Optimization and Systems theory    |   KTH    |

Latest News

Here you can find the latest news about the course.
August 19
The notes for the first class are now available here First class
August 20
The notes for the second class are now available here Second class
August 21
The notes for the third class are now available here Third class New version
August 21
Recommended reading:
Reference material for Lectures 1-3
Partial stochastic realization
Byrnes on Euler
August 25
The notes for the fourth class are now available here Fourth class
August 26
The homework problems are now available Homework
August 27
The notes for the fifth class are now available Fifth class
August 27
Recommended reading:
Important moments in control
Moment problems in the spirit of Krein

Lectures on moment problems in signals, systems and control - summer 2008

SF3881: Lectures on moment problems in signals, systems and control

- A two week intensive mini-course, 3 högskolepoäng

Teacher: Professor Chris Byrnes

The course will be given during two weeks. People who are only interested in coming to the seminars to listen (and interact) are most welcome to do so. For those who would like to get credits the examination will be in the form of exercises. Note that the credits are equivalent to two weeks full time studies, so it will be expected that the students put all their energy on this course and that they finish it in these two weeks.


Beginning with Chebychev’s use of power moments to prove the Central Limit Theorem in the 19th Century, the moment problem has matured from its various special forms to a general class of problems that continues to exert profound influence on the development of analysis and its applications to a wide variety of fields. The crossroads of signals, systems and control are no exception, where moment methods have historically been used in circuit theory, model reduction, optimal control, robust control, signal processing, spectral estimation and stochastic realization theory. Indeed, the moment problem as formulated by Krein et al is a beautiful generalization of several important classical moment problems, including the power moment problem, the trigonometric moment problem and the moment problem arising in Nevanlinna-Pick interpolation.

In this course, we first explore an array of examples, starting with the Chebychev’s calcualtions and with the classical use of moments for a form of model reduction. This naturally leads to the interpretation of a broad range of interpolation problems within the context of the generalized moment problem, in the sense of Krein and Nudel'man. We also review the moment problem as formulated by Markov and its application to time optimal control. Each of these formulations involve a natural constraint on the required form of the solution of the corresponding moment problem, but both make essential use of convexity.

Motivated by classical applications and examples, in both finite and infinite dimensional system theory, we recently formulated another version of the monent problem that we call the moment problem for positive rational measures. The formulation reflects the importance of rational functions in engineering appplications. While this version of the problem is decidedly nonlinear, the basic tools still rely on convexity. In particular, we present a solution to this problem in terms of a convex optimization problem that generalizes the maximum entropy approach used in several classical special cases. We conclude with several applications to problems in signals, systems and control.

The schedule for the classes is as follows.

Lesson Date Day Location
1 18/8 Monday 3733
2 20/8 Wednesday 3721
3 21/8 Thursday 3721
4 25/8 Monday 3721
5 27/8 Wednesday 3721
6 28/8 Thursday 3721

The seminar rooms 3721 and 3733 are located on the 7:th floor at the math department, Lindstedtsv. 25.

This course is sponsored by CIAM and ACCESS.