Calculus of Variations

Course #MA4311

Est.imated Completion Time: 3 months

Overview

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.

Included in degrees & certificates

  • 299

Prerequisites

  • MA2121

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.
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