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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.
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