Synthesis of Zonal Control of Lumped Sources for the Heat Conduction Process

Abstract

The paper studies the problem of synthesis of control of lumped sources for an object with distributed parameters based on discrete observation of the phase state at specific object points. We propose an approach in which the whole phase space at the observed points is preliminarily divided in some way into given subsets (zones). The synthesized controls are selected from the class of piecewise-constant functions, and their current values are determined by a subset of the phase space containing the population of current states of the object at the observed points, at which controls take constant values. Such synthesized controls are called zonal. We give a numerical technique for obtaining optimal values of zonal controls using efficient first-order optimization methods. To this purpose, we derive formulas for the gradient of the objective function in the space of zonal controls.

https://www.bakumathj.org/archive/Vol2No2/j.bmj.033.pdf

“…The formulated problem of synthesizing zonal controls ( 1)-( 6) leads to a finitedimensional optimization problem. For numerical solution to this problem, we propose to use the approach described in [16], [17]. To solve the problem in the case of a simple design of a set of admissible controls V (for example, a parallelepiped, hyper-sphere, polyhedron, etc.…”

“…We have used the principle of zonality of control parameters as the basis of numerical techniques for solving such specific optimization and inverse problems like the problem of optimal placement of production and injection wells and optimal control of their flow rates during the operation of an oil reservoir under the regime of water-driven piston displacement [1], the problem of identifying the hydraulic resistance coefficient under the unsteady flow of viscous fluids through pipelines [5], and problems of feedback control and identification of objects with lumped parameters [2], [3], [4], [13], [14], [16], [17], [18].…”