Engelska

Vid tillräcklig efterfrågan

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.

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.

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

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

Föreläsningar

övningar

Skriftlig tentamen

Muntlig tentamen

Inlämningsuppgifter

Weekly hand-in problems or take-home exam
Students should take an active role in the weekly exercise sessions

Underkänd, godkänd

Linear algebra, Multidimensional analysis, Complex Function theory, Systems and Transforms

 

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

FMA003F

 -06-09

FN1/Anders Gustafsson