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Optimal control in coefficients of boundary value problems with unilateral constraints. (English) Zbl 0544.49005
Let $$\Omega \subset {\mathbb{R}}^ n$$ be a bounded domain with boundary $$\partial \Omega$$, $$\psi_ 1$$, $$\psi_ 2$$, $$\psi$$ be given functions such that the sets $K_ 1\equiv \{z:\quad Z\in \overset \circ W^ 1_ 2(\Omega),\quad \psi_ 1(x)\leq z(x)\leq \psi_ 2(x),\quad x\in \Omega \},$
$K_ 2\equiv \{z:\quad z\in W^ 1_ 2(\Omega),\quad \psi(x)\leq z(x),\quad x\in \partial \Omega \}$ are nonempty and let $U\equiv \{u:\quad u\in L_ 2(\Omega),\quad 0<c_ 1\leq u(x)\leq c_ 2,\quad x\in \Omega \}$ be a set of admissible controls. For given functions $$d\in L_ 2(\Omega)$$ and $$f\in(W^ 1_ 2(\Omega))^*$$ the following problems $$(i=1,2)$$ $J\equiv \int_{\Omega}(z-d)^ 2dx\to \min,\quad u\in U,\quad z\in K_ i,\int_{\Omega}u\cdot(\nabla z,\nabla \eta -\nabla z)dx\geq \ll f,\eta -z\gg \quad \forall \eta \in K_ i$ are discussed. The form of the directional derivative of the functional $$J=J(u)$$ (for $$u\in U)$$ and also for the similar case with a full matrix of second order coefficients) is described. Extensions (by means of G- convergence) of the initial problems and necessary conditions for optimality are given.
Reviewer: U.Raitums

MSC:
 49J40 Variational inequalities 35J85 Unilateral problems; variational inequalities (elliptic type) (MSC2000) 49K20 Optimality conditions for problems involving partial differential equations 49J20 Existence theories for optimal control problems involving partial differential equations 35J25 Boundary value problems for second-order elliptic equations 49J45 Methods involving semicontinuity and convergence; relaxation 54A20 Convergence in general topology (sequences, filters, limits, convergence spaces, nets, etc.)