based models, we follow a Boltzmann approach. We generate (sub)optimal controls signals for the kinetic limit of the multi-agent dynamics, by sampling of the optimal solution of the associated two-agent dynamics. Numerical experiments assess the performance of the proposed sparse design.

Posted in Senza categoria | Comments Off on A Boltzmann approach to mean-field sparse feedback control

Mean field control hierarchy

imag_webGiacomo Albi, Young-Pil Choi, Massimo Fornasier, and Dante Kalise

(01/08/2016  arxiv.org/abs/1608.01728)

In this paper we model the role of a government of a large population as a mean field optimal control problem. Such control problems are constrainted by a PDE of continuity-type, governing the dynamics of the probability distribution of the agent population. We show the existence of mean field optimal controls both in the stochastic and deterministic setting. We derive rigorously the first order optimality conditions useful for numerical computation of mean field optimal controls. We introduce a novel approximating hierarchy of sub-optimal controls based on a Boltzmann approach, whose computation requires a very moderate numerical complexity with respect to the one of the optimal control. We provide numerical experiments for models in opinion formation comparing the behavior of the control hierarchy.

Posted in preprint | Tagged , , , , | Comments Off on Mean field control hierarchy

Recent advances in opinion modeling: control and social influence

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.

Posted in contributed article | Tagged , , | Comments Off on Recent advances in opinion modeling: control and social influence

Discrete and Continuum Modeling of Biological Network Formation

We present an overview of recent analytical and numerical results for the elliptic parabolic system of partial differential equations proposed by Hu and Cai, which models formation of biological transport networks. The model describes the pressure field using a Darcy’s type equation and the dynamics of the conductance network under pressure force effects. Randomness in the material structure is represented by a linear diffusion term and conductance relaxation by an algebraic decay term. We first introduce micro and mesoscopic models and show how they are connected to the macroscopic PDE system. Then, we provide an overview of analytical results for the PDE model, focusing mainly on the existence of weak and mild solutions and analysis of the steady states. The analytical part is complemented by extensive numerical simulations. We propose a discretization based on finite elements and study the qualitative properties of network structures for various parameter values.

Posted in contributed article | Tagged , , | Comments Off on Discrete and Continuum Modeling of Biological Network Formation

Opinion dynamics over complex networks: kinetic modeling and numerical methods

site_artGiacomo Albi, Lorenzo Pareschi, Mattia Zanella

(31/03/2016 arxiv.org/abs/1604.00421)

In this paper we consider the modeling of opinion dynamics over time
dependent large scale networks. A kinetic description of the agents’
distribution over the evolving network is considered which combines an opinion
update based on binary interactions between agents with a dynamic creation and
removal process of new connections. The number of connections of each agent
influences the spreading of opinions in the network but also the way
connections are created is influenced by the agents’ opinion. The evolution of
the network of connections is studied by showing that its asymptotic behavior
is consistent both with Poisson distributions and truncated power-laws. In
order to study the large time behavior of the opinion dynamics a mean field
description is derived which allows to compute exact stationary solutions in
some simplified situations. Numerical methods which are capable to describe
correctly the large time behavior of the system are also introduced and
discussed. Finally, several numerical examples showing the influence of the
agents’ number of connections in the opinion dynamics are reported.

Posted in preprint | Tagged , , , , | Comments Off on Opinion dynamics over complex networks: kinetic modeling and numerical methods

Selective model-predictive control for flocking systems

All1

Giacomo Albi, Lorenzo Pareschi

(16/03/2016 arxiv.org/abs/1603.05012)

In this paper the optimal control of alignment models composed by a large number of agents is investigated in presence of a selective action of the control. As a first step toward a reduction of computational cost, we introduce a model predictive control (MPC) approximation by deriving a numerical scheme with a feedback selective constrained dynamics. Next, in order to cope with the numerical solution of a large number of interacting agents, we introduce a binary interaction algorithm [http://arxiv.org/abs/1203.0721] which is able to simulate efficiently the selective constrained dynamics. Consistency of the algorithm is also shown in the context of standard kinetic theory. Finally, some numerical simulations are reported to show the efficiency of the proposed techniques.

Posted in preprint | Tagged , , , | Comments Off on Selective model-predictive control for flocking systems