Synthesis of Power and Motion Control of Plate Heating Sources and Optimization of Temperature Measurement Points Placement - Hashimov, Vugar (Kìbernetika ta komp'ûternì tehnologìï (Online), 2025)
Introduction. The problem under consideration belongs to the problems of optimal control of lumped sources in systems with distributed parameters. Such problems are described by initial-boundary-value problems with respect to partial derivative equations of various types. The theory of optimal control problems for systems with distributed parameters began to be most actively developed in the 60’s of the last. This was due to the need to problem such important processes as the development of large oil and gas fields and pipeline transportation of hydrocarbon raw materials. Similar problems are relevant in metallurgy, ecology and many other industries. The purpose of the paper. This paper investigates the problem of synthesizing optimal control of moving point heat sources for heating a two-dimensional plate. The powers of the sources and their trajectories of motion, described by ordinary differential equations, are optimized. In addition, in the problem under consideration, the locations of temperature measurement points are also optimized. The necessary optimality conditions for the feedback parameters and the coordinates for setting the measurement points are obtained. The conditions contain formulas for the components of the gradient of the objective functional according to the parameters being optimized. Results. The necessary conditions of optimality of the synthesized feedback parameters and formulas for the components of the gradient of the objective functional on these parameters are obtained. These formulas allow the use of efficient first-order numerical optimization methods to solve the problem of feedback parameter synthesis. The results of computer experiments obtained using first-order numerical optimization methods are presented. Conclusions. An approach to feedback control of the motion and power of lumped sources in systems with distributed parameters is proposed. The problem of controlling moving heat sources used to heat a plate is considered. The powers and control actions on the motion of point sources are determined in the form of proposed dependencies on the results of the taken measurements. The differentiability of the functional with respect to the feedback parameters is shown, and formulas for the gradient of the functional with respect to the synthesized parameters are obtained. The formulas allow us to use efficient numerical methods of first-order optimization and available standard software packages to solve the problem of synthesis of control of lumped sources. Keywords: plate heating, feedback control, moving sources, temperature measurement points, feedback parameters.