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Optimal Control of Ordinary Differential Equations 1

Module MA3312

This Module is offered by Department of Mathematics.

This module handbook serves to describe contents, learning outcome, methods and examination type as well as linking to current dates for courses and module examination in the respective sections.

Module version of SS 2012

There are historic module descriptions of this module. A module description is valid until replaced by a newer one.

Whether the module’s courses are offered during a specific semester is listed in the section Courses, Learning and Teaching Methods and Literature below.

available module versions
SS 2021WS 2020/1WS 2018/9SS 2012WS 2011/2

Basic Information

MA3312 is a semester module in English language at Master’s level which is offered irregular.

This Module is included in the following catalogues within the study programs in physics.

  • Catalogue of non-physics elective courses
Total workloadContact hoursCredits (ECTS)
150 h 45 h 5 CP

Content, Learning Outcome and Preconditions


Necessary conditions (Euler-Lagrange, Legendre-Clebsch), integral constraints, algebraic constraints, differential equations as constraints, control constraints, bang-bang and singular control, application problems

Learning Outcome

After successful completion of the module students are able to understand and apply the basic notions, concepts, and methods of optimal control theory for ordinary differential equations. They master in particular the formulation and the evaluation of the first variation for problems with interior point conditions. They know fundamentals of the different
classes of constraints. They have learned to transform optimal control problems into boundary value problems suitable for numerical treatment. They know how to apply this knowledge to the solution of problems from science and engineering.


MA 2304 Numerical Methods for Ordinary Differential Equations,
MA 2005 Ordinary Differential Equations

Courses, Learning and Teaching Methods and Literature

Courses and Schedule

Learning and Teaching Methods

In the lectures, the contents will be presented in a talk with demonstrative examples, as well as through discussion with the students. The lectures should motivate the students to carry out their own analysis of the themes presented and to independently study the relevant literature. Corresponding to each lecture, practice sessions will be offered, in which exercise sheets and solutions will be available. In this way, students can deepen their understanding of the methods and concepts taught in the lectures and independently check their progress.


blackboard and/or projector


Bryson, Ho: Optimal Control,
Leitmann: The Calculus of Variations and Optimal Control,
Pesch: Schlüsseltechnologie Mathematik,
original research papers

Module Exam

Description of exams and course work

The exam will be in written form (60 minutes). Students demonstrate that they have gained deeper knowledge of definitions, theorems and main mathematical tools of optimal control theory presented in the course and their applicability to problems in science and engineering. The students are in particular expected to be able formulate the complete first variation for a given optimal control problem with interior point conditions and to draw conclusions from that formulation. Students show that they are able to distinguish and handle different classes of constraints.The students are expected to be able to derive the methods, to explain their properties, and to transform them into formulations properly suited for the numerical solution.

Exam Repetition

The exam may be repeated at the end of the semester.

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