RTNS 2007

PRESENTATION
This is the fourth winter school in Dynamical Systems of the DANCE (Dinámica, Atractores y Nolinealidad: Caos y Estabilidad) Spanish network.
COURSES

• Carmen Chicone (University of Missouri)
  An Invitation to Applied Mathematics
• Russell Johnson (University of Firenze)
  Oscillation theory and control of nonautonomous systems
• Enric Fossas-Colet (Universidad Politècnica de Catalunya)
  Variable Structure Systems. Sliding Control Mode

CONTENTS

An Invitation to Applied Mathematics
Carmen Chicone
Abstract An approach to applied mathematics through mathematical modeling with differential equations will be presented in context using historical and modern examples. Students will take a guided through the development and analysis of several mathematical models of real physical processes. The aim of the course is to introduce students to the main issues encountered in applied mathematics; motivate their advanced study of differential equations, mathematical analysis, numerical methods, and science; and stimulate interest in applied mathematics.

1. What is Applied Mathematics?
2. Gravity and Two Body Interaction.
3. How to solve an Ordinary Differential Equation.
4. Conservation of Mass I: Stirred Tanks and Membrane Transport.
5. Conservation of Mass II: Reaction Diffusion.
6. How to solve a Partial Differential Equation.
7. The Watt Governor I: When is a System Stable?
8. The Watt Governor II: What Happens When Design Parameters Change?
9. Projectiles I: Ballistics and Pursuit.
10. Projectiles II: Missiles and Control.

Lecture notes will be provided. Exercises in modeling, analysis, and numerical methods will be suggested. Demonstrations of numerical computations (using Mathematica) will be given. Some concepts from basic differential equations, vector analysis, advanced calculus, and linear algebra will be used. But, no special knowledge beyond the usual background from undergraduate mathematics will be assumed.

Oscilation theory and control of nonautonomous systems
Russell Johnson

1. Nonautonomous Dynamical Systems
   1. Elements of topological dynamics
   2. Elements of ergodic theory
   3. The Bebutov construction
   4. Exponential dichotomies
   5. Lyapunov exponents
   6. Rotation numbers
2. Oscillation and Control
   7. Disconjugacy
   8. The linear regulator problem
   9. The Yakubovich frequency theorem
   10. Minimal attenuation control
   11. Absolute stability

Variable Structure Systems. Sliding Control Mode
Enric Fossas-Colet

1. Introduction. History. Examples. Some specific problems (existence of solutions, non uniqueness,...). Mathematical remarks. The uniqueness problem.
2. Single input, single output systems. Geometric methods. The equivalent control method. Ideal sliding dynamics. Robustness.
3. Examples and exercices. Basic converters in DC-DC, DC-AC operation modes.
4. Multi-input, multi-output systems. A kind of Lyapunov theorem. Changing variables (in the states, in the inputs). Other design methods. Ideal sliding dynamics.
5. Examples and exercices.

There will be two practical sessions on computers.