On relation of feedback control problems with loaded differential equations

We consider a problem of controlling a heating appliance used for heating a heat-transfer agent, which delivers heat into a closed system. To control the process, we use feedback, under which information on the process state used to form the current control values is continuously received from individual points of the appliance with installed temperature sensors. The mathematical model of the controlled process is described by a pointwise loaded first-order hyperbolic equation. In the problem, both the parameters of a synthesized feedback control, and a location of the measurement points. In the paper, formulas for the functional gradient of the problem are obtained, which allow to use numerical first-order optimization methods to solve the problem. The results of computer experiments for numerical solution to the problem on test data are given.