Abstract
The problem of synthesis of the boundary optimal control of the cooling process of media with heat conductive viscosity is investigated. In addition to the distributed parameters, the concentrated parameters act on the system. This is due to the fact that the temperature of the external environment is unknown and varies according to a given law. As a result, the process is described by a system of partial differential equations and ordinary differential equations. In this case, heat transfer occurs at the right end of the rod. This complicates the obtaining of a solution of this boundary-value problem in an explicit form. But it is possible to establish the existence and uniqueness of the solution of the corresponding boundary-value problem for concrete admissible controls. The criterion of quality is a quadratic functional and it is required to build control in the form of feedback. First by the Fourier method, the problem under consideration is formulated in an infinite-dimensional phase space. As a result, the problem of synthesis of optimal control in a functional space is obtained. To solve this problem, the dynamic programming method is used. To do this, let’s introduce the Bellman functional and obtain the Bellman equation, which this functional satisfies. The solution of this equation allows to find the control parameter in the form of a functional defined on the set of the state function. Further, by introducing the corresponding functions, feedback control is constructed for the original problem. Unlike program control, this allows to influence the behavior of the system at any time, that is, to ensure the self-regulation of the process. However, let’s note that the difficulties in solving this problem are connected with the justification of the proposed method. This is established by the investigation of a closed system.
Highlights
If the material does not have a conductive viscosity, the process of heat or cooling transfer is described by a parabolic equation
The difficulty in solving the synthesis problem of optimal control in this case is due to the fact that it is not possible to find an explicit solution of the corresponding boundary-value problem
In a material with a heat conductive viscosity, the process of heat transfer is described by a third-order partial differential equation
Summary
If the material does not have a conductive viscosity, the process of heat or cooling transfer is described by a parabolic equation. The obtained results relate to the problems of optimal control with systems containing elements with distributed parameters, and they can be applied to the process of heating and cooling the material with the minimum energy expenditure. A vector-valued function p(x, t) = (p0 (t), p(x, t)) will be called an admissible control if p0 (t) ∈L2 (0, T), p(x, t) ∈L2 (Q).
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