Wintersemester

Heuristic Algorithms

One-semester course of  lectures and practical studies.

The course taught for the 1^st-year students of Master program “Applied Mathematics”.

The course focuses on the optimal control of linear and non-linear dynamic systems. It’s devoted to general formation principles of necessary and sufficient conditions for tasks optimality of optimal control, issues of solutions existence. On this base dynamic processes of continuous and discrete kinds have been researched.

Course Objective: statement of mathematical tools applied in optimal control theory, set of tasks for optimal control theory and solution methods’ study, as well as acquirement of skills how to use these methods in a specific context implementing practical problems.

Course Tasks:

-          explain the role of mathematical control theory in research of controlled dynamic system;

-          discover different statements of control theory applications;

-          show the core of these tasks and its solution in a specific context;

-          explore formation methods of quality criteria depending on a task’ specifics;

-          examine finding techniques of optimal control and field of its application;

-          make comparative analysis of these methods.

Estimated Results:

Students completed this course have to

Know:

– main definitions and terms of mathematical theory of optimal control;

– basic necessary and sufficient conditions for optimality applied in problem-solving of optimal control;

– numerical methods for optimal control task-solving;

Be able to:

– apply necessary and sufficient conditions solving practical tasks;

– use approximate methods for optimal solution of economic problems;

Understand:

– skills and techniques implementing modern software for optimal control problem-solving;

– research of obtained solutions by analytic, graphic and numeric methods.

Course Content

1 Introduction to Optimal Control (OC)

-   Some examples

-   Problems statement of Optimal Control

-   Admissible control and associated trajectory

-   Optimal Control problems

-   Calculus of Variation problems 

 2 The simplest problem of OC 

-   The necessary condition of Pontryagin

-   The proof  in a particular situation 

-   Sufficient conditions 

-   First generalizations 

-   Initial/final conditions on the trajectory   

-   On minimum problems 

-   The case of Calculus of Variation 

-   Examples and applications 

-   The curve of minimal length

-   A problem of business strategy I 

-   A two-sector model 

-   A problem of inventory and production I

-   Singular and bang-bang controls 

 3 General problems of OC 

-   Problems of Bolza, Mayer and Lagrange

-   Problems with fixed or free final time 

-   Fixed final time 

-   Free final time 

-   The proof of the necessary condition 

-   The Bolza problem in Calculus of Variations

-   Labor adjustment model of Hamermesh

-   Time optimal problem 

-   The classical example of Pontryagin and its boat 

-   Existence and controllability results

-   Infinite horizon problems 

-   The model of  Ramsey 

-   Autonomous problems

-   Current Hamiltonian  in model of optimal  consumption

 4 Constrained problems of  OC

-   The general case 

-   Pure state constraints 

-   Commodity trading 

-   Isoperimetric problems in CoV

-   Necessary conditions with regular constraints 

-   The multiplier ν as shadow price 

-   The foundation of Cartagena 

-   The Hotelling model of socially optimal extraction

 5 OC with dynamic programming 

-   The value function: necessary conditions 

-   The final condition 

-   Bellman's Principle of optimality

-   The Bellman-Hamilton-Jacobi equation 

-   The value function: sufficient conditions 

-   More general problems of  OC 

-   Examples and applications 

-   Infinite horizon problems 

-   A model of optimal consumption  

- Problems with discounting and salvage value

Literatur