Modelling and control of systems

Practical Mathematics
3 year
first or second
Hours per week – 1. or 2. semester:
Content (Syllabus outline)

Nonlinear systems of differential equations:
phase plane, qualitative analysis, equilibrium, nullclines, limit cycles, linearization, stability.
Discrete dynamical systems:
first and second order iterative equations, logistic equation, cycles, kaos.
(project(seminar): money for the future, controlled fishery, spread of disease, wolf/moose system , extinction diabetes mode, etc)
Fundaments of the control of system:
proportional, derivative and integral feedback.
(project(seminar): heating control, fermentation, etc)
Optimal control of systems:
modeling optimal control, choice of objective function, Pontryagin maximus principle, control with minimal energy, minimal time, »bang-bang control«.
(project(seminar): hotrod problem, stopping a space ship, controlling the harmonic oscillator, etc)


J.Farlow, J.E.Hall, J.M.McDill, B.H.West: Differential Equations & Linear Algebra, Prentice Hall, New Jersey, 2002.
F. Križanič: Navadne diferencialne enačbe in variacijski račun, DZS, 1974, str. 457- 484.
D.J.Higham, N.J.Higham: Matlab Guide, SIAM, Philadelphia, 2000.

Objectives and competences

Students will acquire elementary knowledge about regulation and optimal control of linear and non linear systems (discrete and continuous), and some skill how to solve this problems with program Matlab.
They will be able to use the acquired knowledge at posing and resolving problems that appears in practices, such as, mechanics, environment sciences, and economics.

Intended learning outcomes

Knowledge and understanding:
Knowledge and understanding of the basic concepts of control of the systems.
Capacity to connect their mathematical knowledge and programming in Matlab, and to implement developed methods in natural sciences and technics.
Modeling and control of systems is one of the basic subjects necessary to understand mechanics and other subjects of biological, technical and social sciences. Knowledge is necessary in modeling of almost all systems.
Integrating theory and practical applications in solving problems.
Transferable skills:
Posing a problem, selection of a method and its application in solving the problem. Analysis of the results from the cases. Skills in using literature. Knowledge is transmitted to virtually all sciences.

Learning and teaching methods

Lectures, exercises, homeworks, consultations


Type (examination, oral, coursework, project):
positive mark on the project (seminar) is precondition to seed the theoretical part of the exam.
oral exam
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

Jasna Prezelj:
FORSTNERIČ, Franc, IVARSSON, Björn, KUTZSCHEBAUCH, Frank, PREZELJ-PERMAN, Jasna. An interpolation theorem for proper holomorphic embeddings. Mathematische Annalen, ISSN 0025-5831, 2007, bd. 338, hft. 3, str. 545-554. [COBISS-SI-ID 14335065]
PREZELJ-PERMAN, Jasna. A relative Oka-Grauert principle for holomorphic submersions over 1-convex spaces. Transactions of the American Mathematical Society, ISSN 0002-9947, 2010, vol. 362, no. 8, str. 4213-4228. [COBISS-SI-ID 15641433]
PREZELJ-PERMAN, Jasna, SLAPAR, Marko. The generalized Oka-Grauert principle for 1-convex manifolds. Michigan mathematical journal, ISSN 0026-2285, 2011, vol. 60, iss. 3, str. 495-506. [COBISS-SI-ID 16134745]