Infopaket

Generalized optimal control

ДВС.1.04

Вибіркова дисципліна для ОП

First

2022/2023

7 Semester

3

LO 12. Solve individual engineering problems and/or problems arising in at least one subject area: in sociology, economics, ecology, and medicine. LO 19. Collect and interpret relevant data and analyze complexities within their specialization to make judgments, PLO 23.1. To be able to use professionally profiled knowledge, abilities and skills in the field of computational mathematics and informatics for modeling real processes of various nature.

Full-time form

To study the course "Generalized optimal control", a student must know the basic concepts of mathematical analysis, linear algebra, differential equations, probability theory, and functional analysis.

Part 1. Fractional differential equations Definition of fractional integrals and derivatives. Fractional integration and differentiation of a power function. Composition formulas of fractional integrals and derivatives. The concept of random walk with continuous time. Montrol-Weiss formula. Derivation of fractional equations of subdiffusion and superdiffusion. Concept of anomalous diffusion. Fractional Sobolev spaces on a line: construction using the Fourier transform and fractional derivatives. Weak solvability of the subdiffusion equation in fractional Sobolev spaces. Point control problem. Part 2. Extremal problems and optimal control problems Necessary and sufficient conditions for the extremum in finite-dimensional problems without constraints and with constraints of the type of equalities. Elements of differential calculus in normalized spaces. Smooth tasks without limits. Lagrange scheme. Problems of classical calculus of variations: Lagrange's problem and Boltz's problem. A weak extremum in the Lagrange and Boltz problems. Calculus of variations problems with higher derivatives. Dynamic programming method. The Bellman function and its properties. The Bellman equation of the problem of optimal speed. Analytical design of a linear regulator. Pontryagin's maximum principle. Sufficient optimality conditions in the form of the maximum principle. Independent work: Pontryagin's maximum principle in: ● tasks of optimal control; ● problems of optimal speed.

1. Kilbas A.A., Srivastava H.M., Trujillo J.M. Theory and Applications of Fractional Differential Equations. — Amsterdam: Elsevier, 2006 – 523 r. 2. Agranovich M.S. Sobolevskie prostranstva, ikh obobshcheniia i ellipticheskie zadachi v oblastiakh s gladkoi i lipshitsevoi granitsei. — Moskva: Izdatel-stvo MТsNMO, 2013 – 379 s. 3. Alekseev V.M., Tikhomirov V.M., Fomin S.V. Optimal-noe upravlenie. – M.: Nauka, 1979. – 425 s. 4. Iang L. Lektsii po variatsionnomu ischisleniiu i teorii optimal-nogo upravleniia. – M.: Mir, 1974. – 315 s. 5. Perestiuk M.O., Stanzhits-kii O.M., Kapustian O.V. Ekstremal-nі zadachі. Navchal-nii posіbnik. – K.: VPТs Kiїvs-kii unіversitet, 2004. – 50 s. 6. Perestiuk M.O., Stanzhits-kii O.M., Kapustian O.V. Zadachі optimal-nogo keruvannia. Navchal-nii posіbnik. – K.: TVіMS, 2004. – 55 s. 7. Pshenichnyi B.N. Vypuklyi analiz i ekstremal-nye zadachi. – M.: Nauka, 1980. – 318 s.

Lectures, seminars, independent work

Semester assessment: 1. Control work 1: RN 1.1., RN 1.2, RN 2.1, RN 2.2, RN 2.3 ‒ 30 points/18 points. 2. Control work 2: RN 1.3, RN 1.4, RN 2.4, RN 2.5, RN 2.6 ‒ 30 points/18 points. Semester assessment. Work in the semester consists of 2 parts. When assigning points for a part, the following is taken into account: the assessment for the control work - 30 points. The final control is conducted in the form of an exam - 40 points. - learning outcomes that will be evaluated: PH 1.1, PH 1.2, PH 1.3, PH 1.4, PH 2.1, PH 2.2, PH 2.3, PH 2.4, PH 2.5, PH 2.6. - form of conduct: written work. - types of tasks: 4 written tasks (2 theoretical questions and 2 practical tasks).

Ukrainian