Fundamentals of Optimal Control Theory
FSI-SORCompulsoryMaster's (2nd cycle)Acad. year: 2016/2017Winter semester2. year of study4 credits
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.
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.
Mode of delivery
90 % face-to-face, 10 % distance learning
Linear algebra, differential and integral calculus, ordinary differential equations, mathematical programming, calculus of variations.
Recommended optional programme components
Recommended or required reading
Pontrjagin, L. S. - Boltjanskij, V. G. - Gamkrelidze, R. V. - Miščenko, E. F.: Matematičeskaja teorija optimalnych procesov, Moskva, 1961.
Bryson A.E., Ho Y.C.: Applied Optimal Control, Taylor & Francis, USA, 1975.
Víteček, A., Vítečková, M.: Optimální systémy řízení, Ostrava, 1999.
Lee, E. B. - Markus L.: Foundations of optimal control theory, New York, 1967.
Alexejev, V. M. - Tichomirov, V. M. - Fomin, S. V.: Matematická teorie optimálních procesů, Praha, 1991.
Čermák, J.: Matematické základy optimálního řízení, Brno, 1998.
Planned learning activities and teaching methods
The course is taught through lectures explaining the basic principles and theory of the discipline. Exercises are focused on practical topics presented in lectures.
Assesment methods and criteria linked to learning outcomes
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 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
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, attendance at seminars is checked. Lessons are planned according to the week schedules. Absence from seminars may be compensated for by the agreement with the teacher.
Type of course unit
26 hours, optionally
Teacher / Lecturer
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.
13 hours, compulsory
Teacher / Lecturer
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).