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The article proposes an approach to solving the problem of synthesis of motion and power control of lumped sources in systems with distributed parameters. For concreteness, the problem of feedback control of moving heat sources during rod heating is considered. The powers and velocities of point sources are assigned depending on the measured measuring points. Formulas for the linear dependence of the synthesized parameters on the measured temperature values are proposed. The differentiable of the functional with respect to the parameters of the feedback is shown, formulas for the gradient of the functional with respect to the synthesized parameters are obtained. The formulas make it possible to use effective first-order numerical optimization methods and available standard software packages to solve the problem of synthesizing the control of lumped sources.

We note that the proposed approach to the synthesis of control of lumped sources can be used in automatic control systems and automatic control of lumped sources for many other technological processes and technical objects. The objects themselves can be described by other partial differential equations and types of boundary contitions.