Course detail

Mathematical Methods Of Optimal Control

FSI-9MORAcad. year: 2020/2021

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. Bellman's principle of optimality. Pontryagin's maximum principle. Time-optimal control of linear problems. Problems with state constraints. Applications.

Learning outcomes of the course unit

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.


Differential and integral calculus, ordinary differential equations.


Not applicable.

Recommended optional programme components

Not applicable.

Recommended or required reading

Pontrjagin, L. S. - Boltjanskij, V. G. - Gamkrelidze, R. V. - Miščenko, E. F.: Matematičeskaja teorija optimalnych procesov, Moskva, 1961.
Brunovský, P.: Matematická teória optimálneho riadenia, Bratislava, 1980.
Víteček, A., Vítečková, M.: Optimální systémy řízení, Ostrava, 1999.
Lee, E. B. - Markus L.: Foundations of optimal control theory, New York, 1967.
Čermák, J.: Matematické základy optimálního řízení, Brno, 1998.
Alexejev, V. M. - Tichomirov, V. M. - Fomin, S. V.: Matematická teorie optimálních procesů, Praha, 1991.

Planned learning activities and teaching methods

The course is taught through lectures explaining the basic principles and theory of the discipline.

Assesment methods and criteria linked to learning outcomes

Course-unit credit is awarded on the following conditions: Active participation in seminars. 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 written (possibly followed by an 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.

Language of instruction

Czech, English

Work placements

Not applicable.


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.

Specification of controlled education, way of implementation and compensation for absences

Attendance at lectures is recommended. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.

Classification of course in study plans

  • Programme D-APM-K Doctoral, 1. year of study, summer semester, 0 credits, recommended

  • Programme D4P-P Doctoral

    branch D-APM , 1. year of study, summer semester, 0 credits, recommended

  • Programme D-KPI-P Doctoral, 1. year of study, summer semester, 0 credits, recommended

Type of course unit



20 hours, optionally

Teacher / Lecturer


1. The scheme of variational problems and basic task of optimal control theory.
2. Dynamic programming. Bellman's principle of optimality.
3. Maximum principle.
4. Time-optimal control of an uniform motion.
5. Time-optimal control of a simple harmonic motion.
6. Basic properties of optimal controls.
7. Optimal control of systems with a variable mass.
8. Variational problems of flight dynamics.
9. Energy-optimal control problems.
10. Variational problems with state constraints.