Variational analysis in problems on optimization of systems with distributed parameters and vector functions of sets (Q803927)

The paper deals with the abstract optimization problem \(\Phi\) (w)\(\to \inf\), \(w\in W\), \(F(w)=0\), G(w)\(\in \Xi\), \(\Phi: W\to {\mathbb{R}}\), F: \(W\to Y\), G: \(W\to Z\), \(W=\Theta \times U=\{w\}\), \(\Theta \subset X=\{x\}\), \(\Xi =co \Xi\). Here X, Y are Banach spaces, Z is a normed linear space and \(U=\{u\}\) is a topological space. The functions \(\Phi\) (\(\cdot,u)\), F(\(\cdot,u)\), G(\(\cdot,u)\) are supposed to be smooth. The main result is the sufficient regularity condition concerning the mapping P: \(U\to Y\times Z\times {\mathbb{R}}\), \(P(u)=[F(x^ 0,u),G(x^ 0,u),\Phi (x^ 0,u)],\) under which at the stationary point \(w^ 0=(x^ 0,u^ 0)\) necessary optimality conditions of the maximum principle type hold. As a consequence the maximum principle for an elliptic controlled system with distributed parameters and state constraints is established.