# SF2852 Optimal Control Theory 7.5 credits

### Offering and execution

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Select the semester and course offering above to get information from the correct course syllabus and course offering.

## Course information

### Content and learning outcomes

#### Course contents *

Dynamic programming in continuous and discrete time. Hamilton-Jacobi-Bellman equation. Theory of ordinary differential equations. The Pontryagin maximum principle. Linear quadratic optimization. Model predictive control Infinite horizon optimal control problems. Sufficient conditions for optimality. Numerical methods for optimal control problems.

#### Intended learning outcomes *

The overall goal of the course is to provide an understanding of the main results in optimal control and how they are used in various applications in engineering, economics, logistics, and biology.

Measurable goals:

To pass the course, the student should be able to do the following:

• Describe how the dynamic programming principle works (DynP) and apply it to discrete optimal control problems over finite and infinite time horizons,
• Use continuous time dynamic programming and the associated Hamilton-Jacobi-Bellman equation to solve linear quadratic control problems,
• Use the Pontryagin Minimum Principle (PMP) to solve optimal control problems with control and state constraints,
• Use Model Predictive Control (MPC) to solve optimal control problems with control and state constraints. You should also be able understand the difference between the explicit and implicit MPC control and explain their respective advantages.
• Formulate optimal control problems on standard form from specifications on dynamics, constraints and control objective. In addition be able to explain how various control objectives affect the optimal performance.
• Explain the principles behind the most standard algorithms for numerical solution of optimal control problems and use Matlab to solve fairly simple but realistic problems.

• Integrate the tools learnt during the course and apply them to more complex problems.
• Explain how PMP and DynP relates to each other and know their respective advantages and disadvantages. In particular, be able to describe the difference between feedback control versus open loop control and also be able to compare PMP and DynP with respect to computational complexity.
• Combine the mathematical methods used in optimal control to derive the solution to variations of the problems studied in the course.

#### Course Disposition

No information inserted

### Literature and preparations

#### Specific prerequisites *

In general:

150 university credits (hp) including 28 hp in Mathematics,  6 hp in Mathematical Statistics and 6 hp in Optimization. Documented proficiency in English corresponding to English B.

More precisely for KTH students:

Passed courses in calculus, linear algebra, differential equations, mathematical statistics, numerical analysis, optimization.

#### Recommended prerequisites

The prerequisites is a Swedish or foreign degree equivalent to Bachelor of Science of 180 ECTS credits, with at least 45 ECTS credits in mathematics. The students should have documented knowledge corresponding to basic university courses in analysis, linear algebra, numerical analysis, differential equations and transforms, mathematical statistics, and optimization.

#### Equipment

No information inserted

#### Literature

Material from the department of Mathematics.

### Examination and completion

A, B, C, D, E, FX, F

#### Examination *

• TENA - Examination, 7.5 credits, Grading scale: A, B, C, D, E, FX, F

Based on recommendation from KTH’s coordinator for disabilities, the examiner will decide how to adapt an examination for students with documented disability.

The examiner may apply another examination format when re-examining individual students.

#### Other requirements for final grade *

A written examination (TENA; 7,5 hp).
Optional homeworks give bonus credit on the exam.

#### Opportunity to complete the requirements via supplementary examination

No information inserted

#### Opportunity to raise an approved grade via renewed examination

No information inserted

### Further information

#### Course web

Further information about the course can be found on the Course web at the link below. Information on the Course web will later be moved to this site.

Course web SF2852

SCI/Mathematics

Mathematics

#### Education cycle *

Second cycle

No information inserted

#### Contact

Johan Karlsson (johan.karlsson@math.kth.se)

#### Ethical approach *

• All members of a group are responsible for the group's work.
• In any assessment, every student shall honestly disclose any help received and sources used.
• In an oral assessment, every student shall be able to present and answer questions about the entire assignment and solution.

#### Supplementary information

The academic year Autumn16/Spring17 the course is given in Period 4 (Spring17).

The academic year Autumn17/Spring18 the course is not given at all.

The academic year Autumn18/Spring19 the course is given in Period 1 (Autumn18).