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Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2017, Number 50, Pages 30–44 (Mi vtgu616)  

MATHEMATICS

Difference approximation and regularization of the optimal control problem for a parabolic equation with an integral condition

R. K. Tagieva, V. M. Gabibovb

a Baku State University, Baku, Azerbaijan
b Lenkaran State University, Azerbaijan

Abstract: Let a controlled process be described in the region $\mathcal{Q}_T=\{(x,t): 0<x<\ell, 0<t\leqslant T\}$ by the following boundary-value problem for a linear parabolic equation with an integral boundary condition:
\begin{gather*} \frac{\partial u}{\partial t}-\frac{\partial}{\partial x}(k(x,t)\frac{\partial u}{\partial x})+q(x,t)u=f(x,t), (x,t)\in\mathcal{Q}_T,
u(x,0)=\varphi(x), 0\leqslant x\leqslant\ell,
\frac{\partial u}{\partial x}(0, t)=0, 0<t\leqslant T,
k(\ell, t)\frac{\partial u}{\partial x}(\ell, t)=\int_0^{\ell} H(x)\frac{\partial u}{\partial x}(x, t)dx+g(t), 0<t\leqslant T, \end{gather*}
where $\varphi(x)\in W_2^1(0, l)$, $f(x, t)\in L_2(\mathcal{Q}_T)$, $g(t)\in W_2^1(0, T)$, $H(x)\in \mathring{W}_2^1(0,l)$ are given functions, $k(x, t)$, $q(x, t)$ — are control functions, and $u=u(x,t)=u(x,t,\nu)$ — is solution of the boundary value problem, i.e. the process state corresponding to the control $\upsilon$.
We introduce the set of admissible controls
\begin{gather*} V=\{\upsilon=(k(x,t), q(x,t))\in H=W_2^1(\mathcal{Q}_T)\times L_2(\mathcal{Q}_T): 0<\nu\leqslant k(x,t)\leqslant\mu,
| \frac{\partial k(x,t)}{\partial x}|\leqslant \mu_1, | \frac{\partial k(x,t)}{\partial t}|\leqslant\mu_2, |q(x, t)|\leqslant\mu_3 a.e. on \mathcal{Q}_T\}, \end{gather*}
where $\nu, \mu, \mu_1, \mu_2, \mu_3>0$ — are given numbers.
We define the target functional
$$ J(\upsilon)=\int_0^T|u(x, T;\upsilon)-u_T(x)|^2dx, $$
where $u_T(x)\in W_2^1(0, l)$ — the given function.
In the present work, the optimal control problem for a parabolic equation with an integral boundary condition and control coefficients is considered. Estimates of the accuracy of the difference approximations by state and function are established. The process of A. N. Tikhonov’s regularization of the approximations is carried out.

Keywords: optimal control, parabolic equation, integral boundary condition, difference approximation.

DOI: https://doi.org/10.17223/19988621/50/3

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Bibliographic databases:

UDC: 517.977.58
Received: 19.06.2017

Citation: R. K. Tagiev, V. M. Gabibov, “Difference approximation and regularization of the optimal control problem for a parabolic equation with an integral condition”, Vestn. Tomsk. Gos. Univ. Mat. Mekh., 2017, no. 50, 30–44

Citation in format AMSBIB
\Bibitem{TagGab17}
\by R.~K.~Tagiev, V.~M.~Gabibov
\paper Difference approximation and regularization of the optimal control problem for a parabolic equation with an integral condition
\jour Vestn. Tomsk. Gos. Univ. Mat. Mekh.
\yr 2017
\issue 50
\pages 30--44
\mathnet{http://mi.mathnet.ru/vtgu616}
\crossref{https://doi.org/10.17223/19988621/50/3}
\elib{https://elibrary.ru/item.asp?id=30778970}


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