Viscosity solutions and uniqueness for systems of inhomogeneous balance laws
Graziano Crasta Benedetto Piccoli
Discrete & Continuous Dynamical Systems - A 1997, 3(4): 477-502 doi: 10.3934/dcds.1997.3.477
This paper is concerned with the Cauchy problem

$(*) \quad \quad u_t+[F(u)]_x=g(t,x,u),\quad u(0,x)=\overline{u}(x),$

for a nonlinear $2\times 2$ hyperbolic system of inhomogeneous balance laws in one space dimension. As usual, we assume that the system is strictly hyperbolic and that each characteristic field is either linearly degenerate or genuinely nonlinear.
Under suitable assumptions on $g$, we prove that there exists $T>0$ such that, for every $\overline{u}$ with sufficiently small total variation, the Cauchy problem ($*$) has a unique "viscosity solution", defined for $t\in [0,T]$, depending continuously on the initial data.

keywords: continuous dependence. uniqueness Systems of conservation laws
Optimal control of a multi-level dynamic model for biofuel production
Roberta Ghezzi Benedetto Piccoli
Mathematical Control & Related Fields 2017, 7(2): 235-257 doi: 10.3934/mcrf.2017008

Dynamic flux balance analysis of a bioreactor is based on the coupling between a dynamic problem, which models the evolution of biomass, feeding substrates and metabolites, and a linear program, which encodes the metabolic activity inside cells. We cast the problem in the language of optimal control and propose a hybrid formulation to model the full coupling between macroscopic and microscopic level. On a given location of the hybrid system we analyze necessary conditions given by the Pontryagin Maximum Principle and discuss the presence of singular arcs. For the multi-input case, under suitable assumptions, we prove that generically with respect to initial conditions optimal controls are bang-bang. For the single-input case the result is even stronger as we show that optimal controls are bang-bang.

keywords: Optimal control bioprocesses Pontryagin Maximum Principle
Optimal syntheses for state constrained problems with application to optimization of cancer therapies
Benedetto Piccoli
Mathematical Control & Related Fields 2012, 2(4): 383-398 doi: 10.3934/mcrf.2012.2.383
The use of combined therapies to treat cancer is common nowadays and some papers already addressed the relative optimization problems. In particular, it is natural to have state constraints, which usually correspond to bounds on feasible amounts of drugs to be used. The application of Pontryagin Maximum Principle is particularly difficult in such case. Therefore, we resort to sufficient conditions for optimality to achieve results more easily applicable to systems biology models. The approach is developed both for candidate value functions and optimal syntheses. Then it is shown at work on some specific problems in combined cancer therapy.
keywords: Optimal control cancer therapies. state constraint sufficient condition for optimality
Coupling of microscopic and phase transition models at boundary
Mauro Garavello Benedetto Piccoli
Networks & Heterogeneous Media 2013, 8(3): 649-661 doi: 10.3934/nhm.2013.8.649
This paper deals with coupling conditions between the classical microscopic Follow The Leader model and a phase transition (PT) model. We propose a solution at the interface between the two models. We describe the solution to the Riemann problem.
keywords: coupling conditions Follow the Leader conservation laws Cauchy Problem. phase transition model
Sparse stabilization and optimal control of the Cucker-Smale model
Marco Caponigro Massimo Fornasier Benedetto Piccoli Emmanuel Trélat
Mathematical Control & Related Fields 2013, 3(4): 447-466 doi: 10.3934/mcrf.2013.3.447
This article is mainly based on the work [7], and it is dedicated to the 60th anniversary of B. Bonnard, held in Dijon in June 2012.
    We focus on a controlled Cucker--Smale model in finite dimension. Such dynamics model self-organization and consensus emergence in a group of agents. We explore how it is possible to control this model in order to enforce or facilitate pattern formation or convergence to consensus. In particular, we are interested in designing control strategies that are componentwise sparse in the sense that they require a small amount of external intervention, and also time sparse in the sense that such strategies are not chattering in time. These sparsity features are desirable in view of practical issues.
    We first show how very simple sparse feedback strategies can be designed with the use of a variational principle, in order to steer the system to consensus. These feedbacks are moreover optimal in terms of decay rate of some functional, illustrating the general principle according to which ``sparse is better''. We then combine these results with local controllability properties to get global controllability results. Finally, we explore the sparsity properties of the optimal control minimizing a combination of the distance from consensus and of a norm of the control.
keywords: Cucker--Smale model $l_1$-norm minimization local controllability sparse optimal control. consensus emergence sparse stabilization
Benedetto Piccoli
Networks & Heterogeneous Media 2006, 1(1): i-ii doi: 10.3934/nhm.2006.1.1i
To start a new journal in a fast growing scientific panorama is a serious challenge. We decided to start this new adventure because we felt the necessity of having an applied math journal covering an area of great interest and experiencing a big expansion.

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Keep right or left? Towards a cognitive-mathematical model for pedestrians
Mary J. Bravo Marco Caponigro Emily Leibowitz Benedetto Piccoli
Networks & Heterogeneous Media 2015, 10(3): 559-578 doi: 10.3934/nhm.2015.10.559
In this paper we discuss the necessity of insight in the cognitive processes involved in environment navigation into mathematical models for pedestrian motion. We first provide a review of psychological literature on the cognitive processes involved in walking and on the quantitative one coming from applied mathematics, physics, and engineering. Then, we present a critical analysis of the experimental setting for model testing and we show experimental results given by observation. Finally we propose a cognitive model making use of psychological insight as well as optimization models from robotics.
keywords: Mathematical psychology pedestrian behavior multiscale models crowd dynamics experimental settings.
Traffic modeling and management: Trends and perspectives
Alexandre Bayen Rinaldo M. Colombo Paola Goatin Benedetto Piccoli
Discrete & Continuous Dynamical Systems - S 2014, 7(3): i-ii doi: 10.3934/dcdss.2014.7.3i
The present issue of Discrete and Continuous Dynamical Systems -- Series S is devoted to Traffic Modeling and Management. This subject dramatically developed in recent years. On one hand, the successes of the analytical theory of conservation laws have provided new tools to traffic researchers while, on the other hand, the requirements coming from the applications have grown dramatically. Remarkably, two of the papers that opened the way to this decades long development date the same year. In 1995 ``The Unique Limit of the Glimm Scheme'' by A. Bressan (Archive for Rational Mechanics and Analysis, 130, 3, 205--230) gave a basis for several well posedness results for 1D systems of conservation laws. In the same year, ``Requiem for High-Order Fluid Approximations of Traffic Flow'' by C. Daganzo (Transportation Research Part B: Methodological, 29B, 4, 277--287) posed serious criticisms to models studied at that time and started to fix minimal requirements for a traffic model to be seriously considered.

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A continuum-discrete model for supply chains dynamics
Gabriella Bretti Ciro D’Apice Rosanna Manzo Benedetto Piccoli
Networks & Heterogeneous Media 2007, 2(4): 661-694 doi: 10.3934/nhm.2007.2.661
This paper is focused on continuum-discrete models for supply chains. In particular, we consider the model introduced in [10], where a system of conservation laws describe the evolution of the supply chain status on sub-chains, while at some nodes solutions are determined by Riemann solvers. Fixing the rule of flux maximization, two new Riemann Solvers are defined. We study the equilibria of the resulting dynamics, moreover some numerical experiments on sample supply chains are reported. We provide also a comparison, both of equilibria and experiments, with the model of [15].
keywords: networks conservation laws fluid-dynamic models finite difference schemes Supply chains
On fluido-dynamic models for urban traffic
Mauro Garavello Benedetto Piccoli
Networks & Heterogeneous Media 2009, 4(1): 107-126 doi: 10.3934/nhm.2009.4.107
The aim of this paper is to address the following questions: which models, among fluido-dynamic ones, are more appropriate to describe urban traffic? While a rich debate was developed for the complicate dynamics of highway traffic, some basic problems of urban traffic are not always appropriately discussed. We analyze many recent, and less recent, models focusing on three basic properties. The latter are necessary to reproduce correctly queue formation at lights and junctions, and their backward propagation on an urban network.
keywords: fluido-dynamic models car traffic urban traffic networks.

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