Vestnik Tomskogo Gosudarstvennogo Universiteta. Matematika i Mekhanika
 RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Archive Impact factor Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Vestn. Tomsk. Gos. Univ. Mat. Mekh.: Year: Volume: Issue: Page: Find

 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

Full text: PDF file (553 kB)
References: PDF file   HTML file

Bibliographic databases:

UDC: 517.977.58

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}