Existence Theorem for a Weak Solution of the Optimal Feedback Control Problem for the Modified Kelvin–Voigt Model of Weakly Concentrated Aqueous Polymer Solutions
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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.
The research of the first two authors (Theorem 1 on the existence of a weak solution of the feedback control problem) was supported by the Ministry of Science and Higher Education of the Russian Federation, project no. 14.Z50.31.0037. The research of the third author (Theorem 2 on the existence of a solution minimizing the given cost functional) was supported by the Russian Science Foundation, project no. 19-11-00146.
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