MTH 490 Directed Study

• Lecture:

  • Introduction to Optimal control theory and Hamilton-Jacobi equation.
  • Instructor: Son Tu – tuson@msu.edu
  • Wells Hall D227A (F: 3:00 pm - 4:00 pm).

• Syllabus PDF

A presentation by Minh Nguyen

Minh Nguyen received the "Best Presentation Award" at the 21st Math Student Conference. You can view his presentation here.

Goodreads

I enjoy reading expository texts as they refresh and motivate me greatly. For the subject of this course, here are some interesting reads. While not always directly related, I found them to be great for piquing curiosity.

Note: The list is not in any particular order.

• Hynd, R. The Hamilton-Jacobi equation, Then and Now.
  Notices of the American Mathematical Society 68, 9 (2021), 1457–1467 link   ·   pdf

• Evans, L. Partial Differential Equations.
  Short survey on the mordern theory of PDEs link   ·   pdf

Materials

The lecture note will be updated occasionally here.

• Apr 12, 2024:

  • Optimal control and the Dynamic Programming Principle (DPP), cont.
  • Showing the value function is a viscosity solution to the Hamilton-Jacobi equation.

• Apr 04, 2024:

  • Optimal control and the Dynamic Programming Principle (DPP)

• Mar 29, 2024:

  • Rate of convergence in vanishing viscosity using doubling variable method

• Mar 22, 2024: Lipschitz estimate and finite speed of propagation

• Mar 15, 2024: Perron's method

  • Perron's method for static 1st-order equation
  • Perron's method for time-dependent 1st-order equation

• Mar 08, 2024:

  • Showing the distance function is the viscosity solution to the Eikonal equation.
  • Bernstein's method.

• Feb 24, 2024: Comparison principle for time-dependent problem.

• Feb 16, 2024: Consistency of Viscosity solutions and Comparison principle, doubling variable method for static first-order equations
• Feb 09, 2024: Stability of Viscosity solutions
  • Showing the distance function is the solution to the Eikonal equation
  • Showing $D^\pm u(x)$ is dense
• Feb 02, 2024: Subdifferential and Superdifferential
• Jan 26, 2924: Formal definition of Viscosity solution
• Jan 19, 2024: Detailed calculation on the 1D Eikonal and its 2nd-order approximations $$ \begin{cases} |u'(x)|=1 \qquad \text{in}\;(-1,1) \\ u(-1) = u(1) = 0. \end{cases} $$
• Jan 12, 2024: Introduction to the optimal control problems and the PDE for the value function

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