*Course Syllabus for*
# Optimal Control

Optimal styrteori

## FRT105F, 7.5 credits

**Valid from:** Spring 2014

**Decided by:** FN1/Anders Gustafsson

**Date of establishment:** 2014-03-12

## General Information

**Division:** Automatic Control

**Course type:** Third-cycle course

**Teaching language:** English

## Aim

Optimal control is the problem of determining the control function for a dynamical system to minimize a cost related to the system trajectory. The subject has its roots in the calculus of variations but it evolved to an independent branch of applied mathematics and engineering in the 1950s.
The overall goal of the course is to provide an understanding of the main results in calculus of variations and optimal control, and how these results can be used in various applications such as in robotics, finance, economics, and biology

## Goals

*Knowledge and Understanding*

For a passing grade the doctoral student must

- have developed an understanding for:
- The steps leading up to the Euler-Lagrange equation to account for different boundary conditions,
- The main steps in the proof of Pontryagin's Maximum Principle (PMP),
- The bang-bang principle in optimal control,
- The notion of viscosity solutions to the Hamilton-Jacobi-Bellman equation,

*Competences and Skills*

For a passing grade the doctoral student must

- be able to use calculus of variations to solve infinite dimensional optimization problems,
- be able to use PMP to solve optimal control problems,
- be able to use Dynamic Programming (DynP) to solve optimal control problems,
- be able to solve Linear Quadratic Regulator (LQR) problems,
- be able to use computer software to obtain numerical solutions to optimal control problems.

## Course Contents

The Euler-Lagrange equation. Pontryagin's maximum principle. Dynamic programming. The Hamilton-Jacobi-Bellman equation. Finite and infinite horizon optimal control problems. Viscosity solutions to partial differential equations. The Linear quadratic regulator. Necessary and sufficient conditions for optimality. Numerical methods for optimal control problems.

## Course Literature

Liberzon, D.: Calculus of Variations and Optimal Control: A Concise Introduction.

**Types of instruction:** Lectures, seminars, project

**Examination format:** Seminars given by participants

**Grading scale:** Failed, pass

**Examiner:**

## Admission Details

## Course Occasion Information

**Course coordinators:**

**Web page:** http://www.control.lth.se/Education/DoctorateProgram/optimal-control.html