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EB

ONLINE

Cham : Springer International Publishing AG, 2022

1 online resource (238 pages)

ISBN 9783030930158 (electronic bk.)

ISBN 9783030930141

Progress in Nonlinear Differential Equations and Their Applications Ser. ; v.99

Print version: Bayen, Alexandre Control Problems for Conservation Laws with Traffic Applications Cham : Springer International Publishing AG,c2022 ISBN 9783030930141

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.

001896930

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(Au-PeEL)EBL6961667

(MiAaPQ)EBC6961667

(OCoLC)1314612744