1. 
Name of the discipline 
EXTREME OBJECTIVES

2. 
Study course, specialty 
4, Computer mathematics and systems analysis 
3. 
Semester of study 
7 
4. 
Amount of credits 
3 
5. 
Lecturer’s S.N.P. 
Lebedev Andrey Vladimirovich 
6. 
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 finitedimensional spaces; – construct models of extremal problems in finitedimensional and infinitedimensional spaces; – find admissible extremals in the classical variational problem; – use the secondorder conditions for solving problems in the calculus of variations; – use the necessary condition for a strong extremum in variational problems. 
7. 
Prerequisites 
Mathematical analysis Algebra and Number Theory Differential equations Optimization methods in finitedimensional spaces Functional Analysis 
8. 
Contents of the discipline 
Optimization problems in infinitedimensional spaces. The problem of brachistochrone. Introduction to the calculus of variations. History and significance in the development of infinitedimensional 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 infinitedimensional 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 WeierstrassErdman 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. 
9. 
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. Finitedimensional 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 
10. 
Methods of teaching 
Lectures, laboratory practice, SSI (students scientific investigation) 
11. 
Language of teaching 
Russian 
12. 
Knowledge control and requirements 
Tests 
13. 
Certification form 
credit 