Pedestrian Dynamics: Mathematical Theory and Evacuation Control

Introduction Motivation Literature Survey Outline Derivation of Conservation Laws Mass Conservation Momentum Conservation Energy Conservation Combined Equations General Conservation Traffic Models: One Dimensional Case Lighthill-Whitham-Richards Model Payne-Whitham Model Aw-Rascle Model Zhang Model Pedestrian and Control Models in One Dimension Traffic Models: Two-Dimensional Case Two-Dimensional LWR Model Two-Dimensional Payne-Whitham Model Two-Dimensional Aw-Rascle Model Two-Dimensional Zhang Model Conservation Law Solutions Method of Characteristics Classical or Strong Solutions Weak Solutions Scalar Riemann Problem Admissibility Conditions Kruzkov's Entropy Function Well-posedness Oleinik Entropy Condition Scalar Initial-Boundary Problem Traffic Control Scalar Conservation Law Solution Dynamical Systems and C0-Semigroups Optimal Control Optimal Flux Control for Scalar Conservation Law Feedback Control for Scalar Law Advective Feedback Control for Relaxation Systems Wellposedness for Bounded Advection Control Simulations for Advective Control Godunov's Method Simulation Results for Advective Control Conclusions Summary Contributions Future Work

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  Richard Courant,et al.  Supersonic Flow And Shock Waves , 1948 .

[3]  M J Lighthill,et al.  ON KINEMATIC WAVES.. , 1955 .

[4]  W. Rudin.  Principles of mathematical analysis , 1964 .

[5]  Joseph L. Schofer,et al.  A STATISTICAL ANALYSIS OF SPEED-DENSITY HYPOTHESES , 1965 .

[6]  J. Glimm.  Solutions in the large for nonlinear hyperbolic systems of equations , 1965 .

[7]  J. Marsden,et al.  On continuity and smoothness of group actions , 1970 .

[8]  Robert Herman,et al.  Kinetic theory of vehicular traffic , 1971 .

[9]  Michael G. Crandall,et al.  The semigroup approach to first order quasilinear equations in several space variables , 1972 .

[10]  G. Folland.  Introduction to Partial Differential Equations , 1976 .

[11]  Lucien W. Neustadt,et al.  Optimization: A Theory of Necessary Conditions , 1976 .

[12]  J. A. Walker,et al.  Dynamical Systems and Evolution Equations , 1980 .

[13]  W. Greub.  Linear Algebra , 1981 .

[14]  J. Smoller.  Shock Waves and Reaction-Diffusion Equations , 1983 .

[15]  John M. Watts,et al.  Computer models for evacuation analysis , 1987 .

[16]  N. Risebro.  A front-tracking alternative to the random choice method , 1993 .

[17]  Bernardo Cockburn,et al.  On Convergence to Entropy Solutions of A Single Conservation Law , 1996 .

[18]  Felix Otto,et al.  Initial-boundary value problem for a scalar conservation law , 1996 .

[19]  P. Raviart,et al.  Numerical Approximation of Hyperbolic Systems of Conservation Laws , 1996, Applied Mathematical Sciences.

[20]  E. Toro.  Riemann Solvers and Numerical Methods for Fluid Dynamics , 1997 .

[21]  A. V. Fursikov,et al.  Optimal control of distributed systems , 1999 .

[22]  Pushkin Kachroo,et al.  Robust Feedback Control of a Single Server Queueing System , 1999, Math. Control. Signals Syst..

[23]  Hector O. Fattorini,et al.  Infinite Dimensional Optimization and Control Theory: References , 1999 .

[24]  C. Dafermos.  Hyberbolic Conservation Laws in Continuum Physics , 2000 .

[25]  L. Chambers.  Linear and Nonlinear Waves , 2000, The Mathematical Gazette.

[26]  A. Bressan.  Hyperbolic systems of conservation laws : the one-dimensional Cauchy problem , 2000 .

[27]  Michael Schreckenberg,et al.  Pedestrian and evacuation dynamics , 2002 .

[28]  R. LeVeque.  Finite Volume Methods for Hyperbolic Problems: Characteristics and Riemann Problems for Linear Hyperbolic Equations , 2002 .

[29]  Sabiha Amin Wadoo,et al.  Feedback Control Design and Stability Analysis of One Dimensional Evacuation System , 2006, 2006 IEEE Intelligent Transportation Systems Conference.

[30]  Issam S. Strub,et al.  Weak formulation of boundary conditions for scalar conservation laws: an application to highway traffic modelling , 2006 .

[31]  Serge Alinhac,et al.  Hyperbolic Partial Differential Equations , 2009 .