Numerical modelling and optimization (2020/2021)

Course code
Giacomo Albi
Teaching is organised as follows:
Unit Credits Academic sector Period Academic staff

Learning outcomes

The aim of the first module is to deepen the knowledge and skills especially in the modern theory of dynamical systems and give the student a solid appreciation of the deep connections between mathematics and other scientific disciplines, both in terms of the mathematical problems that they inspire and the important role that mathematics plays in scientific research and industry. Mathematical software tools, and others, will be used to implement algorithms for the solution of the real world problems studied during the course. At the end of the course the student is expected to be able to complete professional and technical tasks of a high level in the context of mathematical modelling and computation, both working alone and in groups. In particular the student will be able to write a model of a real problem, to recognise the effective parameters and analyse the model and its possible implications. The second module wants to provide sufficient theoretical and numerical background for the optimal control of dynamical systems. Such problems will be developed by means of real application examples, and recent research studies. At the end of the course students will be able to decide which numerical method is suitable for the solution of some specific optimal control problems. He/She will be able to provide theoretical results on the controllability and stability of certain optimal control problem and numerical methods. He/She will be able to develop his/her own code, and capable choose the appropriate optimization method for each application shown during the course.


The entire course will be available online. In addition, a number of the lessons/all the lessons (see the course
schedule) will be held in-class.

The course presents different differential models and their control with application in biology, economics and robotics.
The analysis of these models will be enached by the study of theoretical aspects, and the development of several computational methods.

The course is divided in two parts, for the detail program refer to the dedicated page

* Modelling of complex and multi-agent systems (swarming, opinion formaton, Network, and (non-)holonomic systems).
* Bifurcation analysis, and geometric control.


*Introduction to Optimal control and numerical methods: direct and indirect methods, dynamic programming, Model-Predictive Control.
* Linear and Nonlinear optimization, KKT conditions, gradient methods, quasi-Newton and Newton methods. Convex optimization.
*Examples and exercises with Matlab/Octave.

The program is in accordance with the ECMI standards (European Consortium for Mathematics in Industry,

Assessment methods and criteria

The student is expected to demonstrate the ability to mathematically formalize and solve models used in several scientific discipline, using, adapting and developing the models and advanced methods discussed during the lectures. To that end the final evaluation will consist in a written and oral exam.

Written exam: Questions and exercises the solution could require the use of computer.
Oral exam: Project and discussion of the written exam with questions. The subject of the project should be decided together with the teacher.

The assessment methods could change according to the academic rules.
The online exam is granted for all the students will require it during the academic year 2020/21.

Reference books
Author Title Publisher Year ISBN Note
A. Bressan, B. Piccoli Introduction to the Mathematical Theory of Control AIMS 2008 1-60133-002-2
Nocedal, Jorge, Stephen Wright Numerical optimization Springer Science & Business Media 2006
Stephen Lynch Dynamical Systems with Applications using Mathematica® (Edizione 1) Birkhäuser 2017 978-3-319-87089-2 Access to the Notebook used in the book
Stephen Lynch Dynamical Systems with Applications using MATLAB® (Edizione 2) Birkhäuser 2014 978-3-319-33041-9 Access to the Matlab files used in the book
Stephen Lynch Dynamical Systems with Applications using Python® (Edizione 1) Birkhäuser 2018 978-3-030-08624-4 Access to the python files used in the book
S. Strogatz Nonlinear Dynamics and Chaos: With Applications to Physics, Biology, Chemistry, and Engineering Westview Press 2010