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

• Superior document: Progress in Nonlinear Differential Equations and Their Applications Series ; v.99
• Author: Bayen, Alexandre.
• TeilnehmendeR: Delle Monache, Maria Laura.; Garavello, Mauro.; Goatin, Paola.; Piccoli, Benedetto.
• Place / Publishing House: Cham : : Springer International Publishing AG,, 2022.; Ã2022.
• Year of Publication: 2022
• Edition: 1st ed.
• Language: English
• Series: Progress in Nonlinear Differential Equations and Their Applications Series
• Physical Description: 1 online resource (238 pages)

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.