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1. |
Name of the discipline |
EXTREMAL PROBLEMS AND VARIATONS CALCULUS |
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2. |
Study course, specialty |
4, Mathematics (economy activity) |
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3. |
Semester of study |
7 |
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4. |
Amount of credits |
4 |
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5. |
Lecturer’s S.N.P. |
Lebedev Andrei Vladimirovich |
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6. |
Aims of the study |
Study of the principal methods of classical variational problems solution. Improvement of professional competence level in optimization problems solution. Further formation of student’s abstract mathematical thinking experience and skills of its application in concrete problems solution along with mathematical culture increase. As a result student has to be able to: — find minimums and maximums of functions defined on finite dimensional spaces; — compile extremal problems models in finite dimensional and infinite dimensional spaces; — find extremals for classical variational problems; — apply second order conditions for variational problems solution; — apply the necessary and sufficient conditions for finding of strong extremums in variational problems. |
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7. |
Prerequisites |
Mathematical analysis (calculus) Algebra and number theory Differential equations Optimization in finite dimensional spaces Functional analysis |
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8. |
Contents of the discipline |
Optimization problems in infinite dimensional spaces. Brachistochrone problem. Introduction to variations calculus. History of variational problems and their meaning for infinite dimensional analysis development. Classical variational problem. Strong and weak extremums in classical variational problem. Differentiation of mappings in normed spaces. The first and second Lagrange variations. Gateau and Freshet derivatives. The general form of the first and second variation of integral functional. The first and second order conditions for extremum of functions defined on normed spaces. The necessary first and second order conditions for local extremum in classical variational problem. Integral and differential Lagrange – Euler equations. The second order variation theory. Legendre necessary condition. Jacoby necessary condition. Jacoby weak minimum sufficient condition for classical variational problem. Weierstrass necessary condition for strong minimum in classical variational problem. Isoperimetric variational problem. Local extremum in isoperimetric variational problem. |
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9. |
Recommended literature |
1. Galeev E.M., Tihomirov V.M. Short course of extremal problems theory, Moscow, MSU, 1989 (in Russian). 2. Alekseev V.M., Galeev E.M., Tihomirov V.M. Collection of problems in optimization. Theory. Examples. Problems. Moscow, Nauka, 1984 (in Russian). 3. Gorokhovik V.V. Finite dimensional optimization problems. Minsk, BSU, 2006 (in Russian). 4. Lebedev A.V., Pindrik O.I. Variational problems (weak and strong extremum). Minsk, BSU, 2012 (in Russian). |
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10. |
Methods of teaching |
Lectures, laboratory practice, SSI (students scientific investigation) |
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11. |
Language of teaching |
Russian, English |
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12. |
Knowledge control and requirements |
Tests, examination mark takes into account running tests marks (coefficient 0.3) |
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13. |
Certification form |
Examination |