Topic outline

  • Lecturer: Harri Ehtamo

    Assistant: Anton von Schantz


    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  \dot{\textbf{x}}(t) = \textbf{a}\,[\,\textbf{x}(t),\textbf{u}(t),t\,], trace the optimal trajectory x*(t) that minimizes the cost J=\Phi\,[\,\textbf{x}(t_0),t_0,\textbf{x}(t_f),t_f\,] + \int_{t_0}^{t_f} \mathcal{L}\,[\,\textbf{x}(t),\textbf{u}(t),t\,] \,\operatorname{d}t.
      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  J[y] = \int_{x_1}^{x_2}  L[x,y(x),y'(x)]\, dx  \, .
    • Dynamic Programming (DP) problem. Find optimal controls u_k (optimal policy) that minimizes the expected cost dp1
      of the discrete stochastic system dp2
      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)