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Loaded equations, arising during synthesis of control of rod heating by moving sources

The article proposes an approach to solving the problem of synthesis of motion and power control of lumped sources. For concreteness, the problem of linear feedback control of moving heat sources during heating of the rod is considered. The powers and motion of point sources, participating in the right-hand side of the differential equation of parabolic type, are determined depending on the measured values of the process state at the points of measurement. As a result, the right-hand side of the differential equation linearly depends on the values of the process state at the given points of the bar. Formulas for the components of the gradient of the functional with respect to the parameters of linear feedback are obtained, which make it possible to use first-order optimization methods for the numerical solution of synthesis problems.