Conservation law models for traffic flow on a network of roads
Alberto Bressan Khai T. Nguyen
The paper develops a model of traffic flow near an intersection, where drivers seeking to enter a congested road wait in a buffer of limited capacity. Initial data comprise the vehicle density on each road, together with the percentage of drivers approaching the intersection who wish to turn into each of the outgoing roads.
    If the queue sizes within the buffer are known, then the initial-boundary value problems become decoupled and can be independently solved along each incoming road. Three variational problems are introduced, related to different kind of boundary conditions. From the value functions, one recovers the traffic density along each incoming or outgoing road by a Lax type formula.
    Conversely, if these value functions are known, then the queue sizes can be determined by balancing the boundary fluxes of all incoming and outgoing roads. In this way one obtains a contractive transformation, whose fixed point yields the unique solution of the Cauchy problem for traffic flow in an neighborhood of the intersection.
    The present model accounts for backward propagation of queues along roads leading to a crowded intersection, it achieves well-posedness for general $L^\infty $ data, and continuity w.r.t. weak convergence of the initial densities.
keywords: Hamilton Jacobi equations Lax type formula. Traffic flows scalar conservation laws network of roads
Nearly optimal patchy feedbacks
Alberto Bressan Fabio S. Priuli
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
A case study in vanishing viscosity
Stefano Bianchini Alberto Bressan
We consider a special $2 \times 2$ viscous hyperbolic system of conservation laws of the form $u_t + A(u)u_{x} = \varepsilon u_{x x}$, where $A(u) = Df(u)$ is the Jacobian of a flux function $f$. For initialdata with smalltotalv ariation, we prove that the solutions satisfy a uniform BV bound, independent of $\varepsilon $. Letting $\varepsilon \to 0$, we show that solutions of the viscous system converge to the unique entropy weak solutions of the hyperbolic system $u_t + f(u)_{x} = 0$. Within the proof, we introduce two new Lyapunov functionals which control the interaction of viscous waves of the same family. This provides a first example where uniform BV bounds and convergence of vanishing viscosity solutions are obtained, for a system with a genuinely nonlinear field where shock and rarefaction curves do not coincide.
keywords: hyperbolic system of conservation laws bounded variation vanishing viscosity approximation
On the blow-up for a discrete Boltzmann equation in the plane
Alberto Bressan Massimo Fonte
We study the possibility of finite-time blow-up for a two dimensional Broadwell model. In a set of rescaled variables, we prove that no self-similar blow-up solution exists, and derive some a priori bounds on the blow-up rate. In the final section, a possible blow-up scenario is discussed.
keywords: Discrete Boltzmann equation asymptotic behaviour and blow-up Broadwell model.
Continuous Riemann solvers for traffic flow at a junction
Alberto Bressan Fang Yu
The paper studies a class of conservation law models for traffic flow on a family of roads, near a junction. A Riemann Solver is constructed, where the incoming and outgoing fluxes depend Hölder continuously on the traffic density and on the drivers' turning preferences. However, various examples show that, if junction conditions are assigned in terms of Riemann Solvers, then the Cauchy problem on a network of roads can be ill posed, even for initial data having small total variation.
keywords: network of roads. Riemann solvers traffic flow
Existence of optima and equilibria for traffic flow on networks
Alberto Bressan Ke Han
This paper is concerned with a conservation law model of traffic flow on a network of roads, where each driver chooses his own departure time in order to minimize the sum of a departure cost and an arrival cost. The model includes various groups of drivers, with different origins and destinations and having different cost functions. Under a natural set of assumptions, two main results are proved: (i) the existence of a globally optimal solution, minimizing the sum of the costs to all drivers, and (ii) the existence of a Nash equilibrium solution, where no driver can lower his own cost by changing his departure time or the route taken to reach destination. In the case of Nash solutions, all departure rates are uniformly bounded and have compact support.
keywords: Traffic network user equilibria. scalar conservation law global optima
Optimal control problems on stratified domains
Alberto Bressan Yunho Hong
We consider a class of optimal control problems defined on a stratified domain. Namely, we assume that the state space $\mathbb{R}^N$ admits a stratification as a disjoint union of finitely many embedded submanifolds $\mathcal{M}_i$. The dynamics of the system and the cost function are Lipschitz continuous restricted to each submanifold. We provide conditions which guarantee the existence of an optimal solution, and study sufficient conditions for optimality. These are obtained by proving a uniqueness result for solutions to a corresponding Hamilton-Jacobi equation with discontinuous coefficients, describing the value function. Our results are motivated by various applications, such as minimum time problems with discontinuous dynamics, and optimization problems constrained to a bounded domain, in the presence of an additional overflow cost at the boundary.
keywords: Hamilton-Jacobi equation Viscosity solution Optimal control theory
A center manifold technique for tracing viscous waves
Stefano Bianchini Alberto Bressan
In this paper we introduce a new technique for tracing viscous travlling profiles. To illustrate the method, we consider a special $2\times 2$ hyperbolic system of conversation laws with viscosity, and show that any solution can be locally decomposed as the sume of 2 viscous travlling proflies. This yields the global existence, stability and uniform BV bounds for every solution with suitably small BV data.
keywords: conservation laws Hyperbolic systems well posedness.
Impulsive control of Lagrangian systems and locomotion in fluids
Alberto Bressan
Aim of this paper is to provide a survey of the theory of impulsive control of Lagrangian systems. It is assumed here that an external controller can determine the evolution of the system by directly prescribing the values of some of the coordinates. We begin by motivating the theory with a couple of elementary examples. Then we discuss the analytical form taken by the equations of motion, and their impulsive character. The following sections review various results found in the literature concerning the continuity of the control-to-trajectory map, the existence of optimal controls, and the asymptotic controllability to a reference state. In the last section we indicate a further application of the theory, to the control of deformable bodies immersed in a fluid.
keywords: Controlled Stability. Key words and phrases: Impulsive Control Lagrangian System
Errata corrige
Alberto Bressan
keywords: stratified domain viscosity solution optimal control. Hamilton-Jacobi equation

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