Control and Optimization of Distributed Systems and Partial Differential Equations 

Disclaimer: This page may be outdated. It is preserved here for informational purposes.
class is likely not to be taught in the years to come.

Practical information

Instructor: Professor Alexandre Bayen

Control Number:CE 291F, ME236, EE291. This class is cross listed between EECS, ME and CE so that students can fit it in their respective curricula.

Course description

This is an introductory course to control and optimization of systems driven by partial differential equations (PDEs). The first part of the class will focus on fundamental techniques to solve these equations both analytically (when possible), and numerically. This part of the course will be accessible to students with exposure to control and no (or very little) exposure to partial differential equations. The techniques presented in class will include separation of variables, spectral decomposition, self-similar solutions, characteristics, complex embedding. The second part of the class will address stability, control and optimization of these PDEs. It will be accessible to students without background in control. Stability will be investigated using spectral analysis. Adjoint-based optimization, Hamilton-Jacobi and differential flatness techniques will be applied for open loop trajectory design. Lyapunov techniques will be devised for stabilization and control. 

The class will put emphasis on networks. Applications (in particular course projects) will include networks of one dimensional systems: water distribution channels, electromagnetic waves in transmission cables, towed cable systems for marine oil exploration, highway systems, oil drilling, mine ventilation networks, blood circulation in vessels. Examples in higher dimensions will include 2D or 3D fluid mechanics, in particular propagation of contaminants in water. 

Here is a list of partial differential equations and corresponding applications that will be covered in class:

  1. The wave equation
  2. The Euler-Bernoulli beam equation (materials)
  3. The heat equation (thermosciences)
  4. The LWR equation for highway traffic (transportation)
  5. The Saint-Venant equations, shallow water equations, Hayami'e equation (hydraulics)
  6. The membrane equation (mechanical engineering, MEMS)
  7. The Telegraph equation (communication)
  8. Maxwell’s equations (electromagnetism)
  9. Vibrating string (acoustics)
  10. Vorticity equation (aerodynamics)
  11. Euler’s equations (fluid mechanics)

An expanded description of the class is available here.

General information

  1. Homeworks will count for 40% of the final grade
  2. There will be a midterm, open-book, and open notes, which will count for 20% of the final grade
  3. There will be one class project. Students are encouraged to bring their own research-related projects. Projects will be suggested to students (see sample list of projects for the prevoius years below), if they need. Students are allowed to team up for projects, if the scope of the project is large. The final project will count for 40% of the finla grade.

Sample projects

Here are a few sample projects final presentations (2014)

  1. Optimal Path Planning under Different Norms in Continuous State Spaces(Nobuaki Fujii and Shuyang Li)
  2. Sensitivity-Based Interval PDE Observer for Battery SOC Estimation (Hector Perez)
  3. Real-time queue length estimation: Implementation and limitations (Michaella Chung and Ziheng Lin)
  4. Dynamic Positioning Control using Hamilton-Jacobi Techniques (Qian Zhong and David Fernandez)
  5. Real-time estimation of travel times (Maxime Baez)
  6. L2 Stabilization of Coupled Viscous Burgers’ Equation (Saad Qadeer and Jean-Baptiste Sibille)
  7. Motion Planning of Water Tank (Zhi Li)
  8. Backstepping Feedback Control of Open-C (Mandy Huo and Sami Malek)
  9. Point Queue Model Validation using NGSIM Data (Avi Hecht and Vishwanath Bulusu)
  10. Optimal charging of a V2G aggregator system (François Debaillon and Caroline Le Floch)
  11. Lane Changing Decision Making Algorithm (Yaoqiong Du)
  12. A new PDE-based approach for construction scheduling and resource allocation (Paul Gabet and Julien Nachef)
  13. Analytical Solution for Two-Dimensional Transport Equation with Application in a Control Problem (Zeshi Zheng)

Here are a few sample projects final presentations (2013)

  1. Oil Spill Simulation in Ocean (Zhi Long Liu and Daniel Shen)
  2. Modeling a Circular Membrane Under Pressure (Kirti Mansukhani and Rachel Nancollas)
  3. Analytic and Numerical Solutions to the Kinematic Wave equation (Sid Feygin and Ziran Zhang)
  4. Modeling the Heat Equation in 3D for Grilling a Steak (Gaetano Andreisek and Michael Nava)
  5. Differential Games among Major Fix-wing UAVs (Zhengyin Qian)
  6. From PDEs to Simulation Games with Cellular Automata (Dan Assouline and Claudia Bongiovanni)
  7. Deciphering the Black‐Scholes‐Merton Partial Differential Equation (Antoine Chaignon and Steeve Bohbot)
  8. Parameter Identification for Multimodal Human Motion Measurement (Aaron Bestick and Robert Matthew)
  9. Crowd dynamic simulation (Camille Potiron and Bruno Burtschell)
  10. Freeway density estimation from velocity measurements (Matt Wright and David Shulman)
  11. Boundary Flow Prediction and Simulation (Myles Iribarne and Cheng-Ju Wu)
  12. Options Pricing Modeling Using Black Scholes Equation and its Extensions (Sean Mitchell and Michael Nill)
  13. Waves through Watermelons (Shuokai Chang and Angela Cheng)
  14. Reachability/Unsafe Region Analysis for Drifters (Matthew Chong)

Here are a few sample projects final presentations (2012)

  1. Application of Piezoelectric Tiles in Traffic Energy Harvesting (Kavin Phongpandecha and Katherine Yip)
  2. Optimizing Chlorine Dosing to Control Bacterial Regrowth in Drinking Water Distribution Systems (Xiao Jun Tang and John Erickson)
  3. Image Segmentation Using Chan-Vese Level Set Method and comparison with other methods (Patrick Ghannage)
  4. Options Pricing (Spencer Lin and Sebastian Ruf)
  5. Stackelberg Routing on Highway Networks (Walid Krichene)
  6. Transfer Equations: An Attempt to Pose an Optimization Problem (Henry Kagey)
  7. Motion Planning of a Water Tank with Differential Flatness and Optimal Control (Dimitar Ho, Brian Grunloh and Nick Jenson)
  8. Input-Output Control of Overhead Cranes (Jia Chang Huang)
  9. A Precompensation Filter for the Telegraph Equation (Tony Dear and Alejandro Schuler)
  10. Inversion of Rayleigh Wave Dispersion Measurements in Geological Media (Chris Sherman)
  11. Optimal Highway Traffic Control using a Velocity-Cell Transmission Model (Alessandro Castagnotto and Nicholas Wong)
  12. Water Flow Estimation in One Open Channel with Data Assimilation (Mohamed Hariri Nokob and Meng Cai)
  13. Behavior of the Lattice Schrodinger Equation (Eric Bourgain-Chang)
  14. Modeling Tree Sap Flow Using PDEs (Justin Beutel, Bryn Pittinger and Brandon Wong)
  15. Mean Field Games (Max Balandat)

Here are a few sample projects final presentations (2010)

  1. Travel time estimation using Kernel Regression (Kevin Allanet and Matthieu Nahoum)
  2. Control of active floating sensors (Leah Anderson and Kevin Weekly)
  3. A General Phase Transition Model - Applications for vehicular traffic (Juan Argote)
  4. Arterial traffic - An integrated approach of distributed systems modeling and statistical models (Aude Hofleitner)
  5. Hydrodynamic Modeling for Experimental Lagrangian Drifter Data (Díaz Ledezma)
  6. Using PDE models to describe mRNA expression pattern dynamics (Insoon Yang)
  7. Modeling of Human Response to Ground Motion using Discrete and Continuous Methods (Jack Reilly and Brenda Dix)
  8. Learning Traffic Flow Dependencies in Highway Networks (Samitha Samaranayake and Sébastien Blandin)
  9. A filtering algorithm to GPS probe vehicles (Timothy Hunter)
  10. Set Point Control of a Thermal Load Population (Travis Walter)
  11. Full Computer Simulation of a Hybrid Test (Catherine Whyte)
  12. Numerical Schemes from the Perspective of Consensus (Yusef Shafi)

Here are a few sample projects final presentations (2009)

  1. A general phase transition model for vehicular traffic (Sebastien Blandin)
  2. Control of Aggregated Power Level of Safety Messages in VANET (Ching-Ling Huang)
  3. System Identification of Distributed Parameter Systems (Claus R Danielson)
  4. Attenuation of turbulent velocity in hot-wire anemometer mea (Ilse Ruiz and Mercado)
  5. Some Analysis on Bus Data (Kayan Nowrouzi)
  6. Using PDEs to Calculate Optimal Foot Motion for Walking (Iris Tien)
  7. Estimation of Occupancy Distribution in Buildings (Mehdi Maasoumy)
  8. Using Reachable Sets to Simulate Dynamic Games (Eugene Li and Bryan Yaya)
  9. Optimization of Experimental Parameters for Comparing Enhanced Geothermal Working Fluids in the Lab (Mario Magliocco)
  10. Creating a Visual, Interactive Representation of Traffic Flow (Eric Mai)
  11. Adjoint-Based Electromagnetic Shape Optimization (Owen Miller)
  12. Variational Formulation of the LWR PDE (Anthony Patire)
  13. Estimation of State Noise for the Ensemble Kalman filter algorithm for 2D shallow water equations. (Aymeric Mellet and Julie Percelay)
  14. Real-time Estimation of Flow States in Open Channels via Lagrangian Sensing (Mohammad Rafiee)
  15. Optimization of Triple Pendulum Bearing Design (Tracy Becker)
  16. Numerical Growth & Branching (Timothy Tresierras)

Here are a few sample projects final presentations (Fall 2007)

  1. Plates Subjected to Moving Loads (Ram Rajagopal, EECS, Branko Kerkez, CEE)
  2. Shock free control of a freeway network, (Ajith Muralidharan, ME, Rene Sanchez, ME)
  3. Reachability Analysis for Hybrid Simulations, (Benjamin Fine, ME)
  4. Analysis of a “Simple” Model of Brain Activity, (Hope Weiss, ME)
  5. 2D implicit surfaces using level set methods, (James Lew, CEE)
  6. Simulation of a Highway using a second order LWR PDE, (Jerry Jariyasunant, CEE)
  7. Option pricing, (Jiangchuan Huang, CEE)
  8. Pricing of Options, (Mathieu Cabannes, CEE)
  9. Using wave propagation properties to identify soil parameters (Moanna Reynau, CEE)

Here are a few sample projects final presentations (Spring 2007)

  1. Safe aerial refueling using Hamilton-Jacobi techniques (Jerry Ding, EECS)
  2. Control of epileptic seizures in the human cortex (Beth Lopour, ME)
  3. Using the viability algorithm to develop a value function for an air traffic control problem (Andrew Tinka, CEE)
  4. Moskowitz surface and fundamental diagram generation (Eric Lew and Shuo Yang, ME)
  5. Parameter identification for soil dynamic systems (Min Chen, CEE)
  6. Active water absorber (Matthiew Carney, ME)
  7. Modeling of single flagellum bacterial motion (Justin Hsia, EECS)
  8. Modeling river dynamics on the Niger river (Emily Kumpel, CEE)
  9. Frequency model in open channel with lateral flow (Qingfang Wu, CEE)

Here are a few sample projects final presentations (2007)

  1. Shear Thickening Fluids (Andrew Armey)
  2. Public Source Identification (Arthur Wiedmer)
  3. A Study on Options Pricing (Raman Bhatia)
  4. Extremum Seeking Using a Unicycle Model (Caroline Chow)
  5. Parameter identification in NMPC algorithm with application to waypoint following by autonomous vehicle (Esten Ingar Grøtli)
  6. Hydromorphology of an Urbanizing Watershed Using Multivariate Elasticity (Julian Fulton)
  7. A Study of Transonic Flow and Airfoils (Huiliang Lui)
  8. Nonlinear Model Predictive Control Applied to Multiple Aircraft Deconflicted Path Planning with Weather Avoidance Constraints (Jessica Pannequin)
  9. Active Water-Wave Absorber (Matthew Carney)
  10. Hybrid Analysis: Error Propagation (Matthew Vaggione)
  11. Cancellation of Acoustic Waves (Trucy Phan)
  12. River Flow Control using the Hayami Model (Tarek Rabbani)
  13. Dynamic sectorization in action 2 (Rami El Mawas)
  14. Multilane Traffic Model and Simulation (Matthew Rosa)
  15. Computation of the Violin Front Response to an Excitation Generated by String Vibrations (Claire Saint-Pierre)
  16. Optimal Control for Vehicle Maneuvering (Timmy Siauw)
  17. Gait Panning for Walk and Roll Robot (Katie Strausser)

Here are a few sample projects final presentations (Spring 2006)

  1. Modeling and Optimization Analysis of Single Flagellum Bacterial Motion (Edgar Lobaton, EECS)
  2. Liquid Phase Boundary Control for Fabrication of Features in Thermoplastic, Micro-Hair Arrays (Jessica Pannequin, Brian Schubert, EECS)
  3. The Generalization and Application of Particular Solutions to Lamb’s Problem (Greg McLaskey, CEE)
  4. Active Control of Suspension Bridges (Patricia Decker, CEE)
  5. Study on Level Set Approach to Image Segmentation (Xu Guan, CEE)
  6. PDE methods for image processing (Andrew Aquila, EECS)
  7. Reachability Analysis for a Lower Extremity Exoskeleton (Kurt Amundson, ME)
  8. Computing the reachability of the LWR Equation (Ram Rajagopal, EECS)
  9. Planar Cell Polarity in Drosophila melanogaster (Anil Aswani, EECS)


For the project, you will be expected to conduct significant work on one of the following topics, or a topic of your choice related to the material covered in class:

  1. Modeling systems with PDEs for control purposes
  2. Algorithm design for control and/or optimization of PDE driven systems
  3. Simulation tools for control of PDE driven systems
  4. Hardware implementation of control and/or optimization algorithms on a PDE driven system

You will first review the literature on the subject you have chosen. If you choose your own research topic, you will be responsible for finding the proper set of articles relevant for your problem. If you choose one suggested project, some references will be provided to you as a basis for further reading. Depending on your topic, you will balance your time between algorithm design, simulation, and/or hardware implementation. In the first weeks of the project, you will be expected to set up clear goals with the instructor, and a plan to achieve these goals. You will meet with the instructor several time to assess the progress made on the project. You will give a short presentation of your project to the class at the end of the semester. 


You will be expected to write a report, to summarize your work. We suggest that you use these LaTeX files to write up your report, but you are free to use any editing software you like. Remember that your report should be written in a way which is understandable for someone who does not have exposure to the field. Number tables and figures sequentially and refer to them in the text of the results section.  Be sure to label all plot axes and tables and show units of measure.  For calculated quantities, report the appropriate number of significant figures. Here is a rough outline of what a good technical report would look like: 

  1. Introduction:  objectives of your project, background and motivation.
  2. Literature review: describe the state of the art in the field; include all proper references, explain where your project fits. 
  3. Problem investigated. Describe the physical system you are modeling, eventually describe the derivation of the model. Pose the problem of controlling the system.
  4. Control. If you are deriving your own control or optimization algorithm, include all derivations. If the derivations are too long, put them in the appendix, in order to have a clear flow in this section. Summarize your theoretical contributions. 
  5. Simulation. If you are designing a simulation tool for control or optimization purposes, describe which algorithms you have used, and how they address your problem. Describe the software implementation, and the validation of the software (for example on model problems).
  6. Hardware implementation. If you are implementing control algorithms on an experimental testbed, describe which algorithms you have used, and how they address your problem. Describe the hardware implementation.
  7. Results:  present the results of your experiments in tabular and/or graphical form, but include text that organizes and describes the results to guide the reader through them. 
  8. Discussion:  discuss the results, compare with theory, comment on the significance of the results, discuss reasons for disagreement, and suggest how the measurements and the experiment could be improved.
  9. Summarize the main results and findings of the experiment.  Nothing new here; just provide a brief restatement and summary of what is already presented in previous sections.
  10. Bibliography: list all references used in the text.
  11. Appendix: include derivations, raw data, calculations, and spreadsheets if appropriate.