The work is devoted to the study of weak solvability of the optimal feedback control problem for the Voigt model with variable density. The proof is based on the approximationtopological approach. First, the solvability of the approximation problem is proved, and then, based on a priori estimates that are independent of the approximation parameter, it is shown that from the sequence of solutions of the approximation problem, a subsequence converging to the solution of the original problem can be chosen. Then it is shown that among weak solutions to the problem there is at least one that gives a minimum to a given cost functional.

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