Construction of a minimizing sequence for the problem of cooling of the given segments of the rod with phase constraint

For the process of controlling the temperature of a thin rod, an optimization problem of cooling its given segments was set. The cooling control was selected so that the phase restriction occurs throughout the exposure. As a result, the heat equation with the control action was transformed into a nonlinear integro-differential equation. The relation between the obtained nonlinear initial-boundary value problem and the standard linear problem of the heat equation, which is solved by the Fourier method, was shown. All this makes it possible to move from the initial distributed problem to the concentrated optimization problem with respect to the Fourier coefficients of the solution of the linear problem. The possibility of its reduction to a finite shortened system was demonstrated for the resulting countable system of differential equations. The algorithm for searching the control coefficients of a shortened problem was given in order to find the optimal control parameters and the optimal value of the quality criterion for shortened problems. It was proved that a minimizing sequence of control parameters was obtained for the initial optimization problem.