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 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.]: Year: Volume: Issue: Page: Find

 Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 2016, Volume 20, Number 1, Pages 54–64 (Mi vsgtu1463)

Differential Equations and Mathematical Physics

On optimal control problem for the heat equation with integral boundary condition

R. K. Tagiyeva, V. M. Gabibovb

a Baku State University, Baku, AZ-1148, Azerbaijan
b Lankaran State University, Lankaran, AZ-4200, Azerbaijan

Abstract: In this paper we consider the optimal control problem for the heat equation with an integral boundary condition. Control functions are the free term and the coefficient of the equation of state and the free term of the integral boundary condition. The coefficients and the constant term of the equation of state are elements of a Lebesgue space and the free term of the integral condition is an element of Sobolev space. The functional goal is the final. The questions of correct setting of optimal control problem in the weak topology of controls space are studied. We prove that in this problem there exist at least one optimal control. The set of optimal controls is weakly compact in the space of controls and any minimizing sequence of controls of a functional of goal converges weakly to the set of optimal controls. There is proved Frechet differentiability of the functional of purpose on the set of admissible controls. The formulas for the differential of the gradient of the purpose functional are obtained. The necessary optimality condition is established in the form of variational inequality.

Keywords: optimal control, heat equation, necessary optimality condition

DOI: https://doi.org/10.14498/vsgtu1463

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Bibliographic databases:

UDC: 517.977
MSC: 49J20, 35K20
Original article submitted 22/XI/2015
revision submitted – 22/I/2016

Citation: R. K. Tagiyev, V. M. Gabibov, “On optimal control problem for the heat equation with integral boundary condition”, Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.], 20:1 (2016), 54–64

Citation in format AMSBIB
\Bibitem{TagGab16} \by R.~K.~Tagiyev, V.~M.~Gabibov \paper On optimal control problem for the heat equation with integral boundary condition \jour Vestn. Samar. Gos. Tekhn. Univ., Ser. Fiz.-Mat. Nauki [J. Samara State Tech. Univ., Ser. Phys. Math. Sci.] \yr 2016 \vol 20 \issue 1 \pages 54--64 \mathnet{http://mi.mathnet.ru/vsgtu1463} \crossref{https://doi.org/10.14498/vsgtu1463} \zmath{https://zbmath.org/?q=an:06964472} \elib{https://elibrary.ru/item.asp?id=26898088} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. T. K. Yuldashev, “Obyknovennoe integro-differentsialnoe uravnenie s vyrozhdennym yadrom i integralnym usloviem”, Vestn. Sam. gos. tekhn. un-ta. Ser. Fiz.-mat. nauki, 20:4 (2016), 644–655
2. R. K. Tagiev, V. M. Gabibov, “Raznostnaya approksimatsiya i regulyarizatsiya zadachi optimalnogo upravleniya dlya parabolicheskogo uravneniya s integralnym usloviem”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2017, no. 50, 30–44
3. R. K. Tagiyev, Sh. I. Maharramli, “Variational method of solving inverse problem for a parabolic equation with integral conditions”, Proceedings of the 6Th International Conference on Control and Optimization With Industrial Applications, v. II, eds. A. Fikret, B. Tamer, Baku State Univ., Inst. Applied Mathematics, 2018, 286–288
4. E. Tabarintseva, “Approximate solving of an inverse problem for a parabolic equation with nonlocal data” (Novosibirsk, Russian Federation; 26–30 August, 2019), OPCS, 2019, 15th International Asian School-Seminar Optimization Problems of Complex Systems (2019), 8880207, 173-178
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