April 2009

Problems of Advanced Optimization


The course is given by Horia Cornean from the Department of Mathematical Sciences.

A short description:
This 3 ECTS course is intended to be a rather elementary introduction to two types of optimization problems; it is mainly addressed to students with a background in engineering, therefore certain proofs are considered too technical to be presented.

We will start with a short review of a number of analysis results for functions of several variables; then we will focus on the Lagrange multiplier method and see how it can be naturally extended in order to arrive in a natural way to the famous Pontryagin Maximum Principle (PMP). We will then make the connection with the Calculus of Variations, where one typically tries to minimise an integral depending on a curve (e.g. the travel time for an aeroplane); the central point there will be the study of Euler equations.

Then the treated sphere of problems will be extended to the general setting of Optimal Control theory, where one has a system of differential equations depending on a (time-dependent) control parameter, which one tries to determine such that the `cost functional' (an integral) is minimised. We will investigate in detail the Maximum Principle including various transversality conditions; the Euler equations in calculus of variations will be shown to be a particular case of PMP. Also sufficient conditions for solutions will be covered. Application to mechanics and economics will be presented, and the linear quadratic cost problem will be a main example; there will room and time for questions and solving of exercises.

Prerequisites: Basic differential and integral calculus together with the theory of ordinary differential equations.


  1. Lawrence C. Evans, University of California at Berkeley: An Introduction to Mathematical Optimal Control Theory Version 0.2. The notes can be downloaded from here.
  2. Jon Johnsen, AAU: Notes on Calculus of Variations. The notes can be downloaded from here.
  3. Horia Cornean, AAU: Basic vector analysis for engineers. Download.
  4. Horia Cornean, AAU: On the maximum principle. Download.