1. SylabUZ
2. Faculty of Exact and Natural Sciences
3. winter term 2023/2024
4. Mathematics - Second-cycle studies leading to MS degree
5. Control Theory 2

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Control Theory 2 - course description

Aim of the course

After the course of “control theory 2” students should be able to solve themselves practical and theoretical problems on the topic of dynamical nonlinear systems and theory of set valued functions.

 

Prerequisites

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Theory of measure and Lebesgue's integral, Control theory 1 

Scope

Lecture:

1. Problems of the theory of optimal control (6 hours).
2.  Controllability and properties of admissible sets of dynamical controlled systems (4 hours).
3.  Support functions and integral maximum rule (2 hours).
4.  Hausdorff metric, continuity of operators (4 hours).
5. Problem of optimal control as a differential inclusion (2 hours).
6. Continuity and measurability of multifunctions (4 hours).
7. Selection problems: minimal, Tschebyshev, barycentric and Steiner point (4 hours).
8. Selections theorems of Michael and Kuratowski Ryll-Nardzewski  (4 hours).
9. Kakutani fixed point theorem (2 hours).
10.  Filipoff theorem  (2 godz.).
11.  Aumann integral and its properties (4 hours).
12. Connections of control problems with theory of differential inclusions and theorems of the existence of solutions of differtential inclusions (3 hours).
13. Viability problem (4 hours).

Classes:

1. Practical problems leading to optimal control problems (2 hours).
2.  Properties of admissible sets in dynamical control systems (2hours).
3. calculation of support functions and applications of integral maximum rule (1 hour).
4. Calculation of Hausdorff metric in given spaces (2hours).
5. Calculation of upper and lower semicontinuity and measurability of given multifunctions  (2 hours).
6. Minimal, Tschebyshev, barycentric and Steiner selections - practical calculations (2 hours).
7. Viability problem and properties of tangent cones (2 hours).
8. Test (2 hours)

Teaching methods

Conventional lecture; problem lecture.
Auditorium exercises – solving standard problems enlightening the significance of the theory, exercises on applications, solving problems.

Learning outcomes and methods of theirs verification

Outcome description	Outcome symbols	Methods of verification	The class form

Assignment conditions

Activity of students during classes. Final grade controlling if student pass the minimal level of learning efects.

The final note of Control theory 2 depends on the note of classes (40%) and the note of the exam (60%).

Student is able to pass the exam if he obtained previously the positive note of classes.

Recommended reading

1. M. Kisielewicz, Differential Inclusions and Optimal Control, PWN – Kluwer Acad. Publ. 1991,
2. J. P.  Aubin, A. Cellina, Differential Inclusions, Springer Verlag 1984,
3. S. Rolewicz,Functional Analysis and control theory, linear systems, Mathematics and its applications, Springer 1987

Further reading

1. 1. J. Zabczyk, Mathematical control theory, an Introduction, Birkhauser 1996

Notes

Modified by dr Ewa Sylwestrzak-Maślanka (last modification: 10-04-2024 16:07)