Coefficient inverse problem of control type for elliptic equations with additional integral condition

R. K. Tagiev ; R. S. Kasimova

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 48 (2017), pp. 17-29

Résumé

Let it be required to minimize the functional $$ J(\upsilon)=\int_0^1\left|u(0,x_2;\upsilon)-\int_0^1H(x_1,x_2)u(x_1,x_2;\upsilon)dx_1 \right|^2dx_2 $$ on solutions $u(x)=u(x;\upsilon)=u(x_1,x_2;\upsilon)$ of the boundary-value problem \begin{gather*} -\sum_{i=1}^2\frac\partial{\partial x_i}\left(\upsilon(x_2)\frac{\partial u}{\partial x_i}\right)+q(x)u=f(x), \quad x\in\Omega,\\ -\upsilon(x_2)\frac{\partial u}{\partial x_1}=g(x), \quad x\in\Gamma_{-1},\\ u(x;\upsilon)=0, \quad x\in\Gamma\setminus\Gamma_{-1}, \end{gather*} corresponding to all admissible controls in the set $$ V=\{\upsilon=\upsilon(x_2)\in W_2^1(0,1): 0\leqslant\upsilon(x_2)\leqslant\mu, |\upsilon'(x_2)|\leqslant\mu_1\text{ п.в. на }(0, 1)\}, $$ where $\Omega=\{x=(x_1, x_2): 0$, $\Gamma_{-1}=\{x=(0,x_2): 0$, $H(x_1,x_2)$, $q(x)$, $f(x)$, $g(x)$ are given functions. In this paper, we consider a coefficient inverse problem of the control type for an elliptic equation with a quality criterion corresponding to an additional integral condition. The questions of correctness of the formulation of the inverse problem of the control type are investigated. The Frechet differentiability of the quality criterion is proved and an expression for its gradient is found. A necessary optimality condition is established in the form of a variational inequality.

Détail

Mots-clés : elliptic equation
Keywords: inverse problem, integral condition.