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Control of an elliptic problem with pointwise state constraints. (English) Zbl 0606.49017
The paper deals with a control problem for an elliptic equation of second order $$Ay=v$$ on $$\Omega$$, $$y=0$$ on $$\partial \Omega$$. The cost functional is of the form $J(v)=\int_{\Omega}(y(v)-y_ 0)^ 2dx+(r/2)\int_{\Omega}v^ 2(x)dx,\quad v\in K$ where K is a convex closed subset of $$L^ 2(\Omega)$$, $$y_ 0\in L^ 2(\Omega)$$. The following control problem is solved: minimize J(v) for $$v\in K$$ and $$| y(v,x)| \leq 1$$ for all $$x\in \Omega.$$
The existence and uniqueness of a solution is proved. Optimality conditions are given and regularity of the optimal solution is investigated.
Reviewer: I.Bock

##### MSC:
 49K20 Optimality conditions for problems involving partial differential equations 35J25 Boundary value problems for second-order elliptic equations 49J20 Existence theories for optimal control problems involving partial differential equations 35B37 PDE in connection with control problems (MSC2000) 35D10 Regularity of generalized solutions of PDE (MSC2000) 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 93C20 Control/observation systems governed by partial differential equations
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