FELIXOVÁ, L. Matematické metody teorie optimálního řízení a jejich užití [online]. Brno: Vysoké učení technické v Brně. Fakulta strojního inženýrství. 2011.

# Posudky

## Posudek vedoucího

### Čermák, Jan

This diploma thesis provides a survey of mathematical methods of optimal control theory and their applications. A special emphasize is put on the problem of energy efficient train control, its analysis and discussions of the exact solution. In my opinion, the aims of this thesis have been quite fulfilled. The original contribution of this work consists especially in finding of an analytical form of the solution, namely in the case, when a quadratic resistance function is involved into the studied mathematical model. The results published in the literature standardly employ numerical methods for its solving, at least for calculations of switching times separating optimal regimes. In this work, the utilization of numerical methods is minimized. I can also express my satisfaction with the author's approach to this work. She was active, conscientious and always tried to prepare this thesis in the best way she knew. Of course, some things could have been done better, e.g. the stylistic level or English of this thesis. However, these imperfections do not change my favourable impression from our cooperation and my general evaluation. On this account, I recommend to accept this thesis.

Dílčí hodnocení
Kritérium Známka Body Slovní hodnocení
Splnění požadavků a cílů zadání A
Postup a rozsah řešení, adekvátnost použitých metod A
Vlastní přínos a originalita B
Schopnost interpretovat dosažené vysledky a vyvozovat z nich závěry A
Využitelnost výsledků v praxi nebo teorii A
Logické uspořádání práce a formální náležitosti B
Grafická, stylistická úprava a pravopis B
Práce s literaturou včetně citací A
Samostatnost studenta při zpracování tématu B
Navrhovaná známka
A

## Posudek oponenta

### Tomášek, Petr

The thesis deals with problems arising in optimal control theory. In particular, there are considered two approaches of solving optimal control problems: Pontryagin's maximum principle and Bellman's principle of optimality. The train time-energy control problems are utilized to illustrate the feasibility of the both approaches. The Pontryagin's approach turns out to be more suitable then the Bellman's one in such problems. The contents illustrates that the structure of the thesis is well built. After the preface and biographies of L.S. Pontryagin and R.E. Bellman the basics of the optimal control theory are introduced with respect to the both considered approaches. Then the train time-energy control problem is formulated and discussed under various assumptions. The Pontryagin's maximum principle is utilized to obtain optimal control modes. The application of the Bellman's approach is discussed at the end of the thesis. Finally, the conclusion summarizes the work and gives several ways of possible generalizations of the previous analysis. The thesis is correct in the mathematical sense but there are some typesetting and language mistakes in the text, e.g. page 10, the 3rd row from bottom: missing dot at "et al" - et al. page 10: There is wrong word order in the last sentence (this mistake appears in several places in the thesis). page 15, the end of the first paragraph: There is missing comma after . page 15, the 13th row: The subject is missing in the sentence. "... of view is possible..." - "... of view it is possible..." page 15, next to the equation (4.1): "...the dot denotes the derivation..." - "...the dot denotes the derivative..." page 16: There should be bold symbol f in the second equation. Moreover, if the argument "(t)" appears at bold u, it is convenient to add the appropriate arguments to other symbols. page 16: Paragraph 4.2 and some following theorems start with the sentence: "In the phase space are given points a,b." It should be reformulated with respect to the word order. page 16: The sentence is no longer slanted after the equation (4.2). in the middle of the page 19: "Now let w be an arbitrary point...". The point mentioned in the next sentence is not the one from the previous sentence. page 20: There is missing minus sign in the left hand side of the equation (4.12) and the previous one. page 25: "formulas" - formulae at the end of page 28: "... will be represented." - "... is realized.". page 34: "From the second equation ..." - "From the first equation..." page 45: "7.4 An illustrating example" - "An illustrative example" page 50, just before eq. (9.2): "so that the cost functional" - "such that the cost functional" page 51, the second equation: There is missing square of the middle term on the right-hand side of the equation. The objectives of the thesis have been fulfilled. The text is well logically built and embodies all attributes associated with the master thesis at the mathematical engineering programme.

Dílčí hodnocení
Kritérium Známka Body Slovní hodnocení
Splnění požadavků a cílů zadání A
Postup a rozsah řešení, adekvátnost použitých metod A
Vlastní přínos a originalita A
Schopnost interpretovat dosaž. vysledky a vyvozovat z nich závěry B
Využitelnost výsledků v praxi nebo teorii A
Logické uspořádání práce a formální náležitosti A
Grafická, stylistická úprava a pravopis B
Práce s literaturou včetně citací A
Navrhovaná známka
A

#### Otázky

• There is given assumption 4\alpha b - a^2>0 without any comment at page 30. What happens if this assumption is not fulfilled? Such a case is not discussed in the following text although a,b are assumed to be arbitrary positive constants (see the first paragraph of section 7).
• At the top of page 17 there is written that f_0 had been introduced in the definition of the functional J. But there are given no assumptions to f_0. Can you formulate the appropriate properties f_0 should have?

eVSKP id 33980