Functional Analysis in System Theory
Funktionalanalys i systemteori

FMA003F, 9 högskolepoäng

Gäller från och med: Autumn 2013
Beslutad av: FN1/Anders Gustafsson
Datum för fastställande: 2014-06-09

Allmänna uppgifter

Avdelning: Mathematics
Kurstyp: Ren forskarutbildningskurs
Undervisningsspråk: English

Syfte

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.

Mål

Kunskap och förståelse

För godkänd kurs skall doktoranden

• 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

Färdighet och förmåga

För godkänd kurs skall doktoranden

• 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

Kursinnehåll

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.

Kurslitteratur

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

Kursens undervisningsformer

Undervisningsformer: Föreläsningar, övningar

Kursens examination

Examinationsformer: Skriftlig tentamen, muntlig tentamen, inlämningsuppgifter. Weekly hand-in problems or take-home exam Students should take an active role in the weekly exercise sessions
Betygsskala: Underkänd, godkänd
Examinator:

Antagningsuppgifter

Förutsatta förkunskaper: Linear algebra, Multidimensional analysis, Complex Function theory, Systems and Transforms

Kurstillfällesinformation

Kontaktinformation och övrigt

Kursansvarig: Andrey Ghulchak <andrey.ghulchak@math.lth.se>