On the optimization formulation of the coefficient inverse problem for a parabolic equation with an additional integral condition

R. K. Tagiev   ; R. A. Kasumov

Vestnik Tomskogo gosudarstvennogo universiteta. Matematika i mehanika, no. 45 (2017), pp. 49-59

Résumé

Let a controlled process be described in $\mathcal{Q}_T = \{(x,t) \in R^2 : 0 < x <\ell, 0 < t < T\}$ by the following initial-boundary value problem for a linear parabolic equation: \begin{gather*} u_t-(k(x,t)u_x)_x+q(x,t)u=f(x,t), \quad (x,t)\in\mathcal{Q}_T,\\ u\mid_{t=0}=\varphi(x), \quad 0\leqslant x\leqslant\ell,\\ u_x\mid_{x=0}=u_x\mid_{x=\ell}=0, \quad 0<t\leqslant T. \end{gather*} Here $\ell, T>0$ are given numbers $f (x,t)\in L_2 (\mathcal{Q}_T)$, $\varphi(x)\in W_2^1(0,\ell)$ are given functions, $k(x,t)$, $q(x,t)$ are unknown coefficients, $\upsilon(x,t) = (k(x,t),q(x,t))$ is a control, $u = u(x,t)= u(x,t;\upsilon)$ is the solution to the boundary value problem corresponding to the control $\upsilon = \upsilon(x,t)$. Let us introduce a set of admissible controls \begin{gather*} V = \{\upsilon(x,t) = (k(x,t),q(x,t)) \in H = W_2^1(\mathcal{Q}_T)\times L_2 (\mathcal{Q}_T): 0 < v \leqslant k(x,t)\leqslant \mu,\\ |k_x (x,t)|\leqslant\mu_1,\ |k_t (x,t)| \leqslant\mu_2,\ 0 \leqslant q_0 \leqslant q(x,t) \leqslant q_1 \text{ п.в.на } \mathcal{Q}_T\}, \end{gather*} where $\mu\geqslant v> 0$, $\mu_1, \mu_2 > 0$, $q_1 \geqslant q_0 \geqslant 0$ are given numbers. Let us state the following coefficient inverse problem of optimal control type: among all the admissible controls $\upsilon(x,t)=(k(x,t),q(x,t))\in V$, find the controls $\upsilon_*(x,t)=(k_*(x,t),q_*(x,t))\in V$ minimizing the functional $$ J(\upsilon)=\int_0^T\left|\int_0^{\ell}K(x, t)u(x,t;\upsilon)dx-E(t) \right|^2dt $$ where $K(x,t)$, $E(t)$ are known functions, $\upsilon= \upsilon(x,t)$ is a control $u = u(x,t) = u(x,t;\upsilon)$ is a generalized solution to the boundary value problem from $V_2^{1,0} (\mathcal{Q}_T)$ corresponding to the control $\upsilon = \upsilon(x, t)$ is a given set. In the present work, the optimization formulation of the coefficient inverse problem for a parabolic equation with an additional integral condition is considered. The questions of correctness of the optimization formulation of the inverse problem are investigated. The differentiability of the objective functional is proved and the formula for its gradient is found. A necessary condition of optimality is found in the form of a variational inequality.

Détail

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

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