Existence Theorem for a Weak Solution of the Optimal Feedback Control Problem for the Modified Kelvin–Voigt Model of Weakly Concentrated Aqueous Polymer Solutions

A theorem on the existence of a weak solution of the optimal feedback control problem for the modified Kelvin–Voigt model of weakly concentrated aqueous polymer solutions is proved. The proof is based on 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 multivalued right-hand side. At the second step, the resulting inclusion is approximated by an operator inclusion with better properties. Then the existence of solutions for this inclusion is proved by applying a priori estimates of solutions and the degree theory for a class of multivalued mappings. At the third step, it is shown that the sequence of solutions of the approximation inclusion contains a subsequence that converges weakly to the solution of the original inclusion. Finally, it is proved that, among the solutions of the considered problem, there exists at least one minimizing a given cost functional.