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Control Problems for Conservation Laws with Traffic Applications : Modeling, Analysis, and Numerical Methods.

Bibliographic Details
Main Author: Bayen, Alexandre.
Other Authors: Delle Monache, Maria Laura., Garavello, Mauro., Goatin, Paola., Piccoli, Benedetto.
Format: eBook
Language:English
Published: Cham : Springer International Publishing AG, 2022.
Edition:1st ed.
Series:Progress in Nonlinear Differential Equations and Their Applications Series
Subjects:
Electronic books.
Online Access:Click to View
Table of Contents:
  • Intro
  • Contents
  • List of Figures
  • List of Tables
  • 1 Introduction
  • 2 Boundary Control of Conservation Laws Exhibiting Shocks
  • 2.1 Introduction
  • 2.2 Boundary Controls for Smooth Solutions Co-authored by Amaury Hayat
  • 2.3 The Attainable Set
  • 2.3.1 The Scalar Case with a Single Control
  • 2.3.2 The Burgers' Equation with Two Controls
  • 2.3.3 Temple Systems on a Bounded Interval
  • 2.3.4 General Systems on a Bounded Interval
  • 2.4 Lyapunov Stabilization of Scalar Conservation Laws with Two Boundaries
  • 2.4.1 Approximation of Solutions via Piecewise Smooth Functions
  • 2.4.2 Lyapunov Functional
  • 2.4.3 Control Space and Lyapunov Stability
  • 2.4.4 Greedy Controls
  • 2.4.5 Lyapunov Asymptotic Stability
  • 2.4.6 Nonlocal Controls
  • 2.4.7 Numerical Examples
  • 2.5 Mixed Systems PDE-ODE
  • 2.5.1 Examples
  • 2.6 Bibliographical Notes
  • 3 Decentralized Control of Conservation Laws on Graphs
  • 3.1 Introduction
  • 3.2 Control Acting at Nodes Through the Riemann Solver
  • 3.2.1 The Setting of the Problem
  • 3.2.2 The Main Result
  • 3.2.3 Example of Family of Riemann Solvers
  • 3.2.3.1 The Riemann Solver RS1
  • 3.2.3.2 The Riemann Solver RS2
  • 3.2.3.3 The Riemann Solver RS3
  • 3.3 Modeling Signalized Intersections
  • 3.3.1 The Hamilton-Jacobi Representation of Signal Models
  • 3.3.2 When Spillback Is Absent
  • 3.3.3 When Spillback Is Present and Sustained
  • 3.4 Control for a Freeway Model
  • 3.4.1 Freeway Model
  • 3.4.2 Optimal Control Problem
  • 3.4.3 Numerical Example
  • 3.5 Optimal Control on Boundary and Flux Constraint
  • 3.5.1 Optimal Control Problems
  • 3.6 Optimization of Travel Time on Networks via Local Distribution Coefficients
  • 3.6.1 Optimization of Simple Networks
  • 3.6.2 Simulations of Two Urban Networks
  • 3.6.3 Emergency Management
  • 3.7 Bibliographical Notes
  • 4 Distributed Control for Conservation Laws.
  • 4.1 Introduction
  • 4.2 Riemann Solver Semigroup and Stability
  • 4.2.1 Classical Riemann Solver Semigroup Solutions
  • 4.2.2 Stability of the Standard Riemann Semigroup
  • 4.3 Needle-Like Variations for Variable Speed Limit
  • 4.3.1 Variable Speed Limit: Control Problem
  • 4.3.2 Needle-Like Variations
  • 4.3.3 Three Different Control Policies
  • 4.3.3.1 Instantaneous Policy
  • 4.3.3.2 Random Exploration Policy
  • 4.3.3.3 Gradient Method
  • 4.3.4 Numerical Results
  • 4.3.4.1 Godunov Scheme for Hyperbolic PDEs
  • 4.3.4.2 Velocity Policies
  • 4.3.4.3 Simulations
  • 4.4 Discrete-Optimization Methods for First Order Models
  • 4.4.1 Traffic Flow Network Modeling
  • 4.4.1.1 Coupling Conditions at Junctions
  • 4.4.1.2 Boundary Conditions
  • 4.4.2 Optimization Problem for VSL and Ramp Metering
  • 4.4.2.1 Variable Speed Limits
  • 4.4.2.2 Ramp Metering
  • 4.4.3 Numerical Simulations
  • 4.4.3.1 Optimization Approach
  • 4.4.3.2 Numerical Results
  • 4.5 Discrete-Optimization Methods for Second Order Models
  • 4.5.1 The Aw-Rascle Model on Networks
  • 4.5.1.1 Coupling and Boundary Conditions
  • 4.5.2 Numerical Simulations for Aw-Rascle on Network with Control
  • 4.5.2.1 Numerical Method
  • 4.5.2.2 Numerical Results
  • 4.5.2.3 Capacity Drop
  • 4.5.2.4 Coordinated Speed Control and Ramp Metering
  • 4.6 Bibliographical Notes
  • 5 Lagrangian Control of Conservation Laws and Mixed Models
  • 5.1 Introduction
  • 5.2 PDE-ODE Models for Moving Bottlenecks
  • 5.2.1 A Macroscopic Model with Space Dependent Flux
  • 5.2.2 PDE-ODE Models with Flux Constraint
  • 5.2.3 A PDE-ODE Model for Vehicle Platooning
  • 5.3 Numerical Methods for Moving Bottlenecks
  • 5.3.1 A Coupled Godunov-ODE Scheme for Model (5.1)
  • 5.3.2 A Conservative Scheme for Non-Classical Solutions to the PDE-ODE Models with Flux Constraint
  • 5.4 Traffic Management by Controlled Vehicles.
  • 5.4.1 Field Experiments
  • 5.4.2 Numerical Experiments
  • 5.5 Bibliographical Notes
  • 6 Control Problems for Hamilton-Jacobi Equations Co-authored by Alexander Keimer
  • 6.1 Introduction
  • 6.2 Strong Solutions
  • 6.2.1 The Bounded Domain Case
  • 6.3 Generalized Solutions
  • 6.3.1 Piecewise Affine Initial and Boundary Datum
  • 6.3.2 Piecewise Affine Initial Datum
  • 6.3.3 Piecewise Affine Left Hand Side Boundary Datum
  • 6.3.4 Compatibility Conditions
  • 6.4 Optimization with Hamilton-Jacobi Equations
  • 6.5 Bibliographical Notes
  • A Conservation and Balance Laws and Boundary Value Problems
  • A.1 Basic Definitions
  • A.2 BV Functions
  • A.3 The Method of Characteristics
  • A.4 Weak Solutions
  • A.5 Entropy Admissible Solutions
  • A.5.1 Kruzkov Entropy Condition
  • A.6 The Riemann Problem
  • A.6.1 The Scalar Case
  • A.6.1.1 The Riemann Problem for a Strictly Convex Flux
  • A.6.1.2 The Riemann Problem for a Concave Flux
  • A.6.2 The System Case
  • A.7 The Cauchy Problem
  • A.7.1 Wave-Front Tracking for the Scalar Case
  • A.7.2 The System Case
  • A.8 Boundary Conditions for Scalar Conservation Laws
  • A.8.1 The Left Boundary Condition for the Riemann Problem
  • A.8.2 The Right Boundary Condition for the Riemann Problem
  • B Models for Vehicular Traffic and Conservation Laws on Networks
  • B.1 Lighthill-Whitham-Richard Model for vehicular Traffic on Networks
  • B.2 Dynamics at Simple Junctions
  • B.2.1 Two Incoming and One Outgoing Roads
  • B.2.2 One Incoming and Two Outgoing Roads
  • B.2.3 Two Incoming and Two Outgoing Roads
  • B.3 Constructing Solutions on a Network
  • Bibliography
  • Index.

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