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

R. K. Tagiev   ; V. M. Gabibov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 50 (2017), pp. 30-44

Résumé

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}\left(k(x,t)\frac{\partial u}{\partial x}\right)+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,\\ \left| \frac{\partial k(x,t)}{\partial x}\right|\leqslant \mu_1, \left| \frac{\partial k(x,t)}{\partial t}\right|\leqslant\mu_2, |q(x, t)|\leqslant\mu_3\text{ 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.

Détail

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

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