Funktionalanalys i systemteori

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

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

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.

*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

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.

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

**Types of instruction:** Lectures, exercises

**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**Grading scale:** Failed, pass**Examiner:**

**Assumed prior knowledge:** Linear algebra, Multidimensional analysis, Complex Function theory, Systems and Transforms

**Course coordinator:** Andrey Ghulchak `<andrey.ghulchak@math.lth.se>`