 2020-2021
Master Cyber-Physical Systems

# Syllabus

This course presents the main numerical tools for the study and the analysis of continuous-time dynamical systems. Different classes of dynamical systems will be considered among ordinary differential equations, differential-algebraic equations, and differential equations with delay in the first part of the course. Numerical algorithms to solve initial value problems and two point boundary value problems will be presented as well as their stability properties emphasizing their pros and cons. The last part of the course will be an introduction to the reachability analysis of dynamical systems based on interval analysis methods. Practical exercices on computers using Python and C++ will help understanding the presented algorithms.

This course is made of two parts:

• Part 1: classical numerical methods for various kinds of dynamical systems
• Part 2: set-based methods to rugourously enclose solution of dynamical systems

# Teachers

• Alexandre Chapoutot, U2IS
• Julien Alexandre dit Sandretto, U2IS

# Numerical methods for dynamical systems

## Lecture 1

Introduction on numerical simulation methods. In particular, a quick overview of the simulation engine of Simulink will be presented.

## Lecture 2

Initial value problem of ODE: one-step methods

## Lecture 3

Initial value problem of ODE: muti-step methods

## Lecture 4

Introduction to stability analysis for numerical methods for ODE. Introduction to DAE theory and numerical methods to sovle initial value problems for DAE

## Lecture 5

Discontinuous dynamical systems. Introduction to two point boundary value problems for ODE. Introduction to delay differential equations and their numerical solutions.

# Guaranteed numerical integration

## Lecture 1

Introduction to interval analysis

## Lecture 2

Set-based numerical integration of ODE

## Lecture 3

Set-based numerical integration of DAE

TBD

# Software

• For practical work of Part 1, Python should be used

• For practical work of Part 2, DynIBEX will be used. It is a C++ library combining interval constraint solvers with numerical guranteed integration of ODE/DAE