The existence theorem for weak solution of optimal feedback control problem for the modified Kelvin-Voigt model of weakly concentrated aqueous polymer solutions

In this paper the existence theorem on weak solution of the optimal feedback control problem for the modified Kelvin-Voigt model of weakly concentrated aqueous polymer solutions. The proof is carried out on the basis of an approximation-topological approach to the study of fluid dynamic problems. At the first step, the considered feedback control problem is interpreted as an operator inclusion with a multi-valued right-hand side. In the second step, the resulting inclusion is approximated by an operator inclusion with better properties. Then, on the basis of a priori estimates of solutions and the degree theory of a class of multi-valued mappings, the existence of solutions for this inclusion is proved. In the third step, it is shown that from the sequence of solutions of the approximation inclusion one can extract a subsequence that converges weakly to the solution of the original inclusion. Then it is proved that among the solutions of the considered problem there is a solution that gives a minimum to a given quality functional.