Optimal control in coefficients of boundary value problems with unilateral constraints
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.
scientific article; zbMATH DE number 3866051