COURSE OUTLINE

IOE 600/EECS 600

Function Space Methods in System Theory

Fall Term 1998

Professor Robert L. Smith

Department of Industrial

and Operations Engineering

Office Hours: W 1-3, 2895 IOE Building

Phone: (734) 763 2060

Email: rlsmith@umich.edu

COURSE DESCRIPTION:

This is a mathematics course on the theory and methods of functional analysis for the modeling and analysis of systems. Topics include vector spaces, normed linear and Hilbert spaces, linear operators, optimization of functionals, and duality.

TEXT:

Luenberger, David G., Optimization by Vector Space Methods, John Wiley & Sons, NY, 1997

Exam 1 (Tu Oct 27) 35%

Exam 2 (Th Dec 10) 35%

Homework 30%

Introduction 1.1-1.4

Linear Spaces 2.1

Vector spaces 2.2-2.5

Normed linear spaces 2.6

Topology 2.7-2.9 ,2.13, 2.15

lp and Lp spaces 2.10

Banach spaces 2.11-2.12

Method of Successive Approximations 10.1-10.2

Linear Operators 6.1

Space of bounded linear operators 6.2

Inverse linear operators 6.3

Newton's Method 10.3

Hilbert Spaces 3.1

Projection Theorem 3.2-3.3

Orthogonality and the Gram-Schmidt Procedure 3.4-3.5

Normal equations and Fourier series 3.6-3.9

Minimum norm problems 3.10-3.12

Dual Spaces 5.1

Spaces of linear functionals 5.2-5.3

Hahn-Banach theorem (Extension form) 5.4

Riez-Representation theorem 5.5

Second dual spaces and weak convergence 5.6,5.10

Alignment and orthogonal complements 5.7

Minimum norm problem and its applications 5.8-5.9

Hahn-Banach theorem (Geometric form) and its applications 5.11-5.12

Optimization of Functionals (if time permits) 7.1

Gateaux and Frechet Derivatives 7.3

Conditions for an optimum 7.4

Application to the calculus of variations 7.5

Conjugate Duality (if time permits)

Convex Functionals and their properties 7.8-7.9

Conjugate functionals 7.10

Fenchel duality 7.12

COURSE POLICIES:

1) Homework: Students are allowed to work in groups on homework. However each student is individually responsible for expressing their answers in their own terms. Also you may not acquire, read, or otherwise utilize answers from solutions handed out in previous terms. Homework is due at the beginning of class one week after it is assigned.

2) Exams: Please note the exam times above. Valid excuses for failing to meet an exam are personal illness or illness in your immediate family. You must observe the Honor Code with respect to examinations and all other aspects of this course.