State constrained patchy feedback stabilization
Fabio S. Priuli
Mathematical Control & Related Fields 2015, 5(1): 141-163 doi: 10.3934/mcrf.2015.5.141
We construct a patchy feedback for a general control system on $\mathbb{R}^d$ which realizes practical stabilization to a target set $\Sigma$, when the dynamics is constrained to a given set of states $S$. The main result is that $S$--constrained asymptotically controllability to $\Sigma$ implies the existence of a discontinuous practically stabilizing feedback. Such a feedback can be constructed in ``patchy'' form, a particular class of piecewise constant controls which ensure the existence of local Carathéodory solutions to any Cauchy problem of the control system and which enjoy good robustness properties with respect to both measurement errors and external disturbances.
keywords: Asymptotic controllability patchy feedback state constraint robustness. stabilization
A destination-preserving model for simulating Wardrop equilibria in traffic flow on networks
Emiliano Cristiani Fabio S. Priuli
Networks & Heterogeneous Media 2015, 10(4): 857-876 doi: 10.3934/nhm.2015.10.857
In this paper we propose a LWR-like model for traffic flow on networks which allows to track several groups of drivers, each of them being characterized only by their destination in the network. The path actually followed to reach the destination is not assigned a priori, and can be chosen by the drivers during the journey, taking decisions at junctions.
    The model is then used to describe three possible behaviors of drivers, associated to three different ways to solve the route choice problem: 1. Drivers ignore the presence of the other vehicles; 2. Drivers react to the current distribution of traffic, but they do not forecast what will happen at later times; 3. Drivers take into account the current and future distribution of vehicles. Notice that, in the latter case, we enter the field of differential games, and, if a solution exists, it likely represents a global equilibrium among drivers.
    Numerical simulations highlight the differences between the three behaviors and offer insights into the existence of equilibria.
keywords: networks multi-population model Nash equilibrium. source-destination model Wardrop equilibrium Traffic multi-path model multi-commodity model
Nearly optimal patchy feedbacks
Alberto Bressan Fabio S. Priuli
Discrete & Continuous Dynamical Systems - A 2008, 21(3): 687-701 doi: 10.3934/dcds.2008.21.687
The paper is concerned with a general optimization problem for a nonlinear control system, in the presence of a running cost and a terminal cost, with free terminal time. We prove the existence of a patchy feedback whose trajectories are all nearly optimal solutions, with pre-assigned accuracy.
keywords: Robustness. Optimization Discontinuous Feedback Control

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