Course detail

Fundamentals of Optimal Control Theory

FSI-SOR-AAcad. year: 2025/2026

The course familiarises students with basic methods used in the modern control theory. This theory is presented as a remarkable example of the interaction between practical needs and mathematical theories. Also dealt with are the following topics:
Optimal control. Pontryagin's maximum principle. Time-optimal control of linear problems. Problems with state constraints. Singular control. Applications.

Language of instruction

English

Number of ECTS credits

5

Mode of study

Not applicable.

Guarantor

prof. RNDr. Jan Čermák, CSc.

Department

Institute of Mathematics (ÚM)

Entry knowledge

Linear algebra, differential and integral calculus, ordinary differential equations, mathematical programming, calculus of variations.

Rules for evaluation and completion of the course

Course-unit credit is awarded on the following conditions: Active participation in seminars. Fulfilment of all conditions of the running control of knowledge.
Examination: The examination tests the knowledge of definitions and theorems (especially the ability of their application to the given problems) and practical skills in solving of examples. The exam is oral part.
Grading scheme is as follows: excellent (90-100 points), very good
(80-89 points), good (70-79 points), satisfactory (60-69 points), sufficient (50-59 points), failed (0-49 points). The grading in points may be modified provided that the above given ratios remain unchanged.
Attendance at lectures is recommended, attendance at seminars is obligatory and checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Aims

The aim of the course is to explain basic ideas and results of the optimal control theory, demonstrate the utilized techniques and apply these results to solving practical variational problems.
Students will acquire knowledge of basic methods of solving optimal control problems. They will be made familiar with the construction of mathematical models of given problems, as well as with usual methods applied for solving.

Study aids

Not applicable.

Prerequisites and corequisites

Not applicable.

Basic literature

Bryson, A.E. - Ho, Y.Ch. :Applied Optimal Control: Optimization, Estimation and Control, Taylor & Francis Group, New York, 1975. (EN)
Howlett, P.G. - Pudney,P.J.: Energy-Efficient Train Control,Springer, London, 1995.. (EN)

Recommended reading

Geering H.-P.: Optimal Control with Engineering Applications, Springer-Verlag Berlin and Heidelberg GmbH & Co. KG, 2007 (EN)
Lee, E. B. - Markus L.: Foundations of optimal control theory, New York, 1967. (EN)
Pontrjagin, L. S. - Boltjanskij, V. G. - Gamkrelidze, R. V. - Miščenko, E. F.: Matematičeskaja teorija optimalnych procesov, Moskva, 1961. (EN)

Classification of course in study plans

  • Programme N-AIM-A Master's 2 year of study, summer semester, compulsory
  • Programme N-MAI-A Master's 1 year of study, summer semester, compulsory
  • Programme N-MAI-P Master's 1 year of study, summer semester, compulsory

Type of course unit

 

Lecture

26 hod., optionally

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. The scheme of variational problems and basic task of optimal control theory.
2. Maximum principle.
3. Time-optimal control of an uniform motion.
4. Time-optimal control of a simple harmonic motion.
5. Basic results on optimal controls.
6. Variational problems with moving boundaries.
7. Optimal control of systems with a variable mass.
8. Optimal control of systems with a variable mass (continuation).
9. Singular control.
10. Energy-optimal control problems.
11. Variational problems with state constraints.
12. Variational problems with state constraints (continuation).
13. Solving of given problems.

Exercise

13 hod., compulsory

Teacher / Lecturer

prof. RNDr. Jan Čermák, CSc.

Syllabus

1. The general scheme of variational problems demonstrated by examples.
2. The basic task of optimal control theory demonstrated by examples.
3. Time-optimal control of an uniform motion demonstrated by examples.
4. Time-optimal control of a simple harmonic motion demonstrated by examples.
5. Linear time-optimal control problems with fixed boundaries.
6. Linear time-optimal control problems with moving boundaries.
7. Optimal control of systems with a variable mass demonstrated by examples.
8. Optimal control of systems with a variable mass demonstrated by examples (continuation).
9. Optimal control of systems with a variable mass demonstrated by examples (continuation).
10. Problem of an energy optimal control of a train.
11. Nonlinear programming problems in optimal control problems.
12. Variational problems with state constraints.
13. Variational problems with state constraints (continuation).