# American Institute of Mathematical Sciences

September  2019, 8(3): 473-488. doi: 10.3934/eect.2019023

## Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation

 South Ural State University, Institute of Natural Sciences and Mathematics, 454080, Chelyabinsk, Lenin av, 76, Russian Federation

Received  January 2018 Revised  November 2018 Published  September 2019 Early access  May 2019

Fund Project: The work was supported by Act 211 Government of the Russian Federation, contract 02.A03.21.0011.

The paper presents sufficient conditions for existence of an optimal control of solutions to a non-autonomous degenerate operator-differential evolution equation. We construct families of operators that solve this equation, as well as classical and strong solutions of the multipoint initial-final problem for the equation. We show that there exists a solution of an optimal control problem for a given operator-differential equation with a multipoint initial-final condition. The paper, in addition to the introduction and the bibliography, contains five sections. The first three parts contain information about the solvability of the multipoint initial-final problem for a non-autonomous equation. The fourth section presents the main result of the article; that is, a theorem on existence of optimal control of solutions to a multipoint initial-final problem. In the fifth part, the optimal control problem for the non-autonomous modified Chen – Gurtin model with the multipoint initial-final condition is investigated on the basis of the obtained abstract results.

Citation: Minzilia A. Sagadeeva, Sophiya A. Zagrebina, Natalia A. Manakova. Optimal control of solutions of a multipoint initial-final problem for non-autonomous evolutionary Sobolev type equation. Evolution Equations & Control Theory, 2019, 8 (3) : 473-488. doi: 10.3934/eect.2019023
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##### References:
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