Construction of Control with Constraints for Nonlinear Systems with Coefficients Depending on the Control Object State
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Journal of Mathematical Sciences, Vol. 250, No. 1, October, 2020
CONSTRUCTION OF CONTROL WITH CONSTRAINTS FOR NONLINEAR SYSTEMS WITH COEFFICIENTS DEPENDING ON THE CONTROL OBJECT STATE Z. N. Murzabekov ∗ al-Farabi Kazakh National University 71, al-Farabi Ave, Almaty 050040, Republic of Kazakhstan [email protected]
G. A. Mirzakhmedova al-Farabi Kazakh National University 71, al-Farabi Ave, Almaty 050040, Republic of Kazakhstan [email protected]
UDC 517.977.55
We consider an optimal control problem on a finite time-interval for a three-sector economic control object. We reduce the economic system to an optimal control problem for a nonlinear system with coefficients independent of the control object state and find a nonlinear synthesizing control based on the feedback principle and certain constraints on control. The results obtained for the nonlinear system are used to construct the control parameters in the mathematical model of a three-sector economic control object. We find an optimal distribution between the labor and investment resources satisfying the balance relations. Bibliography: 3 titles. Illustrations: 2 figures.
In [1, 2], Lagrange multipliers are used to study optimal control problems for technical systems and the linearized system of an economic cluster. In this paper, we consider an economic system and transform it to an optimal control problem for a class of nonlinear systems with coefficients depending on the control object state. We transform the nonlinear differential equation describing the original control system to a system of linear structure, but with parameters depending on the state. Synthesizing the control, we use the nonlinear quadratic functional which allows us to construct the Riccati matrix equation with parameters independent of the control object state. This approach is a basis of the synthesis of nonlinear optimal control systems. We propose a combined method based on constructing a nonlinear feedback, which allows us to represent the sought control as a synthesizing control depending on the state of the nonlinear system and current time. Moreover, owing to this method, it is possible to take into account constraints on the control values. ∗
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Translated from Problemy Matematicheskogo Analiza 104, 2020, pp. 69-74. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2501-0076
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The results obtained for nonlinear systems are used to construct the control parameters of a three-sector economic control object on a finite time interval. We note that the share of labor and investment resources are variable in all the three economy sectors.
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The Three-Sector Economic Control Object Model
We consider the optimal control problem for a three-sector economic control object model consisting of the following sectors: i = 0 (material), i = 1 (fund-creator), i = 2 (consumer). The mathematical model consists of the following three components [3]: (a) three differential equations governing the dynamics of the capital-labor ra
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