The course is aimed at students in mathematics at the comprehensive as well as the technical universities. Solid knowledge of linear algebra, e.g. Linear Algebra and Its Applications by David Lay, and calculus, e.g., Calculus by James Stuart or Calculus: a complete course by Robert Adams, are essential.

**Aim of the course**

The course aims at students in pure and applied mathematics with an interest in applications of (linear) algebra and ordinary differential equations. The purpose of the course is to introduce the students to basic concepts and more advanced notions of the mathematical theory of systems.

**Description**

Mathematical systems theory is concerned with problems related to dynamic phenomena in interaction with their environment. These problems include:

- modelling. Obtaining a mathematical model that reflects the main features. A mathematical model may be represented by difference or differential equations, but also by inequalities, algebraic equations and logical constraints.\
- Analysis and simulation of the mathematical model.
- Prediction and estimation.
- Control. By choosing inputs or, more general, by imposing additional constraints on some of the variables, the system may be influenced so as to obtain certain desired behavior. Feedback is an important example of control.

The main objects of study in this course are systems modeled by linear time-invariant differential equations. We start with a treatment of the theory of algebraic representation of dynamical systems using polynomial matrices. The main tool is the Euclidean algorithm applied to matrices of real polynomials. The main result is a complete characterization of all representations of a given system.

Several other representations are introduced along with their relations. Important examples of such representation are input-output representations that reveal that some variables may be unrestricted by the equations, and state space representations that visualize the separation of past and future, also referred to as the Markov property.

Controllability and observability are important system theoretic concepts. A controllable system has the property that a desired future behavior can always be obtained, independent of the past behavior, provided that this future behavior is compatible with the laws of the system.

Observability means that the complete behavior may be reconstructed from incomplete observations. The theory of controllability and observability forms one of the highlights of the course. Stability can be an important and desirable property of a system. Stabilization by static or dynamic feedback is one of the key features of Systems and Control. In the pole placement theorem linear algebraic methods and the notion of controllability are used in their full strength. The theorem, loosely speaking, says that in a controllable system the dynamical behavior can be changed as desired, in terms of characteristic values, by using appropriate feedback. It forms one of the most elegant results of the course and indeed of the field of Systems and Control.

**Schedule**

13 September: Utrecht

20 September: Utrecht

27 September Utrecht

4 October: Utrecht

11 October: Utrecht

18 October: Utrecht

25 October: Utrecht

1 November: Utrecht

8-12 November: Intensive Week in Twentee

22 November: Presentations, Utrecht

6 December: Exam, Utrecht

10 January: Retake, Utrecht.

- Docent: Jan Willem Polderman
- Docent: Felix Schwenninger
- Docent: Stephan Trenn