No. 5 (264), issue 11Pages 13 - 24
THE OPTIMAL CONTROL OVER SOLUTION OF THE INITIAL-FINISH VALUE PROBLEM FOR THE BOUSSINESQUE-L'OVE EQUATIONA.A. Zamyshlyaeva, O.N. Tsyplenkova
Of concern is the optimal control problem for the Sobolev type equation of second order with relatively polynomially bounded operator pencil. The theorem of existence and uniqueness of strong solutions of initial-finish problem for abstract equation is proved. The sufficient and, in the case when infinity is a removable singularity of the A-resolvent operator pencil, the necessary conditions for optimal control existence and uniqueness of such solutions are found. The initial-finish problem for the Boussinesque - L'ove equation, which describes the longitudinal vibrations of an elastic rod, is investigated. We use the ideas and methods developed by G.A. Sviridyuk and his disciples. The proof of the existence and uniqueness of optimal control theorem is based on the theory of optimal control developed by J.-L. Lions. Full text
- Sobolev-type equations, relatively polynomially bounded operator pencil, strong solutions, optimal control.
- 1. Zagrebina S.A. The Initial-Finish Problem for the Navier - Stokes Linear System [Nachal'no-konechnaya zadacha dlya lineynoy sistemy Nav'e - Stoksa]. Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya 'Matematicheskoe modelirovanie i programmirovanie', 2011, no. 4 (221), issue 7, pp. 35 - 39.
2. Sviridyuk, G.A., Fedorov V.E. Linear Sobolev Type Equations and Degenerate Semigroups of Operators. Utrecht; Boston; K'oln; Tokyo, VSP, 2003.
3. Keller A.V. Numerical Solution of the Start Control for a System of Equations of Leontief Type [Chislennoe reshenie zadachi startovogo upravleniya dlya sistemy uravneniy leont'evskogo tipa]. Obozrenie prikladnoy i promyshlennoy matematiki, 2009, vol. 16, issue 2, pp. 345 - 346.
4. Manakova N.A. The Optimal Control Problem for the Oskolkov Nonlinear Filtration Equation. Differential Equations, 2007, vol. 43, no. 9, pp. 1213 - 1221.
5. Sviridyuk G.A., Zagrebina S.A. The Showalter-Sidorov Problem as a Phenomena of the Sobolev-type Equations. J. News of Irkutsk State University. Series 'Mathematics', 2010, vol. 3, no. 1, pp. 51 - 72.
6. Sviridyuk G.A., Zamyshlyaeva A.A. The Phase Space of a Class of Linear Higher-Order Sobolev Type Equations. Differential Equations, 2006, vol. 42, no. 2, pp. 269 - 278.
7. Zamyshlyaeva A.A. The Initial-Finish Value Problem for the Boussinesque - L'ove Equation [Nachal'no-konechnaya zadacha dlya uravneniya Bussineska-Lyava]. Vestnik Yuzhno-Ural'skogo gosudarstvennogo universiteta. Seriya 'Matematicheskoe modelirovanie i programmirovanie', 2011, no. 37 (254), issue 10, pp. 22 - 29.
8. Lions Zh.-L. Оptimal'noe upravlenie sistemami, opisyvaemymi uravneniyami s chastnymi proizvodnymi [Optimal Control of Systems Described by Equations with Partial Derivatives]. Moscow, Мir, 1972.
9. Zamyshlyaeva, A.A. The Phase Space of a Class of Linear Sobolev Type Equations of the Second Order [Fazovye prostranstva odnogo klassa lineynykh uravneniy sobolevskogo tipa vtorogo poryadka]. Vychislitel'nye tekhnologii, 2003, vol. 8, no. 4, pp. 45 - 54.