MA4311 Calculus of Variations

First and second order tests, Lagrange multipliers, Euler-Lagrange equation, non-smooth solutions, optimization with constraints, Weierstrass condition, optimal control of ODE systems, Pontryagin maximum principle. Applications may include: control and dynamical systems, estimation, weak formulations, Hamilton's variational principle, or others depending on the interests of the students.

Prerequisite

MA2121

Lecture Hours

4

Lab Hours

0

Course Learning Outcomes

• Apply first and second order tests to solve unconstrained optimization problems.
• Apply Lagrange multipliers to solve optimization problems with equality constraints.
• Apply KKT conditions to solve general optimization problems with both equality and inequality constraints.
• Design deep neural networks using various types of activation functions and architectures.
• Use a deep learning package to train neural networks for regression problems.
• Apply techniques of the calculus of variations to derive the Euler-Lagrange equation.
• Minimize a cost functional with a fixed end.
• Solve problems with end-points not fixed.
• Optimize a cost functional with nonsmooth solutions.
• Solve isoperimetric problems and optimization with constraints.
• 2. Apply Pontryagin Maximum Principle to find optimal control for simple examples.
• 4. Find time optimal control for linear systems.
• Derive the Riccati equation for the Linear Quadratic Regulator problem (LQR).
• Solve LQR problems using the Riccati equation.