Lecturer: Harri Ehtamo
Assistant: Anton von Schantz
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.
Teaching: Lectures (24h) and exercise sessions (24h)
Assessment methods: Exam, homework, attending exercises and lectures, and project work.
Grading scale: Pass or 0-5
Study material: Lecture slides and exercises. Additional reading:
- D. E. Kirk: Optimal Control Theory. Prentice Hall, 1970 (2004). (<- the main book)
- D. P. Bertsekas: Dynamic Programming and Optimal Control, vol 1(and 2). Athena Scientific, 1995
- M. L. Kamien and N. L. Schwartz: Dynamic Optimization - The calculus of variations and optimal control in economics and management, 2nd edition. North Holland, 1991.
Language of instruction: English
Prerequisites: 1st and 2nd years math, recommended MS-C2105 Optimoinnin perusteet (or equivalent)