Optimal Control of Sources with Feedback with Optimization of the Placement of Measurement Points Along Given Trajectories

In this paper, we study the problem of synthesizing the optimal control of the power of point-wise heat sources for heating a two-dimensional plate. The current power values are determined depending on the plate temperature values measured by the measuring devices. At the same time, these devices move along predefined individual trajectories. The movement velocities of the devices depend on the results of the current measurements. Necessary optimality conditions are obtained for the feedback parameters determined by the current values of the source powers and the velocities of the measuring devices. The conditions contain formulas for the gradient components of the objective functional with respect to the optimized feedback parameters. The results of computer experiments obtained with the use of first-order numerical optimization methods are presented.