In the paper, optimal control problem for cooling process with minimum energy in materials with heat conducting viscosity is considered. To solve approximately the considered problem, a finite-dimensional approximation for the solution of the corresponding boundary value problem in the form of truncated Fourier series is constructed and an integral representation for the coefficients of this series is obtained. This yields a system of integral equations with respect to control parameters. So,

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... e problem is reduced to finding a minimum norm function from these moment relations. Applying the theorem on orthogonal decomposition of a normalized space, every approximation of control parameter and the corresponding value of a functional in analytic form is finded.