summary (Разработка численно-аналитического метода и алгоритма решения задачи оптимального управления (на примере трехсекторной инвестиционной экономической модели)), страница 2

The process of solving this problem is based on the use ofthe Pontryagin maximum principle. The maximum condition is used to determine thegeneral structure of optimal controls. For the control functions of a prescribed structurewith arbitrarily many switchings, the analytic representation is obtained for the functionsof state and the so-called conjugate variables which, by their theoretical meaning, areLagrange multipliers in the original extremum problem with constraints. The obtainedanalytical results permit constructing an algorithm for determining the control functionsand the corresponding state functions which satisfy the general system of relations consisting5of necessary conditions for extremum and the constraints of the original optimal controlproblem.

The constructed algorithm is realized as a program package. This programproduct, using the prescribed initial parameters of the model, allows one numericallyto analyze a rather wide class of theoretically possible control functions and to determinethe controlled processes satisfying necessary conditions and constraints.Basic results presented to be defended1. Development of a general numerical-analytical method for analysis of optimalcontrol problems and its realization as an example of the posed control problem.2. Statement of a new mathematical optimal control problem in the framework ofa dynamical model of three-sector economics.3.

Development of the numerical algorithm for solving systems comprising necessaryconditions and the constraints of the original optimal control problem.4. Development of a program realizing this numerical algorithm.Scientific novelty of the workThe following scientific results are obtained in the work:1. A new numerical-analytical method for solving the optimal control problem isdeveloped.

This method permits analyzing the system of relations comprising necessaryconditions and the constraints of the original problem and numerically determining theadmissible extremals.2. A new meaningful mathematical optimal control problem is posed and justifiedin the framework of the dynamical model of three-sector economics.

This problem is solvedby a method based on the Pontryagin maximum principle.3. The system comprising necessary conditions and the constraints of the originaloptimal control problem is analytically analyzed. Explicit analytic representations of thestate functions and conjugate variables are obtained for a class of control functions whosestructure is determined by methods based on the maximum principle.4.

An algorithm is constructed for determining controlled processes which areadmissible extremals in the optimal control problem under study. This algorithm can beused in a wide class of control problems, where the conjugate equations can, in general,depend on the state functions of the model.5. A program realizing this algorithm is developed. For a given set of inputparameters of the model, this program permits determining numerical and graphicalrepresentations of controlled processes which are admissible extremals in the originaloptimal control problem.6General research conclusionsThis dissertation work is a study of a specific fundamental problem of mathematicaleconomics, namely, of the optimal control problem in a macroeconomic dynamical model.The study itself has a profound complex character and consists of two stages.

At the firststage, the analytic methods related to the contemporary mathematical theory of optimalcontrol are used. The main point at the second stage is the numerical algorithm whichpermits determining one or several solutions of a very complicated system of equationscomprising necessary conditions for extremum and the constraints of the original optimalcontrol problem. This algorithm is completely original. Moreover, it is rather generaland can be used to solve various optimal control problems, not only in economics.

Thedeveloped algorithm is realized as the program package which, for prescribed initialparameters of the model, permits determining specific controlled processes suspected tobe optimal, i.e., admissible extremals in the original optimal control problem.AcknowledgmentsThe authors especially expresses her thanks to her scientific adviser, cand. phys.math. sci., associate professor Shnurkov P.V. for active participation in her research andthe help in solving the scientific organization processes, and for his constant attention andhighly qualified guidance during her work on the dissertation.List of publications in the topic of the dissertationThe main points of the dissertation are presented in the author’s publications inthe leading reviewed scientific journals recommended by the Higher Attestation Commissionof the Russian Federation Ministry of Education and Science:1. Zasypko V.V., Optimal control of investments in a closed dynamical model ofthree-sector economics: mathematical statement of the problem and general analysis basedon the maximum principle.

// Vestnik MGTU im. N.E. Bauman. Ser. Estestv. Nauki. –2014. – № 2, pp. 101–115, 0.71 a.sh. (with Shnurkov P.V.; author’s personal contribution– 0.35 a.sh.).2. Zasypko V.V., Analytical study of problems of investment optimal control in aclosed dynamical model of three-sector economics. // Vestnik MGTU im. N.E. Bauman.Ser. Estestv.

Nauki. – 2014. – № 4, pp. 101–120., 0.81 a.sh. (with Shnurkov P.V.; author’spersonal contribution – 0.4 a.sh.).3. Zasypko V.V., Development of an algorithm for solving the problem of investmentoptimal control in a closed dynamical model of three-sector economics. // Informatika iee Primeneniya, – 2016. – 10, № 1, pp. 82–95. (with Shnurkov P.V., Belousov V.V., and7Gorshenin A.K.).and in other publications of the author:4.

Pisarenko V.V., Control of investments of the fund lending sector in the dynamicalmodel of three-sector economics. // Survey of Applied and Industrial Mathematics. Vol. 18.no. 4. Scientific reports. XII All-Russia Symposium on Applied and Industrial Mathematics.2011. Sochi. pp. 654–655, 0.12 a.sh.5. Pisarenko V.V., Control of investments of the fund lending sector in the dynamicalmodel of three-sector economics. // Book of abstracts. All-Russia Conference “AppliedProbability Theory and Theoretical Informatics”. – Moscow: IPI RAN, 2012. – pp.

88–90,0.14 a.sh. (with Shnurkov P.V.; author’s personal contribution – 0.07 a.sh.).6. Pisarenko V.V., Control of investments of the fund lending sector in the dynamicalmodel of three-sector economics. // Book of abstracts. International Conference “ProbabilityTheory and Its Applications"dedicated to B.V. Gnedenko on the occasion of his 100thbirthday, – Moscow: LENAND, 2012. – pp. 269–270, 0.12 a.sh.

(with Shnurkov P.V.;author’s personal contribution – 0.06 a.sh.).7. Zasypko V., Trajectory analysis of control process for optimal control of investmentsin the model of a three-sector economy. // Book of abstracts. XXXI International Seminaron Stability Problems for Stochastic Models and VII International Workshop “AppliedProblems in Theory of Probabilities and Mathematical Statistics Related to Modelingof Information Systems"and International Workshop “Applied Probability Theory andTheoretical Informatics". – Moscow: IPI RAN, 2013. – pp. 111–113, 0.1 a.sh.

(withShnurkov P.V.; author’s personal contribution – 0.05 a.sh.).8.