## Topic outline

### Lecturer: Harri Ehtamo

##### Assistant: Anton von Schantz

NOTE!

The first lecture is on Wed 16.1.2019 14:15-16:00 in U5,

and the first exercise session on Tue 22.1.2019 14:15-16:00 in U7.

Exam times:

Tue 9.4.2019 13-16

Fri 31.5.2019 13-16

#### Why this course?

This course examines dynamic (aka multistage) optimization models. They capture many relevant real-life problems: scheduling, route planning, solving optimal strategies for games, inventory control, investment problems, machine repair, text processing, dna sequence matching, stopping problems, airplane/rocket flight path optimization, minimum time/effort problems, optimal fishery management, saving/consumption optimization, optimal feedback controllers for plants and regulator problems and so on.

The models that are examined are

• Optimal control problem. Find control u(t) that makes the system  trace the optimal trajectory x*(t) that minimizes the cost .
Function a describes how the system behaves at state x(t) at time t under control u(t). The cost function J consists of start and end point costs and running cost that is given by function L that may depend on the state x(t) and control u(t).
• Calculus of variations. Find continuous/differentiable curve y(x) that is extremum for
• Dynamic Programming (DP) problem. Find optimal controls u_k (optimal policy) that minimizes the expected cost
of the discrete stochastic system
f_k describes how the system evolves to the next state x_k+1 when the state is x_k, control u_k is chosen and there is stochastic disturbance is w_k. The cost function is given by g_k. This is discrete version of the optimal control problem.

#### Practical matters

Teaching: Lectures (24h) and exercise sessions (24h)

Assessment methods: Exam (100%), extra points from homework and exercises

Grading scale: 0-5

Study material: Lecture slides and exercises. Additional reading:

Language of instruction: English

Prerequisites: 1st and 2nd years math, recommended MS-C2105 Optimoinnin perusteet (or equivalent)