Course Syllabus for

# Functional Analysis in System Theory Funktionalanalys i systemteori

## FMA003F, 9 credits

Valid from: Autumn 2013
Decided by: FN1/Anders Gustafsson
Date of establishment: 2014-06-09

## General Information

Division: Mathematics
Course type: Third-cycle course
Teaching language: English

## Aim

System theory uses often linear models to describe and optimize dynamic processes. The main goal of the course is to introduce linear systems as abstract linear operators and to give knowledge about basic notions and methods in functional analysis that are used to study and solve optimization problems for such operators i normed spaces. The course develops also an ability to mathematical abstraction that makes it easier to see similarities between different problems, and is suitable for diverse applications, such as control theory, signal processing etc.

## Goals

Knowledge and Understanding

For a passing grade the doctoral student must

• know and understand the main notions for normed vector spaces and linear operators
• have understanding of the relation between linear systems and operators, especially how different properties of linear systems can be reformulated for operators and vice versa
• have knowlidge about typical vector spaces that are the most popular in applications, and their dual spaces.
• have understanding for the min-max duality principle and the main conditions for it
• be able to explain the teory basics on the oral examination

Competences and Skills

For a passing grade the doctoral student must

• be able to rewrite a linear system as operator and vice verca
• be able to calculate adjoint and inverse to linear systems
• be able to reformulate a particular problem as an abstract optimization problem in the course, calculate the dual problem and use the alignment principle to find solutions
• be able to use the notions to solve problems within the course frame

## Course Contents

Normed vector spaces, Banach/Hilbert spaces. Linear operators, adjoint and invers operator. Linear systems as operators, adjoint systems, stability. Quadratic optimization-problems for linear systems. Causal and time-invariant systems, Hankel/Toeplitz operators, transfer function. Topological vector spaces, linear functionals, dual space. Weak topologies. Optimization in Banach/Hilbert spaces. Min-max theorem and duality. Minimum norm theorems. Nehari theorem and other extremal problems in Hardy spaces. Hahn-Banach theorem and separation of convex sets. Convex analysis in normed spaces.

## Course Literature

The course literature is a compilation from several books in functional analysis and optimization (available) as well as the lecture notes

## Instruction Details

Types of instruction: Lectures, exercises

## Examination Details

Examination formats: Written exam, oral exam, written assignments. Weekly hand-in problems or take-home exam Students should take an active role in the weekly exercise sessions