Topics in Dynamic Modeling and Optimal Control

BEA511 Topics in Dynamic Modeling and Optimal Control

  • Topics


    Theoretical Topics

    The objective is to provide students with a capability to formulate and analyze problems in different fields of economics and management science using the tools of optimal control theory. Deterministic and stochastic theory will be presented. Continuous time problems are emphasized. Applied examples (applications) constitute the main part. Classical Hamiltonian formulation with focus on the Maximum Principal (MP) including different constraining relations and their transversality conditions. Modern formulation applying the value function concept through Dynamic Programming (DP) and its associated Hamilton-Jacobi-Bellman equation (HJB) are introduced. to facilitate and bridge the gap to the course BEA514- Topics in numerical optimization.

    Stochastic optimal control problems are incorporated in this part.

    The relations between MP and DP formulations are discussed. The main focus is put on producing feedback solutions from a classical Hamiltonian formulation. Interpretations of theoretical concepts are emphasized, e.g. that the Hamiltonian is the shadow price on time.

    Differential games are introduced.

    The ideas of Equivalent Representation and The Principal of Extension are introduced.

    Finite and infinite time horizons are treated. Relaxation of the optimality concept is introduced through the notion of "Catching-Up optimality", which may apply if the classical value becomes infinite.

    Applied Topics

    Among others, we study applications such as Ramsey's growth model, production and storage planning, advertising, management of (non-) renewable resources, Pigouvian taxation and pollution control, road planning, maintenance and sale and allocation of private wealth on consume, secure and risky investments. Real options are presented in a generic setting.

    Topics will be lectured in the following sequence:

    Short summary of difference- and differential equations and stability analysis

    Basic concepts and ideas. Global optimality. Shadow prices and theoretical and practical interpretations of basic notions e.g. transversality conditions.

    Introductions to applied control problems. Necessary and sufficient conditions.

    Discounting and present value formulations. Extensions of optimality in infinite horizon problems.

    Constrained problems: State and mixed restrictions. Bang-bang and singular controls.

    Dynamic programming (DP) and the Hamilton-Jacobi-Bellman (HJB) equation.

    Summary of stochastic processes. Stochastic feedback control. Merton´s example

    Differential games - open and closed loop policies

    1. Numerical solution algorithms for HJB: The probability approach.

  • Learning outcome

    Learning outcome

    After successfully completing the course, the candidates should be able to


    analyze and evaluate potential dynamic and stochastic effects on economic quantities and resources depending on policy choices

    identify practical limitations of present day numerical solution approaches



    review, assess and utilize relevant scientific papers addressing dynamic optimization

    formulate and model operational management tasks evolving in time

    identify potential intrinsic deterministic chaos in the formulated model



    take part in and manage interdisciplinary research involving dynamic modelling and decisions tasks in an operational setting

    analyze models with respect to dynamic as well as structural stability

    identify potential or implicit conserved quantities (conservation laws)

    formulate and analyze problems in different fields of economics and management science applying the tools of classical and modern theories of optimal control and the calculus of variation.

  • Teaching


    Topics/papers will be partly lectured by course responsible and partly presented for discussion in class by students

  • Required prerequisites

    Required prerequisites

    Knowledge of medium advanced calculus and some familiarity with differential and difference equations and introductory probability theory

  • Requirements for course approval

    Requirements for course approval


    Activity in class 2) Two individual assignments

  • Assessment


    Based on: 1. Activity in class, 2. Two assignments, 3. Short oral examination at the end of the course

  • Grading Scale

    Grading Scale


  • Computer tools

    Computer tools

    The use of high-level programing in Maple and MatLab will be an integrated part of the course.

  • Semester



  • Literature


    Main topics and applications are presented in part two of the textbook "Dynamic Optimization: The theory of Variations and Optimal Control in Economics and Management" by Morton I. Kamien and Nancy L. Schwartz in the series "A series of Volumes in dynamic economics: Theory and applications" volume 4.

    Additional topics are given in lecture notes and selected journal articles.


ECTS Credits
Teaching language

Course responsible

Leif Kristoffer Sandal