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On an optimal domain with respect to $$(x,t)$$ for a parabolic-type equation. (English. Russian original) Zbl 0863.49005
Differ. Equations 31, No. 7, 1216-1217 (1995); translation from Differ. Uravn. 31, No. 7, 1265-1266 (1995).
The problem to be solved in the short paper is to find the optimal pair $$\{\Omega^*,y^*(x,t)\}$$ of admissible domains $$\Omega\subset R^2$$ and solutions $$y(x,t)$$ to a boundary value problem for the equation $$Ay=f(x,t)$$ with a parabolic operator $$A$$ so that the functional $J(\Omega^*,y^*)= \min_{\Omega,y} |y-y^0|L_2(\Omega)|^2$ under a given function $$y^0(x,t)$$. The main result is a theorem about the existence of $$\{\Omega^*,y^*\}$$ in question.
##### MSC:
 49J20 Existence theories for optimal control problems involving partial differential equations 93C20 Control/observation systems governed by partial differential equations 35K20 Initial-boundary value problems for second-order parabolic equations