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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