# Optimal Control of Solutions to the Multipoint Initial-Final Problem for Nonstationary Relatively Bounded Equations of Sobolev Type

M.A. Sagadeeva, A.D. BadoyanWe study the problem of optimal control of solutions to an operator-differential equation, which is not solved with respect to the time derivative, together with a multipoint initial-final condition. In this case, one of the operators in the equation is multiplied by a scalar function of time. By the properties of the operators involved, the stationary equation has analytical resolving group. We construct a solution to the multipoint initial-final problem for the nonstationary equation. We show that a unique optimal control of solutions to this problem exists.Full text

Apart from the introduction and bibliography, the article consists of three sections. The first section provides the essentials of the theory of relatively p-bounded operators. In the second section we construct a strong solution to the multipoint initial-final problem for nonstationary Sobolev-type equations. The third section contains our proof that there exists a unique optimal control of solutions to the multipoint initial-final problem.

- Keywords
- optimal control; multipoint initial-final problem; Sobolev-type equations; relatively bounded operator.
- References
- 1. Favini A., Yagi A. Degenerate Differential Equations in Banach Spaces. New York, Basel, Hong Kong, Marcel Dekker, Inc, 1999. 236 p.

2. Demidenko G.V., Uspenskii S.V. Partial Differential Equations and Systems not Solvable with Respect to the Highest-Order Derivative. New York, Basel, Hong Kong, Marcel Dekker, Inc, 2003. 239 p.

3. Sviridyuk G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht, Boston, Koln, VSP, 2003. 216 p. DOI: 10.1515/9783110915501

4. Al'shin A.B., Korpusov M.O., Sveshnikov A.G. Blow-up in Nonlinear Sobolev Type Equations. Berlin, de Gruyter, 2011. 648 p.

5. Zagrebina S.A. The Multipoint Initial-Finish Problem for the Stochastic Barenblatt-Zheltov-Kochina Model. Bulletin of the South Ural State University. Series: Computer Technologies, Automatic Control, Radio Electronics, 2013, vol. 13, no. 3, pp. 5-11. (in Russian)

6. Sagadeeva M.A., Badoyan A.D. The Optimal Control over Solutions of Special Form of Nonstacionary Sobolev Type Equations in Relatively Spectral Case. Vestnik Magnitogorskogo gosudarstvennogo universiteta. Matematika [Bulletin of Magnitogorsk State University. Mathematics], 2013, no. 15, pp. 68-80. (in Russian)

7. Sagadeeva M.A., Badoyan A.D. The Problem of Optimal Control over Solutions of the Nonstationary Barenblatt-Zheltov-Kochina Model. Bulletin of the South Ural State University. Series: Computer Technologies, Automatic Control, Radio Electronics, 2014, vol. 14, no. 2, pp. 5-11.

8. Zagrebina S., Sagadeeva M. The Generalized Splitting Theorem for Linear Sobolev type Equations in Relatively Radial Case. Izvestiya Irkutskogo gosudarstvennogo universiteta. Seriya: Matematika [The Bulletin of Irkutsk State University. Series: Mathematics], 2013, vol. 7, pp. 19-33.

9. Keller A.V. Relatively Spectral Theorem. Vestnik Chelyabinskogo gosudarstvennogo universiteta. Seriya Matematika. Mekhanika [Bulletin of the Chelyabinsk State University. Series of Mathematic and Mechanic], 1996, no. 1 (3), pp. 62-66. (in Russian)