7 semester


Name of the discipline  



Study course,


4, Computer mathematics and systems analysis


Semester of study



Amount of credits                     



Lecturer’s S.N.P.

Lebedev Andrey Vladimirovich


Aims of the study

Study of the basic methods for solving classical variational problems. Increase the level of professional competence in solving optimization problems. Further development of students’ skills in abstract mathematical thinking and the ability to apply it in specific tasks, increase their mathematical culture.

As a result of the study, the student should be able to:

– find the minimum and maximum points for functions defined on finite-dimensional spaces;

– construct models of extremal problems in finite-dimensional and infinite-dimensional spaces;

– find admissible extremals in the classical variational problem;

– use the second-order conditions for solving problems in the calculus of variations;

– use the necessary condition for a strong extremum in variational problems.



Mathematical analysis

Algebra and Number Theory

Differential equations

Optimization methods in finite-dimensional spaces

Functional Analysis


Contents of the discipline

Optimization problems in infinite-dimensional spaces. The problem of brachistochrone.

Introduction to the calculus of variations. History and significance in the development of infinite-dimensional analysis.

The simplest variational problem.

Strong and weak extrema in the simplest variational problem

Variations of the objective functional of the simplest variational problem.

Differentiation of functions and mappings defined on normed spaces. Definitions of the first and second variations of Lagrange. Derivatives of Gato and Frechet.

 General view of the first and second variations of the integral functional.

Necessary conditions for a local minimum in the simplest variational problem.

Conditions for a local minimum of first and second order for functions defined on normed spaces. Singularities of sufficient conditions for a local minimum in infinite-dimensional spaces. Necessary conditions of the first and second order for the local minimum of the simplest variational problem in terms of variations of the target functional.

Integral and differential Euler equations. The Weierstrass-Erdman condition. Hilbert’s theorem. Theory of the second variation. A necessary condition for Legendre. Necessary condition for Jacobi.

A sufficient condition for a weak local minimum in the simplest variational problem.

The use of a sufficient Jacobi condition for a weak local minimum in the simplest variational problems.

A necessary Weierstrass condition for a strong minimum in the simplest variational problem.

Isoperimetric variational problem.

The local minimum in the isoperimetric variational problem.


Recommended literature

1. Galeev EM, Tikhomirov VM A short course in the theory of extremal problems. – Moscow, Moscow State University, 1989.

2. Alekseev VM, Galeev EM, Tikhomirov VM Collection of tasks on optimization. Theory. Examples, Tasks. Study Guide: – Moscow, Science, 1984.

3. Gorokhovik V.V. Finite-dimensional optimization problems. – Minsk, 2006.

4. Lebedev AV, Pindrik OI Variational problems (strong and weak extremum). Method. Allowance

5. Bakhtin VI, Ivanishko IA, Lebedev AV, Pindrik OI The Lagrange principle. Method. allowance


Methods of teaching

Lectures, laboratory practice, SSI (students scientific investigation)


Language of teaching



Knowledge control and requirements



Certification form